mafelap 2013 - brunel university...

MAFELAP

2013

Conference on the Mathematicsof Finite Elements and Applications

10–14 June 2013

Abstracts

MAFELAP 2013

The organisers of MAFELAP 2013 are pleased toacknowledge the nancial support given to theconference by the Institute of Mathematics andits Applications (IMA) in the form of IMA Stu-dentships.

Contents of the MAFELAP 2013 Abstracts

Invited, parallel and mini-symposium order

Invited talks

Finite Element Methods in Coastal Ocean Modeling: Successes and Challenges

Clint N DawsonZIENKIEWICZ LECTURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

Forty years of the Crouzeix-Raviart element

Susanne C. BrennerBABUSKA LECTURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

Advances in reducing the mesh burden in computational mechanics applications tofracture and surgical simulation

Stephane P.A. Bordas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2

Fields, control fields, and commuting diagrams in isogeometric analysis

Annalisa Buffa, Giancarlo Sangalli and Rafael Vazquez. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4

The inf-sup constant of the divergence

Martin Costabel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5

Finite element methods for surface PDEs

Charles M. Elliott. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6

A stochastic collocation approach to PDE-constrained optimization for random dataidentification problems

Max Gunzburger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7

Time Domain Integral Equations for Computational Electromagnetism

Peter Monk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-8

The Emergence of Predictive Computational Science: Validation and Verification ofComputational Models of Complex Physical Systems

J. Tinsley Oden. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9

What is the Largest Finite Element System that can be Solved Today?

Ulrich Ruede. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-10

i

Double complexes and local bounded cochain projections

Ragnar Winther. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1-10

Talks in parallel sessions

Verification of functional a posteriori error estimates for obstacle problem

Petr Harasim and Jan Valdman. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-2

Convergence of hp-FEM in Three Dimensional Computation of Thermoelectric Effects

Razi Abdul-Rahman and Sarah Kamaludin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-3

Discontinuous Galerkin time stepping schemes combined with local projection stabi-lization methods applied to transient Stokes problems: stability and convergence

Naveed Ahmed and Gunar Matthies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-4

Anisotropic mesh adaptation for the Ambrosio-Tortorelli model: application to quasi-static crack propagation

Marco Artina, Massimo Fonrasier, Simona Perotto and Stefano Micheletti. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-5

A posteriori error analysis for time-dependent Stokes equations

Eberhard Bansch, Johan Jansson, Fotini Karakatsani and Charalambos Makri-dakis

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-6

Non-Conforming Finite Element Methods for the Obstacle Problem

Carsten Carstensen and Karoline Kohler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-7

On the numerical simulation of well stability

Philippe R B Devloo, Erick Raggio Slis Santos and Diogo Lira Cecılio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-8

Estimation of discretization and algebraic error via quasi-equilibrated fluxes for dis-continuous Galerkin methods

Vıt Dolejsı, Ivana Sebestova and Martin Vohralık. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2-9

Condensing the spectral element method for time domain wave problems

Dugald B Duncan and Mark Payne. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-10

Local mass conservation of Stokes finite elements

Daniele Boffi, Nicola Cavallini, Francesca Gardini and Lucia Gastaldi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-11

Enriching a Hankel Basis by Ray Tracing in the Ultra Weak Variational Formulation

C. J. Howarth, Simon Chandler-Wilde, Stephen Langdon and P.N. Childs.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12

ii

Error estimates for nonlinear convective and singularly perturbed problems in finiteelement methods

Vaclav Kucera. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-14

A domain decomposition method with an optimized penalty parameter

Chang-Ock Lee and Eun-Hee Park. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-15

The Partition of Unity Method for the 3D elastic wave problems in the high frequencydomain

M. Mahmood, O. Laghrouche, A. El-Kacimi and J. Trevelyan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16

On the numerical treatment of essential boundary conditions within positivity-preservingfinite element methods for convection-dominated transport problems

Matthias Moller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-18

Algebraic Flux Correction in a Partial Differential-Algebraic Framework

Julia Niemeyer and Bernd Simeon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-19

Approximation of eddy currents in an axisymmetric unbounded domain

Pilar Salgado and Virginia Selgas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-21

A uniform convergence analysis of three-step Taylor Galerkin finite element monotoneiterative domain-decomposition scheme for singularly perturbed problems

Vivek Sangwan and B. V. Rathish Kumar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22

With a hierarchical error indicator toward anisotropic mesh refinement

Rene Schneider. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-23

Computational Aspects in Smooth Approximation of Data

Karel Segeth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-23

iii

Talks in Mini-Symposium

A priori finite element error estimates in optimal control

A priori error estimates for finite element methods for H(2,1)-elliptic equations

Thomas Apel, Thomas G. Flaig and Serge Nicaise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

Crank-Nicolson and Stormer-Verlet discretization schemes for optimal control problemswith parabolic partial differential equations

Thomas Apel and Thomas G. Flaig. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-3

Error estimates for Dirichlet control problems in polygonal domains

Thomas Apel, Mariano Mateos, Johannes Pfefferer and Arnd Rosch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4

Boundary concentrated FEM for optimal control problems

Sven Beuchler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5

Error estimates for the velocity tracking problem using duality arguments

Konstantinos Chrysafinos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5

Convergence and error analysis of a numerical method for the identification of matrixparameters in elliptic PDEs

Klaus Deckelnick and Michael Hinze. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6

Optimal Control of Biharmonic Operator

Stefan Frei, Rolf Rannacher and Winnifried Wollner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7

An interior penalty method for distributed optimal control problems governed by thebiharmonic operator

Thirupathi Gudi, Neela Nataraj and Veeranjaneyulu Sadhanala. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-7

A priori error estimates for parabolic optimal control problems with point controls

Dmitriy Leykekhman and Boris Vexler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-8

Optimal error estimates for finite element discretization of elliptic optimal control prob-lems with finitely many pointwise state constraints

Dmitriy Leykekhman, Dominik Meidner and Boris Vexler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9

iv

Verification of optimality conditions and discretization error estimates

Martin Naß and Arnd Rosch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10

On discretized nonconvex elliptic optimal control problems with pointwise state con-straints

Ira Neitzel, Johannes Pfefferer and Arnd Rosch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10

Sparse Elliptic Control Problems in Measure Spaces: Regularity and FEM Discretiza-tion

Konstantin Pieper and Boris Vexler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11

Optimal boundary control problems in energy spaces

Olaf Steinbach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11

Finite element methods for fourth order variational inequalities arising from ellipticoptimal control problems

Li-yeng Sung. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12

Analysis and applications of boundary element methods

The BEM++ boundary element library and applications

S.R. Arridge, T. Betcke, M. Schweiger and W. Smigaj. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2

A recursive integral equations approach for electromagnetic scattering by biperiodicmultilayer gratings

Beatrice Bugert and Gunther Schmidt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3

Black-Box Preconditioning of FEM/BEM matrices by H-matrix techniques

Markus Faustmann, Jens Markus Melenk and Dirk Praetorius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4

One-equation FEM-BEM coupling for elasticity problems

Michael Feischl, Thomas Fuhrer, Michael Karkulik and Dirk Praetorius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5

An axiomatic approach to optimality of adaptive algorithms with applications to BEM

Michael Feischl and Dirk Praetorius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7

v

A reduced basis boundary element method for a class of parameterized electromagneticscattering model

M. Ganesh, J. S. Hesthaven and B. Stamm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8

Retarded potential boundary integral equations for sound radiation in a half-space

Heiko Gimperlein. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9

Analysis of a non-symmetric coupling of Interior Penalty DG and BEM

Norbert Heuer and Francisco-Javier Sayas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-9

Adaptive nonconforming boundary element methods

Norbert Heuer and Michael Karkulik. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10

Parallel BEM-Based Methods

Dalibor Lukas, Michal Merta, Lukas Maly, Petr Kovar and Tereza Kovarova. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-11

On the quasi-optimal convergence in FEM-BEM coupling

Jens Markus Melenk, Dirk Praetorius and B. Wohlmuth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-12

On the Ellipticity of Coupled Finite Element and One-Equation Boundary ElementMethods for Boundary Value Problems

Gunther Of and Olaf Steinbach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13

Radial Basis Functions with Applications to Elasticity

Sergej Rjasanow and Richards Grzhibovskis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14

Stokes flow about a collection of slip solid particles

A. Sellier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-15

Boundary Element Methods for Acoustic Resonance Problems

Gerhard Unger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-16

BEM Based Shape Optimization Using Shape Calculus and Multiresolution Analysis

Jan Zapletal, Kosala Bandara, Fehmi Cirak, Gunther Of and Olaf Steinbach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-17

vi

Boundary-Domain Integral Equations

Localized boundary-domain integral equations approach for Dirichlet and Robin prob-lems of the theory of piezo-elasticity for inhomogeneous solids

Otar Chkadua. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2

Numerics and spectral properties of boundary domain integral and integro-differentialoperators in 3D

Richards Grzhibovskis and Sergey E. Mikhailov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3

Spectral properties and perturbations of boundary-domain integral equations

Sergey E. Mikhailov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4

Acoustic scattering by inhomogeneous anisotropic obstacle: Boundary-domain integralequation approach

David Natroshvili. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6

Computational Micromagnetics

magnum.fe: A micromagnetic finite-element code based on FEniCS

Claas Abert. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-2

Multiscale simulation of magnetic nanostructures

Florian Bruckner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-3

Finite element and boundary element method in magnetic spin transport and magnetichybrid structures

Gino Hrkac, Marcus Page and Dieter Suess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-4

Coupling and numerical integration of LLG

Marcus Page and Dirk Praetorius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-5

A nonlocal parabolic and hyperbolic model for type-I superconductors

Karel Van Bockstal and Marian Slodicka. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-6

vii

Computational challenges in Discontinuous Galerkin methods

Energy stability for discontinuous Galerkin approximation of a problem in elasotody-namics

Paola F. Antonietti, Blanca Ayuso de Dios, Ilario Mazzieri and Alfio Quarteroni. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-2

Staggered discontinuous Galerkin methods for Maxwell’s equations

Eric Chung. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-3

Efficient Discontinuous Galerkin method for meteorological applications

Andreas Dedner and Robert Klofkorn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-4

Discontinuous Galerkin Methods for Phase Field Models of Moving Interface Problems

Xiaobing Feng. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5

Discontinuous Galerkin methods for non-linear interface problems

Emmanuil H. Georgoulis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5

On the convergence of adaptive discontinuous Galerkin methods

Thirupathi Gudi and Johnny Guzman. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-6

A cochain complex for interior penalty methods: error estimates and multigrid throughdifferential relations

Guido Kanschat and Natasha Sharma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7

Generalized DG-Methods for highly indefinite Helmholtz problems

Jens Markus Melenk, Asieh Parsania and Stefan A. Sauter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-7

Convergence of High Order Methods for the Miscible Displacement Problem

Beatrice Riviere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-8

HP-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretiza-tions of advection-dominated flows

Jaap van der Vegt and Sander Rhebergen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-9

Mixed hp-DGFEM for Linear Elasticity in 3D

Thomas P. Wihler and Marcel Wirz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-10

viii

Discontinuous Galerkin methods in fluid flows

Hybridizable Discontinuous Galerkin Methods for the incompressible Oseen and Navier-Stokes equations

Aycil Cesmelioglu, Bernardo Cockburn, Ngoc Cuong Nguyen and Jaime Peraire. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2

Commuting diagrams for the TNT elements on cubes

Bernardo Cockburn and Weifeng Qiu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-2

Development and validation of a discontinuous Galerkin wave prediction model

Ethan Kubatko and Angela Nappi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-3

Coupling of Stokes and Darcy Flows using Discontinuous Galerkin and Mimetic FiniteDifference Method

Konstantin Lipnikov, Danail Vassilev and Ivan Yotov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-4

Local Discontinuous Galerkin Method for Inkjet Drop Formation and Motion

Tatyana Medvedeva, Onno Bokhove and Jaap van der Vegt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-5

Space-time (H)DG methods for incompressible flows

Sander Rhebergen, Bernardo Cockburn and Jaap van der Vegt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-5

A Local Discontinuous Galerkin Method for the Propagation of Phase Transition inSolids

Lulu Tian, Yan Xu, J.G.M. Kuerten and Jaap van der Vegt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-6

Elliptic Eigenvalue Problems: Recent Developments in Theoryand Computation

Guaranteed lower bounds for eigenvalues

Carsten Carstensen and Joscha Gedicke. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-2

Adaptive path-following method for nonlinear PDE eigenvalue problems

Carsten Carstensen, Joscha Gedicke, V. Mehrmann and Agnieszka Miedlar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3

Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems

Carsten Carstensen, Dietmar Gallistl and Mira Schedensack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3

ix

Computation of ground states of Schrodinger operator with large magnetic fields

Monique Dauge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-4

Finite element analysis of a non-self-adjoint quadratic eigenvalue problem

Christian Engstrom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-5

Solving an elliptic eigenvalue problem via automated multi-level sub-structuring andhierarchical matrices

Peter Gerds and Lars Grasedyck. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-6

Auxiliary subspace error estimation for high-order finite element eigenvalue approxi-mations

Stefano Giani, Luka Grubisic, Harri Hakula and Jeffrey S Ovall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-7

Kato’s square root theorem as a basis for relative estimation theory of eigenvalueapproximations

Stefano Giani, Luka Grubisic, Agnieszka Miedlar and Jeffrey S Ovall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-8

High precision verified eigenvalue estimation for elliptic differential operator over polyg-onal domain of arbitrary shape

Xuefeng Liu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9

Spectral analysis for a mixed finite element formulation of the elasticity equations

Salim Meddahi, David Mora and Rodolfo Rodrıguez. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-10

Finite Elements for Elliptic Eigenvalue Problems in the Preasymptotic Regime

Stefan A. Sauter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-10

Accurate Computations of Matrix Eigenvalues with Applications to Differential Oper-ators

Qiang Ye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-11

Error Estimation and adaptive modelling

Reduced basis finite element heterogeneous multiscale method for quasilinear problems

Yun Bai. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-2

x

Error Estimation and Adaptive Modeling for Viscous Incompressible Flows

Paul T. Bauman. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-3

Goal-Oriented Error Estimation and Adaptivity for the Time-Dependent Low-MachNavier-Stokes Equations

Varis Carey and Paul T. Bauman. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-3

Adaptive inexact Newton methods with a posteriori stopping criteria for nonlineardiffusion PDEs

Alexandre Ern and Martin Vohralık. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-4

Tree approximation versus AFEM

Francesca Fierro, Alfred Schmidt and Andreas Veeser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-5

Contraction and Optimal Convergence of a Goal-Oriented Adaptive Finite ElementMethod

Ricardo H. Nochetto, A.J. Salgado and K.G. van der Zee. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-6

Finite Element Methods for Convection-Dominated Problems

A computable error bound for a 3-dimensional convection-diffusion-reaction equation

Mark Ainsworth, Alejandro Allendes, Gabriel R. Barrenechea and Richard Rankin

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-2

Augmented Taylor-Hood Elements for Incompressible Flow

Daniel Arndt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-3

A nonlinear dissipation to avoid local oscillations for the finite element approximationof the convection-diffusion equation

Joan Baiges and Ramon Codina. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-4

Investigations of a FEM-FCT scheme applied to a 1D model problem

Gabriel R. Barrenechea, Volker John and Petr Knobloch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-5

A posteriori error estimation in stabilized discretizations of stationary convection-diffusion-reaction problems

Markus Bause and Kristina Schwegler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-6

xi

Robust error estimates in weak norms with application to implicit large eddy simulation

Erik Burman. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-7

Anisotropic Local Projection Stabilization in Streamline and Crosswind Directions

Helene Dallmann and Gert Lube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-8

On Superconvergence for Higher-Order FEM in Convection-Diffusion Problems

Sebastian Franz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-9

An adaptive SUPG method for evolutionary convection-diffusion equations

Javier de Frutos, Bosco Garcıa-Archilla and Julia Novo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-9

SUPG finite element method for PDEs in time-dependent domains

Sashikumaar Ganesan and Shweta Srivastava. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-10

Stabilization of convection-diffusion problems by Shishkin mesh simulation

Bosco Garcıa-Archilla. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-11

A robust SUPG norm a posteriori error estimator for stationary convection-diffusionequations

Volker John and Julia Novo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-11

Velocity-pressure reduced order models for the incompressible Navier–Stokes equations

Volker John. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-12

A Finite Element Method for a Noncoercive Elliptic Convection Diffusion Problem

Klim Kavaliou and Lutz Tobiska. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-12

On the Role of the Helmholtz Decomposition in Mixed Methods for IncompressibleFlows and a New Variational Crime

Alexander Linke. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-13

A two-level local projection stabilisation on uniformly refined triangular meshes

Gunar Matthies and Lutz Tobiska. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-14

A Flux-Corrected Transport method based on local projection stabilization for non-stationary transport problems

Friedhelm Schieweck and Dmitri Kuzmin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-15

xii

Towards Anisotropic Quality Tetrahedral Mesh Generation

Hang Si. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-16

A local projection stabilization method for finite element approximation of a magne-tohydrodynamic model

Benjamin Wacker and Gert Lube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-17

Finite Element Methods for Multiphysics Problems

A stabilized finite volume element formulation for sedimentation-consolidation pro-cesses

Raimund Burger, Ricardo Ruiz-Baier and Hector Torres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-2

Double layer potential boundary conditions for the Hybridizable Discontinuous Galerkinmethod

Zhixing Fu, Norbert Heuer and Francisco-Javier Sayas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-3

A linear finite element scheme for the stochastic Landau–Lifshitz–Gilbert equation

Beniamin Goldys, Kim-Ngan Le and Thanh Tran. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-4

A decoupled preconditioning technique for a mixed Stokes-Darcy model

Antonio Marquez, Salim Meddahi and Francisco-Javier Sayas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-5

Conforming and divergence-free Stokes elements

Michael Neilan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-5

An exactly divergence-free finite element method for a generalized Boussinesq problem

Ricardo Oyarzua and Dominik Schotzau. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-6

hp-Time-Discontinuous Galerkin for Pricing American Put Options

Ernst P. Stephan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-6

Finite Elements in Nonlinear Spaces

Subdivision Method for the Canhan-Helfrich model

Jingmin Chen, Sara Grundel, Robert Kusner, Thomas Yu and Andrew Zigerelli. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-2

xiii

On Potts and Blake-Zisserman functionals for manifold-valued data

Laurent Demaret, Martin Storath and Andreas Weinmann. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-3

B-Spline quasiinterpolation of manifold-valued data

Philipp Grohs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-4

Intrinsic discretization error bounds for geodesic finite element approximations of el-liptic minimization problems

Hanne Hardering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-4

Simulation of Q-tensor fields with constant orientational order parameter in the theoryof uniaxial nematic liquid crystals

Alexander Raisch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-5

Finite elements for problems with singularities

Eigenvalue problems in a non-Lipschitz domain

Gabriel Acosta and Marıa Gabriela Armentano. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-2

Anisotropic mesh refinement in polyhedral domains: error estimates with data in L2(Ω)

Thomas Apel, Ariel Lombardi and Max Winkler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-3

Strong convergence for Gauss’ law with edge elements

Patrick Ciarlet, Haijun Wu and Jun Zou. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-4

hp-Adaptive FEM Based on Continuous Sobolev Embeddings

Thomas Fankhauser, Thomas P. Wihler and Marcel Wirz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-5

Mapping and regularity results for Schroedinger operators with inverse square poten-tials

Eugenie Hunsicker, Hengguang Li, Victor Nistor and Vile Uski. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-6

Finite Element Method for Schroedinger operators with inverse square potentials

Eugenie Hunsicker, Hengguang Li, Victor Nistor, Jorge Sofo and Vile Uski. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-7

Linear Finite Elements may be only First-Order Pointwise Accurate on AnisotropicTriangulations

Natalia Kopteva. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-8

xiv

hp finite element methods for singularly perturbed transmission problems

Serge Nicaise and Christos Xenophontos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-9

Foundations of isogeometric analysis

Towards isogeometric analysis for compressible flow problems and unstructured meshes

R. Abgrall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-2

Arbitrary-degree Analysis-suitable T-splines

L. Beirao da Veiga, Annalisa Buffa, Giancarlo Sangalli and Rafael Vazquez. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-3

Isogeometric Analysis and Non-matching Domain Decomposition Methods

Michel Bercovier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-5

Implementation of high order impedance boundary conditions in isogeometric methods

Annalisa Buffa, Luca Di Rienzo and Rafael Vazquez. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-6

A Computational Cost Analysis of Isogeometric Analysis

Nathan Collier, Lisandro Dalcin, David Pardo, Maciej Paszynski and Victor Calo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-7

Mixed Isogeometric Collocation Methods for the Stokes Equations

John A. Evans, Dominik Schillinger, Rene Hiemstra and Thomas J.R. Hughes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-8

Algebraic Multilevel Preconditioning in Isogeometric Analysis

Krishan Gahalaut and Satyendra Tomar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-9

Guaranteed and sharp a-posteriori error estimates in isogeometric analysis

S.K. Kleiss and Satyendra Tomar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-10

Local refinements in IgA based on hierarchical generalized B-splines

Carla Manni, Francesca Pelosi and Hendrik Speleers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-11

Efficient assembly method for isogeometric discretizations

Angelos Mantzaflaris and Bert Juttler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-12

xv

Comparison of boundary element method discretisation technologies for acoustic anal-ysis

Robert N. Simpson, Michael A. Scott, Matthias Taus, Derek C. Thomas andHaojie Lian

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-13

Splines on triangulations in isogeometric analysis

Hendrik Speleers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-14

Approximation Properties of Singular Parametrizations in Isogeometric Analysis

Thomas Takacs and Bert Juttler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-15

Adaptive Hierarchical B-Splines for Local Refinement in Isogeometric Analysis

Anh-Vu Vuong and Bernd Simeon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-15

Global and local error estimates for problems with singularitiesor low regularity

Error analysis of discontinuous Galerkin methods for the Stokes problem under minimalregularity

Santiago Badia, Ramon Codina, Thirupathi Gudi and Johnny Guzman. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-2

Optimality of an adaptive FEM for controlling local energy errors

Alan Demlow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-2

Optimal error estimates for the parabolic problem in L∞(Ω;L2([0, T ])) norm

Dmitriy Leykekhman and Boris Vexler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-3

A posteriori estimation of hierarchical type for a Schrodinger operator with inversesquare potential

Hengguang Li and Jeffrey S Ovall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-4

Localized pointwise estimates for the fully nonlinear Monge-Ampere equation

Michael Neilan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-5

Robust Localization of the Best Error with Finite Elements in the Reaction-DiffusionNorm

Francesca Tantardini, Andreas Veeser and Rudiger Verfurth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-6

xvi

High order finite element methods: A mini symposium cele-brating Leszek Demkowicz’s contributions

FEM with discrete transparent boundary conditions for the Cauchy problem for theSchrodinger equation on the whole axis

Alexander Zlotnik. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-2

High Order FEM for Wave Propagation: Like it or lump it

Mark Ainsworth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-3

Commuting Quasi interpolants for T-Spline Spaces

Annalisa Buffa, Giancarlo Sangalli and Rafael Vazquez. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-4

A PDE-constrained optimization approach to the discontinuous Petrov-Galerkin methodwith a trust region inexact Newton-CG solver

Tan Bui-Thanh and Omar Ghattas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-5

A posteriori error control for DPG methods

Carsten Carstensen, Leszek Demkowicz and Jay Gopalakrishnan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-6

Improved stability estimates for the hp-Raviart-Thomas projection operator on quadri-laterals

Alexey Chernov and Herbert Egger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-7

DPG Method for Wave Propagation Problems, A Better Understanding

Leszek Demkowicz, Jay Gopalakrishnan, Jens Markus Melenk, Ignacio Muga andDavid Pardo

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-8

A space-time multigrid method for high order time discretizations

Martin Gander, Martin Neumuller and Olaf Steinbach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-9

Partial expansion of a Lipschitz domain and some applications

Jay Gopalakrishnan and Weifeng Qiu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-9

Dispersive and Dissipative Errors in the DPG Method with Scaled Norms for HelmholtzEquation

Jay Gopalakrishnan, Ignacio Muga and Nicole Olivares. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-10

xvii

Adaptive and hybridized Hermite methods for initial-boundary value problems

Thomas Hagstrom, Daniel Appelo and Ronald Chen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-11

On hp-Boundary Layer Sequences

Harri Hakula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-12

Discontinuous Galerkin hp-BEM with quasi-uniform meshes

Norbert Heuer and Salim Meddahi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-13

Two-Grid hp–Adaptive Discontinuous Galerkin Finite Element Methods for Second–Order Quasilinear Elliptic PDEs

Paul Houston. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-14

New Hybrid Discontinuous Galerkin Methods

Youngmok Jeon and Eun-Jae Park. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-15

Godunov SPH Methods for Simulating Complex Flows with Free Surfaces Over RapidlyChanging Natural Terrains

Dinesh Kumar, E. B. Pitman and A. K. Patra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-16

Application of hp Finite Elements to the Accurate Computation of Polarisation Tensorsfor the Eddy Current Problem

P.D. Ledger and W.R.B. Lionheart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-17

Recent advances in finite element simulation of electromagnetic wave propagation inmetamaterials

Jichun Li. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-18

hp-FEM for singular perturbations with multiple scales

Jens Markus Melenk and Christos Xenophontos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-19

hp Adaptive Finite Element Methods Based on Derivatives Recovery and Supercon-vergence

Hieu Nguyen and Randolph E. Bank. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-19

Comparison of different finite element models and methods for the Girkmann Shell-Ring Problem

Antti H. Niemi and Julien Petit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-20

xviii

Pyramidal finite elements

Nilima Nigam, Argyrios Petras and Joel Phillips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-20

Application of the adaptive finite element method to numerical simulations of arteries

Waldemar Rachowicz and Adam Zdunek. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-21

Discontinuous Petrov-Galerkin Methods for Incompressible Flow: Stokes and Navier-Stokes

Nathan V. Roberts, Leszek Demkowicz and Robert Moser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-22

Preconditioning for high order Hybrid DG Methods

Joachim Schoberl and Christoph Lehrenfeld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-23

Application of the fully automatic hp-FEM to elastic-plastic problems

Marta Serafin and Witold Cecot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-24

A new error analysis for Crank-Nicolson Galerkin FEMs for a generalized nonlinearSchrodinger equation

Jilu Wang. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-25

B-spline FEM approximation of wave equation

Hongrui Wang and Mark Ainsworth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-25

The Low-storage Curvilinear Discontinuous Galerkin Method

T. Warburton. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-26

A novel formulation for nearly inextensible and nearly incompressible finite hyperelas-ticity

Adam Zdunek, Waldemar Rachowicz and T. Eriksson. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17-27

Innovative compatible and mimetic discretizations for partialdifferential equations

Basic Principles of Virtual Element Methods

L. Beirao da Veiga, Franco Brezzi, Andrea Cangiani, Gianmarco Manzini, L.D. Mariniand Alessandro Russo

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-2

xix

Mimetic discretizations of elliptic problems

Gianmarco Manzini. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-3

A Virtual Element Method with high regularity

L. Beirao da Veiga and Gianmarco Manzini. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-4

The Virtual Element Method for general second-order elliptic operators on polygonaland polyhedral meshes

Franco Brezzi, L. Donatella Marini and Alessandro Russo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-5

Nonsmooth initial data error estimates for the finite volume element method for aparabolic problem

Panagiotis Chatzipantelidis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-6

Convection dominated discontinuous Galerkin multiscale method

Daniel Elfverson and Axel Malqvist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-7

MHM Method for Advective-Reactive Dominated Models

Christopher Harder, Diego Paredes and Frederic Valentin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-8

Trefftz-DG methods for wave propagation: hp-version and exponential convergence

Andrea Moiola, Ralf Hiptmair, Ilaria Perugia and Christoph Schwab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-9

The discrete maximum principle in the family of mimetic finite difference discretizations

Daniil Svyatskiy, Konstantin Lipnikov and Gianmarco Manzini. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-10

A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems

Marco Verani. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-11

Integrodifferential Relations in Direct and Inverse Problems ofMathematical Physics

Norm-Optimal Iterative Learning Control for a Heating Rod Based on the Method ofIntegro-Differential Relations

Harald Aschemann, Dominik Schindele and Andreas Rauh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-2

Variational Formulations of Inverse Dynamical Problems in Linear Elasticity

Georgy Kostin and Vasily Saurin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-3

xx

Design and Experimental Validation of Control Strategies for a Spatially Two-DimensionalHeat Transfer Process Based on the Method of Integro-Differential Relations

Andreas Rauh, Luise Senkel and Harald Aschemann. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-4

Integro-Differential Relations in Linear Elasticity: Static Case

Vasily Saurin and Georgy Kostin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-6

Large scale computing with applications

Adaptive Asynchronous Parallel Calculations at Petascale using Uintah

Martin Berzins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-2

Accelerator-friendly parallel adaptive mesh refinement

Carsten Burstedde, Lucas C. Wilcox, Georg Stadler and Donna Calhoun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-3

Total efficiency of core components in Finite Element frameworks

Markus Geveler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-4

Massive parallel simulation of water and solute transport in porous media

Olaf Ippisch, Markus Blatt and Jorrit Fahlke. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-4

Scalable Parallel Multilevel Solution of Elliptic Problems

Peter K. Jimack, Mark A. Walkley and Jianfei Zhang. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-5

Thoughts on general purpose finite element libraries and hybrid programming

Guido Kanschat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-6

Patching Adaptivity for Large Scale Problems - A New Lightweight Adaptive Schemeand its Application in Computational Electrocardiology

Dorian Krause, Rolf Krause, Thomas Dickopf and Mark Potse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-6

Fast and Scalable Elliptic Solvers for Anisotropic Problems in Geophysical Modelling

Eike Mueller, Robert Scheichl and Eero Vainikko. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-7

Parallel Incompressible Flow Simulations using Divergence-Free Finite Elements

Tobias Neckel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-8

xxi

On Large-Scale Mechanics Simulations with the Parallel Toolbox

Aurel Neic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-9

Recent Developments in NGSolve for Distributed and Many-Core Parallel Computing

Joachim Schoberl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-10

Analysis of adaptive space-time finite elements for parabolic problems

Kunibert G. Siebert. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-10

Scalable solvers for elliptic problems discretized by adaptive high-order finite elements

Georg Stadler, Tobin Isaac, Hari Sundar, Carsten Burstedde and Omar Ghattas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-11

Algebraic multilevel preconditioning in H(curl) and H(div) space

Satyendra Tomar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-11

Low Rank Tensor Based Numerical Methods

Adaptive methods based on tensor representations of coefficient sequences and theircomplexity analysis

Markus Bachmayr and Wolfgang Dahmen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-2

Black Box Approximation Strategies in the Hierarchical Tensor Format

Jonas Ballani and Lars Grasedyck. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-3

Alternating minimal energy methods for linear systems in higher dimensions. Part II:Faster algorithm and application to nonsymmetric systems

Sergey V. Dolgov and Dmitry V. Savostyanov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-4

hp-DG time stepping for high-dimensional evolution problems with low-rank tensorstructure

Vladimir Kazeev. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-5

Hartree-Fock eigenvalue solver using tensor-structured two-electron integrals

Venera Khoromskaia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-6

Super-fast solvers for PDEs discretized in the quantized tensor spaces

Boris Khoromskij. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-7

xxii

Alternating minimal energy methods for linear systems in higher dimensions. Part I:SPD systems

Dmitry V. Savostyanov and Sergey V. Dolgov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-8

Mathematical and statistical modeling in biology

Acoustic Localisation of Coronary Artery Stenosis: Wave Propagation in Soft TissueMimicking Gel

H. Thomas Banks, Malcolm J. Birch, Mark P. Brewin, Steve E. Greenwald, Shuhua Hu,Zackary Kenz, Carola Kruse, Dwij Mehta, Simon Shaw and John R. Whiteman

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-2

Efficient numerical methods for coupled PDE-ODE systems: An application in inter-cellular signaling

Thomas Carraro, Elfriede Friedmann and Daniel Gerecht. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-3

Modeling and inverse problem considerations for a viscoelastic tissue model

Zackary Kenz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-3

High Order Space-Time Finite Element Schemes for the Dynamics of Viscoelastic SoftTissue

Carola Kruse, Simon Shaw, John R. Whiteman, H. Thomas Banks, Zackary Kenz,Shuhua Hu, Steve E. Greenwald, Mark P. Brewin and Malcolm J. Birch

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-4

New advances in a posteriori error estimation

Computable error bounds for finite element approximation on non-polygonal domains

Mark Ainsworth and Richard Rankin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-2

Guaranteed and robust error bounds for singularly perturbed problems in arbitrarydimension

Mark Ainsworth and Tomas Vejchodsky. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-3

Instance optimality for the maximum strategy

Lars Diening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-4

A framework for robust a posteriori error control in unsteady nonlinear advection-diffusion problems

Vıt Dolejsı, Alexandre Ern and Martin Vohralık. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-5

xxiii

Quasi-optimal AFEM for non-symmetric operators

Michael Feischl, Thomas Fuhrer and Dirk Praetorius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-6

A posteriori error estimates for the wave equation

Omar Lakkis, Emmanuil H. Georgoulis and Charalambos Makridakis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-7

On Mathematical Methods Generating Fully Reliable A Posteriori Estimates for Non-linear Boundary Value Problems

Sergey Repin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-8

Adaptive finite elements for PDE constrained optimal control problems

Kunibert G. Siebert. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-8

Non-Standard Finite Elements and Solvers in Solid Mechanics

A new coarse space for FETI-DP in the context of almost incompressible elasticity

Sabrina Gippert, Axel Klawonn and Oliver Rheinbach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-2

Nonlinear FETI-DP and BDDC Methods

Axel Klawonn, Martin Lanser and Oliver Rheinbach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-2

LSFEM for geometrically and physically nonlinear elasticity problems

Benjamin Muller, Gerhard Starke, Jorg Schroder, Alexander Schwarz and Karl Steeger

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-3

An Approach to Adaptive Coarse Spaces in FETI-DP Methods

Oliver Rheinbach, Axel Klawonn and Patrick Radtke. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-3

Geodesic Finite Elements

Oliver Sander. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-4

Aspects on mixed least-squares finite elements for hyperelastic problems

Alexander Schwarz, Karl Steeger, Jorg Schroder, Gerhard Starke and Benjamin Muller

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-5

On isogeometric finite elements in solid mechanics and vibrational analysis

Bernd Simeon and Oliver Weeger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-6

xxiv

Momentum Balance in First-Order System Finite Element Methods for Elasticity

Gerhard Starke. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24-7

Novel Methods for Time-Harmonic Wave Equations

Analysis of a Cartesian PML approximation to acoustic scattering problems in Rn

James H. Bramble and Joseph E. Pasciak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-2

A high frequency BEM for scattering by a class of nonconvex obstacles

Simon Chandler-Wilde, David Hewett, Stephen Langdon and Ashley Twigger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-3

A high frequency boundary element method for scattering by two-dimensional screens

Simon Chandler-Wilde, David Hewett, Stephen Langdon and Ashley Twigger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-4

Solving the steady-state ab-initio laser theory with FEM

Sofi Esterhazy, Matthias Liertzer, Jens Markus Melenk and Stefan Rotter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-5

How should one choose the shift for the shifted Laplacian to be a good preconditionerfor the Helmholtz equation?

Martin Gander, I. G. Graham and E. A. Spence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-6

Hybrid numerical-asymptotic approximation for high frequency scattering by penetra-ble convex polygons

Samuel Groth, David Hewett and Stephen Langdon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-7

Analysis of preconditoners for Helmholtz equation using Pesudospectrum

Antti Hannukainen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-9

A domain decomposition preconditioner for mixed hybrid infinite elements

Martin Huber, Lothar Nannen and Joachim Schoberl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-10

Improving the Shifted Laplace Preconditioner by Multigrid Deflation

A. H. Sheikh, D. Lahaye and C. Vuik. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-11

xxv

Numerical Methods for Parabolic Equations

Energy conservative/dissipative approximations of nonlinear evolution problems

Charalambos Makridakis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26-2

A posteriori error analysis for dG in time ALE formulations

Andrea Bonito, Irene Kyza and Ricardo H. Nochetto. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26-3

Discontinuous Galerkin Approximation of porous Fisher-Kolmogorov Equations

Fausto Cavalli, Giovanni Naldi and Ilaria Perugia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26-4

On adaptive discontinuous Galerkin methods for parabolic problems

Emmanuil H. Georgoulis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26-5

The hp-adaptive Galerkin time stepping method for nonlinear differential equationswith finite time blow up

Barbel Janssen and Thomas P. Wihler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26-5

Maximum-norm strong approximation rates for noisy reaction-diffusion equations

Omar Lakkis, G.T. Kossioris and M. Romito. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26-6

A new approach to error analysis of fully discrete finite element methods for nonlinearparabolic equations

Buyang Li and Weiwei Sun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26-7

Numerical Methods for Reaction-Transport Equations with Ap-plications in Medicine

Finite element analysis of the mechano-chemical regulation of wound contraction insurgical wounds

Etelvina Javierre, Clara Valero, Maria Jose Gomez-Benito and Jose Manuel Garcia-Aznar

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-2

Presentation of results of finite-element analyses on a two-dimensional mechanochem-ical model for dermal wound healing

D.C. Koppenol and Fred J. Vermolen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-3

xxvi

Mathematical modelling and numerical simulations of actin dynamics in the eukaryoticcell

Anotida Madzvamuse, Uduak George and Angelique Stephanou. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-4

Analyzing the Treatment of a Bacterial Infection in a Wound Using Oxygen Therapy

Richard Schugart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-5

A Semi–Stochastic Model for the Immune Response System

Fred J. Vermolen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-6

Multiscale models of tumor cells: from in-vitro aggregates to in-vivo vascularized tu-mors

Irene Vignon-Clementel, Nick Jagiella and Dirk Drasdo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-7

Numerical methods for contact and other geometrically non-linear problems

Parallel solution of contact shape optimization problems with Coulomb friction basedon domain decomposition

P. Beremlijski, Tomas Brzobohaty, Tomas Kozubek and Alexandros Markopoulos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-2

Parallel solution of elasto-plastic problems

Martin Cermak and Michal Merta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-3

Convergence analysis for multilevel variance estimators in Multilevel Monte CarloMethods and application for random obstacle problems

Alexey Chernov and Claudio Bierig. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-4

Scalable algorithms and conditioning of constraints arising from variationally consistentdiscretization of contact problems

Zdenek Dostal, Tomas Kozubek and Oldrich Vlach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-5

Parallel solution of contact problems based on TFETI

Zdenek Dostal, Tomas Brzobohaty, Tomas Kozubek, Alexandros Markopoulosand Oldrich Vlach

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-6

Local averaging of contact with non matching meshes

Guillaume Drouet and Patrick Hild. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-8

xxvii

BE/FE Approximation of higher order for nonsmooth problems, effective quadrature,and time discretization by implicit Runge-Kutta methods

Joachim Gwinner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-9

A discretization for dynamic large deformation contact problems of nonlinear hypere-lastic continua

Ralf Kornhuber, Oliver Sander and Jonathan Youett. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-10

Parallel Level Set Methods for Large Deformation Contact Problems

Rolf Krause, Valentina Poletti, Roberto Croce and Petr Kotas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-11

Error estimators for a partially clamped plate problem with boundary elements

Matthias Maischak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-12

Optimal active-set and spectral algorithms for the solution of 3D contact problemswith anisotropic friction

Lukas Pospısil, Zdenek Dostal and Tomas Kozubek. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-13

hp-adaptive FEM with biorthogonal basis functions for elliptic obstacle problems

Andreas Schroder and Lothar Banz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-14

High order BEM for frictional contact problems

Ernst P. Stephan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28-15

Numerical methods for fully nonlinear elliptic equations

Pseudo transient continuation and time marching methods for Monge-Ampere typeEquations

Gerard Awanou. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-2

General full discretizations for center manifolds, here for fully nonlinear equations andFEMs

Klaus Bohmer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-2

Numerical Solution of Monge-Ampere Equation on Domains Bounded by PiecewiseConics

Oleg Davydov and Abid Saeed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-3

xxviii

Discontinuous Galerkin Finite Element Differential Calculus and Applications

Xiaobing Feng. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-3

A Finite Element Method for Hamilton-Jacobi-Bellman equations

Max Jensen and Iain Smears. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-4

Adaptivity and fully nonlinear problems

Omar Lakkis and Tristan Pryer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-4

Finite element methods for the Monge-Ampere equation

Michael Neilan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29-5

Numerical modeling of flow in subsurface reservoirs

Multiphase flow through porous media: An adaptive control volume finite elementmethod

Peyman Mostaghimi, James R. Percival, Brendan S. Tollit, Stephen J. Neeth-ling, Gerard J. Gorman, Matthew D. Jackson, Christopher C. Pain and JeffersonL.M.A. Gomes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-2

Multipoint flux domain decomposition time-splitting methods on general grids

Andres Arraras, Laura Portero and Ivan Yotov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-3

An optimization approach to large scale simulations of fluid flows in fractured mediawith finite elements on nonconforming grids

Stefano Berrone, Sandra Pieraccini and Stefano Scialo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-4

Pressure jump interface law for the Stokes-Darcy coupling: Confirmation by directnumerical simulations

Thomas Carraro and Christian Goll. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-5

Efficient Bayesian uncertainty quantification of subsurface flow models using nestedsampling and sparse polynomial chaos surrogates

Ahmed H. Elsheikh, Mary F. Wheeler and Ibrahim Hoteit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-6

High-order cut-cell techniques for numerical upscaling in porous media

Christian Engwer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-7

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Adjoints of finite element models

Patrick E. Farrell and Simon W. Funke. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-8

A global Jacobian method for mortar discretizations of nonlinear porous media flows

Benjamin Ganis, Mika Juntunen, Gergina Pencheva, Mary F. Wheeler and Ivan Yotov

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-9

Direct numerical simulation of two-phase flow at the pore scale

Ali Q Raeini, Branko Bijeljic and Martin J Blunt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-10

Modeling flow with nonplanar fractures

Gurpreet Singh, Omar Al-Hinai, Gergina Pencheva and Mary F. Wheeler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-11

Computational Environments for Energy and Environmental Modeling in Porous Media

Mary F. Wheeler. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-11

Multiscale domain decomposition methods for porous media flow coupled with geome-chanics

Ivan Yotov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30-12

PDEs on Surfaces

Unfitted finite element methods using bulk meshes for surface partial differential equa-tions

Klaus Deckelnick, Charles M. Elliott and Tom Ranner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-2

Discontinuous Galerkin methods for surface PDEs

Andreas Dedner, Pravin Madhavan and Bjorn Stinner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-3

Pattern formation in morphogenesis on evolving biological surfaces: Theory, numericsand applications

Anotida Madzvamuse, Raquel Barreira, Charles M. Elliott, Ammon J. Meir andNecibe Tuncer

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-4

An ALE ESFEM for solving PDEs on evolving surfaces

Vanessa Styles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-5

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Numerical Simulations of Chemotaxis-Driven PDEs on surfaces

Stefan Turek and Andriy Sokolov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31-6

Sensitivity analysis and optimization for fluid-structure inter-action problems

Parameter Estimation in Fluid-Structure Interaction and Subsurface Flows

Ahmed H. Elsheikh, Thomas Richter, Mary F. Wheeler and Thomas Wick. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32-2

Towards optimal control of large deformation FSI problems including contact and topol-ogy change

Thomas Richter and Thomas Wick. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32-3

Calculation of sensitivities for fluid-structure interactions

Thomas Wick and Winnifried Wollner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32-3

Stochastic finite elements and PDEs

A posteriori error estimation for stochastic Galerkin FEMs

Alex Bespalov, Catherine Powell and David Silvester. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-2

Weak truncation error estimates for elliptic PDEs with lognormal coefficients

Julia Charrier and Arnaud Debussche. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-3

Adaptive Anisotropic Spectral Stochastic Methods for Uncertain Scalar ConservationLaws

Alexandre Ern, Olivier Le Maıtre and Julie Tryoen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-4

Sparse adaptive tensor Galerkin approximations of stochastic PDE-constrained controlproblems

Angela Kunoth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-5

Finite element approximation of the Cahn-Hilliard-Cook equation

Stig Larsson. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-5

Exploring Emerging Manycore Architectures for Uncertainty Quantification ThroughEmbedded Stochastic Galerkin Methods

Eric Phipps, H. Carter Edwards, Jonathan Hu and Jakob T. Ostien. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-6

xxxi

Adaptive, Sparse Quadratures for Bayesian Inverse Problems

Claudia Schillings and Christoph Schwab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-7

Multilevel Markov chain Monte Carlo algorithms for uncertainty quantification in sub-surface flow

Aretha Teckentrup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33-7

Superconvergence in DG: analysis and recovery

A New Lax-Wendroff Discontinuous Galerkin Method with Superconvergence

Wei Guo, Jianxian Qiu and Jing-Mei Qiu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34-2

Energy norm error estimation for averaged discontinuous Galerkin methods in onespace dimension

Ferenc Izsak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34-2

Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering: Practical Considera-tions When Applied to Visualization

Robert M. Kirby. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34-3

Error Estimation for the Discontinous Galerkin Method Applied to Hyperbolic Con-servation Laws

Lilia Krivodonova and Noel Chalmers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34-4

Computationally Efficient Boundary Filtering Using Smoothness-Increasing Accuracy-Conserving (SIAC) Methods

Xiaozhou Li. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34-5

Superconvergence of a HDG method for fractional diffusion problems

Kassem Mustapha and Bernardo Cockburn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34-6

Post-processing discontinuous Galerkin solutions to Volterra integro-differential equa-tions: Analysis and Simulations

Jennifer K. Ryan and Kassem Mustapha. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34-7

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Time-Domain Boundary Integral Equations

A fully discrete Kirchhoff formula based on CQ and Galerkin BEM

Lehel Banjai, Antonio Laliena and Francisco-Javier Sayas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-2

Time-domain FEM/BEM coupling

Lehel Banjai, Volker Gruhne, Christian Lubich and Francisco-Javier Sayas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-3

Convolution-in-time approximations of TDBIEs

Penny J Davies and Dugald B Duncan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-4

Quadrature Schemes and Adaptivity for 2D Time Domain Boundary Element Methods(TD-BEM)

Matthias Glafke and Matthias Maischak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-5

Solving the heat equation with a fast multipole Galerkin boundary element method

Michael Messner, Johannes Tausch, Martin Schanz and Wolfgang Weiss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-6

A Generalized Convolution Quadrature with Variable Time Stepping

Stefan A. Sauter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-7

Adaptive methods for retarded boundary integral equations

Stefan A. Sauter and Alexander Veit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-7

BEM for Parabolic Phase Phange Problems with Moving Interfaces

Johannes Tausch and Elizabeth Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-8

A hybrid approach to the time marching solution of Maxwell’s equations

Daniel S. Weile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-9

Using space-time Galerkin stability theory to define a robust collocation method fortime-domain boundary integral equations in electromagnetics

Elwin van ’t Wout, Duncan R. van der Heul, Harmen van der Ven and Kees Vuik. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35-10

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1 Abstracts of talks of invited speakers

FINITE ELEMENT METHODS IN COASTAL OCEAN MODELING:SUCCESSES AND CHALLENGES

Clint N Dawson

The University of Texas at Austin,Department of Aerospace Engineering and Engineering Mechanics,

College of Engineering, 1 University Station C0600, Austin TX 78712, [email protected]

The coastal ocean contains a diversity of physical and biological processes, often oc-curring at vastly different scales. In this talk, we will outline some of these processesand their mathematical description. We will then discuss how finite element methodsare used in coastal ocean modeling and recent research into improvements to thesealgorithms. We will also highlight some of the successes of these methods in simulat-ing complex events, such as hurricane storm surges. Finally, we will outline severalinteresting challenges which are ripe for future research.

FORTY YEARS OF THE CROUZEIX-RAVIART ELEMENT

Susanne C. Brenner

Center for Computation and Technology and Department of Mathematics,Louisiana State University, Baton Rouge, LA 70803, USA

[email protected]

The Crouzeix-Raviart element, which is the simplest example of a nonconforming finiteelement, was introduced in a 1973 paper on the Stokes equation. In this talk I willdiscuss old and new results related to the Crouzeix-Raviart element that illustrate themethodologies for handling various aspects of nonconforming finite element methods,such as a priori and a posteriori error analyses, and the design and analysis of fastsolvers.

1-1

ADVANCES IN REDUCING THE MESH BURDEN INCOMPUTATIONAL MECHANICS APPLICATIONS TO

FRACTURE AND SURGICAL SIMULATION

Stephane P.A. Bordas a

Institute of Mechanics & Advanced Materials (IMAM),School of Engineering, Cardiff University,

Queen’s Buildings, The Parade, CARDIFF CF24 3AA, Wales, [email protected], [email protected]

This presentation will address recent advances in enriched numerical methods to sim-plify the treatment of evolving discontinuities in the field variables or their derivatives:cracks or material interfaces; and to treat geometrically intricate domains and theirevolution

The presentation will be composed of three parts:

1. advances in numerical methods aiming at simplifying the treatment of complexgeometries;

Two competing approaches coexist in the literature to simplify the solution ofpartial differential equations over domains of complex and/or evolving geome-tries. One focuses on streamlining the transition between computer aided design(CAD) data and the solution of problems over the corresponding domains. Anexample of this is isogeometric analysis [3] where the geometry description andthe approximation of the field variables are tied, thus enabling an exact treatmentof the boundary as well as simplifying eventual geometric design iterations.

The second approach follows an orthogonal direction, where the geometry is un-coupled from the field variable discretisation, e.g. embedded boundary methodssuch as the structured extended finite element method of [2].

We will present results emanating from both lines of thought: isogeometric analysis of three-dimensional structures [8] and embedded interfaces for complex ge-ometries including sharp edges and vertices [6]. Some of these methods werereviewed in [4] and a tutorial on isogeometric analysis and associated implemen-tation aspects (MATLAB) is provided in [7].

2. advances in enriched formulations for evolving discontinuities such as cracks

The extended finite element method (XFEM) was introduced in [1] with, as abasis, the partition of unity enrichment method of [5]. The XFEM enables thesimulation of evolving discontinuities without or with minimal remeshing.

We present recent developments in the area and focus on tackling difficultiesassociated with the control of the conditioning number of the system, the controlof the error in quantities of interest. We also discuss results on the smoothed(extended) finite element method.

3. applications of those methods to problems in multi-crack growth and brain surgerysimulation.

1-2

Finally, as an application, we will present a simple method to grow several hun-dreds of cracks in two-dimensions in order to predict their growth and coalescencein brittle materials using the XFEM.

We will also present recent results permitting the simulation of cutting and con-tact during brain surgery simulation at 30 frames per second using an implicittime integration method and a hybrid, asynchronous CPU/GPU solver.

We finish the presentation by conclusions and propositions for future work.

Acknowledgements

Stephane P.A. Bordas wishes to thank the organisers of MAFELAP for their kind in-vitation. He is grateful to the European Research Council for funding the researchpresented (ERC Stg grant agreement No. 279578: “RealTCut Towards real time mul-tiscale simulation of cutting in non-linear materials with applications to surgical sim-ulation and computer guided surgery”).

References

[1] Ted Belytschko and T Black. Elastic crack growth in finite elements with minimalremeshing. International journal for numerical methods in engineering, 45(5):601–620, 1999.

[2] Ted Belytschko, Chandu Parimi, Nicolas Moes, N Sukumar, and Shuji Usui. Struc-tured extended finite element methods for solids defined by implicit surfaces. In-ternational journal for numerical methods in engineering, 56(4):609–635, 2003.

[3] Thomas JR Hughes, John A Cottrell, and Yuri Bazilevs. Isogeometric analysis:Cad, finite elements, nurbs, exact geometry and mesh refinement. Computer meth-ods in applied mechanics and engineering, 194(39):4135–4195, 2005.

[4] H Lian, SPA Bordas, R Sevilla, and RN Simpson. Recent developments incad/analysis integration. arXiv preprint arXiv:1210.8216, 2012.

[5] Jens Markus Melenk and Ivo Babuska. The partition of unity finite element method:basic theory and applications. Computer methods in applied mechanics and engi-neering, 139(1):289–314, 1996.

[6] Mohammed Moumnassi, Salim Belouettar, Eric Bechet, Stephane Bordas, DidierQuoirin, and Michel Potier-Ferry. Finite element analysis on implicitly defined do-mains: An accurate representation based on arbitrary parametric surfaces. Com-puter methods in applied mechanics and engineering, 200(5):774–796, 2011.

[7] Vinh Phu Nguyen, Robert N Simpson, Stephane Bordas, and Timon Rabczuk.An introduction to isogeometric analysis with matlab\ textsu perscript \ tex-tregistered implementation: Fem and xfem formulations. arXiv preprintarXiv:1205.2129, 2012.

[8] MA Scott, RN Simpson, JA Evans, S Lipton, SPA Bordas, TJR Hughes, andTW Sederberg. Isogeometric boundary element analysis using unstructured t-splines. Computer Methods in Applied Mechanics and Engineering, 254:197221,2012.

1-3

FIELDS, CONTROL FIELDS, AND COMMUTINGDIAGRAMS IN ISOGEOMETRIC ANALYSIS

Annalisa Buffa1, Giancarlo Sangalli2 and Rafael Vazquez1

1 IMATI CNR “E. Magenes”, Via Ferrata 1, 27100 Pavia, [email protected]

2 Universita degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy

The numerical discretization of equations enjoying a relevant geometric structure, is oneof the most interesting challenge of numerical analysis for PDEs and several results havebeen obtained in the last decade. On the one hand, discrete schemes have to preservethe geometric structure of the underlying PDEs in order to avoid spurious behaviors,instability or non-physical solutions: e.g., in electromagnetics, numerical schemes hasto be related with a discrete De Rham complex. On the other hand, especially in viewof high frequency computations, numerical schemes need to be efficient and accurate.

In this talk, I will present splines approximations of the vector fields and I willdiscuss the properties of the spline discretization of the De Rham complex. I will showthe relation between the spline complex and the topology of the knot mesh and of thecontrol net. I will address the construction of canonical basis for spline spaces of vectorfields, and possibly discuss the extension to NURBS.

I will finish my talk with a few numerical results.

References

[1] A. Buffa, G. Sangalli, R. Vazquez, Isogeometric Methods for Computa-tional Electromagnetics: B-spline and T-spline discretizations, J. Comput. Phys.,submitted.

[2] A. Buffa, J. Rivas, G. Sangalli, R. Vazquez, Isogeometric Discrete Differ-ential Forms in Three Dimensions, SIAM J. Numer. Anal. 49 (2011), pp. 818-842.

[3] A. Buffa, G. Sangalli, R. Vazquez, Isogeometric analysis in electromagnet-ics: B-splines approximation , Comput. Methods Appl. Mech. Engrg. 199 (2010),no. 17-20, pp. 11431152.

1-4

THE INF-SUP CONSTANT OF THE DIVERGENCE

Martin Costabel

IRMAR, Universite de Rennes 1, Rennes, [email protected]

The inf-sup condition for the divergence, under different disguises variously attributedto Lions, Necas, Babuska-Aziz or Ladyzhenskaya, Babuska and Brezzi, states the posi-tivity of the inf-sup constant of a domain Ω, or equivalently, the existence of a boundedright inverse of the divergence operator mapping the Sobolev space H1

0 (Ω) to the spaceof functions square integrable on Ω with mean value zero. The condition has beenknown to hold for bounded Lipschitz domains for half a century, and its close relationto Korn’s second inequality, to the spectrum of the Schur complement of the Stokessystem and to the century-old Cosserat eigenvalue problem has been known for a verylong time, too. Yet, despite this venerable history and despite its permanent place inthe spotlight because of its importance for stability and efficiency estimates in fluiddynamics, the dependence of the constant on the domain is a subject where still manybasic questions remain open. Others have recently found answers or have seen progresstowards answers. The talk will trace some of this progress and present results obtainedtogether with Monique Dauge and in collaboration with Christine Bernardi and VivetteGirault.

1-5

FINITE ELEMENT METHODS FOR SURFACE PDES

Charles M. Elliott

Mathematics Institute and Centre for Scientific Computing,University of Warwick, Coventry CV4 7AL, UK

[email protected]

Surface partial differential equations arise in a wide variety of applications:- surfactantsin two phase flow, pattern formation on growing biological surfaces, surface phase sep-aration, diffusion along fractures in porous media, modelling of biomembranes etc.They are examples of partial differential equations on manifolds. As such they providea remaining challenge within the general subject of the numerical analysis of partialdifferential equations. The framework is essentially geometric because the domain inwhich the equation holds is curved. They are linked naturally to the geometric equa-tions for surfaces such as the minimal surface equation, motion by mean curvatureand Willmore flow. In this talk we consider finite element methods for approximatingthe solution of partial differential equations on surfaces. We focus on surface finiteelements on triangulated surfaces, implicit surface methods using level set descriptionsof the surface, unfitted finite element methods and diffuse interface methods. In orderto formulate the methods we need geometric analysis and, in the context of evolvingsurfaces, transport formulae. A wide variety of equations and applications are covered.Some ideas of the numerical analysis will be presented along with illustrative numeri-cal examples. The topic is also the subject of a mini-symposium at this conference. Iwill report on work primarily with Gerd Dziuk. Other collaborators are Klaus Deck-elnick, Tom Ranner, Bjorn Stinner, Chandrashekar Venkataramaran, Vanessa Stylesand Anotida Madzvamuse.

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A STOCHASTIC COLLOCATION APPROACH TOPDE-CONSTRAINED OPTIMIZATION FOR

RANDOM DATA IDENTIFICATION PROBLEMS

Max Gunzburger

Department of Scientific Computing, Florida State University,Tallahassee FL 32306-4120, USA

[email protected]

We present a scalable, embarrassingly parallel mechanism for optimal identification ofstatistical moments (mean value, variance, covariance, etc.) or even the whole prob-ability distribution of input random data, given the probability distribution of someresponse (quantity of physical interest) of a system of partial differential equations(PDEs). The stochastic inverse problem can be described by an objective functionalconstrained by a system of stochastic PDEs. Several identification objectives are dis-cussed that either minimize the expectation of a tracking cost functional or minimizethe difference of desired statistical quantities in the appropriate norms. The distributedparameters/controls can be either deterministic or stochastic. Given an objective, weprove the existence of an optimal solution, establish the validity of the Lagrange multi-plier rule, and obtain a stochastic optimality system of equations. To characterize datawith moderately large amounts of uncertainty, we introduce a novel stochastic param-eter identification algorithm that integrates an adjoint-based deterministic algorithmwith the sparse grid stochastic collocation finite element method. This allows for de-coupled, moderately high-dimensional, parameterized computations of the stochasticoptimality system, where at each collocation point deterministic analyses and tech-niques can be utilized. The rigorously derived error estimates for the fully discreteproblems are described and used to compare the efficiency of the method with thatof several other techniques. Numerical examples illustrate the theoretical results anddemonstrate the distinctions between the various stochastic identification objectives.

This is joint work with Catalin Trenchea (University of Pittsburgh) and ClaytonWebster (Oak Ridge National Laboratory).

1-7

TIME DOMAIN INTEGRAL EQUATIONS FORCOMPUTATIONAL ELECTROMAGNETISM

Peter Monk

Mathematical Sciences, University of Delaware, Newark DE 19716, [email protected]

Scattering problems for Maxwell’s equations can be solved in the frequency or timedomain. In the frequency domain both finite element and boundary integral methodsare in common use, and their relative strengths and weaknesses are well understood.In contrast, in the time domain the principal technique is the finite difference timedomain method (or the Discontinuous Galerkin Method). However, time domain in-tegral equations have become much more popular in recent years, although they stillrepresent a considerable coding challenge. This can be mitigated by using the convo-lution quadrature approach (CQ) [see C. Lubich, On the multistep time discretizationof linear initial-boundary value problems and their boundary integral equations, Nu-mer. Math., 67 (1994), pp. 365389.], together with a boundary Galerkin methodin space and efficient integral equation software such as BEM++ [see W. Smigaj, S.Arridge, T. Betcke, J. Phillips, M. Schweiger, “Solving Boundary Integral Problemswith BEM++”, ACM Trans. Math. Software]. The result is a convenient and robustmethod.

I shall outline the CQ method applied to Maxwell’s equations using the problem ofcomputing waves scattered by a penetrable object as a model problem. After discussingsome theoretical aspects such as error estimates and other properties of the scheme,I shall present numerical results computed using the recently developed time domaintoolbox within the BEM++ library.

1-8

THE EMERGENCE OF PREDICTIVE COMPUTATIONAL SCIENCE:VALIDATION AND VERIFICATION OF COMPUTATIONAL MODELS

OF COMPLEX PHYSICAL SYSTEMS

J. Tinsley Oden

Institute for Computational Engineering and Sciences (ICES)The University of Texas at Austin, Texas, USA

[email protected]

Predictive Science is understood to be the scientific discipline concerned with the useof mathematical and computational models to forecast physical events. It embracesthe processes of model selection, calibration, validation, verification, and their use inforecasting features of physical events with quantified uncertainty.

It is argued that the principles of predictive science and the tools it employs arerooted in the philosophical foundations of science itself and that they evoke a needfor reviewing exactly how scientific knowledge is obtained and how it is interpreted ina statistical setting. We adopt a Bayesian framework to discuss these issues and todescribe methods of statistical calibration, model plausibility, validation, and solutionof stochastic systems. We explore several areas fundamental to contemporary compu-tational science: coarse-gaining and validation of molecular models, maximum entropymethods, experimental design based on Shannon information theoretics, virtual valida-tion of models, model bias or inadequacy, and model selection. Finite element methodsenter in the analysis of macroscale models of validation experiments. Examples fromnanomanufacturing are discussed.

1-9

WHAT IS THE LARGEST FINITE ELEMENT SYSTEMTHAT CAN BE SOLVED TODAY?

Ulrich Ruede

University Erlangen-Nuremberg, Department of Computer Science 10,Cauerstraße 11, D-91058 Erlangen, Germany

[email protected]

Top supercomputers have progressed beyond the Peta-Scale, i.e. they are capable toperform in excess of 1015 operations per second and their main memory approachesthat scale. However, this performance can only be achieved with massively parallelsystems that have several hundred thousand processor cores, posing severe challengesto algorithm and software development. The fastest FE solvers, such as multigridmethods, scale linearly in numerical complexity with modest constants, but there hasbeen a debate whether they do not suffer from sequential bottlenecks. In this talk wewill present our experience in implementing multigrid solvers for FE problems with upto a trillion (1012) unknowns. This is e.g. enough to discretize the whole volume ofplanet for simulating the the Earth mantle convection problem with a global resolutionof about 1km. The compute times are then around 1 minute for computing a singleflow field, so that the algorithm can still be used reasonably within an implicit timestepping procedure.

DOUBLE COMPLEXES AND LOCALBOUNDED COCHAIN PROJECTIONS

Ragnar Winther

Centre of Mathematics for Applications (CMA), University of Oslo.CMA c/o Dept of Mathematics, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway

[email protected]

The construction of bounded projections is a key tool for establishing stability of variousfinite element methods. In this talk we discuss a new construction of projectionsfrom HΛk, the space of differential k forms which belong to L2 and whose exteriorderivative also belongs to L2, to finite dimensional subspaces consisting of piecewisepolynomial differential forms defined on a simplicial mesh of the domain. Thus, theirdefinition requires less smoothness than assumed for the definition of the canonicalinterpolants based on the degrees of freedom. Moreover, these projections commutewith the exterior derivative and are bounded in the HΛk norm independent of themesh size. Unlike some other recent work in this direction, the projections are alsolocally defined in the sense that they are defined by local operators on overlappingmacroelements, in the spirit of the Clement interpolant. A double complex structureis introduced as a key tool to carry out the construction.

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2: Parallel session talks

2 Abstracts of talks in parallel sessions

2-1


VERIFICATION OF FUNCTIONAL A POSTERIORIERROR ESTIMATES FOR OBSTACLE PROBLEM

Petr Harasim and Jan Valdmana

Centre of Excellence IT4Innovations,VSB-Technical University of Ostrava, Czech Republic

[email protected],

We verify functional a posteriori error estimates proposed by S. Repin for the case ofan obstacle problem. Quality of a numerical solution obtained by the finite elementmethod is compared with the exact solution (obtained in 1D case) to demonstrate thesharpness of the studied error estimated. Extention to 2D case will be also reported.

2-2


CONVERGENCE OF HP -FEM IN THREE DIMENSIONALCOMPUTATION OF THERMOELECTRIC EFFECTS

Razi Abdul-Rahmana and Sarah Kamaludinb

School of Mechanical Engineering, Universiti Sains Malaysia, Penang, [email protected], [email protected]

Solutions to coupled-field problems like thermoelecric effects are traditionally computedwith a partitioned-based approach where data exchanges between respective domainsmust be carefully treated to ensure convergence of solutions. Alternatively, a monolithicframework to describe the coupled fields is arguable better amenable to adaptive finiteelement solutions. A difficulty arises for many cases where the resulting system matricesare nonsymmetrical and negative definite. Resorting to nonlinear solvers is costlyfor adaptive solutions. Nevertheless, under certain assumptions of the thermoelectriceffects, it is possible to model them as a linear, symmetric system of coupled PDEs.We present such an approach following the framework which is based on the Onsagersreciprocity theorem elaborated in [C. Vokas and M. Kasper. Adaptation in multinatureproblems. In: Proceedings of the 13th International IGTE Symposium on NumericalField Calculation in Electrical Engineering, Graz, 2008]. In our implementation, onlythe constitutive relations involving the Peltier and Seebeck effects are considered torelate heat and electric potential. By assuming negligible nonlinear effects of Thomsoneffect and Joule heating, a linearization of the constitutive thermoelectric equationswith respect to a reference temperature is possible with application of the Onsagersreciprocity theorem.

Computational performance and convergence of hp-adaptive solutions in three di-mensional problems based on the framework is of interest. The hp-adaptivity is imple-mented to simultaneously reduce the discretization errors of the two physical natures,i.e., temperature and electric potential, in a similar manner as in a single nature prob-lem. An a posteriori error estimator based on element-wise residuals and jumps atelement boundaries is used. In the current implementation, h or p-extension is trig-gered if any of the two natures has not reached a prescribed error limit. Hence, dueto the direct formulation of the coupled problem, the local solution of one nature maybe over-evaluated relative to the other one. Nevertheless, the convergence of errors inenergy norm for both natures is found to be reasonble. Our implementation stronglysuggests that the thermoelectric problems in three dimensions may well be computedwith the hp-FEM so that accurate results can be achieved without an excessive use ofcomputational resources when treating them as nonlinear problems, for example as in[J. L. Perez-Aparicio, R. L. Taylor and D. Gavela. Finite element analysis of nonlinearfully coupled thermoelectric materials. Computational Mechanics, 40: 35-45, 2007].Thus, it allows fast hp-FEM computations while preserving accuracy within a range ofpractical problems.

2-3


DISCONTINUOUS GALERKIN TIME STEPPING SCHEMESCOMBINED WITH LOCAL PROJECTION STABILIZATION

METHODS APPLIED TO TRANSIENT STOKES PROBLEMS:STABILITY AND CONVERGENCE

Naveed Ahmeda and Gunar Matthies

Universitat Kassel, Fachbereich 10 Mathematik und Naturwissenschaften,Institut fur Mathematik, Heinrich-Plett-Straße 40, 34132 Kassel, Germany

[email protected]

We will consider finite element methods for solving transient Stokes problems in thecase of equal order interpolation of velocity and pressure. Since these pairs do notsatisfy an inf-sup condition, a spatial stabilization of the pressure is needed. We willapply a stabilization term based on the one-level version of local projection method.Since projection space and ansatz space are defined on the same mesh, no couplingof degrees of freedom not belonging to the same mesh cell is introduced. This is incontrast to the continuous interior penalty method, the subgrid scale modeling, theorthogonal subgrid method and the two-level local projection stabilization method.

Our main interest lies in the combination of local projection methods in space anddiscontinuous Galerkin methods of degree k in time. We will derive the unconditionalstability of the method and give error estimates for the semi discrete and for the fullydiscrete problem.

Our numerical examples will show that dG(k) methods are accurate of order k + 1in the L2(L2)-norm for velocity and pressure. The dG-norm which consists of theintegrated LPS-norm and the jumps at the discrete time points shows a convergenceof order k + 1/2 in time. At the discrete time points, the error in the velocity issuperconvergent of order 2k + 1 for dG(k). By means of a simple post-processing, thesolutions of dG(k) can be improved such that the convergence order k + 2 is obtainedfor the L2(L2)-norm of the velocity. At the discrete time points, the post-processingprovided for dG(k) a superconvergence of order 2k + 1 also for the pressure.

The choice of stabilization parameters plays a critical role in the success of stabilizedmethods. Our numerical studies will suggest how to choose stabilization parameters.

We also verify for higher order time discretization that, for small time steps, thepressure is stable for initial data that are not discretely divergence free.

The research was supported by DFG through grant MA 4713/2− 1.

2-4


ANISOTROPIC MESH ADAPTATION FOR THEAMBROSIO-TORTORELLI MODEL: APPLICATION

TO QUASI-STATIC CRACK PROPAGATION

Marco Artina1a, Massimo Fonrasier1b, Simona Perotto2c and Stefano Micheletti2d

1Faculty of Mathematics, Technische Universitat Munchen,Boltzmannstrasse 3, 85748, Garching, Germany

[email protected], [email protected],

2MOX - Modeling and Scientific Computing,Dipartimento di Matematica ”F.Brioschi”, Politecnico di Milano,

Piazza Leonardo da Vinci 32, I-20133 Milano, [email protected], [email protected]

The minimization of the Mumford-Shah functional represents a very challenging issuesince it is non-smooth and non-convex. This functional characterizes several problems.In particular, we are interested in the Francfort-Marigo model in the context of thequasi-static fracture propagation.

To numerically approximate the problem, one needs first to Γ-approximate thenon-smooth energy, which depends on the displacement and on its discontinuity set,by using a smoother version (i.e. the model proposed by Ambrosio and Tortorelli)where a smooth indicator function identifies the discontinuity set. Then, we resort toan adaptive finite element approach based on piecewise linear elements. Nevertheless,similarly to early work by Chambolle et al. but differently from recent approaches bySuli et al. where isotropic meshes are used, in this work we investigate how anisotropicmeshes can lead to significant improvements in terms of the balance between accuracyand complexity. Indeed, the main advantage achievable is a significant reduction ofthe number of elements to capture with good confidence the expected fracture path.The employment of anisotropic grids allows us to shortly follow the propagation of thefracture by refining it only in a very thin neighborhood of the crack.

In this talk, we first present the derivation of a novel anisotropic a posteriori errorestimator driving the mesh adaptation for the approximation of the Ambrosio-Tortorellimodel. Then, we provide several numerical results which corroborate the accuracy aswell as the computational saving led by an anisotropic mesh adaptation procedure.

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A POSTERIORI ERROR ANALYSISFOR TIME-DEPENDENT STOKES EQUATIONS

Eberhard Bansch1, Johan Jansson2,Fotini Karakatsani3 and Charalambos Makridakis4

1Chair of Applied Mathematics III,University of Erlangen-Nuremberg, Cauerstr. 11 91058 Erlangen, Germany,

[email protected]

2BCAM - Basque Center for Applied Mathematics,Mazarredo 14, 48009 Bilbao Basque Country, Spain,

[email protected]

3Department of Mathematics and Statistics,University of Strathclyde, 16 Richmond Street, Glasgow G1 1XQ, United Kingdom,

[email protected]

4Department of Applied Mathematics,University of Crete, L. Knosou GR 71409 Heraklion, Greece,

[email protected]

We derive residual-based a posteriori error estimates of optimal order for fully discreteapproximations of time-dependent Stokes problem. The time discretization uses thebackward Euler method and the spatial discretization uses finite element spaces thatare allowed to change in time. The a posteriori error estimates are derived by applyingthe reconstruction technique.

References

[1] E. Bansch, J. Jansson, F. Karakatsani, Ch. Makridakis, On the a posteriori errorcontrol of time dependent Stokes equations, preprint (2013).

[2] F. Karakatsani, Ch. Makridakis, A posteriori estimates for approximations of timedependent Stokes equations, IMA J. Numer. Anal. 27 (2007) 741–764.

[3] O. Lakkis, Ch. Makridakis, Elliptic reconstruction and a posteriori error estimatesfor fully discrete linear parabolic problems. Math. Comp., 75 (2006) 1627-1658.

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NON-CONFORMING FINITE ELEMENT METHODSFOR THE OBSTACLE PROBLEM

Carsten Carstensen and Karoline Kohler

Institut fur Mathematik, Humboldt-Universitat zu Berlin,Unter den Linden 6, D-10099 Berlin, Germany

[email protected]

In this talk we will present a priori and a posteriori error analysis for the non-conformingCrouzeix-Raviart finite element method (FEM). We consider general obstacles χ ∈H2(Ω) on bounded polygonal domains Ω ⊆ R2 and non-homogeneous Dirichlet bound-ary conditions. Under standard regularity assumptions u ∈ H2(Ω) the Crouzeix-Raviart finite element solution allows for a best-approximation result for the gradientsplus additional terms which converge linearly as the maximal mesh size approacheszero. Reliable and efficient control over the error follows from residual based a poste-riori error analysis. The design of a discrete Lagrange multiplier leads to a guaranteedupper bound for the exact error. The Crouzeix-Raviart FEM allows for the computa-tion of a guaranteed lower bound for the exact minimal energy, which in turn can beemployed to present an alternative a posteriori error estimate. Numerical experimentsinvestigate the practical performance of a related adaptive algorithm and explore theaccuracy of upper and lower energy bounds.

2-7


ON THE NUMERICAL SIMULATION OF WELL STABILITY

Philippe R B Devloo1, Erick Raggio Slis Santos2 and Diogo Lira Cecılio3

1 Departamento de Estruturas - Faculdade de Engenharia Civil - Unicamp, [email protected]

2 CENPES - Petrobras, Brazil

3 Departamento de Engenharia de Petroleo -Faculdade de Engenharia Mecanica - Unicamp, Brazil

A finite element simulation is presented for analysing the stability of excavated wellsthrough the use of the Sandler DiMaggio elastoplastic model. In a first step the geolog-ical rock is stressed to the effective in-situ stress state. The stress state of the excavatedwell is obtained by a gradual stress state transfer at the boundary of the well from theoriginal. The resulting stress state shows plastic behaviour in the region around thewellbore. The finite element mesh is hp-refined around the wellbore to enhance theprecision of the elastoplastic simulation.

The second invariant of the plastic deformation tensor is USED as damage criterionto decide on material collapse. Material collapse is represented numerically by theremoval of elements.

In order to apply adaptive methods to elastoplastic simulations, general purposeprocedures were developed to transfer the elastoplastic deformation history from onemesh to another.

The Sandler DiMaggio elastoplastic material model was implemented in a generalpurpose object oriented framework which computes the elastoplastic response of amaterial point based on the definition of the yield surface and hardening law. Thetangent matrix of the stress strain relation is computed consistently by the use ofautomatic differentiation.

Unstable wells show increase of plastic deformation as the material is removed.Stable wells reduce localized plastic deformation as the well geometry is modified.

This model represents advances in terms of well stability analyses in the excavationand production phases.

2-8


ESTIMATION OF DISCRETIZATION AND ALGEBRAIC ERRORVIA QUASI-EQUILIBRATED FLUXES FOR DISCONTINUOUS

GALERKIN METHODS

Vıt Dolejsı1a, Ivana Sebestova1b and Martin Vohralık2

1Charles University in Prague, Faculty of Mathematics and Physics,Sokolovska 83, 186 75 Praha 8, Czech Republic.

[email protected],[email protected]

2INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, [email protected]

We present the generalization of guaranteed and locally efficient a posteriori errorestimates based on quasi-equilibration of fluxes reconstruction for interior penalty dis-continuous Galerkin methods on simplicial meshes. The estimation newly involves thealgebraic error arising from the inexactness in solving underlying algebraic problem. Itis measured via an algebraic error flux reconstruction. Therefore, the flux reconstruc-tion as the total error consists of individual components: discretization and algebraicerror reconstructions.

We focus on exploiting the main advantage of DGM, namely decoupling of the globalproblem into element-wise problems. Firstly, variable polynomial degree of the approx-imate solution is permitted. Secondly, hanging nodes are allowed in our setting and theflux reconstruction is constructed in broken Raviart–Thomas–Nedelec space avoidingenforcing continuity of normal traces as well as constructing of matching submeshes.Moreover, following an approach proposed in [1], the algebraic component is treated byperforming some additional steps of the iterative algebraic solver and subsequently bycomparing the discretization error at two iteration steps of linear algebra computation.Such a construction is inexpensive however only results in quasi-equilibration. Further,we point out that the factual construction of considered reconstructions is not neededand as such evaluation of estimators is not costly.

Using derived estimates, we propose stopping criteria for iterative algebraic solversand subsequently an adaptive strategy for solving a linear elliptic equation. Finally,we present numerical results demonstrating good prediction of distribution of bothdiscretization and algebraic components and significant computational savings in com-parison with classical approaches.

References

[1] V. Dolejsı, I. Sebestova, and M. Vohralık. Discretization and algebraic error esti-mation by equilibrated fluxes for discontinuous Galerkin methods on nonmatchinggrids. In preparation.

2-9


CONDENSING THE SPECTRAL ELEMENT METHODFOR TIME DOMAIN WAVE PROBLEMS

Dugald B Duncana and Mark Payneb

Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, [email protected], [email protected]

The Spectral Element Method (SEM) is widely used in the approximation of time do-main linear wave propagation problems posed in second order form, e.g. the acousticand elastic waves equations. At its heart is a clever combination of node-based poly-nomial approximation within each element with a quadrature rule using the internalnode locations as quadrature points to give a diagonal mass matrix, hence making thescheme explicit. The clever part is in choosing the node locations to give high orderaccuracy and good stability properties. In 1d, the SEM consists of one set of explicitdifference equations on three time levels at “main nodes” located at the ends of theelements and linked to points in the elements around them, and other sets of equationsat “internal nodes” which are not so widely linked, sometimes only to points withintheir own element. In 2d, nodes on element edges have internal element linkages inone direction and wider linkages in the other and this generalises further in 3d. All ofthis is explained comprehensively in Cohen’s book “Higher-order numerical methodsfor transient wave equations” (Springer 2002).

In this talk we present a different view of the SEM obtained after condensing outthe internal nodes to leave a multilevel scheme on the main nodes only. We show theresult, perhaps surprising, that the relatively popular SEMs are actually equivalentto little-used (indeed deprecated) multilevel “symmetric schemes” and so they havethe same properties. We show that recasting the SEM in this form can make some ofthe analysis easier, we compare the efficiency of implementing the two approaches andderive improved SEM-type schemes using the condensed form to guide the process.

2-10


LOCAL MASS CONSERVATION OF STOKES FINITE ELEMENTS

Daniele Boffi1a, Nicola Cavallini1b Francesca Gardini1c and Lucia Gastaldi2

1Dipartimento di Matematica, Univerista di Pavia, Pavia, [email protected], [email protected]

[email protected]

2Dipartimento di Ingegneria Civile, Architettura, Territorio,Ambiente e di Matematica, Universita di Brescia, Brescia, Italy.

[email protected]

It is well-known that in the approximation of the solution of the Stokes problem theincompressibility constraint is imposed in a weak sense, so that the divergence freecondition is not imposed exactly at the discrete level, unless the divergence of thevelocity space is contained in the pressure space.

In this talk we discuss the stability of some Stokes finite elements, which allow fora more local enforcement of the divergence free condition. In particular, we consider amodification of Hood-Taylor and Bercovier-Pironneau schemes which consists in addingpiecewise constant functions to the pressure space. This enhancement, which had beenalready used in the literature, is driven by the goal of achieving an improved massconservation at element level. We prove the inf-sup condition for the enhanced spacesin a general setting and we present some numerical tests which confirm the stabilityproperties and the improvement in the local mass conservation (see [2]).

Moreover, we analyze the pressure enhancement modification in the framework oflowest order stabilized finite elements as well (see [1]). The main consequence of theanalysis is that the enhancement is poorly effective in the case of low order elements andnon smooth data. In particular, if the polynomial order of the velocity space is not highenough (quadratic in 2D or cubic in 3D), then the pressure enhancement requires anappropriate stabilization involving the pressure jumps along the interelements. First ofall, we observe that such stabilization destroys the local nature of the mass conservationproperty. Moreover, the stabilization term introduces a consistency error with reducedrate of convergence in case of non-smooth pressure solutions. This drawback applies,for instance, to the stabilized P1−P0 element or to the enhanced version of the popularP1−P1 stabilized element. In order to circumvent this phenomena we introduce a newfinite element that combines the feasible characteristics of the P1−P1 stabilized elementand mass conservation properties of the Bercovier-Pironneau element.

One of the motivations for this research is the application of the proposed schemes tothe finite element Immersed Boundary Method for the approximation of fluid-structureinteraction problems (see [3]).

References

[1] D. Boffi, N. Cavallini, F. Gardini, L. Gastaldi: “Stabilized stokes element and localmass conservation”, Bollettino U.M.I., (9) V (2012), no. 3, 543-573.

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[2] D. Boffi, N. Cavallini, F. Gardini, L. Gastaldi: “Local mass conservation of Stokesfinite elements”, J. Sci. Comput., 52 (2012), no. 2, 383-400.

[3] D. Boffi, N. Cavallini, F. Gardini, L. Gastaldi: “Immersed boundary method: per-formance analysis of popular finite element spaces”, in:Coupled Problems 2011, pp.1-12, M. Papadrakakis, E. Onate and B. Schrefler (Eds). Cimne.

ENRICHING A HANKEL BASIS BY RAY TRACINGIN THE ULTRA WEAK VARIATIONAL FORMULATION

C. J. Howarth1a, Simon Chandler-Wilde1, Stephen Langdon1 and P.N. Childs.2

1University of Reading, Reading, [email protected]

2Schlumberger Gould Research, Cambridge, UK.

The Ultra Weak Variational Formulation (UWVF) is a new generation finite elementmethod for approximating time harmonic acoustic and electromagnetic wave propaga-tion. The UWVF assumes wave like behaviour on each element, but otherwise allowsflexibility in the approximation space. We exploit this flexibility by combining thenumerical method with ray tracing solutions, in order to find accurate solutions at alower computational cost than the standard UWVF.

Time harmonic acoustic wave scattering is modelled in 2D by the Helmholtz prob-lem where Ω is a polygonal domain with boundary Γ. The UWVF approximation takesthe form of a linear combination of basis functions upon each element Ωk. Basis func-tions are required to solve the homogeneous Helmholtz equation, so incorporating theoscillatory behaviour of the solution. Much current literature uses an equally spacedplane wave basis on each element. As an alternative, we instead use a Hankel basis:cylindrical waves originating from source points yk,l /∈ Ωk. The choice of yk,l allowsflexibility in both the direction and the level of curvature of the basis function over theelement.

At high frequencies a ray model gives a good understanding of the direction ofpropagation of a wave. We use the ideas of ray tracing to find a good a-priori choiceof basis function. Consider a domain Ω enclosing a smooth, convex scatterer with nostraight edges, the boundary of which we denote by Γ. Let the incident field ui be aplane wave and the wavenumber κ be constant in Ω: ray directions perpendicular tothe wavefronts are parallel, until they reach the scatterer, reflect, and continue in astraight line. For any given point in the illuminated region x ∈ Ω, by tracing the raythrough this point back, we can find the point of interaction z ∈ Γ where the incidentray hits the scatterer, and the angle of reflection θr.

For points inside an element Ωk, we consider the local scattered field to be originat-ing from a single centre of curvature, which we can find by considering the intersectionof rays from points which are close to one another, x and x0. If we extend the raysthat travel through x and x0 back through the scatterer, they will cross at some pointP either within or on the opposite side of the scatterer. Taking the limit as x → x0,we take this intersection point as the centre of curvature xC.

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The UWVF can be extended to incorporate the ray traced directions and centres ofcurvature into the Hankel basis, using just two basis functions per element: one a singleplane wave representing the incident field and the other a point source centred at xC,found by the ray tracing algorithm representing the scattered field. Using just the tworay traced basis functions per element for a circular scatterer, for κ = 80, we get an L2

relative error of under 9% using 0.4 degrees of freedom per wavelength. If only a moregeneral idea of the wave interaction is needed rather than high accuracy, perhaps asan initial guess of state, then this approach suggests potential computational savingscompared to more standard methods. To achieve higher accuracy we require morebasis functions per element. In addition to including the ray traced basis representingthe incident field direction, we also include further directions, equally spaced around acircle of radius R 1 (thus simulating plane waves), together with a final point sourcecentered at the center of curvature xC given by our ray tracing algorithm. Includingthe ray traced basis functions leads to a reduction in the overall number of degrees offreedom required to achieve a given level of accuracy.

2-13


ERROR ESTIMATES FOR NONLINEAR CONVECTIVEAND SINGULARLY PERTURBED PROBLEMS

IN FINITE ELEMENT METHODS

Vaclav Kucera

Department of Numerical Mathematics, Charles University in Prague,Sokolovska 83, Praha 8, 186 75, Czech republic.

[email protected]

We are concerned with the analysis of the discontinuous Galerkin (DG) and standardconforming finite element methods applied to the nonstationary convective or singularlyperturbed convection-diffusion problem defined in Ω ⊂ Rd:

a)∂u

∂t+ div f(u) = ε∆u+ g in Ω× (0, T ),

b) u∣∣ΓD×(0,T )

= uD, ε∂u

∂n

∣∣ΓN×(0,T )

= gN ,

c) u(x, 0) = u0(x), x ∈ Ω.

We derive apriori error estimates in the L∞(L2)-norm which are uniform with re-spect to ε → 0 and are valid even for the limiting case ε = 0. For various explicitschemes, such an error analysis was presented in a series of papers starting with [Q.Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontin-uous Galerkin methods for scalar conservation laws, 2004]. There the argument relieson a nonstandard estimate of the convective terms using E-fluxes and a mathematicalinduction argument with respect to time. Thus the technique cannot be directly ap-plied to estimates for the method of lines (no discrete structure with respect to time)and implicit discretizations (not enough apriori information about the solution on thenext time level). In the present work, we circumvent these obstacles:Method of lines. Here we apply two different techniques. First, we use the so-calledcontinuous mathematical induction [Y. R. Chao, A note on Continuous mathemati-cal induction, 1919] instead of standard mathematical induction used for the explicitschemes. This is a technique that we shall also use in the implicit case. Alternatively,we prove the same result using a nonlinear variant of Gronwall’s inequality. We provethat the latter technique has no discrete counterpart for implicit time discretizations.Implicit time discretization. First, we prove that for the implicit scheme, the de-sired error estimates cannot be proved only from the error equation and the consideredestimates of its individual terms. To obtain more information about the problem, weintroduce a continuation eh : [0, T ]→ L2(Ω) of the error enh, n = 0, · · · , N constructedby means of a suitable modification of the discrete problem. We prove that estimatesfor this continuated solution directly imply estimates for the original implicit solution.The fact that eh is continuous with respect to time and that it relates to the struc-ture of the problem allows us to prove estimates for eh via continuous mathematicalinduction.

A principal artefact of the technique is that we obtain a rather restrictive CFL-likecondition even in the case of an implicit time discretization and that the result is not

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valid for the lowest order approximation degrees (we need p > 1 +d/2). The techniquecan be straightforwardly generalized to much more complicated problems.

A DOMAIN DECOMPOSITION METHOD WITHAN OPTIMIZED PENALTY PARAMETER

Chang-Ock Lee1 and Eun-Hee Park2

1 Department of Mathematical Sciences, KAIST, Daejeon, South [email protected]

2 Division of Computational Mathematics,National Institute for Mathematical Sciences, Daejeon, South Korea

[email protected]

The dual-primal finite element tearing and interconnecting (FETI-DP) method [1] is adomain decomposition method, which enforces the continuity across the subdomain in-terface by Lagrange multipliers. A dual iterative substructuring method with a penaltyterm was introduced in the previous works by the authors [2, 3], which is a variant ofthe FETI-DP method in terms of the way to deal with the continuity on the interface.The proposed method imposes the continuity not only by using Lagrange multipliersbut also by adding a penalty term which consists of a positive penalty parameter η anda measure of the jump across the interface. Due to the penalty term, the proposed it-erative method has a better convergence property than the standard FETI-DP methodin the sense that the condition number of the resultant dual problem is bounded by aconstant independent of the subdomain size and the mesh size.

In this talk, we will discuss a further study for a dual iterative substructuringmethod with a penalty term in terms of its convergence analysis and practical efficiency.On one hand, the convergence studies in [2, 3] rule out the case when a relatively small ηis taken. On the other hand, the condition number estimate without any size limitationon η will be presented. Moreover, based on the close relationship between the FETI-DPmethod and the proposed method, which results from the condition number estimate,it is shown that a penalty parameter chosen in a certain range, called a nearly optimalrange of η, is sufficient to accelerate the convergence of the dual iteration. In addition,inner preconditioners for subdomain problems will be discussed for an improvement ofpractical efficiency.

References

[1] C. Farhat and M. Lesoinne and K. Pierson. A scalable dual-primal domain de-composition method. Numer. Linear Algebra Appl., 7 (2000), pp. 687–714.

[2] C.-O. Lee and E.-H. Park. A dual iterative substructuring method with a penaltyterm. Numer. Math., 112 (2009), pp. 89–113.

[3] C.-O. Lee and E.-H. Park. A dual iterative substructuring method with a penaltyterm in three dimensions. Comput. Math. Appl., 64 (2012), pp. 2787–2805.

2-15


THE PARTITION OF UNITY METHOD FOR THE 3D ELASTICWAVE PROBLEMS IN THE HIGH FREQUENCY DOMAIN

M. Mahmood1a, O. Laghrouche1b, A. El-Kacimi1c and J. Trevelyan2

1Institute for Infrastructure and Environment,Heriot-Watt University Edinburgh EH14 4AS, UK

[email protected], [email protected],[email protected]

2School of Engineering and Computing Sciences,Durham University Durham DH1 3LE, UK

[email protected]

Key words: PUFEM, finite element method, plane waves, elastic waves, 3D.

A growing research has been developed on wave numerical modelling in different fieldse.g. in seismolog, geophysics, soil mechanics and biomedical ultrasound. The relatedproblems are modeled using mainly Helmhotz, Maxwell’s and Navier’s elastic waveequations depending on wave propagation media and type of application.

A number of different numerical methods have been used to solve the elastic waveequations but among the most commonly used is finite element method (FEM), dueto e.g. its flexibility in handling complex geometries, its ability to handle differenttypes of media etc.. However, the use of standard finite element method for solvingacoustic or elastic problems, in medium and high frequency domain, becomes limited interms of memory capacity and computationally very expensive in terms of CPU time.Our object is to develop finite elements, for three dimensional elastic wave problems,capable of containing many wavelengths per nodal spacing. This will be achieved byapplying the plane wave basis decomposition to the 3D elastic wave equation. Theseelements will allow us to relax the traditional requirement of around ten nodal pointsper wavelength and therefore solve elastic wave problems without refining the mesh ofthe computational domain at each frequency. The accuracy and effectiveness of theproposed technique will be determined by comparing solutions for selected problemswith available analytical solutions and/or to high resolution numerical solutions usingconventional finite elements. The method of plane wave basis decomposition used todevelop wave finite elements for the two-dimensional elastic wave equation [1-2] will beextended to three dimensions. The governing equation is a vector equation and multiplewave speeds are present for any given frequency. In an infinite elastic medium, there aretwo different types of wave propagating simultaneously, the dilatation or compressionwave (P), and the distortional or shear wave (S). The application of the Helmholtzdecomposition theorem to the displacement field yields a scalar wave equation forthe P-wave potential and a vector wave equation for the S-wave potential. The twowave equations are independent but the boundary conditions depend on both P-waveand S-wave potentials, thus coupling the associated scalar P-wave and vector S-waveequations.

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References

[1] El Kacimi A and Laghrouche O.: Numerical Modelling of Elastic Wave propagationin Frequency Domain by the Partition of Unity Finite Element Method. Interna-tional Journal for Numerical Methods in Engineering, 77: 1646–1669, 2009.

[2] El Kacimi A and Laghrouche O.: Improvement of PUFEM for the numerical so-lution of high frequency elastic wave scattering on unstructured triangular meshgrids. International Journal for Numerical Methods in Engineering, 84: 330–350,2010.

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ON THE NUMERICAL TREATMENT OF ESSENTIALBOUNDARY CONDITIONS WITHIN POSITIVITY-PRESERVINGFINITE ELEMENT METHODS FOR CONVECTION-DOMINATED

TRANSPORT PROBLEMS

Matthias Moller

Institute of Applied Mathematics (LS III),TU Dortmund University of Technology,

Vogelpothsweg 87, D-44227 Dortmund, [email protected]

The design of positivity-preserving numerical schemes for convection-dominated flowproblems has been a topic of active research for many years. There is thus a largenumber of publications on stabilized finite element methods for the convection-diffusionequation such as SUPG, LPS or SOLD schemes. Their common aim is to reduce orat best prevent the generation of nonphysical undershoots and overshoots near steepgradients. While an improper implementation of essential boundary conditions mayalso cause spurious oscillations this aspect is hardly addressed in the literature.

In this talk, the algebraic flux correction (AFC) methodology [2] is revisited andgeneralized to the case of weakly imposed essential boundary conditions [1]. In essence,the classical continuous Galerkin discretization of the convection-diffusion equation isconverted into a non-oscillatory high-resolution scheme by adding artificial dissipationwhich is adaptively removed in regions where the solution is smooth with the aid ofa flux limiter. All required information is extracted from the finite element systemmatrix which is manipulated based on rigorous mathematical constraints. We suggesta generalization of these positivity criteria which account for the additional contribu-tion of boundary integral terms to the discretized transport operator responsible forthe weak imposition of boundary conditions. An alternative approach to prescribingDirichlet boundary values (in strong sense) without violating positivity preservationwas suggested in [3]. In this talk, both approaches are compared by presenting numer-ical examples for the two-dimensional convection-diffusion equation.

References

[1] Y. Bazilevs, T.J.R. Hughes: Weak imposition of Dirichlet boundary conditions influid mechanics. Computers & Fluids, 36(1), 2007, 1226.

[2] D. Kuzmin: Algebraic flux correction I. Scalar conservation laws. Chapter 6 in:D. Kuzmin, R. ohner, S. Turek: Flux-Corrected Transport, Springer, 2nd edition2012, 145–192.

[3] J. Niemeyer, B. Simeon: On finite element methodflux correctedtransport stabilization for advection-diffusion problems in a par-tial differential-algebraic framework. Preprint. URN: http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:hbz:386-kluedo-34442, 2013.

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ALGEBRAIC FLUX CORRECTION IN A PARTIALDIFFERENTIAL-ALGEBRAIC FRAMEWORK

Julia Niemeyera and Bernd Simeon

Felix-Klein Centre for Mathematics, University of Kaiserslautern,Paul-Ehrlich Straße 31, D-67663 Kaiserslautern, Germany,

[email protected]

Applications like coupled multiphysics problems, domain decomposition or the magne-tohydrodynamic equations include constraints or invariants on the partial differentialequation system. The partial differential-algebriac framework allows a general handlingby appending these constraints by means of Lagrange multipliers.If an advection-dominated flow problem is part of the coupled system, the finite ele-ment method is known to tend to produce unphysical oscillations. Therefore a suitablestabilization method needs to be included. During the last decades a stabilization onthe algebraic level by flux correction labeled as algebraic flux correction (AFC) hasbeen introduced [1] and improved [2].The main goal of this talk is the extension of the AFC method to the partial differential-algebraic framework. In short, to derive a stable numerical solution means to avoidbirth and growth of unphysical local extrema on the one hand and to suppress globalover- and undershoots on the other hand. While the prevention of unphysical localextrema can be derived by modifications of the system matrices, the used time inte-gration method needs to guarantee positivity preserving in every single timestep suchthat no global over- and undershoots arise.The numerical solution of a differential-algebraic equation (DAE), e.g., stemming fromthe semi-discretization of coupled problems with the finite element method, is not thesame as for an ordinary differential equation (ODE). Particulary in the most generalcase the positivity preservation property of a time integrator in the ODE case cannotbe transferred to the DAE case [3].In this talk we concentrate on DAE systems stemming from a finite element discretiza-tion. After a brief introduction to DAEs we will show that the one-step θ−schemeremains positivity preserving when applied to a DAE under certain conditions. TheFinite Element Method - Flux Corrected Transport algorithm will be modified to han-dle DAEs [4] and we show some first simulation results.

References

[1] D. Kuzmin, S. Turek: Flux correction tools for finite elements. J. Comput. Phys.175, 525–558 (2002).

[2] D. Kuzmin: Algebraic flux correction I. Scalar conservation laws. Chapter 6 in:D. Kuzmin, R. Lohner, S. Turek: Flux-Corrected Transport, Springer, 2nd edition2012, 145–192.

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[3] A.-K. Baumann, V. Mehrmann: Numerical integration of positive lineardifferential-algebraic systems. Preprint 2012-02, Institut fur Mathematik, TUBerlin, 2012

[4] J. Niemeyer, B. Simeon: On Finite Element MethodFlux Corrected TransportStabilization for Advection-Diffusion Problems in a Partial Differential-AlgebraicFramework. Preprint. URN: http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:hbz:386-kluedo-34442

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http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:hbz:386-kluedo-34442

http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:hbz:386-kluedo-34442


APPROXIMATION OF EDDY CURRENTS IN ANAXISYMMETRIC UNBOUNDED DOMAIN

Pilar Salgado1 and Virginia Selgas2

1Departamento de Matematica Aplicada, Escola Politecnica Superior,Universidad de Santiago de Compostela, 27002 Lugo (Spain)

[email protected]

2Departamento de Matematicas, Escuela Politecnica de Ingenierıa,Universidad de Oviedo 33203 Gijon (Spain)

[email protected]

The time-harmonic eddy current model arises in physical and industrial applicationssuch as the modeling of induction furnaces (see, for instance, [1]), where the physicalgeometry is axisymmetric and unbounded. To approximately solve this axisymmetricproblem, we propose and analyze a symmetric FEM and BEM coupling method interms of a magnetic vector potential. Moreover, we show that our formulation is well-posed, and also propose a discretization that leads to a convergent Galerkin scheme;see [3] and the references given therein. We finally develop some numerical techniquesto implement this method in a MatLab code and show its applicability to simulate anindustrial problem; see Fig. 1.

Fig. 1: Numerical results for the simulation of a small induction furnace composed bya graphite crucible containing silicon in its interior and a four-turns coil. Both figuresshow the approximation of the current density: the one on the left focus in the crucible,whereas that on the right shows the whole furnace.

References

[1] A. Bermudez, D. Gomez, M.C. Muniz, P. Salgado, R. Vazquez. Numerical simu-lation of a thermo-electromagneto-hydrodynamic problem in an induction heatingfurnace. Applied Numerical Mathematics 59 (2009), 2082-2104.

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[2] A. Bermudez, C. Reales, R. Rodrıguez, P. Salgado, Numerical analysis of a finiteelement method for the axisymmetric eddy current model of an induction furnace.IMA Journal of Numerical Analysis 30 (2010), 654-676.

[3] V. Selgas, A symmetric BEM-FEM method for an axisymmetric eddy current prob-lem. Submitted.

A UNIFORM CONVERGENCE ANALYSIS OF THREE-STEPTAYLOR GALERKIN FINITE ELEMENT MONOTONE

ITERATIVE DOMAIN-DECOMPOSITION SCHEME FORSINGULARLY PERTURBED PROBLEMS

Vivek Sangwan1 and B. V. Rathish Kumar2

1School of Mathematics and Computer Applications, Thapar University, India,[email protected]

2Department of Mathematics and Statistics, IIT Kanpur, Kanpur, [email protected]

Singularly perturbed problems appear in almost all the branches of science and engi-neering. Singularly perturbed problems are characterized by a small parameter multi-plied by the highest order derivative terms. As this small parameter, generally calledsingularly perturbed parameter, approaches to zero, sharp boundary or internal layersevolve in the solution. Conventional methods fail to capture these layers. We needspecial techniques to capture these layers. Fitted mesh methods and fitted operatormethods are two principle approaches which are used to find the approximate solutionof singularly perturbed problems. We have used both the techniques in our study ofapproximate solutions of these problems. The present contribution concerns the sta-bility and uniform convergence results for a nonlinear singularly perturbed problemusing three-step Taylor Galerkin finite element scheme via monotone iterative domaindecomposition algorithm for nonoverlappping domains. Shishkin mesh has been usedfor domain discretization. The monotone convergence of the monotone iterative al-gorithm and monotone iterative domain decomposition algorithm has been proved byshowing that the sequence of upper or lower solutions generated by the iterative schemeconverges to the approximate solution of the discretized problem. Thus the uniformconvergence of the proposed nonoverlapping domain decomposition scheme has beenestablished under three-step Taylor Galerkin finite element framework and is shownto be of order ∆t3 + ∆x. Numerical experiments have been carried out to depict therobustness and efficiency of the proposed scheme in capturing very sharp boundarylayers.

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WITH A HIERARCHICAL ERROR INDICATOR TOWARDANISOTROPIC MESH REFINEMENT

Rene Schneider

TU Chemnitz, Fakultat fur Mathematik, Chemnitz, Germany,[email protected]

We propose a new approach to adaptive mesh refinement. Instead of considering localmesh diameters and their adaption to solution features, we propose to evaluate thebenefit of possible refinements in a direct fashion, and to select the most profitablerefinements. The resulting refinement guide can be seen as a hierarchical edge-errorindicator. We demonstrate that based on this approach a directional refinement oftriangular elements can be achieved, allowing arbitrarily high aspect ratios.

However, only with the help of edge swapping and/or node removal (directional un-refinement) near optimal performance can be achieved for strongly anisotropic solutionfeatures. With these ingredients even re-alignment of the mesh with arbitrary errordirections is achieved. Numerical experiments demonstrate the utility of the proposedanisotropic refinement strategy.

COMPUTATIONAL ASPECTS IN SMOOTHAPPROXIMATION OF DATA

Karel Segeth

Technical University of Liberec, Liberec, Czech [email protected]

In the seventies, Talmi and Gilat introduced a way of data approximation calledsmooth. Our concern in the paper is to show mathematical properties of this pro-cedure applied to the exact interpolation as well as to the fitting of data when, at thesame time, we take into account the smoothness of the approximation curve and itsderivatives. This curve is the solution of a variational problem with constraints thatrepresent the interpolation conditions at nodes. Some procedures known in numericalanalysis, e.g. spline approximation, can be considered a special case of the smoothapproximation.In particular, we are concerned with the choice of the system exp(ikx) for the basisfunctions of the smooth approximation. This is the case when the evaluation of cer-tain infinite series needed in the approximation process can be transformed into thecomputation of Fourier integrals. We present results of some 1D numerical examplesthat show advantages and drawbacks of this procedure.This research has been supported by the Institute of Novel Technologies and AppliedInformatics, Faculty of Mechatronics, Informatics and Interdisciplinary Studies, Tech-nical University of Liberec.

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3: Mini-Symposium: A priori finite element error estimates in optimal control

3 Mini-Symposium: A priori finite element error

estimates in optimal control

Organiser: Thomas Apel

3-1


A PRIORI ERROR ESTIMATES FOR FINITE ELEMENT METHODSFOR H(2,1)-ELLIPTIC EQUATIONS

Thomas Apel1a, Thomas G. Flaig1b and Serge Nicaise2

1Institut fur Mathematik und Bauinformatik,Universitat der Bundeswehr Munchen, D-85579 Neubiberg, Germany

[email protected] [email protected]

2LAMAV, Institut des Sciences et Techniques de Valenciennes,Universite de Valenciennes et du Hainaut Cambresis, B.P. 311, 59313 Valenciennes

Cedex, [email protected]

The convergence of finite element methods for linear elliptic boundary value problemsof second and forth order is well understood. In our talk we will introduce finite elementapproximations of some linear semi-elliptic boundary value problem of mixed order on atwo dimensional rectangular domain Q. The equation is of second order in one directionand forth order in the other. We establish a regularity result and estimates for the finiteelement error of conforming approximations of this equation. The finite elements inuse have a tensor product structure, in one dimension we use cubic Hermite elements,in the other dimension Lagrange elements of order k = 1, 2, 3. For these elements weprove the error bound O(h2 + τ k) in the energy norm and O

((h2 + τ k)(h2 + τ)

)in the

L2(Q)-norm.This type of equations appears in the optimal control of parabolic partial differential

equations if one eliminates the control and the state (or the adjoint state) in the firstorder optimality conditions.

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CRANK-NICOLSON AND STORMER-VERLET DISCRETIZATIONSCHEMES FOR OPTIMAL CONTROL PROBLEMS WITH

PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

Thomas Apela and Thomas G. Flaigb

Institut fur Mathematik und Bauinformatik,Universitat der Bundeswehr Munchen, 85577 Neubiberg, Germany

[email protected], [email protected]

In this talk we present the discretization of optimal control problems with parabolicpartial differential equations. In particular we discuss the problem

minα

2‖y(·, T )− yD‖2

H +β

2

∫ T

0

‖y − yd‖2Hdt+

ν

2

∫ T

0

‖u‖2Hdt

s.t. yt + Ay = u in Ω× (0, T ],

∂y

∂n= 0 on ∂Ω× (0, T ],

y(·, 0) = v in Ω,

with a second order elliptic operator A and H = L2(Ω).For the temporal discretization we focus on a family of Crank-Nicolson schemes with

different discretizations for the state y and the adjoint state p so that discretizationand optimization commute. One of the schemes can be explained as the Stormer-Verlet scheme. So we can interpret the method in the context of geometric numericalintegration. The Hamiltonian structure of the parabolic optimal control problem isalso discussed.

Further we point out that two schemes may also be obtained as a Galerkin methodwith quadrature. If we use linear finite elements for the spatial discretization, we canprove second order convergence in space and time for one of the schemes. Finally wepresent numerical examples where second order convergence in time is observed.

References

[1] Thomas Apel and Thomas G. Flaig. Crank-Nicolson schemes for optimal con-trol problems with evolution equations. SIAM Journal on Numerical Analysis,50(3):1484–1512, 2012.

[2] Thomas G. Flaig. Discretization strategies for optimal control problems withparabolic partial differential equations. PhD thesis, Universitat der BundeswehrMunchen, submitted, 2013.

3-3


ERROR ESTIMATES FOR DIRICHLET CONTROLPROBLEMS IN POLYGONAL DOMAINS

Thomas Apel1, Mariano Mateos2, Johannes Pfefferer3 and Arnd Rosch4

1Institut fur Mathematik und Bauinformatik,Universitat der Bundeswehr Munchen, 85577 Neubiberg, Germany

[email protected]

2Departamento de Matematicas, E.P.I. de Gijon,Universidad de Oviedo, Campus de Gijon, 33203 Gijon, Spain

[email protected]

3Institut fur Mathematik und Bauinformatik,Universitat der Bundeswehr Munchen, 85577 Neubiberg, Germany

[email protected]

4Fachbereich Mathematik, Universtat Duisburg-Essen,Forsthausweg 2, 47057 Duisburg, Germany

[email protected]

We study a control constrained Dirichlet optimal control problem governed by an el-liptic equation posed on a domain with polygonal boundary. We admit both the casesof a convex or a non-convex domain.

First, we make a detailed study of the regularity of the solution near the corners.Despite the non-convexity of the domain we are able to prove that the optimal controlis a continuous function. We can even precise its regularity by using classical Sobolevspaces W 1,p(Γ), where p > 2 depends on the greatest convex angle of the domain.

The regularity of the state and of the adjoint state is also studied, both in theframework of classical Sobolev spaces and of weighted Sobolev spaces.

Finally, we obtain error estimates for the finite element approximation of the prob-lem. We discretize both control and state by means of continuous piecewise linearfunctions on quasi-uniform meshes. The order of convergence depends on the regular-ity of the data and on the angles of the domain.

Some numerical experiments are presented to illustrate the theoretical estimates.

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BOUNDARY CONCENTRATED FEM FOROPTIMAL CONTROL PROBLEMS

Sven Beuchler

Institute for Numerical Simulation, University of Bonn, Bonn, [email protected]

We investigate the discretization of optimal boundary control problems for ellipticequations on two-dimensional polygonal domains by the boundary concentrated finiteelement method. We prove that the discretization error ‖u∗ − u∗h‖L2(Γ) decreases likeN−1, where N is the total number of unknowns. This makes the proposed methodfavorable in comparison to the h-version of the finite element method, where the dis-cretization error behaves like N−3/4 for uniform meshes. Moreover, we present analgorithm that solves the discretized problem in almost optimal complexity. The talkis complemented with numerical results.

The presentation is the result of collaborations with K. Hofer (Bonn), D. Wachsmuth,J.-E. Wurst (Wuerzburg).

ERROR ESTIMATES FOR THE VELOCITY TRACKINGPROBLEM USING DUALITY ARGUMENTS

Konstantinos Chrysafinos

Department of Mathematics,National Technical University of Athens, Athens, Greece.

[email protected],

An optimal control problem related to the velocity tracking problem of Navier-Stokesflows is considered. The main goal is to minimize the tracking functional using dis-tributed controls with control constraints. The schemes under consideration are basedon a discontinuous time-stepping approach combined with standard conforming finiteelements for the spacial discretization. Special emphasis will be placed to the role ofduality arguments in developing error estimates, and to the role of the related adjointequation which exhibits very interesting structural properties.This is a joint work with Eduardo Casas, University of Cantabria, Santander, Spain.

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CONVERGENCE AND ERROR ANALYSIS OF A NUMERICALMETHOD FOR THE IDENTIFICATION OF MATRIX

PARAMETERS IN ELLIPTIC PDES

Klaus Deckelnick1 and Michael Hinze2

1Institut fur Analysis und Numerik, Otto–von–Guericke–Universitat Magdeburg,Universitatsplatz 2, 39106 Magdeburg, Germany

[email protected]

2Schwerpunkt Optimierung und Approximation, Universitat Hamburg,Bundesstraße 55, 20146 Hamburg, Germany

[email protected]

In this talk we present and analyze a numerical method for solving the inverse problemof identifying the diffusion matrix in an elliptic PDE from distributed noisy measure-ments. We use a regularized least squares approach in which the state equations aregiven by a finite element discretization of the elliptic PDE. The unknown matrix param-eters act as control variables and are handled with the help of variational discretizationas introduced in [M. Hinze, A variational discretization concept in control constrainedoptimization: the linear-quadratic case, Comput. Optim. Appl. 30, 45–61 (2005)]. Fora suitable coupling of Tikhonov regularization parameter, finite element grid size andnoise level we are able to prove L2–convergence of the discrete solutions to the uniquenorm–minimal solution of the identification problem; corresponding convergence ratescan be obtained provided that a suitable projected source condition is fulfilled. Finally,we present a numerical experiment which supports our theoretical findings.

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OPTIMAL CONTROL OF BIHARMONIC OPERATOR

Stefan Frei1a, Rolf Rannacher1b and Winnifried Wollner2

1University of Heidelberg, Department of Mathematics,INF 293/294, 69120 Heidelberg, [email protected],[email protected]

2University of Hamburg, Department of Mathematics,Bundesstr 55, 20146 Hamburg, Germany

[email protected]

In this talk a priori error estimates are derived for the finite element discretization ofoptimal distributed control problems governed by the biharmonic operator. The stateequation is discretized in primal mixed form using continuous piecewise biquadraticfinite elements, while piecewise constant approximations are used for the control. Theerror estimates derived for the state variable as well as that for the control are order-optimal on general unstructured meshes. However, on uniform meshes not all errorestimates are optimal due to the low-order control approximation. All theoreticalresults are confirmed by numerical tests.

AN INTERIOR PENALTY METHOD FOR DISTRIBUTEDOPTIMAL CONTROL PROBLEMS GOVERNED

BY THE BIHARMONIC OPERATOR

Thirupathi Gudi, Neela Nataraja and Veeranjaneyulu Sadhanala

Department of Mathematics, IIT Bombay, Powai, Mumbai 400076. [email protected]

http://www.math.iitb.ac.in/~neela

In the past few years, C0 interior penalty methods have been attractive for solving thefourth order problems. In this work, a C0 interior penalty method has been proposedand analyzed for an optimal control problem governed by the biharmonic operator. Thestate equation is discretized using continuous piecewise quadratic finite elements whilepiecewise constant approximations are used for the control variable. Error estimates arederived for both the state and control variables. Theoretical results are demonstratedby numerical experiments.

3-7

http://www.math.iitb.ac.in/~neela


A PRIORI ERROR ESTIMATES FOR PARABOLIC OPTIMALCONTROL PROBLEMS WITH POINT CONTROLS

Dmitriy Leykekhman1 and Boris Vexler2

1Department of Mathematics, University of Connecticut, [email protected]

2Technische Universitat Munchen, Faculty of Mathematics,Boltzmannstraße 3, 85748 Garching b. Munich, Germany

[email protected]

In this talk, we consider the following optimal control problem:

minq,u

J(q, u) :=1

2

∫ T

0

‖u(t)− u(t)‖2L2(Ω)dt+

α

2

∫ T

0

|q(t)|2dt

subject to the second order parabolic equation

ut(t, x)−∆u(t, x) = q(t)δx0 , (t, x) ∈ I × Ω,

u(t, x) = 0, (t, x) ∈ I × ∂Ω,

u(0, x) = 0, x ∈ Ω

and subject to pointwise control constraints

qa ≤ q(t) ≤ qb a. e. in I.

Here I = [0, T ], Ω ⊂ R2 is a convex polygonal domain, x0 ∈ IntΩ fixed, and δx0 isthe Dirac delta function. The parameter α is assumed to be positive and the desiredstate u fulfills u ∈ L2(I;L∞(Ω)). The control bounds qa, qb ∈ R∪ ±∞ fulfill qa < qb.Such problems are challenging due to low regularity of the state equation. We usethe standard continuous piecewise linear approximation in space and the first orderdiscontinuous Galerkin method in time to approximate the problem numerically. De-spite low regularity of the state equation, we obtain almost optimal h2 +k convergencerate for the control in L2 norm, without any relationship between the size of the spacediscretization h and the time steps k. The main ingredients of the analysis are sharpregularity results and the new global and local fully discrete a priori pointwise in spaceand L2 in time error estimates the parabolic problems.

3-8


OPTIMAL ERROR ESTIMATES FOR FINITE ELEMENTDISCRETIZATION OF ELLIPTIC OPTIMAL CONTROL PROBLEMS

WITH FINITELY MANY POINTWISE STATE CONSTRAINTS

Dmitriy Leykekhman1, Dominik Meidner2a and Boris Vexler2b

1University of Connecticut, Department of Mathematics,196 Auditorium Road, Storrs, CT 06269-3009,USA

[email protected]

2Chair of Optimal Control,Technische Universitat Munchen, Faculty of Mathematics,Boltzmannstraße 3, 85748 Garching b. Munich, Germany


In this talk, we consider the following model elliptic optimal control problem withfinitely many state constraints on a convex smooth domain Ω in two and three dimen-sions:

Minimize1

2‖u− ud‖2

L2(Ω) +α

2‖q‖2

L2(Ω)

subject to the state equation

−∆u = q in Ω,

u = 0 on ∂Ω

and to finitely many pointwise state constraints

u(xi) = bi i = 1, 2, . . . , n,

Here, x1, x2, . . . , xn ∈ Ω are mutually distinct given points.Such problems are challenging due to low regularity of the adjoint variable. For

the discretization of the problem we consider continuous linear elements on quasi-uniform and graded meshes separately. Our main result establishes optimal a priorierror estimates for the control, state, and the Lagrange multiplier on the two mentionedtypes of meshes, see following table.

Proved orders of convergence on quasi-uniform / graded meshes

Dimension Error in control Error in state Error in Lagrange multiplier

2 1 / 2 2 / 2 2 / 23 1

2/ 2 1 / 2 1 / 2

In particular, in three dimensions the optimal second order convergence rate for allthree variables is possible only on properly refined graded meshes.

The derived estimates are illustrated by a numerical example.

3-9


VERIFICATION OF OPTIMALITY CONDITIONSAND DISCRETIZATION ERROR ESTIMATES

Martin Naßa and Arnd Roschb

Faculty of Mathematics, University of Duisburg-Essen, [email protected], [email protected]

Optimal control of a semilinear elliptic partial differential equation is a nonconvex op-timization problem. Hence second-order sufficient conditions are needed to ensurelocal optimality. Such conditions allow to derive a priori error estimates for FE-discretizations.

However, this strategy has an essential drawback. The second-order condition hasto be satisfied in the exact solution, but only a numerical approximation of the exactsolution is available. Consequently it is impossible to check the second-order sufficientcondition.

In this talk we present another strategy. We require only a coercivity conditionfor the numerical solution which can be checked numerically. This is the main tool toshow discretization error estimates for a FE-discretization.

ON DISCRETIZED NONCONVEX ELLIPTIC OPTIMAL CONTROLPROBLEMS WITH POINTWISE STATE CONSTRAINTS

Ira Neitzel1, Johannes Pfefferer2 and Arnd Rosch3

1 Technische Universitat Munchen Centre for Mathematical Sciences,M17 Boltzmannstr. 3 D-85748 Garching b. Munich, Germany

[email protected]

2 Universitat der Bundeswehr Munchen, Institut fur Mathematik und BauinformatikFakultat fur Bauingenieur- und Vermessungswesen D-85577 Neubiberg, Germany

[email protected]

3 Universitat Duisburg-Essen, Fachbereich Mathematik,Forsthausweg 2 D-47057 Duisburg, Germany

[email protected]

We discuss properties of finite-element-discretized optimal control problems subject topointwise state constraints and a semilinear elliptic state equation. We focus on twoaspects of the discretized problem. First, we discuss the convergence of discrete locallyoptimal controls to associated solutions of the continuous problem. We prove a rate ofconvergence depending on the discretization parameter h with the help of second ordersufficient conditions (SSC) for the original problem. Second, we discuss how these SSCfor the continuous problem transfer to the discrete level. This is motivated by the factthat for instance the proof of convergence of the SQP method or properties like localuniqueness of solutions rely on SSC.

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SPARSE ELLIPTIC CONTROL PROBLEMS IN MEASURE SPACES:REGULARITY AND FEM DISCRETIZATION

Konstantin Piepera and Boris Vexlerb

Chair of Optimal Control,Technische Universitat Munchen, Faculty of Mathematics,Boltzmannstraße 3, 85748 Garching b. Munich, Germany

[email protected] [email protected]

In this talk we consider an optimal control problem governed by an elliptic equation,where the control variable lies in a measure space. This formulation leads to a sparsestructure of the optimal control, which provides among other things an elegant way toattack problems of optimal source placement. We discuss the functional analytic settingof the problem under consideration and the regularity issues of the optimal solution.Moreover, we present a discretization concept and prove a priori error estimates forthe discretization error, which significantly improve the estimates from the literature.Numerical examples for problems in two and three space dimensions illustrate ourresults.

OPTIMAL BOUNDARY CONTROL PROBLEMS IN ENERGY SPACES

Olaf Steinbach

Institut fur Numerische Mathematik, TU Graz,Steyrergasse 30, A 8010 Graz, Austria

[email protected]

We consider Dirichlet boundary control problems subject to second order partial differ-ential equations where the set of admissible functions is a subset of the related energyspace, i.e. of H1/2(Γ). An equivalent norm is induced by the Steklov–Poincare oper-ator which realizes the Dirichlet to Neumann map for a given Dirichlet control. Thereduced formulation results in a variational inequality for the unknown control, whichis equivalent to a partial differential equation with bilateral constraints of Signorinitype on the boundary. We discuss finite and boundary element discretizations of theoptimality system and we present related a priori error estimates both in the energynorm in H1/2(Γ) and in L2(Γ). Applications involve the stationary and transient heatequation, and the boundary control of the stationary Navier–Stokes equations.

This talk is based on joint work with L. John, A. Kimeswenger, G. Of, andT. X. Phan.

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FINITE ELEMENT METHODS FOR FOURTH ORDERVARIATIONAL INEQUALITIES ARISING FROM

ELLIPTIC OPTIMAL CONTROL PROBLEMS

Li-yeng Sung

Department of Mathematics and Center for Computation and Technology,Louisiana State University, LA 70803, USA.

[email protected]

In this talk we will discuss finite element methods for elliptic optimal control problemswith pointwise state constraints formulated as fourth order variational inequalities.This is joint work with S.C. Brenner and Y. Zhang.

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4: Mini-Symposium: Analysis and applications of boundary element methods

4 Mini-Symposium: Analysis and applications of

boundary element methods

Organisers: Martin Schanz and Olaf Stein-bach

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THE BEM++ BOUNDARY ELEMENTLIBRARY AND APPLICATIONS

S.R. Arridge1a, T. Betcke2b, M. Schweiger1c and W. Smigaj2d

1Department of Computer Science, University College London, [email protected], [email protected],

2Department of Mathematics, University College London, [email protected], [email protected]

BEM++ is an open source Galerkin boundary element library for the solution of sys-tems of boundary integral equations. Its core is developed in C++ with a convenientPython interface on top of it. It offers support for the fast assembly and approxi-mate LU decomposition of boundary integral operators via hierarchical matrices, andconnects to the Trilinos library for iterative solvers.

Currently, BEM++ supports Laplace, Helmholtz, and Maxwell problems. In thistalk we give an introduction to the library, and demonstrate various application ex-amples, including layered problems in optical tomography, and convolution quadraturemethods for time-domain problems.

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A RECURSIVE INTEGRAL EQUATIONS APPROACH FORELECTROMAGNETIC SCATTERING BYBIPERIODIC MULTILAYER GRATINGS

Beatrice Bugert1,2,a and Gunther Schmidt1

1 Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

2 Berlin Mathematical School, Technical University Berlin, Berlin, Germanya [email protected]

Scattering theory has numerous applications in micro-optics like the construction ofholographic films, optical storage disks and antireflective coatings. Many of theseoptical devices are implemented by a multilayered structure. We study the specialcase of electromagnetic scattering by biperiodic multilayered structures modeled bya finite number of smooth, non-selfintersecting surfaces Σj, j = 0, . . . , N , which are2π-periodic in both x1- and x2-direction and separate regions Gj ⊂ R3 filled withmaterials of constant electric permittivity εj and constant magnetic permeability µj.The scattering of a time-harmonic plane wave Ei incident on the top layer Σ0 of themultilayered structure from G0 is computed by solving

curl curl Ej − κ2jEj = 0 in Gj∈JN

0,

nj ×(E1 −

(E0 − Ei

))= 0, nj ×

(µ−1

1 E1 − µ−10

(E0 + Ei

))= 0 on Σ0,

nj × (Ej+1 − Ej) = 0, nj ×(µ−1j+1Ej+1 − µ−1

j Ej

)= 0 on Σj∈J ,

where J = 1, . . . , N − 1, JN0 = J ∪ 0, N, and, additionally making sure that theoutgoing wave condition at infinity is satisfied. For the study of the above electromag-netic scattering problem, we propose a recursive integral equation algorithm followingthe scheme of [3], in which the equivalent problem for oneperiodic multilayered struc-tures was treated. The combined use of a Stratton-Chu integral representation and anelectric potential ansatz yields a singular integral equation on each interface. Theseequations arise from eachother via recursion from the bottom to the top interface lead-ing to a recursive algorithm. The idea for this potential ansatz is taken from [2]. Weestablish necessary and sufficient conditions such that the existence of solutions arisingfrom the recursive integral equation algorithm imply that the electromagnetic scatter-ing problem is solvable. This result follows in particular from the Fredholm propertiesof one of the occuring operators in the algorithm, which can be established using thetechniques in [1] and the results in [2]. Following [4], it is also possible to find conditionsensuring the uniqueness of solutions of the electromagnetic scattering problem.

References

[1] T. Arens, Scattering by biperiodic layered media: the integral equation approach,Habilitation thesis, KIT, Karlsruhe, 2010.

[2] M. Costabel and F. Le Louer, On the Kleinman-Martin integral equation methodfor electromagnetic scattering by a dielectric body, SIAM J. Appl. Math., 71 (2011),

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pp. 635–656.

[3] G. Schmidt, Conical diffraction by multilayer gratings: A recursive integral equa-tions approach, Preprint of the Weierstrass Institute, 1601 (2011).

[4] G. Schmidt, Integral methods for conical diffraction, Preprint of the WeierstrassInstitute, 1435 (2009).

BLACK-BOX PRECONDITIONING OF FEM/BEM MATRICESBY H-MATRIX TECHNIQUES

Markus Faustmanna, Jens Markus Melenkb and Dirk Praetoriusc

Institute for Analysis and Scientific Computing,Vienna University of Technology, Vienna, Austria.


Various compression techniques for matrices such as H-matrices have been developedin the past to store dense matrices and realize the matrix-vector-multiplication withlog-linear (or even linear) complexity. One particular strength of H-matrices is thatthe H-format includes some arithmetics that provides the (approximate) inversion,(approximate) LU-decomposition etc.

In our talk, we establish that the H-matrix format is rich enough to permit goodapproximations of inverse FEM (for various boundary conditions) and BEM matrices.The question of approximating the inverses of system matrices in the H-format byapproximate H-inversion or H-LU factorization has previously, due to the method ofproof, only been studied in the context of FEM and for Dirichlet boundary conditions[3, 4]. A main feature of our analysis is that we work in a fully discrete setting and thusavoid any additional projection errors. Therefore, we get approximations of arbitraryaccuracy, and we show exponential convergence in the block rank.

One possible application is the black-box preconditioning of the FEM/BEM systemsin iterative solvers by use of an H-LU decomposition. Numerically it has been observedthat such an approach works well in practice [5]. In our talk, we give a mathematicalunderpinning to these observations.

Moreover, we consider matrices arising from the FEM-BEM coupling. Our resultsfor the FEM case (stabilized Neumann problem) and the BEM case (simple-layer op-erator) can be used to derive an efficient block diagonal preconditioner that is basedon the H-LU decomposition in the corresponding blocks.

References

[1] M.Faustmann, J.M.Melenk, D.Praetorius: H-matrix approximability of the inversesof FEM matrices, work in progress.

[2] M.Faustmann, J.M.Melenk, D.Praetorius: Existence of H-matrix approximants tothe inverse of BEM matrices, work in progress.

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[3] M. Bebendorf, W. Hackbusch: Existence of H-matrix approximants to the inverseFE-matrix of elliptic operators with L∞-coefficients, Numer. Math., 95 (2003), 1–28.

[4] S. Borm: Approximation of solution operators of elliptic partial differential equa-tions by H- and H2-matrices. Numer. Math., 115 (2010), 165–193.

[5] L. Grasedyck: Adaptive Recompression of H-matrices for BEM, Computing, 74(2005), 205–223.

ONE-EQUATION FEM-BEM COUPLINGFOR ELASTICITY PROBLEMS

Michael Feischl1, Thomas Fuhrer1, Michael Karkulik2 and Dirk Praetorius1

1Institute for Analysis and Scientific Computing,Vienna University of Technology, Austria

[email protected]

2Faculdad de Mathematicas, Pontificia Unversidad Catolica de Chile

We consider a transmission problem in elasticity with a nonlinear material behaviorin the bounded interior domain, which can be rewritten by means of the symmetriccoupling as well as non-symmetric one-equation coupling methods, such as the Johnson-Nedelec coupling. Problems arise when trying to prove solvability of the Galerkindiscretization, because the space of rigid body motions is contained in the kernel of theLame operator.

In this talk, which is based on the recent preprint [3], we present how to extend theideas of implicit stabilization, developed for Laplace-type transmission problems in [1],to elasticity problems. We introduce modified equations which are fully equivalent (atthe continuous as well as at the discrete level) to the original formulations. Solvabilityof the discrete modified problems, however, hinges on a condition on the discretizationspace, which states that the space is rich enough to tackle the rigid body motions. Weprove that this condition is satisfied for regular triangulations, if the boundary elementspace contains the piecewise constants.

Our analysis extends the works [2, 4, 5, 6]. Unlike [2], we avoid any assumption onthe mesh-size. Unlike [4], we avoid the use of an interior Dirichlet boundary. Unlike [5,6], we avoid any pre- and postprocessing steps as well as the numerical solution ofadditional boundary value problems.

References

[1] Aurada, Feischl, Fuhrer, Karkulik, Melenk, Praetorius: Classical FEM-BEM cou-pling methods: nonlinearities, well-posedness, and adaptivity. Comput. Mech., pub-lished online first, (2012).

[2] Carstensen, Funken, Stephan: On the adaptive coupling of FEM and BEM in 2d-elasticity. Numer. Math., 77 (2012), 187–221.

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[3] Feischl, Fuhrer, Karkulik, Praetorius: Stability of symmetric and nonsymmetricFEM-BEM couplings in nonlinear elasticty. ASC Report 52, TU Wien (2012).

[4] Gatica, Hsiao, Sayas: Relaxing the hypotheses of Bielak-MacCamy’s BEM-FEMcoupling. Numer. Math., 120 (2012), 465–487.

[5] Of, Steinbach: Is the one-equation coupling of finite and boundary element methodsalways stable? Berichte aus dem Institut fur Numerische Mathematik, TU Graz, 6(2011).

[6] Steinbach: On the stability of the non-symmetric bem/fem coupling in linear elas-ticity. Comput. Mech., published online first, (2012).

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AN AXIOMATIC APPROACH TO OPTIMALITY OFADAPTIVE ALGORITHMS WITH APPLICATIONS TO BEM

Michael Feischla and Dirk Praetoriusb

Vienna University of Technology,Institute for Analysis and Scientific Computing, Vienna, Austria


Based on recent joint work [Carstensen-Feischl-Page-Praetorius, 2013+], we considerabstract h-adaptive algorithms of the form

solve −→ estimate −→ mark −→ refine

Given a triangulation T , we assume that we can compute some approximation UT ofthe exact solution u. Starting from an initial triangulation T0 and given an accuracyε > 0, the adaptive algorithm aims to iteratively refine a minimal number of elementsof T0, T1, T2, . . . such that for some N > 0 it holds

‖u− UTN‖ ≤ ε.

Analyzing the existing literature on h-adaptive algorithms, we extract a set of axiomswhich are sufficient —and in some cases even necessary— to obtain convergence withquasi-optimal rates

‖u− UT ‖ ≤ C(#T )−s,

with s > 0 as large as theoretically allowed by the structure of the problem (e.g. s = 1/dfor lowest-order FEM for the Poisson problem in Rd).

The abstract analysis developed is then employed to study applied problems likethe weakly- as well as the hyper-singular integral equation

V u = (1/2 +K)g resp. Wu = (1/2−K ′)ψ on Γ,

where the approximations UT are computed via the boundary element method (BEM)and the adaptive algorithm is driven by some weighted residual error estimator η` from[Carstensen-Maischak-Stephan, 2000] (see [Feischl-Karkulik-Melenk-Praetorius, 2013]).Using the abstract results, we are able to prove quasi-optimal convergence rates for theBEM of both equations. Since the abstract approach applies also to the finite elementmethod (FEM), our next step is to study the coupling of FEM and BEM.

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A REDUCED BASIS BOUNDARY ELEMENT METHODFOR A CLASS OF PARAMETERIZED

ELECTROMAGNETIC SCATTERING MODEL

M. Ganesh1, J. S. Hesthaven2 and B. Stamm3

1 Colorado School of Mines, Golden, CO 80401, [email protected]

2 Brown University, Providence, RI 02912, [email protected]

3 UPMC University of Paris 06 and CNRS [email protected]

We consider a parameterized multiple scattering wave propagation model in three di-mensions. The parameters in the model describe the location, orientation, size, shape,and number of scattering particles as well as properties of the input source field suchas the frequency, polarization, and incident direction. The need for fast and efficient(online) simulation of the interacting scattered fields under parametric variation of themultiple particle surface scattering configuration is fundamental to several applicationsfor design, detection, or uncertainty quantification.

For such dynamic parameterized multiple scattering models, the standard dis-cretization procedures are prohibitively expensive due to the computational cost as-sociated with solving the full model for each online parameter choice. In this work,we propose an iterative offline/online reduced basis approach for a boundary elementmethod to simulate a parameterized system of surface integral equations reformulationof the multiple particle wave propagation model.

The approach includes (i) a greedy algorithm based computationally intensive offlineprocedure to create a selection of a set of a snapshot parameters and the constructionof an associated reduced boundary element basis for each reference scatterer and (ii) aninexpensive online algorithm to generate the surface current and scattered field of theparameterized multiple wave propagation model for any choice of parameters within theparameter domains used in the offline procedure. Comparison of our numerical resultswith experimentally measured results for some benchmark configuration demonstratethe power of our method to rapidly simulate the interaction of scattered wave fieldsunder parametric variation of the overall multiple particle configuration.

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RETARDED POTENTIAL BOUNDARY INTEGRAL EQUATIONSFOR SOUND RADIATION IN A HALF-SPACE

Heiko Gimperlein

Department of Mathematical Sciences, University of Copenhagen,Universitetsparken 5, 2100 Copenhagen O, Denmark,

[email protected]

Motivated by the sound radiation of tires, we discuss a time-dependent boundary in-tegral formulation for the wave equation in R3

+ outside a bound domain Ω. Using aGreen’s function which satisfies the acoustic boundary conditions on ∂R3

+, the exte-rior problem reduces to an integral equation on ∂Ω ∩ R3

+. It is solved by Galerkinapproximation and analytical convolution in time. We investigate the properties ofthe relevant boundary integral operators and deduce a priori and a posteriori errorestimates for the numerical solution. Also computational aspects will be considered.

ANALYSIS OF A NON-SYMMETRIC COUPLINGOF INTERIOR PENALTY DG AND BEM

Norbert Heuer1 and Francisco-Javier Sayas2

1Facultad de Matematicas, Pontificia Universidad Catolica de Chile, Santiago, [email protected]

2University of Delaware, Newark, Delaware, [email protected]

We present an analysis of a non-symmetric coupling of interior penalty discontinuousGalerkin and boundary element methods in two and three dimensions. Main resultsare discrete coercivity of the method, and thus unique solvability, and quasi-optimalconvergence. The proof of coercivity is based on a localized variant of the variationaltechnique from Sayas. This localization gives rise to terms which are carefully an-alyzed in fractional order Sobolev spaces, and by using scaling arguments for rigidtransformations.

Heuer acknowledges support by FONDECYT project 1110324.

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ADAPTIVE NONCONFORMING BOUNDARY ELEMENT METHODS

Norbert Heuera and Michael Karkulikb

Facultad de Matematicas,Pontificia Universidad Catolica de Chile, Santiago, Chile.


Nonconforming approximations to functions can offer various advantages. Usually, non-conforming discretizations for the approximate solution of PDE drop certain continuityrequirements, which makes them flexible as well as easier to implement than their con-forming counterparts. For example, the approach by M. Crouzeix and P.-A. Raviart [1]uses discontinuous basis functions, and inter-element continuity is imposed weakly byenforcing edge or face jumps to have vanishing integral mean. This approach, and var-ious others, have been developed and analyzed rigorously for finite element methods.The situation is less developed in BEM. In [3], the approach of Crouzeix and Raviartis employed for solving the Laplacian’s hypersingular integral equation

Wu = f

on a screen Γ. It is shown that uniform mesh refinement yields a convergence rateO(h1/2) if u ∈ H1(Γ).

The aim of this talk is to present first extensions of this approach, regarding adap-tivity. First, we present numerical examples which lead to the conjecture that, contraryas to usual expectations, imposing a higher regularity on u does not increase the con-vergence rate for uniform refinement. According to the Strang lemma, this is due tothe consistency error, and we discuss briefly why it cannot be expected to convergewith a higher rate. Hence, the optimal order for this method seems to be O(h1/2).However, if the solution has a reduced regularity, i.e. u ∈ Hs(Γ) for 1/2 < s < 1, theconvergence rate for uniform refinement is reduced to O(hs−1/2). The second aim of ourtalk is to present an a posteriori error estimator, based on the h−h/2 methodology [2],which can be employed in an adaptive algorithm. Numerical examples indicate thatadaptivity recovers the optimal rate.

References

[1] M. Crouzeix, P.-A. Raviart: Conforming and nonconforming finite elementmethods for solving the stationary Stokes equations. I, Rev. Francaise Automat. In-format. Recherche Operationelle Ser. Rouge 7 (1973), 33–75.

[2] S. Ferraz-Leite, D. Praetorius: Simple a posteriori error estimators for theh-version of the boundary element method, Computing 83 (2008), 135–162.

[3] N. Heuer, F. J. Sayas: Crouzeix-Raviart boundary elements, Numer. Math. 112(2009), 381–401.

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PARALLEL BEM-BASED METHODS

Dalibor Lukasa, Michal Mertab, Lukas Malyc, Petr Kovard and Tereza Kovarovae

Department of Applied Mathematics & IT4Innovations,VSB-Technical University of Ostrava, Czech [email protected], [email protected],

[email protected], [email protected], [email protected]

In the first part of our contribution we propose a method of a parallel distribution ofdensely populated matrices arising in boundary element discretizations of partial differ-ential equations. In our method the underlying boundary element mesh is decomposedinto N submeshes. Then the related N ×N submatrices are assigned to N concurrentprocesses to be assembled. We additionally require each process to hold exactly one di-agonal submatrix, since its assembling is typically most time consuming when applyingfast boundary elements. We obtain a class of such optimal parallel distributions of thesubmeshes and submatrices by cyclic decompositions of undirected complete graphs.The resulting algorithm enjoys parallel computational scalability O(1/N) and memoryscalability O(1/

√N). This is documented by numerical experiments up to 3 millions of

boundary elements and 133 processors. Hierarchical matrices with the adaptive crossapproximation as well as the fast multipole method are employed.

In the second part a boundary-element counterpart of the domain decompositionvertex solver is proposed and tested for a 2-dimensional Poisson’s equation. While thestandard theory has been developed only for triangular or quadrilateral subdomains,where harmonic base functions are available, the practical mesh-partitioners generatecomplex polygonal subdomains. We aim at bridging this gap. We construct the coarsesolver on general polygonal subdomains so that the local coarse stifness matrix isapproximated by a boundary-element discretization of the Steklov-Poincare operator.The efficiency of our approach is documented by a substructuring into L-shape domains.

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ON THE QUASI-OPTIMAL CONVERGENCEIN FEM-BEM COUPLING

Jens Markus Melenk1a, Dirk Praetorius1b and B. Wohlmuth2

1Institut fur Analysis und Scientific Computing,Technische Universitat Wien, AustriaWiedner Hauptstr. 8-10, A-1040 Wien


2M2 Zentrum Mathematik Technische Universitat Munchen,Boltzmannstr. 3, D-85748 Garching, Germany

[email protected]

We consider the symmetric coupling of FEM and BEM, which involves two field vari-ables, namely, one variable defined in the volume Ω and one on its boundary ∂Ω; thelatter represents the exterior domain. The classical convergence theory of symmetricFEM-BEM coupling studies their joint approximation and shows a best approximationproperty for this joint approximation. However, typical discretizations feature differ-ent approximation powers for the two field variables: The approximation power of thespace used to discretize the boundary variable is often better by h1/2 than that forthe volume variable. In this talk, we show that this better approximation power ofthe space for the boundary variable can translate into better convergence rates for theboundary variable under suitable regularity assumptions.

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ON THE ELLIPTICITY OF COUPLED FINITE ELEMENTAND ONE-EQUATION BOUNDARY ELEMENT METHODS

FOR BOUNDARY VALUE PROBLEMS

Gunther Ofa and Olaf Steinbachb

Institute of Computational Mathematics,Graz University of Technology, Graz, Austria.


We present the extension of recent results on the stability of the Johnson–Nedelec cou-pling of finite and boundary element methods to the case of boundary value problems.

In [1, 2, 3] the case of a free–space transmission problem was considered, and suf-ficient and necessary conditions are stated which ensure the ellipticity of the bilinearform for the coupled problem. The proof was based on the relation of the energieswhich are related to both the interior and exterior problem.

When considering boundary value problems for both interior and exterior problems,additional estimates to bound the energy for the solutions of related subproblems arerequired. Moreover, several techniques for the stabilization of the coupled formulationsare analyzed. Applications involve boundary value problems with either hard or softinclusions, exterior boundary value problems, and macro–element techniques.

References

[1] G. Of, O. Steinbach. Is the one–equation coupling of finite and boundary elementmethods always stable? ZAMM - Journal of Applied Mathematics and Mechanics(2013) published online.

[2] F.–J. Sayas. The validity of Johnson–Nedelec’s BEM–FEM coupling on polygonalinterfaces. SIAM J. Numer. Anal. 47 (2009) 3451–3463.

[3] O. Steinbach. A note on the stable one–equation coupling of finite and boundaryelements. SIAM J. Numer. Anal. 49 (2011) 1521–1531.

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RADIAL BASIS FUNCTIONS WITHAPPLICATIONS TO ELASTICITY

Sergej Rjasanowa and Richards Grzhibovskisb

Department of Mathematics, Saarland University, [email protected], [email protected]

Radial basis functions (RBFs) have become increasingly popular for the constructionof smooth interpolant s : Rn → R through a set o f N scattered, pairwise distinctdata points. In the first part of the talk we introduce the RBF’s [1] and discuss theirproperties. The second part of the talk is devoted to the reconstruction of the three-dimensional metal sheet surfaces obtained via incremental forming techniques. [2]. Inthis application, the data comes from optical measurements of sheet metal parts. Thetop and the bottom surfaces of the part are measured in a fixed frame of reference,and a distribution of thickness along the part is sought. In the third part of the talk,a boundary integral formulation for a mixed boundary value problem in linear elasto-statics with a conservative right hand side is considered [3]. A meshless interpolantfor the scalar potential of the volume force density is constructed by means of radialbasis functions. An exact particular solution to the Lame system with the gradient ofthis interpolant as the right hand side is found. Thus, the need of approximating theNewton potential is eliminated. The procedure is illustrated on numerical examples.

References

[1] H. Wendland. Scattered Data Approximation. Cambridge Monographs on Appliedand Computational Mathematics, Cambridge University Press, 2005.

[2] M. Bambach, R. Grzhibovskis, G. Hirt, and S. Rjasanow. Adaptive cross ap-proximation for surface reconstruction based on radial basis functions. Journal ofEngineering Mathematics, 62:149–160, 2008.

[3] H. Andra, R. Grzhibovskis, and S. Rjasanow. Boundary Element Method for linearelasticity with conservative body forces. In T. Apel and O. Steinbach, editors,Advanced Finite Element Methods and Applications, number 66 in Lecture Notesin Applied and Computational Mechanics, pages 275–297. Springer-Verlag, Berlin-Heidelberg-NewYork, 2012.

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STOKES FLOW ABOUT A COLLECTIONOF SLIP SOLID PARTICLES

A. Sellier

LadHyx. Ecole polytechnique, 91128 Palaiseau Cedex, [email protected]

We consider a collection of N ≥ 1 arbitrarily-shaped solid slip particles immersed ina Newtonian liquid with uniform viscosity µ > 0 and density ρ > 0. Each particlePn with center of mass On and smooth surface Sn experiences a rigid-body migrationwith prescribed translational velocity U(n) (here, the velocity of its attached point On)and angular velocity Ω(n). The resulting flow about the cluster has pressure field p andvelocity field u with typical magnitude V > 0. For particles with length scale a theReynolds number is Re = ρV a/µ. Assuming henceforth that Re = ρV a/µ 1 makesit possible to neglect all inertial effects and to consider that the flow (u, p) in the liquiddomain D is governed by the steady Stokes problem

µ∇2u = ∇p and ∇.u = 0 in D, (1)

(u, p)→ (0, 0) far from the cluster. (2)

Of course, (1)-(2) must be supplemented with relevant boundary conditions on eachparticle surface. In practice, these conditions depend upon the nature of the fluidand/or of those surfaces. For instance, for a rarefied gas in the continuum regime or aliquid near a solid hydrophobic or lyophobic surface one allows the fluid to flow overthe surface which is then called a slip surface. Here, we consider slip particles andadopt the following widely-employed and so-called Navier slip conditions

u(M) = U(n) + Ω(n) ∧OnM + λn σ.n− (n. σ.n)n/µ on each Sn (3)

where n is the unit normal on Sn directed into the liquid and λn ≥ 0 the slip length ofthe surface Sn.

This work will successively examine for the well-posed problem (1)-(3) the followingissues:

(i) Derive regularized boundary-integral equations for the unknown surface tractionf = σ.n exerted on the cluster’s surface by the flow (u, p).

(ii) Implement a boundary element technique to efficiently and accurately invert theencountered boundary-integral equations and calculate the net hydrodynamic force andtorque exerted on each migrating slip particle.

(iii) Test the advocated procedure against both analytical results (single slip sphere)and numerical results (single slip spheroid and a 2-sphere cluster) previously obtainedin the literature using quite different methods.

(iv) Present and discuss new numerical results for several clusters made of slipand/or non-slip particles.

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BOUNDARY ELEMENT METHODS FORACOUSTIC RESONANCE PROBLEMS

Gerhard Unger

Institute of Computational Mathematics,Graz University of Technology, Graz, Austria.

[email protected]

We characterize acoustic resonances as eigenvalues of boundary integral operator eigen-value problems and use boundary element methods for their numerical approximation.Eigenvalue problem formulations for resonance problems which are based on standardboundary integral equations exhibit additional eigenvalues which are not resonancesbut eigenvalues of a related ”interior” eigenvalue problem. In practical computationsit is for some typical applications hard to extract the resonances when using standardboundary integral formulations. In this talk we present regularized combined bound-ary integral formulations which only exhibit resonances as eigenvalues. We providea comprehensive numerical analysis of the boundary element approximations of theseeigenvalue problem formulations where general results of the discretization of eigenvalueproblems for holomorphic Fredholm operator-valued functions are used [O. Steinbach,G. Unger: Convergence analysis of a Galerkin boundary element method for the Dirich-let Laplacian eigenvalue problem. SIAM J. Numer. Anal.,50 (2012), 710–728]. Finallywe present some numerical examples.

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BEM BASED SHAPE OPTIMIZATION USING SHAPECALCULUS AND MULTIRESOLUTION ANALYSIS

Jan Zapletal1, Kosala Bandara2a, Fehmi Cirak2b, Gunther Of3c and Olaf Steinbach3d

1Department of Applied Mathematics, VSB – Technical University of Ostrava, CzechRepublic,

[email protected],

2Department of Engineering, University of Cambridge,[email protected], [email protected],

3Institute of Computational Mathematics, Graz University of Technology,[email protected], [email protected]

In the talk we present shape optimization based on the gradient information obtainedby the shape calculus in the continuous setting. Since the shape of an object is givenby its boundary and the shape gradient of the cost functional presented only involvesevaluation of the Neumann data of the state and its adjoint, the boundary elementmethod provides a suitable scheme for solving the associated boundary value problems.

Together with the shape calculus we use the multiresolution analysis and performthe optimization on different levels of the corresponding surface mesh. This approachserves as a globalization strategy and in connection with mesh smoothening preventsending up with badly shaped solutions.

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5: Mini-Symposium: Boundary-Domain Integral Equations

5 Mini-Symposium: Boundary-Domain Integral Equa-

tions

Organiser: Sergey Mikhailov

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LOCALIZED BOUNDARY-DOMAIN INTEGRAL EQUATIONSAPPROACH FOR DIRICHLET AND ROBIN PROBLEMS OF THE

THEORY OF PIEZO-ELASTICITY FOR INHOMOGENEOUS SOLIDS

Otar Chkadua

Mathematical Institute of I.Javakhishvili Tbilisi State University,2 University str.,Tbilisi, Georgia

and Sokhumi State University, 9 A.Politkovskaia str.,Tbilisi, [email protected]

We consider the three–dimensional Dirichlet and Robin boundary-value problems(BVPs) of piezo-elasticity for anisotropic inhomogeneous solids and develop the gen-eralized potential method based on the localized parametrix method. Using Green’srepresentation formula and properties of the localized layer and volume potentials wereduce the Dirichlet and Robin BVPs to the localized boundary-domain integral equa-tions (LBDIE) systems.

First we establish the equivalence between the original boundary value problemsand the corresponding LBDIE systems. Afterwards, we establish that the localizedboundary-domain integral operators obtained belong to the Boutet de Monvel alge-bra and with the help of the Vishik-Eskin theory, based on the factorization method(Wiener-Hopf method), we investigate corresponding Fredholm properties and proveinvertibility of the localized operators in appropriate function spaces.

This is a joint work with Sergey Mikhailov (Brunel University of London, UK) andDavid Natroshvili (Tbilisi Technical University, Georgia).

Acknowledgements: This research was supported by grant No. EP/H020497/1:“Mathematical analysis of Localized Boundary-Domain Integral Equations forVariable-Coefficient Boundary Value Problems” from the EPSRC, UK.

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NUMERICS AND SPECTRAL PROPERTIES OF BOUNDARYDOMAIN INTEGRAL AND INTEGRO-DIFFERENTIAL

OPERATORS IN 3D

Richards Grzhibovskis1 and Sergey E. Mikhailov2

1 Department of Mathematics, Saarland University, [email protected]

2 Department of Mathematical Sciences, Brunel University London, [email protected]

An elliptic boundary value problem with variable coefficients, where no fundamentalsolution is explicitly available, can be still reduced to a system of Boundary-DomainIntegral or Integro-Differential Equations, BDI(D)Es, based on a parametrix (Levifunction), cf. [3, 1, 4].

In this study, developing results of [2], we consider collocation discretization ofBDI(D)E systems for the ”stationary difusion” problems with variable scalar-valuedcoefficient. In contrast to boundary integral formulation, a volume discretization isnecessary even when the right hand side is zero, and the discretised layer potentials,volume potential and the remainder potential operators produce fully populated matri-ces. Two ways of avoiding prohibitively expensive second order complexity and storagerequirements for the fully populated matrices are discussed. The first one is based onhierarchical matrix technique in conjunction with the adaptive cross approximation(ACA) procedure. The second way is related to a localized parametrix, which leadsto localised BDI(D)Es that are reduced by discretisation to sparse matrices. We com-ment on the implementation details and report the results of numerical experiments inthree-dimensional domains.

Some of the considered non-localised BDI(D)E systems are of the second kind andsolved by fast converging Neumann iterations, which is related with favourable spectralproperties of the operators. We report the spectral properties of the correspondingdiscrete systems and compare them with the theoretical estimates. Note that thenumerical spectral properties of some BDI(D)Es in 2D domains were presented in [5].

References

[1] O. Chkadua, S. E. Mikhailov, and D. Natroshvili. Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, I: Equivalenceand invertibility. Journal of Integral Equations and Applications, 21(4):499–543,2009.

[2] R. Grzhibovskis, S. Mikhailov, and S. Rjasanow. Numerics of boundary-domainintegral and integro-differential equations for BVP with variable coefficient in 3D.Computational Mechanics, Nr. 51, DOI: 10.1007/s00466-012-0777-8, pages 495–503,2013.

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[3] S. E. Mikhailov. Localized boundary-domain integral formulations for problemswith variable coefficients. Engineering Analysis with Boundary Elements, 26:681–690, 2002.

[4] S. E. Mikhailov. Analysis of united boundary-domain integro-differential and inte-gral equations for a mixed BVP with variable coefficient. Math. Methods in AppliedSciences, 29:715–739, 2006.

[5] S. E. Mikhailov and N. A. Mohamed. Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with variable coefficient. Interna-tional Journal of Computer Mathematics, 89(11):1488–1503, 2012.

SPECTRAL PROPERTIES AND PERTURBATIONS OFBOUNDARY-DOMAIN INTEGRAL EQUATIONS

Sergey E. Mikhailov

Department of Mathematics, Brunel University, London, [email protected]

The Dirichlet and Neumann BVPs for a variable-coefficient PDE on three-dimensionalinterior and exterior domains with compact boundaries are reduced to some directBoundary-Domain Integro-Differential Equations (BDIDEs) of the second kind. Thenthe obtained BDIDEs are analysed in the weighted Sobolev (Beppo Levi type) functionspaces more suitable for exterior domains and coinciding with the standard Sobolevspaces in interior domains. Equivalence of the BDIDEs to the original BVPs and theinvertibility of the BDIDE operators are analysed. When the operators are not notinvertible, they are still Fredholm with zero index, and we perturb them with somefinite-dimensional operators, making the perturbed operators invertible and deliveringa solution to of the original BVP. The spectral properties of the BDIDEs are thenstudied and some explicit conditions on the coefficient behaviour are given, insuringthat the BDIE operator spectrum belongs to the unit circle and the Neumann iterationscan be used to solve the corresponding BDIE systems. The analysis employs themethods developed in [1] - [6].

References

[1] Mikhailov, S. E., Finite-dimensional perturbations of linear operators and some applica-tions to boundary integral equations. Engineering Analysis with Boundary Elements, 23,805–813 (1999).

[2] Mikhailov, S. E., Localized boundary-domain integral formulations for problems withvariable coefficients. Eng. Analysis with Boundary Elements, 26, 681–690 (2002).

[3] Chkadua, O., Mikhailov, S. E., Natroshvili, D., Analysis of direct boundary-domain inte-gral equations for a mixed BVP with variable coefficient, I: Equivalence and invertibility.Journal of Integral Equations and Applications, 21(4), 499–543 (2009).

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[4] Mikhailov, S. E., Analysis of united boundary-domain integro-differential and integralequations for a mixed BVP with variable coefficient. Math. Methods in Applied Sciences,29, 715–739 (2006).

[5] Chkadua, O., Mikhailov, S. E., Natroshvili, D., Analysis of segregated boundary-domainintegral equations for variable-coefficient problems with cracks. Numerical Methods forPartial Differential Equations, 27(1), 121–140 (2011).

[6] Chkadua, O., Mikhailov, S. E., Natroshvili, D., Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains. Anal-ysis and Applications, 11(4), 2013 (to appear).

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ACOUSTIC SCATTERING BY INHOMOGENEOUSANISOTROPIC OBSTACLE: BOUNDARY-DOMAIN

INTEGRAL EQUATION APPROACH

David Natroshvili

Georgian Technical University,Department of Mathematics, Tbilisi, Georgia

[email protected]

We consider the time-harmonic acoustic wave scattering by a bounded layered anisotro-pic inhomogeneity embedded in an unbounded anisotropic homogeneous medium. Thematerial parameters and the refractive index are assumed to be discontinuous across theinterfaces between the inhomogeneous interior and homogeneous exterior regions. Thecorresponding mathematical problems are formulated as boundary-transmission prob-lems for a second order elliptic partial differential equation of Helmholtz type withdiscontinuous variable coefficients. We show that the boundary-transmission prob-lems with the help of localized potentials can be reformulated as a localized boundary-domain integral equations (LBDIE) systems and prove that the corresponding localizedboundary-domain integral operators (LBDIO) are invertible.

First we establish the equivalence between the original boundary-transmission prob-lems and the corresponding LBDIE systems which plays a crucial role in our anal-ysis. Afterwards, we establish that the localized boundary domain integral opera-tors obtained belong to the Boutet de Monvel algebra of pseudo-differential operators.And finally, applying the Vishik-Eskin theory based on the factorization method (theWiener-Hopf method) we investigate Fredholm properties of the LBDIOs and provetheir invertibility in appropriate function spaces. This invertibility property impliesthen existence and uniqueness results for the LBDIE systems and the correspondingoriginal boundary-transmission problems.

Beside a pure mathematical interest these results can be applied in constructingand analysis of numerical methods for solution of the LBDIE systems and thus thescattering problems in inhomogeneous anisotropic media.

This is a joint work with Sergey Mikhailov (Brunel University of London, UK) andOtar Chkadua (Tbilisi State University, Georgia).

Acknowledgements: This research was supported by grant No. EP/H020497/1:“Mathematical analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems” from the EPSRC, UK.

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6: Mini-Symposium: Computational Micromagnetics

6 Mini-Symposium: Computational Micromagnet-

ics

Organisers: Dirk Praetorius and ThomasSchrefl

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MAGNUM.FE: A MICROMAGNETIC FINITE-ELEMENTCODE BASED ON FENICS

Claas Abert

Institut fur Angewandte Physik und Zentrum fur Mikrostrukturforschung,Universitat Hamburg, Jungiusstr. 11, D-20355 Hamburg, Germany

[email protected]

Micromagnetism is the theory of choice for the description of magnetization dynamicson the nanometer scale. In contrast to domain theory this semiclassical continuumtheory is able to resolve the structure of domain walls and describe switching pro-cesses. Also it is well suited for the treatment with numerical algorithms as opposedto quantum mechanical ab initio calculations.

In this talk we present the complete micromagnetic finite-element code magnum.fe.magnum.fe is developed using the recently released general-purpose finite element pack-age FEniCS. Due to the high level of abstraction, the code is both very readable andextendable. Hence it delivers a good starting point for the implementation and testingof novel finite-element algorithms.

Along with an overview over the software, we present a fully implicit and linearscheme for the integration of the Landau-Lifshitz-Gilbert equation, that is implementedin magnum.fe.

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MULTISCALE SIMULATION OF MAGNETIC NANOSTRUCTURES

Florian Bruckner

Institute of Solid State Physics,Vienna University of Technology, Vienna, Austria.

[email protected]

Due to the ongoing miniaturization of modern magnetic devices, like GMR sensors,magnetic write heads, spintronic devices and so on, micromagnetic simulations gainmore and more importance, since they are an essential tool to understand the behaviorof magnetic materials in the nanometer scale. Using numerical simulations allows tooptimize the micro-structure of such devices or to test new concepts prior to performingexpensive experimental tests.

A method is presented which allows to extend the micromagnetic model by ad-ditional macroscopic parts which are described in an averaged sense. In contrast tothe microscopic parts which are described by Landau-Lifshitz-Gilbert (LLG) equa-tions, these macroscopic parts are based on classical magnetostatic Maxwell equations,which could be extended to a full Maxwell description in a straight-forward way. Theaveraged description using Maxwell equations allows to overcome the upper bound forthe discrete element sizes, which is intrinsic to the micromagnetic models, since thedetailed domain structure of the ferromagnetic material needs to be resolved.

Combining microscopic and macroscopic models and solving the correspondingequations simultaneously provides a multiscale method, which allows to handle prob-lems of a dimension, which would otherwise be far out of reach. A basic prerequisitefor the application of the method, is that microscopic and macroscopic parts can beseparated into disjoint regions. The performance of the implemented algorithm isdemonstrated by the simulation of the transfer curve of a magnetic recording readhead, as it is built into current hard drives.

Independent of the former approach the use of parallelized algorithms allow thehandle larger problems. A shared-memory-parallelization of hierarchical matrices isdemonstrated, since these require a large amount of the storage consumption as well asof the computation time for typical simulations. For the setup of the matrices a nearlyperfect parallelization could be reached, whereas for the matrix-vector-multiplicationthe computation time stagnates at a few computation cores, due to the restricted mainmemory bandwidth.

References

[1] F. Bruckner, C. Vogler, M. Feischl, D. Praetorius, B. Bergmair, T. Huber, M.Fuger, D. Suss, “3D FEM-BEM-coupling method to solve magnetostatic Maxwellequations”, J. Magn. Magn. Mater., 324 (2012), 1862-1866.

[2] F. Bruckner, M. Feischl, T. Fuhrer, P. Goldenits, M. Page, D. Praetorius, D. Suess,2012), “Multiscale modeling in micromagnetics: well-posedness and numerical in-tegration”, arXiv:1209.5548 [math.NA].

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[3] F. Bruckner, C. Vogler, B. Bergmair, T. Huber, M. Fuger, D. Suss, M. Feischl, T.Fuhrer, M. Page, D. Praetorius, “Combining micromagnetism and magnetostaticMaxwell equations for multiscale magnetic simulations”, commited to J. Magn.Magn. Mater., (2013).

FINITE ELEMENT AND BOUNDARY ELEMENTMETHOD IN MAGNETIC SPIN TRANSPORT

AND MAGNETIC HYBRID STRUCTURES

Gino Hrkac1, Marcus Page2 and Dieter Suess2

1College of Engineering, Mathematics and Physical Sciences,University of Exeter, Devon, UK

[email protected]

2Vienna University of Technology, Austria

In this paper we investigate the difference in the Slonzewski spin torque approach andthe change of spin accumulation based on a diffusion equation derived by Zhang, Levyand Fert and at an interface of a magnetic to a non-magnetic material, in a micromag-netic framework, by considering two cases; first a spin-accumulation due to current andsecond due to field excitation; refereed to as spin-pumping. In spin polarized transportmodels, the magnetization was assumed to be constant / pinned in one of the layersand the polarization was introduced as an extra field that is part of the effective field.Spin diffusion and interfacial effects were neglected but it was shown that these effectsare important in magnetoresistance experiments. Zhang et al. introduced a modelfor the relaxation of a coupled spin-magnetization system, where they considered theone-dimensional case.We start from a spin dynamics equation that is derived from a Boltzman transportequation and couple it to the Land Lifshitz Gilbert equation. We do a comparablestudy between the diffusion approach and the Slonzewski approach that is derivedfrom circuit theory. In the first case of our study we assume a constant electric currentand mapped the spin-accumulation profiles over the interface and for the second casewe omitted the electric current and assume only an applied field and a time varyingmagnetization. This leads to a reduced expression of the spin accumulation current inthe diffusion equation, meaning that the spin accumulation is not fully absorbed in theferromagnet and unabsorbed transverse spin accumulation diffuses over the interfaceinto the metal layer and leads to a modified spin-torque.Also we discuss the effect of symmetric and asymmetric spin torque effects on thevortex oscillations in a spin-valve system in the presence of two vortices and its effecton the frequency and linewidth. We will show that the mutal interaction between twovortices, considering a non-linear current distribution, pseudo-diffusion model, can leadto non-linear oscillation regimes.

The authors gratefully acknowledge financial support from the Royal Society UKand the WWTF.

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COUPLING AND NUMERICAL INTEGRATION OF LLG

Marcus Pagea and Dirk Praetoriusb



In our talk, we give an overview on our recent preprints [arXiv:1209.5548], [arXiv:1303.4009], and [arXiv:1303.4060]. These are concerned with the numerical integrationof extended formulations of the Landau-Lifshitz-Gilbert equation (LLG), where thecoupling of LLG with other PDEs is analyzed.

We extend a recent algorithm of [Alouges 2008] who considered the exchange-onlyformulation of LLG and introduced and analyzed an implicit numerical integratorwhich only requires the solution of one linear system per time step. Moreover, theproposed integrator is unconditionally convergent, i.e. no coupling of spatial mesh-sizeh and time step-size k are imposed for stability. The convergence proof is constructivein the sense that it also proves existence of weak solutions.

Independently, [Goldenits-Praetorius-Suess 2011] and [Alouges-Kritsikis-Toussaint2012] included the explicit integration of uniaxial anisotropy, demagnetization, andapplied external fields (i.e. the effective field is extended, yet linear) into the overallAlouges-type integrator. As before, they proved unconditional convergence.

In the above preprints, we further extend the Alouges integrator to show its full po-tential. By exploiting an abstract framework, we can cover general field contributionsthat might be nonlinear, non-local, and/or time-dependent. Applications include mul-tiscale modelling with LLG, coupling of LLG to the full Maxwell system or some eddy-current regime, or even to the conservation of momentum equation to include magne-tostrictive effects. Moreover, for the coupling of LLG with time-dependent PDEs, onefocus is on the decoupling of the respective time integration. For the Maxwell-LLGsystem, for instance, only two linear systems have to be solved per time step, one forthe LLG part and one for the Maxwell part. Even in this general setting, we still proveunconditional convergence, and our proof also provides the existence of weak solutions.

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A NONLOCAL PARABOLIC AND HYPERBOLIC MODELFOR TYPE-I SUPERCONDUCTORS

Karel Van Bockstala and Marian Slodickab

Research Group NaM2, Department of Mathematical Analysis,Ghent University, Galglaan 2, 9000 Ghent, Belgium


A vectorial nonlocal linear parabolic and hyperbolic problem with applications in su-perconductors of type-I is studied. A superconductive material of type-I occupies abounded domain Ω ⊂ R3 with a Lipschitz continuous boundary ∂Ω. The full Maxwell’sequations (δ = 1) and quasi-static Maxwell’s equations (δ = 0) for linear materials areconsidered. They can be written as

∇×H = J + δ∂tD = J + δε∂tE; (1)

∇×E = −∂tB = −µ∂tH .

The current density J is supposed to be the sum of a normal and a superconductingpart, that is J = Jn +J s. The normal density current Jn is required to satisfy Ohm’slaw Jn = σE. The nonlocal representation of the superconductive current J s byEringen [1] is considered. This representation identifies the state of the superconductor,at time t, with the magnetic field H(·, t) and is given by the linear functional

J s(x, t) =

∫Ω

σ0 (|x− x′|) (x−x′)×H(x′, t)dx′ =: −(K0?H)(x, t), (x, t) ∈ Ω×(0, T ).

Taking the curl of (1) results into the following parabolic (δ = 0) and hyperbolic (δ = 1)integro-differential equation

δεµ∂ttH + σµ∂tH +∇×∇×H +∇× (K0 ?H) = 0. (2)

A new convolution kernel is derived, namely ∇× J s = −K ?H when H is divergencefree. The positive definiteness of the kernel K is shown. Taking into account theprevious consideration, equation (2) can be rewritten as

δεµ∂ttH + σµ∂tH −∆H +K ?H = 0.

The well-posedness of both problems is discussed under low regularity assumptionsand the error estimates for various time-discrete schemes (based on backward Eulerapproximation) are established.

References

[1] A.C. Eringen. Electrodynamics of memory-dependent nonlocal elastic continua. J.Math. Phys., 25:3235–3249, 1984.

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7: Mini-Symposium: Computational challenges in Discontinuous Galerkin methods

7 Mini-Symposium: Computational challenges in

Discontinuous Galerkin methods

Organisers: Paola Antonietti, Paul Houstonand Ilaria Perugia

7-1


ENERGY STABILITY FOR DISCONTINUOUS GALERKINAPPROXIMATION OF A PROBLEM IN ELASOTODYNAMICS

Paola F. Antonietti1, Blanca Ayuso de Dios2, Ilario Mazzieri3 and Alfio Quarteroni4,5

1 MOX, Dipartimento di Matematica,Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133, Milano, Italy

[email protected]

2 Centre de Recerca Matematica, UAB Science Faculty,08193 Bellaterra, Barcelona, Spain

[email protected]

3 MOX, Dipartimento di Matematica,Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133, Milano, Italy

[email protected]

4 MOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo daVinci 32, 20133 Milano, Italy

5 CMCS, Ecole Polytechnique Federale de Lausanne (EPFL),Station 8, 1015 Lausanne, Switzerland

[email protected]

We introduce a family of semidiscrete-DG methods for the approximation of a generalelastodynamics problem. We discuss the energy stability and present the a-priori erroranalysis of the methods. We also propose some schemes that preserve the total energyof the system and present some numerical experiments that verify the theory.

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STAGGERED DISCONTINUOUS GALERKIN METHODSFOR MAXWELL’S EQUATIONS

Eric Chung

Department of Mathematics,The Chinese University of Hong Kong, Hong Kong SAR.

[email protected]

In this talk, a new type of staggered discontinuous Galerkin methods for the threedimensional Maxwell’s equations is presented. The spatial discretization is based onstaggered Cartesian grids so that many good properties are obtained. First of all, ourmethod has the advantages that the numerical solution preserves the electromagneticenergy and automatically fulfills a discrete version of the Gauss law. Moreover, themass matrices are diagonal, thus time marching is explicit and is very efficient. Ourmethod is high order accurate and the optimal order of convergence is rigorously proved.It is also very easy to implement due to its Cartesian structure and can be regardedas a generalization of the classical Yee’s scheme as well as the quadrilateral edge finiteelements. Furthermore, a superconvergence result, that is the convergence rate isone order higher at interpolation nodes, is proved. Numerical results are shown toconfirm our theoretical statements, and applications to problems in unbounded domainswith the use of PML are presented. A comparison of our staggered method and non-staggered method is carried out and shows that our method has better accuracy andefficiency.

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EFFICIENT DISCONTINUOUS GALERKIN METHODFOR METEOROLOGICAL APPLICATIONS

Andreas Dedner1 and Robert Klofkorn2

1Mathematics Institute and Centre for Scientific Computing,University of Warwick, Coventry CV4 7AL, UK

[email protected],

2Computational Math Group, UCAR, Boulder, [email protected]

In this talk we will introduce a dynamic core for local weather predication based on theDiscontinuous Galerkin method. The method is implemented within the Dune softwareframework (www.dune-project.org). It allows the simulation of the compressible multispecies Navier Stokes equations on general 3D meshes in parallel. Special mechanismsare included to allow the simulation of advection dominated flow and for includinglocal grid adaptation. We will demonstrate that the code allows for efficient, highlyscalable parallel simulations.

To achieve this goal we have developed a stabilization mechanism for the advectiondominated case which works on general unstructured meshes. For the diffusion operatorwe have developed a highly efficient discretization approach. To prove the effectivenessand efficiency of our Dune base numerical core, we compare it with the dynamicalcore of the COSMO local area model. COSMO is an operational code, used by manyweather services. This comparison was performed in cooperation with the GermanWeather Service. We compare the performance of the very different cores based on anumber of standard meteorological benchmarks. We will demonstrate that the Dunecore shows a higher numerical convergence rate. The high subscale resolution of theDG method means that Dune produces a lower error when fixing the grid resolutionor the number of degrees of freedom.

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DISCONTINUOUS GALERKIN METHODS FOR PHASE FIELDMODELS OF MOVING INTERFACE PROBLEMS

Xiaobing Feng

Department of Mathematics, The University of Tennessee,Knoxville, TN 37996, U.S.A.

[email protected]

This talk is concerned with some new convergence results for interior penalty discon-tinuous Galerkin (IPDG) approximations of two types of phase field models which aredescribed respectively by the 2nd order Allen-Cahn equations and the 4th order Cahn-Hilliard equations. The main result to be discussed is the convergence of the numericalinterfaces to the sharp interfaces of the limit models of moving interface problems(namely, the mean curvature flow and the Hele-Shaw flow) as both the numerical meshparameters and the phase field parameter (called the interaction length) tend to zero.The crux for establishing the result is to derive, by a nonstandard technique, errorestimates for the IPDG solutions which blows up only polynomially (instead of expo-nentially) in the reciprocal of the phase field parameter. This is a jointly work withYukun Li of the University of Tennessee at Knoxville.

DISCONTINUOUS GALERKIN METHODS FORNON-LINEAR INTERFACE PROBLEMS

Emmanuil H. Georgoulis

Department of Mathematics, University of Leicester,University Road, Leicester LE1 7RH, United Kingdom

[email protected]

A discontinuous Galerkin (dG) method for the numerical solution of initial/boundaryvalue multi-compartment partial differential equation (PDE) models, interconnectedwith interface conditions, is analysed. The study of interface problems is motivatedby models of mass transfer of solutes through semi-permeable membranes. The caseof fast reactions is also included. More specifically, a model problem consisting of asystem of semilinear parabolic advection-diffusion-reaction partial differential equationsin each compartment with only local Lipschitz conditions on the nonlinear reactionterms, equipped with respective initial and boundary conditions, is considered. Generalnonlinear interface conditions modelling selective permeability, congestion and partialreflection are applied to the compartment interfaces. An interior penalty dG methodfor this problem is analysed both in the space-discrete and in fully discrete settings forthe case of, possibly, fast reactions. The a priori analysis shows that the method yieldsoptimal a priori bounds, provided the exact solution is sufficiently smooth. Numericalexperiments indicate agreement with the theoretical bounds. The talk is based on jointwork with Andrea Cangiani (Leicester) and Max Jensen (Durham).

7-5


ON THE CONVERGENCE OF ADAPTIVEDISCONTINUOUS GALERKIN METHODS

Thirupathi Gudi1 and Johnny Guzman2

1Department of Mathematics,Indian Institute of Science Bangalore, 560012 India

[email protected]

2Division of Applied Mathematics, Brown University, Providence RI 02912, USA

Establishing the convergence and optimality of adaptive finite element methods hasbeen an important and active research topic for more than a decade. In the study, oneof the key step is to establish a contraction property of the form:

‖eh‖h + βηh ≤ ρ (‖eH‖H + βηH) ,

where eh and ηh (or eH and ηH) are the exact error and the error estimator, respectively,on the triangulation Th (or TH). The contraction property is proved for the standardfinite element method, for the nonconforming finite element method and for the mixedfinite element methods in the literature. Also a contraction property is proved for thesymmetric interior penalty method in [Karakashian and Pascal, SINUM, 45 (2007),pp. 641–665], [Hoppe, Kanschat and Warburton, SINUM, 47 (2009), pp. 534–550] and[Bonito and Nochetto, SINUM, 48 (2010), pp. 734–771]. The common issue withthese three articles is that the contraction property is derived assuming the penaltyparameters are sufficiently large (i.e. larger than what is needed for stability of themethod). In this talk, we present contraction properties for various symmetric weaklypenalized discontinuous Galerkin methods only assuming that the penalty parametersare large enough to guarantee stability of the method. For example, in the case of theLDG method the stabilizing parameters only have to be positive. This is achieved by anew marking strategy that uses an auxiliary solution obtained by post-processing thediscontinuous Galerkin solution.

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A COCHAIN COMPLEX FOR INTERIOR PENALTYMETHODS: ERROR ESTIMATES AND MULTIGRID

THROUGH DIFFERENTIAL RELATIONS

Guido Kanschata and Natasha Sharmab

IWR, Ruprecht-Karls-Universitat Heidelberg,Im Neuenheimer Feld 368, 69120 Heidelberg, Germany.

[email protected],[email protected]

We show that the recently developed divergence-conforming methods for the Stokesproblems have discrete stream functions. These stream functions in turn solve a contin-uous interior penalty problem for biharmonic equations. The equivalence is establishedfor the most common methods in two dimensions based on interior penalty terms. Weshow that this relation can be exploited to transfer results on multigrid methods andon error estimates between the two schemes. Through the numerical results, we willillustrate the efficiency of the methods obtained from these relations.

GENERALIZED DG-METHODS FOR HIGHLYINDEFINITE HELMHOLTZ PROBLEMS

Jens Markus Melenk1, Asieh Parsania2 and Stefan A. Sauter3

1Institute fur Analysis und Scientific Computing,Technische Universitat Wien, Wien, Austria.

[email protected]

2Institut fur Mathematik, Universitat Zurich, Zurich, [email protected]

3Institut fur Mathematik, Universitat Zurich, Zurich, [email protected]

We develop a stability and convergence theory for the DG-formulation of a highlyindefinite Helmholtz problem. The theory covers conforming as well as nonconform-ing generalized finite element methods. In contrast to conventional Galerkin methodswhere a minimal resolution condition is necessary to guarantee the unique solvability,we prove that the DG-formulation admits a unique solution under much weaker con-ditions. As an application we present the error analysis for the hp-version of the finiteelement method explicitly in terms of the mesh width h, polynomial degree p and wavenumber k. It is shown that the optimal convergence order estimate is obtained underthe conditions that kh/

√p is sufficiently small and the polynomial degree p is at least

O(log k).

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CONVERGENCE OF HIGH ORDER METHODS FORTHE MISCIBLE DISPLACEMENT PROBLEM

Beatrice Riviere

Department of Computational and Applied Mathematics, Rice University,6100 Main Street, Houston, Texas, 77005, USA.

[email protected]

Miscible displacement flow is one important part of enhanced oil recovery. A poly-meric solvent is injected in the reservoir and it mixes with the trapped oil. Accuratesimulation of the displacement of the fluid mixture in heterogeneous media is neededto optimize oil production.

The miscible displacement is mathematically modeled by a pressure equation (ellip-tic) and a concentration equation (parabolic) that are coupled in a nonlinear fashion.The convergence analysis of numerical methods applied to the miscibledisplacement ischallenging because the diffusion-dispersion coefficient in the concentration equation isunbounded.

In this work, we formulate and analyze discontinuous Galerkin in time methodscoupled with several finite element methods (including mixed finite elements and dis-continuous Galerkin) for the miscible displacement. The diffusion-dispersion coefficientis not assumed to be bounded. Other coefficients in the problem are not assumed tobe smooth. Convergence of the numerical solution is obtained using a generalizationof the Aubin-Lions compactness theorem. The Aubin-Lions theorem is not applicablesince the numerical approximations are discontinuous in time. Numerical exampleswith varying order in space and time are also given.

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HP-MULTIGRID AS SMOOTHER ALGORITHM FOR HIGHERORDER DISCONTINUOUS GALERKIN DISCRETIZATIONS

OF ADVECTION-DOMINATED FLOWS

Jaap van der Vegt1 and Sander Rhebergen2

1University of Twente, Department of Applied Mathematics7500 AE, Enschede, The Netherlands

[email protected]

2University of Oxford, Mathematical Institute24-29 St Giles’, Oxford, OX1 3LB, [email protected]

Higher order accurate space-time discontinuous Galerkin discretizations are well suitedfor free boundary problems since they remain conservative on moving and deformingmeshes and are well suited for solution adaptive and parallel computations. The space-time DG discretization results, however, in an implicit discretization in time, whichrequires the solution of a (non)linear algebraic system. In this presentation the recentlydeveloped hp-Multigrid as Smoother (hp-MGS) algorithm for higher order accurate DGdiscretizations of advection-dominated flows will be discussed. The main feature of thisalgorithm is that it uses semi-coarsening h-multigrid as smoother for p-multigrid. Thedevelopment of the hp-MGS algorithm strongly relies on a detailed multilevel Fourieranalysis of the full multigrid algorithm, which provides the essential information tooptimize the coefficients in the semi-implicit Runge-Kutta smoother. In addition, thismultilevel analysis gives detailed information on the spectrum and operator norms ofthe error transformation operator. In this presentation we will consider both hexahedraland prismatic space-time elements. The newly developed multigrid algorithm will bedemonstrated on various problems with thin boundary layers that require significantlystretched meshes.

References

[1] J.J.W. van der Vegt and S. Rhebergen, HP-multigrid as smoother algorithm forhigher order discontinuous Galerkin discretizations of advection dominated flows.Part I. Multilevel Analysis, Journal of Computational Physics, Vol. 231, pp. 7537-7563, 2012.

[2] J.J.W. van der Vegt and S. Rhebergen, HP-multigrid as smoother algorithm forhigher order discontinuous Galerkin discretizations of advection dominated flows.Part II. Optimization of the Runge-Kutta smoother, Journal of ComputationalPhysics, Vol. 231, pp. 7564-7583, 2012.

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MIXED HP -DGFEM FOR LINEAR ELASTICITY IN 3D

Thomas P. Wihlera and Marcel Wirzb

Mathematics Institute, University of Bern, [email protected], [email protected]

We consider mixed hp-discontinuous Galerkin FEM for linear elasticity in axiparallelpolyhedral domains inR3. In order to resolve possible corner, edge, and corner-edge sin-gularities, anisotropic geometric edge meshes consisting of hexahedral elements are ap-plied. We discuss inf-sup stability results on both the continuous as well as the discretelevel. In addition, under certain realistic assumptions (for analytic data) on the regular-ity of the exact solution and based on an hp-interpolation analysis from [D. Schotzau,C. Schwab, and TPW, hp-dGFEM for second-order elliptic problems in polyhedraII: Exponential convergence, accepted for publication in SINUM], we prove that theproposed DG schemes converge at an exponential rate in terms of the fifth root ofthe number of degrees of freedom [TPW and MW, Mixed hp-discontinuous GalerkinFEM for linear elasticity and Stokes flow in three dimensions, Math. Models MethodsAppl. Sci. 22 (2012), no. 8]. A number of numerical experiments will illustrate thetheory.

7-10

8: Mini-Symposium: Discontinuous Galerkin methods in fluid flows

8 Mini-Symposium: Discontinuous Galerkin meth-

ods in fluid flows

Organisers: Aycil Cesmelioglu and SanderRhebergen

8-1


HYBRIDIZABLE DISCONTINUOUS GALERKIN METHODS FOR THEINCOMPRESSIBLE OSEEN AND NAVIER-STOKES EQUATIONS

Aycil Cesmelioglu1, Bernardo Cockburn2, Ngoc Cuong Nguyen3a and Jaime Peraire3b

1Department of Mathematics and Statistics, Oakland University, Rochester, MI, [email protected]

2School of Mathematics, University of Minnesota, Minneapolis, MN, [email protected]

3Department of Aeronautics and Astronautics,Massachusetts Institute of Technology, Cambridge, MA, US


The aim of this work is to analyze hybridizable discontinuous Galerkin (HDG) methodsfor the incompressible stationary Navier-Stokes problem. First, we study HDG methodsfor the Oseen equations. We show optimal convergence for the velocity, its gradient andthe pressure by using same degree polynomial approximations for all the unknowns.Furthermore, after a postprocessing, we obtain a H(div)-conforming, divergence-freevelocity which converges with an additional order. We show numerical examples tovalidate the theoretical convergence rates. Finally, we discuss the extension of theseresults to the Navier-Stokes case using a sequence of Oseen approximations.

COMMUTING DIAGRAMS FOR THE TNT ELEMENTS ON CUBES

Bernardo Cockburn1 and Weifeng Qiu2

1School of Mathematics, University of Minnesota

2Department of Mathematics, City University of Hong [email protected]

We present commuting diagrams for the de Rham complex for new elements definedon cubes which use tensor product spaces. The distinctive feature of these elementsis that, in sharp contrast with previously known results, they have the TiNiest spacescontaining Tensor product spaces of polynomials of degree k, hence their acronym TNT.In fact, the local spaces of the TNT elements differ from the standard tensor productspaces by spaces whose dimension is a small number independent of the degree k. Suchnumber is 7 (number of vertices of the cube minus one) for the space associated withthe divergence operator, 18 (number of faces of the cube times the number of verticesof a face minus one) for the space associated with the curl operator, and 12 (numberof edges of the cube times the number of vertices of an edge minus one) for the spaceassociated with the gradient operator.

8-2


DEVELOPMENT AND VALIDATION OF A DISCONTINUOUSGALERKIN WAVE PREDICTION MODEL

Ethan Kubatkoa and Angela Nappib

Department of Civil, Environmental and Geodetic Engineering,The Ohio State University, Columbus, OH, [email protected], [email protected]

In this talk, we present the development and validation of a discontinuous Galerkin(DG) method for a wave prediction model known as the GLERL–Donelan wave model.Originally formulated at the Canadian Centre for Inland Waters and the US NationalOceanic and Atmospheric Administration’s Great Lakes Environmental Research Lab-oratory (GLERL), the GLERL–Donelan model is a relatively simple parametric wavemodel that has historically formed the basis of the US National Weather Service’s GreatLakes wave forecasts. In contrast to spectral and most other parametric wave models,which solve the so-called spectral action balance equation, the GLERL–Donelan wavemodel is based on the conservation of total wave momentum. This formulation avoidsthe need to solve the action balance equation over a (usually large) set of discretefrequency components, which can make spectral-based models prohibitively expensivefrom a computational perspective. The development and numerical implementation ofa DG method for the solution of the GLERL–Donelan wave model will be discussed,and model validation results will be presented and compared to a spectral-based, third-generation wave model in terms of accuracy and computational cost.

8-3


COUPLING OF STOKES AND DARCY FLOWS USINGDISCONTINUOUS GALERKIN AND MIMETIC

FINITE DIFFERENCE METHOD

Konstantin Lipnikov1, Danail Vassilev2 and Ivan Yotov3

1Applied Mathematics and Plasma Physics Group, Theoretical Division,Mail Stop B284, Los Alamos National Laboratory, Los Alamos, NM 87545, USA,

[email protected]

2Mathematics Research Institute, College of Engineering,Mathematics and Physical Sciences, University of Exeter,

North Park Road, Exeter, EX4 4QF, [email protected]

3Department of Mathematics,301 Thackeray Hall, University of Pittsburgh, Pittsburgh, PA 15260, USA,

[email protected]

We present a numerical method for coupling Stokes and Darcy flows based on discontin-uous Galerkin (DG) elements for Stokes and mimetic finite difference (MFD) methodsfor Darcy. Both methods are locally mass conservative and can handle irregular grids.The MFD methods are especially suited for flow in heterogeneous porous media, asthey provide accurate approximation for both pressure and velocity and can handlediscontinuous coefficients as well as degenerate and non-convex polygonal elements.We develop DG polygonal elements for Stokes, allowing for coupled discretizations onpolygonal grids. Optimal convergence is obtained for the coupled numerical methodand confirmed computationally.

8-4


LOCAL DISCONTINUOUS GALERKIN METHOD FORINKJET DROP FORMATION AND MOTION

Tatyana Medvedevaa, Onno Bokhove and Jaap van der Vegt

1University of Twente, Department of Applied Mathematics7500 AE, Enschede, The Netherlands

[email protected]

In inkjet printing accurate control of droplet formation is crucial to obtain a highprinting quality. In order to investigate this complex process the one-dimensionalmodel proposed by Eggers and Dupont [1] was discretized using a Local DiscontinuousGalerkin method. Viscosity and free surface terms are included in this discretization.Since the resulting discrete system is very stiff an implicit time integration method isused in combination with a Newton method.

The newly developed LDG method is verified with detailed computations of thegrowth rate of the free surface and other exact solutions. Also, simulation of thedroplet formation in inkjet will be presented.

References

[1] J. Eggers, T.F. Dupont, Drop formation in a one-dimensional approximation of theNavier-Stokes equation, J. Fluid Mech.262 (1994) 205-221.

SPACE-TIME (H)DG METHODS FOR INCOMPRESSIBLE FLOWS

Sander Rhebergen1, Bernardo Cockburn2 and Jaap van der Vegt3

1Mathematics Institute, University of Oxford, [email protected]

2School of Mathematics, University of Minnesota, [email protected]

3Department of Applied Mathematics, University of Twente, [email protected]

In this talk I will discuss a new Discontinuous Galerkin (DG) method, namely the Hy-bridizable DG (HDG) method. We recently extended the HDG method to a space-timeformulation allowing efficient and accurate computations on deforming grids/domains.I will introduce the method for the Incompressible Navier-Stokes (INS) equations. Re-sults and efficiency of the space-time HDG method for the INS equations on deformingdomains will then be compared to those of the space-time DG method.

8-5


A LOCAL DISCONTINUOUS GALERKIN METHOD FOR THEPROPAGATION OF PHASE TRANSITION IN SOLIDS

Lulu Tiana, Yan Xub, J.G.M. Kuertenc and Jaap van der Vegtd

Mathematics of Computational Science, Department of Applied Mathematics,University of Twente, Enschede, The [email protected], [email protected],


In this presentation, we will discuss a local discontinuous Garlerkin (LDG) finite ele-ment method for the solution of a hyperbolic-elliptic system modeling the propagationof phase transition in solids. Viscosity and capillarity terms are added to select thephysically relevant solution. The L2−stability of the LDG method is proven for ba-sis functions of arbitrary polynomial order. In addition, using a priori error analysis,it is proven that the LDG discretization converges at optimal order if the solution issufficiently smooth. Also, results of a linear stability analysis to determine the timestep are presented. To obtain a reference exact solution we solved a Riemann prob-lem for a trilinear strain-stress relation using a kinetic relation to select the uniqueadmissible solution. This exact solution contains both shocks and phase transitions.The LDG method is demonstrated by computing several model problems representingphase transition in solids and in fluids with a Van der Waals equation of state. Theresults show the convergence properties of the LDG method.

8-6

9: Mini-Symposium: Elliptic Eigenvalue Problems: Recent Developments in Theory and Computation

9 Mini-Symposium: Elliptic Eigenvalue Problems:

Recent Developments in Theory and Computa-

tion

Organisers: Stefano Giani, Luka Grubisicand Jeffrey Ovall

9-1


GUARANTEED LOWER BOUNDS FOR EIGENVALUES

Carsten Carstensen1 and Joscha Gedicke2

1Institut fur Mathematik, Humboldt-Universitat zu Berlin,Unter den Linden 6, 10099 Berlin, Germany

andDepartment of Computational Science and Engineering, Yonsei University,

120–749 Seoul, Korea,[email protected]

2Institut fur Mathematik, Humboldt-Universitat zu Berlin,Unter den Linden 6, 10099 Berlin, Germany,

[email protected]

This talk introduces fully computable two-sided bounds on the eigenvalues of theLaplace operator on arbitrarily coarse meshes based on some approximation of the cor-responding eigenfunction in the nonconforming Crouzeix-Raviart finite element spaceplus some postprocessing. The efficiency of the guaranteed error bounds involves theglobal mesh-size and is proven for the large class of graded meshes. Numerical examplesdemonstrate the reliability of the guaranteed error control even with inexact solve ofthe algebraic eigenvalue problem. This motivates an adaptive algorithm which moni-tors the discretisation error, the maximal mesh-size, and the algebraic eigenvalue error.The accuracy of the guaranteed eigenvalue bounds is surprisingly high with efficiencyindices as small as 1.4.

9-2


ADAPTIVE PATH-FOLLOWING METHOD FORNONLINEAR PDE EIGENVALUE PROBLEMS

Carsten Carstensen1, Joscha Gedicke2, V. Mehrmann3a and Agnieszka Miedlar3b

1Humboldt-Universitat zu Berlin, Unter den Linden 6, 10099 Berlin, Germanyand Department of Computational Science and Engineering,

Yonsei University, 120–749 Seoul, [email protected]

2Humboldt-Universitat zu Berlin, Unter den Linden 6, 10099 Berlin, [email protected]

3Technische Universitat Berlin,Institut fur Mathematik, MA 4-5, Strasse des 17. Juni 136, 10623 Berlin, Germany.


In this talk we introduce a new approach combining the adaptive finite element methodwith the homotopy method to determine the eigenpairs for problems arrising in acous-tic field computations with proportional damping. The presented adaptive homotopyapproach emphasizes the need of the multi-way adaptation based on different errors,i.e., the homotopy, the discretization and the iteration error. All our statements areillustrated with several numerical examples.

ADAPTIVE NONCONFORMING CROUZEIX-RAVIARTFEM FOR EIGENVALUE PROBLEMS

Carsten Carstensena, Dietmar Gallistlb and Mira Schedensack c

Department of Mathematics, Humboldt-Universitat zu Berlin, [email protected], [email protected],

[email protected]

The nonconforming eigenvalue approximation is of high practical interest because itallows for guaranteed upper and lower eigenvalue bounds and for a convenient computa-tion via a consistent diagonal mass matrix in 2D. The first main result is a comparisonwhich states equivalence of the error of the nonconforming eigenvalue approximationwith its best-approximation error and its error in a conforming computation on thesame mesh. The second main result is optimality of an adaptive algorithm for the ef-fective eigenvalue computation for the Laplace operator with optimal convergence ratesin terms of the number of degrees of freedom relative to the concept of a nonlinear ap-proximation class. The analysis includes an inexact algebraic eigenvalue computationon each level of the adaptive algorithm which requires an iterative algorithm and acontrolled termination criterion.

9-3


COMPUTATION OF GROUND STATES OF SCHRODINGEROPERATOR WITH LARGE MAGNETIC FIELDS

Monique Dauge

IRMAR, Universite de Rennes 1, Campus de Beaulieu, Rennes, [email protected]

The Schrodinger operator with magnetic field takes the form

PA = (i∇+ A)2

where A is a vector field. This operator PA is set on a domain Ω of Rd (d = 2 or 3)and completed by natural boundary conditions (Neumann). Denote it by PΩ,A. Theground states of PΩ,A are the eigenpairs (λ, ψ)

(i∇+ A)2ψ = λψ in Ω and (i∂n + n ·A)ψ = 0 on ∂Ω

associated with the lowest eigenvalues λ. If Ω is bounded, PΩ,A is positive self-adjointwith compact resolvent. If Ω is simply connected, its eigenvalues depend only on themagnetic field B defined as

B = curl A.

The eigenvectors corresponding to two different instances of A for the same B arededuced from each other by a gauge transform.

The ground states of PΩ,A for large B are related to the solutions of the linearizedequations of Ginzburg-Landau for the determination of critical fields for which super-conductivity arises. Introducing a (small) parameter h and setting

Ph,Ω,A = (ih∇+ A)2 with Neumann b.c. on ∂Ω,

we get the relationPh,Ω,A = h2PΩ,A/h

linking the problem with large magnetic field to the semi-classical limit h→ 0.If B is constant (and non-zero), the eigenvectors concentrate at the boundary as

h → 0 with the length scale√h in the normal direction. In dimension d = 2, the

concentration takes place around points of maximal curvature. If Ω has convex cor-ners, they produce even stronger concentration of eigenvectors and, in general, theeigenvectors ψ = ψh have a two-scale structure of the form as h→ 0

ψh(x) = Ψ(x− x0√

h

)eiΦ(x)/h, x→ x0 (x0 corner).

The rapidly oscillating phase Φ(x)/h makes it difficult to compute accurately the eigen-pairs as h → 0. We will show on examples that uniform p-version of finite elementsperform well and is able to capture interesting intertwining behavior of eigenvalues inthe presence of symmetries.

This talk is based on references [1, 2] and work in progress [3].

9-4


References

[1] V. Bonnaillie-Noel and M. Dauge, Asymptotics for the low-lying eigenstatesof the Schrodinger operator with magnetic field near corners, Ann. Henri Poincare,7 (2006), pp. 899–931.

[2] V. Bonnaillie-Noel, M. Dauge, D. Martin, and G. Vial, Computationsof the first eigenpairs for the Schrodinger operator with magnetic field, Comput.Methods Appl. Mech. Engrg., 196 (2007), pp. 3841–3858.

[3] V. Bonnaillie Noel, M. Dauge, and N. Popoff, Polyhedral bodies in largemagnetic fields.

FINITE ELEMENT ANALYSIS OF A NON-SELF-ADJOINTQUADRATIC EIGENVALUE PROBLEM

Christian Engstrom

Department of Mathematics and Mathematical Statistics,Umea University, Sweden

[email protected]

In this talk we present Galerkin spectral approximation theory for non-self-adjointquadratic operator polynomials with periodic coefficients. The main applications arecomplex band structure calculations in metallic photonic crystals, periodic waveguides,and metamaterials. The spectral problem is equivalent to a non-compact block operatormatrix and norm convergence is shown for a block operator matrix having the samegeneralized eigenvectors as the original operator. Convergence rates of finite elementdiscretizations are considered and numerical experiments with the p-version and theh-version of the finite element method confirm the theoretical convergence rates

9-5


SOLVING AN ELLIPTIC EIGENVALUE PROBLEM VIAAUTOMATED MULTI-LEVEL SUB-STRUCTURING

AND HIERARCHICAL MATRICES

Peter Gerdsa and Lars Grasedyckb

Institut fur Geometrie und Praktische Mathematik,RWTH Aachen University, Aachen, Germany.


To solve an elliptic eigenvalue problem we combine the automated multi-level sub-structuring (or short AMLS) method [1, 3, 4] with the concept of hierarchical matrices(or short H-matrices) [2]. AMLS is a sub-structuring method which projects the dis-cretized eigenvalue problem in a small subspace. A reduced eigenvalue problem hasto be computed which delivers approximate solutions of the original problem. Severalpractical examples show that the AMLS method can be much faster than the commonlyused shift-invert block Lanczos algorithm.

Whereas the AMLS method is very effective in the two-dimensional case, the AMLSmethod is getting very expensive in the three-dimensional case, due to the fact that itcomputes the reduced eigenvalue via dense matrix operations.

But here hierarchical matrices can help. H-matrices are a data-sparse approxima-tion of dense matrices which e.g. result from the discretisation of the inverse of ellipticpartial differential operators. The main advantage of H-matrices is that they allowmatrix algebra in almost linear complexity.

In this talk we present how the AMLS method is combined with the H-matricesand how the reduced eigenvalue problem is computed by the fast H-matrix algebra.Beside the discretisation error two additional errors occur, the projection error of theAMLS method and the error caused by the H-matrix approximation. These errors arecontrolled by several parameters. The influence of these parameters will be investigatedin examples. Furthermore we will show in examples that we can compute the reducedeigenvalue problem in the three-dimensional case in almost linear complexity.

References

[1] Jeffrey K. Bennighof and R. B. Lehoucq. An automated multilevel substructur-ing method for eigenspace computation in linear elastodynamics. SIAM J. Sci.Comput., 25(6):2084–2106 (electronic), 2004.

[2] Steffen Borm, Lars Grasedyck, and Wolfgang Hackbusch. Introduction to hierar-chical matrices with applications. Engineering Analysis with Boundary Elements,27(5):405 – 422, 2003.

[3] Weiguo Gao, Xiaoye S. Li, Chao Yang, and Zhaojun Bai. An implementation andevaluation of the AMLS method for sparse eigenvalue problems. ACM Trans. Math.Software, 34(4):Art. 20, 28, 2008.

9-6


[4] Chao Yang, Weiguo Gao, Zhaojun Bai, Xiaoye S. Li, Lie-Quan Lee, Parry Hus-bands, and Esmond Ng. An algebraic substructuring method for large-scale eigen-value calculation. SIAM J. Sci. Comput., 27(3):873–892 (electronic), 2005.

AUXILIARY SUBSPACE ERROR ESTIMATION FOR HIGH-ORDERFINITE ELEMENT EIGENVALUE APPROXIMATIONS

Stefano Giani1, Luka Grubisic2, Harri Hakula3 and Jeffrey S Ovall4

1School of Engineering and Computing Sciences, University of Durham, [email protected]

2Department of Mathematics, University of Zagreb, [email protected]

3Department of Mathematics and Systems Analysis, Aalto University, [email protected]

4Department of Mathematics, University of Kentucky, [email protected]

Given a bounded open set Ω ⊂ Rd (d = 2, 3), we consider high-order (p and hp) finiteelement approximations of eigenvalues and their corresponding invariant subspaces forpositive and self-adjoint second-order elliptic operators on a Hilbert space H ⊂ H1(Ω).The eigenvalue problems are posed in the variational framework, with energy inner-product B(·, ·) and corresponding energy norm ||| · |||. Given an m-dimensional (m“small”) subspace S of approximate eigenfunctions in the (p or hp) finite element spaceV , our error estimates are based on the sines of the m principle angles, as measuredin the energy inner-product, between S and an associated space S. These quantities,which we call approximation defects, provide an ideal measure of how “far” S is frombeing invariant in H, and are used for assessing both eigenvalue and invariant subspaceapproximation errors.

Computable estimates of the approximation defects are obtained from an associatedfamily of boundary value problems by means of auxiliary subspace error estimation:given u ∈ H and its B-projection u ∈ V , an approximate error function ε ≈ u − u iscomputed as the B-projection of u− u onto an auxiliary subspace W—in the manner oftraditional hierarchical basis error estimation, but with potentially more exotic choicesof W . We discuss appropriate choices of W based on V , and provide correspondingreliability analysis; efficiency is trivial in this context.

Experiments employing high-order polynomial spaces on meshes with both trian-gular and quadrilateral elements illustrate the practical performance, in terms of con-vergence and effectivities, of the proposed method.

9-7


KATO’S SQUARE ROOT THEOREM AS A BASIS FOR RELATIVEESTIMATION THEORY OF EIGENVALUE APPROXIMATIONS

Stefano Giani1, Luka Grubisic2, Agnieszka Miedlar3 and Jeffrey S Ovall4

1School of Engineering and Computing Sciences, University of Durham, [email protected]

2Department of Mathematics, University of Zagreb, [email protected]

3Technische Universitat Berlin, Institut fur Mathematik, [email protected]

4Department of Mathematics, University of Kentucky, [email protected]

We present new residual estimates based on Kato’s square root theorem for spec-tral approximations of diagonalizable non-self-adjoint differential operators of diffusion-convection-reaction type. These estimates are incorporated as part of an hp-adaptivefinite element algorithm for practical spectral computations. We present a posteriorierror estimates both for eigenvalues as well as eigenfunctions and prove that they arereliable. We demonstrate the efficiency of the proposed approach on a collection ofbenchmark examples.

9-8


HIGH PRECISION VERIFIED EIGENVALUE ESTIMATION FORELLIPTIC DIFFERENTIAL OPERATOR OVER

POLYGONAL DOMAIN OF ARBITRARY SHAPE

Xuefeng Liu

Research Institute for Science and Engineering, Waseda University, [email protected]

This talk aims to propose a framework to provide high precision verified bounds forthe leading eigenvalues of the self-adjoint elliptic differential operator over polygonaldomain. This framework is based on several fundamental preceding research results[1-4]. To deal with the singularity of eigen-function in the case that the domain has anre-entrant corner, the method given in [1] is adopted. The high precision bounds are ob-tained by applying the Lehmann-Goerisch theorem [3,4] and the homotopy method[2]along with HP-FEM. Such kind of high precision eigenvalue estimation can be used togive sharp bounds for the interpolation error constants. It can also help to investigatethe solution existence for boundary value problems of semi-linear elliptic differentialequations.

References

[1] X. Liu, S. Oishi, Verified eigenvalue evaluation for Laplacian over polygonal do-main of arbitrary shape, to appear in SIAM Journal on Numerical Analysis, 2013.

[2] M. Plum, Bounds for eigenvalues of second-order elliptic differential operators,The Journal of Applied Mathematics and Physics(ZAMP), 42(6):848-863, 1991.

[3] N.J. Lehmann, Optimale eigenwerteinschließungen, Numerische Mathematik ,5(1):246-272, 1963.

[4] H. Behnke, F. Goerisch, Inclusions for eigenvalues of selfadjoint problems,Topics in Validated Computations (ed.J. Herzberger), North-Holland, Amsterdam,pp.277-322, 1994.

9-9


SPECTRAL ANALYSIS FOR A MIXED FINITE ELEMENTFORMULATION OF THE ELASTICITY EQUATIONS

Salim Meddahi1, David Mora2 and Rodolfo Rodrıguez3

1 Departamento de Matematicas, Facultad de Ciencias,Universidad de Oviedo, Oviedo, Spain.

[email protected]

2Departamento de Matematica, Universidad del Bıo-Bıo, Concepcion, [email protected]

3CI2MA, Departamento de Ingenierıa Matematica,Universidad de Concepcion, Concepcion, Chile.

[email protected]

This work deals with the approximation of the linear elasticity eigenvalue problemformulated in terms of the stress tensor and the rotation. This is achieved by con-sidering a mixed variational formulation in which the symmetry of the stress tensoris imposed weakly. We show that a discretization of the mixed eigenvalue elasticityproblem with reduced symmetry based on the lowest order Arnold-Falk-Winther ele-ment, provides a correct approximation of the spectrum and prove quasi-optimal errorestimates. Finally, we report some numerical experiments.

FINITE ELEMENTS FOR ELLIPTIC EIGENVALUE PROBLEMSIN THE PREASYMPTOTIC REGIME

Stefan A. Sauter

Institut fur Mathematik, Universitat Zurich,Winterthurerstr 190, CH-8057 Zurich, Switzerland

[email protected]

Convergence rates for finite element discretisations of elliptic eigenvalue problems inthe literature usually are of the form: If the mesh width h is fine enough then theeigenvalues resp. eigenfunctions converge at some well-defined rate. In our talk, wewill analyse the maximal mesh width h0 - more precisely the minimal dimension of afinite element space - so that the asymptotic convergence estimates hold for h < h0.This mesh width will depend on the size and spacing of the exact eigenvalues, thespatial dimension and the local polynomial degree of the finite element space. We willshow the results of some numerical experiments concerning a) the convergence of theeigenfunctions and - values, b) the convergence of the eigenvalue multigrid method toinvestigate the sharpness of the theoretical results.

This work is in collaboration with L. Banjai and S. Borm.

9-10


ACCURATE COMPUTATIONS OF MATRIX EIGENVALUES WITHAPPLICATIONS TO DIFFERENTIAL OPERATORS

Qiang Ye

Department of Mathematics, University of Kentucky,Lexington, Kentucky 40506-0027, USA

[email protected]

For matrix eigenvalue problems arising in discretizations of differential operators, itis usually smaller eigenvalues that well approximate the eigenvalues of the differentialoperators and are of interest. However, for ill-conditioned matrices, smaller eigenval-ues computed are expected to have low relative accuracy. In this talk, we present ourrecent works on high relative accuracy algorithms for computing eigenvalues of diago-nally dominant matrices. We present an algorithm that computes all eigenvalues of asymmetric diagonally dominant matrix to high relative accuracy. We further considerusing the algorithm in an iterative method for a large scale eigenvalue problem and weshow how smaller eigenvalues of finite difference discretizations of differential operatorscan be computed accurately.

9-11

10: Mini-Symposium: Error Estimation and adaptive modelling

10 Mini-Symposium: Error Estimation and adap-

tive modelling

Organisers: Paul T Bauman and Kris vander Zee

10-1


REDUCED BASIS FINITE ELEMENT HETEROGENEOUSMULTISCALE METHOD FOR QUASILINEAR PROBLEMS

Yun Bai

ANMC-MATHICSE-SB, Ecole Polytechnique Federale de Lausanne, [email protected]

In this talk, we introduce a new multiscale method for quasilinear homogenization prob-lems, that combines the finite element heterogeneous multiscale method (FE-HMM)with reduced basis (RB) techniques based on an offline-online strategy. The FE-HMMfor quasilinear multiscale problems [1], relies on a large number of micro problems thatneed to be computed in each iteration of the Newton method. As in addition macro andmicro meshes need to be refined simultaneously, the FE-HMM for quasilinear problemcan be costly. In contrast in the RB-FE-HMM, only a small number of micro problemsselected by a rigorous a posteriori error estimator need to be computed [2]. We showthat thanks to the new a posteriori error estimator, the result of [3] can be extended toquasilinear problem. A priori error estimates and convergence of the Newton methodcan be established.

Joint work with A. Abdulle and G. Vilmart.

References

[1] A. Abdulle and G. Vilmart, Analysis of the finite element heterogeneous multiscalemethod for nonmonotone elliptic homogenization problems. To appear in Math.Comp., 2013.

[2] A. Abdulle, Y. Bai and G. Vilmart, Reduced basis finite element heterogeneous mul-tiscale method for quasilinear elliptic homogenization problems. Preprint, submittedfor publication, 2013.

[3] A. Abdulle and Y. Bai, Reduced basis finite element heterogeneous multiscale methodfor high-order discretizations of elliptic homogenization problems. J. Comput. Phys.,231(21) (2012) 7014-7036.

10-2


ERROR ESTIMATION AND ADAPTIVE MODELINGFOR VISCOUS INCOMPRESSIBLE FLOWS

Paul T. Bauman

Institute for Computational Engineering and SciencesThe University of Texas at Austin

201 E. 24th St., Stop C0200, Austin, TX 78712, [email protected]

In this work, we consider a surrogate model to approximate solutions to the steady,incompressible Navier-Stokes equations. The surrogate model is constructed by replac-ing the Navier-Stokes model by the Stokes model in some region of the computationaldomain and by coupling the two models along an interface. Modeling error is incurreddue to the approximation. We construct an estimate of the error, based on quantitiesof interest, and an adaptive modeling strategy to reduce the error by adjusting theposition of the interface between the two models. We present two-dimensional numer-ical experiments to demonstrate the effectiveness of both the error estimator and theadaptive strategy. This is joint work with Timo van Opstal, Serge Prudhomme, andHarald van Brummelen.

GOAL-ORIENTED ERROR ESTIMATION ANDADAPTIVITY FOR THE TIME-DEPENDENTLOW-MACH NAVIER-STOKES EQUATIONS

Varis Careya and Paul T. Bauman

Institute for Computational Engineering and Sciences,The University of Texas at Austin

201 E. 24th St., Stop C0200, Austin, TX 78712, [email protected]

We present a goal-oriented algorithm for error control as well as spatial, temporal andmodel adaptivity, targeting the low-mach compressible Navier-Stokes equations. Thealgorithm, using the GRINS computational framework, is illustrated for both stationaryand non-stationary problems. Issues related to stabilization, linearization, and modelchoice are highlighted in the former case, while the additional interplay between storage,efficiency, and numerical accuracy of the forward and adjoint solutions is examined innon-stationary case.

10-3


ADAPTIVE INEXACT NEWTON METHODS WITH A POSTERIORISTOPPING CRITERIA FOR NONLINEAR DIFFUSION PDES

Alexandre Ern1 and Martin Vohralık2

1Universite Paris-Est, CERMICS, Ecole des Ponts ParisTech,77455 Marne la Vallee cedex 2, France

[email protected]


We consider nonlinear algebraic systems resulting from numerical discretizations ofnonlinear partial differential equations of diffusion type. To solve these systems, someiterative nonlinear solver, and, on each step of this solver, some iterative linear solverare used. We derive adaptive stopping criteria for both iterative solvers. Our criteriaare based on an a posteriori error estimate which distinguishes the different errorcomponents, namely the discretization error, the linearization error, and the algebraicerror. We stop the iterations whenever the corresponding error does no longer affectthe overall error significantly. Our estimates also yield a guaranteed upper bound onthe overall error at each step of the nonlinear and linear solvers. We prove the (local)efficiency and robustness of the estimates with respect to the size of the nonlinearityowing, in particular, to the error measure involving the dual norm of the residual. Ourdevelopments hinge on equilibrated flux reconstructions and yield a general framework.We show how to apply this framework to various discretization schemes like finiteelements, nonconforming finite elements, discontinuous Galerkin, finite volumes, andmixed finite elements; to different linearizations like fixed point and Newton; and toarbitrary iterative linear solvers. Numerical experiments for the p-Laplacian illustratethe tight overall error control and important computational savings achieved in ourapproach. More details on the overall approach, analysis, and results can be foundin [1, 2].

References

[1] A. Ern and M. Vohralık, Adaptive inexact Newton methods with a posterioristopping criteria for nonlinear diffusion PDEs. HAL Preprint 00681422 v2, 2012.

[2] A. Ern and M. Vohralık, Adaptive inexact Newton methods: a posteriori errorcontrol and speed-up of calculations, SIAM News, 46(1), 1, 2013.

10-4


TREE APPROXIMATION VERSUS AFEM

Francesca Fierro1, Alfred Schmidt2 and Andreas Veeser3

1Dipartimento di Matematica, Universita degli Studi di Milano, Italy,[email protected],

2Zentrum fur Technomathematik, Universitat Bremen, Germany,[email protected],

3Dipartimento di Matematica, Universita degli Studi di Milano, Italy,[email protected]

Adaptive finite elements methods (AFEM) are an efficient tool for the solution of partialdifferential equations. We consider the so-called h-variant and assume that bisectionis used for the mesh refinement. In this case, if the exact solution is known, the treeapproximation algorithm of P. Binev and R. DeVore offers the opportunity to computenear best meshes at a cost that is linear in the number of bisections. After derivinga simpler local error indicator for this algorithm, it is easy to implement wheneverbisection is available.

In this talk we will report on numerical studies of AFEM that use the aforemen-tioned near best meshes as benchmarks.

10-5


CONTRACTION AND OPTIMAL CONVERGENCE OF AGOAL-ORIENTED ADAPTIVE FINITE ELEMENT METHOD

Ricardo H. Nochetto1, A.J. Salgado1 and K.G. van der Zee2

1Dept. Mathematics and Institute for Physical Science and Technology,University of Maryland, College Park, MD, USA

2Multiscale Engineering Fluid Dynamics, Dept. Mechanical Engineering,Eindhoven University of Technology, Eindhoven, Netherlands

[email protected]

We focus on a goal-oriented adaptive finite element method for an elliptic boundary-value problem, following the estimation and marking strategy proposed by Mommerand Stevenson [1]. In their pioneering work, they proved that the adaptive algorithmconverges with an optimal rate, that is, the error in the output quantity of interestconverges at a rate which is twice that of the usual energy-norm rate. Several extensionsof [1] have recently appeared, for example, a different marking strategy is consideredin [2], while nonsymmetric elliptic problems are considered in [3].

In the current contribution, we reconsider the original goal-oriented adaptive algo-rithm, and we provide a novel contraction result. Based on this result we re-establishthe proof of optimal convergence. Several numerical experiments are presented thatsupport our findings.

References

[1] Mommer, Stevenson, A goal-oriented adaptive finite element method with conver-gence rates, SIAM J Numer Anal 47-2, (2009), pp 861-886

[2] Becker, Estecahandy, Trujillo, Weighted marking for goal-oriented adaptive finiteelement methods, SIAM J Numer Anal 49-6, (2011), pp 2451-2469

[3] Holst, Pollock, Convergence of goal-oriented adaptive finite element methods fornonsymmetric problems, (2011), arXiv:1108.3660v3

10-6

11: Mini-Symposium: Finite Element Methods for Convection-Dominated Problems

11 Mini-Symposium: Finite Element Methods for

Convection-Dominated Problems

Organisers: Volker John, Petr Knobloch andJulia Novo

11-1


A COMPUTABLE ERROR BOUND FOR A 3-DIMENSIONALCONVECTION-DIFFUSION-REACTION EQUATION

Mark Ainsworth1, Alejandro Allendes2, Gabriel R. Barrenechea3 and Richard Rankin4

1 Division of Applied Mathematics, Brown University,182 George Street, Providence, RI 02912, USA.

Mark [email protected]

2 Departamento de Matematica, Universidad Tecnica Federico Santa Marıa,Av. Espana 1680, Casilla 110-V Valparaıso, Chile.

[email protected]

3 Department of Mathematics and Statistics, University of Strathclyde,26 Richmond Street, Glasgow G1 1XH, Scotland.

[email protected]

4 Computational and Applied Mathematics Department, Rice University,6100 Main Street, MS-134 Houston, TX 77005-1892, USA.

[email protected]

Fully computable upper bounds are developed for the discretisation error measured inthe natural (energy) norm for convection-reaction-diffusion problems in three dimen-sions. The upper bounds are genuine upper bounds in the sense that the numericalvalue of the estimated error exceeds the actual numerical value of the true error re-gardless of the coarseness of the mesh or the nature of the data for the problem. Allconstants appearing in the bounds are fully specified. Examples show the estimator tobe reliable and accurate even in the case of complicated three dimensional problems.

11-2


AUGMENTED TAYLOR-HOOD ELEMENTSFOR INCOMPRESSIBLE FLOW

Daniel Arndt

Institute for Numerical and Applied Mathematics,University of Goettingen, [email protected]

It is well-known that in finite element discretizations of incompressible flow problemsthe numerical solution is in general not pointwise solenoidal, unless the divergence ofthe velocity ansatz space is contained in the pressure ansatz space. Although manyturbulence models are based on the conservation of mass, in discretizations often finiteelements are used that are not divergence free.

In this talk, we consider an augmented Taylor-Hood pair that improves conservationof mass. The modification is given by adding elementwise constant functions to thepressure space. We examine the inf-sup stability of this discretization for quadrilateraland hexahedral meshes. Furthermore, convergence results for a Stokes problem withdiscontinuous pressure are presented. As a test case for the Navier-Stokes equationswe consider a turbulent channel flow and examine how the improved mass conservationinfluences the quality of the numerical solution using a turbulence model by Verstappen.

11-3


A NONLINEAR DISSIPATION TO AVOID LOCAL OSCILLATIONSFOR THE FINITE ELEMENT APPROXIMATIONOF THE CONVECTION-DIFFUSION EQUATION

Joan Baigesa and Ramon Codinab

Universitat Politecnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, [email protected], [email protected]

When diffusion is small compared to convection, the Galerkin finite element approxi-mation of the convection-diffusion equation suffers from numerical instabilities. Thisis a well known and, in fact, possibly the first concept one learns when approximatingflow problems using finite elements. Several remedies have been devised along the yearsto overcome this problem, starting with the introduction of purely artificial dissipationin the von Neumann line and leading to several stabilization methods used nowadays.We also employ one of such methods, which consists in adding a stabilizing term to theGalerkin ones depending on the residual of the equation to be solved. Our formulationcan be framed within the variational multi scale concept introduced by T.J.R. Hughesin 1995.

Stabilization methods, and in particular the one we favor, are intended to provideglobal stability to the discrete finite element problem. This in particular means thatone can show stability and convergence of the discrete solution in global norms, i.e.,norms that involve the L2 norm of some terms over all the computational domain.The global instabilities displayed by the Galerkin approximation are thus avoided but,nevertheless, local oscillations may still remain and, in fact, do appear in the neigh-borhood of sharp gradients of the discrete solution. This is due to the fact that localcontrol and particularly control in the L∞ norm cannot be achieved.

In order to avoid the appearance of local instabilities several methods have also beenproposed over the years. In essence, they all rely on the introduction of a nonlineardissipation close to where there are sharp gradients of the approximate solution. Thesedissipations can be motivated in different ways, physical or analytical, in the form ofexplicit dissipation or by introducing limiters to the discrete solution.

The aim of this work is to present a nonlinear dissipation to avoid local oscillationsin the sense described. Succinctly, it has the form of the artificial dissipation thatguarantees that no oscillations may appear but multiplied by a factor that dependson the discrete solution, thus making the formulation nonlinear. This factor is takenas the projection orthogonal to the finite element space of the gradient of the discreteunknown, normalized by the gradient itself. After describing the formulation, we showthat the resulting nonlinear problem is well posed and present several numerical ex-amples which show that the method succeeds in removing local oscillations, nonlinearconvergence is satisfactory and is less over diffusive far from sharp layers than othermethods designed with the same objective.

11-4


INVESTIGATIONS OF A FEM-FCT SCHEMEAPPLIED TO A 1D MODEL PROBLEM

Gabriel R. Barrenechea1, Volker John2 and Petr Knobloch3

1Department of Mathematics and Statistics,University of Strathclyde, Glasgow, Scotland,

[email protected],

2WIAS Berlin and Free University of Berlin, Germany,[email protected],

3Faculty of Mathematics and Physics,Charles University in Prague, Czech Republic,

[email protected]

It is well known that Galerkin finite element discretizations are not appropriate for thenumerical solution of convection dominated problems since the approximate solutionsare typically globally polluted by spurious oscillations of unacceptable magnitude, see,e.g., [3]. Among various remedies, the FEM-FCT schemes proved to be rather efficient[1]. However, for this type of methods, there are no theoretical investigations concern-ing existence, uniqueness and convergence of approximate solutions available in theliterature.

We consider the algebraic flux correction scheme described in [2] and apply it to asteady one-dimensional convection–diffusion equation. This leads to a nonlinear differ-ence scheme whose properties are investigated in detail. In particular, we demonstratethat this problem is generally not solvable and we discuss various improvements.

References

[1] V. John and E. Schmeyer. Finite element methods for time-dependentconvection–diffusion–reaction equations with small diffusion. Comput. MethodsAppl. Mech. Engrg., 198:475–494, 2008.

[2] Dmitri Kuzmin. Algebraic flux correction for finite element discretizations of cou-pled systems. In M. Papadrakakis, E. Onate, and B. Schrefler, editors, Proceedingsof the Int. Conf. on Computational Methods for Coupled Problems in Science andEngineering, pages 1–5. CIMNE, Barcelona, 2007.

[3] H.-G. Roos, M. Stynes, and L. Tobiska. Robust Numerical Methods for SingularlyPerturbed Differential Equations. Convection–Diffusion–Reaction and Flow Prob-lems. 2nd ed. Springer-Verlag, Berlin, 2008.

11-5


A POSTERIORI ERROR ESTIMATION IN STABILIZEDDISCRETIZATIONS OF STATIONARY

CONVECTION-DIFFUSION-REACTION PROBLEMS

Markus Bausea and Kristina Schweglerb

Faculty of Mechanical Engineering, Helmut Schmidt University,University of the Federal Armed Forces Hamburg,

Holstenhofweg 85, 22043 Hamburg, [email protected], [email protected]

The reliable numerical approximation of convection-diffusion-reaction problems

b · ∇u−∇ · (A∇u) + r(u) = f (1)

with small diffusion A is still a challenging task. Eq. (1) is considered as a prototypemodel for more sophicated equations of practical interest. For its numerical solutionstabilized methods are used that aim to introduce a correct amount of artificial diffusionin regions with sharp inner or boundary layers or complicated structures where impor-tant phenomena take place; cf., e.g., [M. Bause, K. Schwegler, Analysis of stabilizedhigher-order finite element approximation of nonstationary and nonlinear convection-diffusion-reaction equations, Comput. Methods Appl. Mech. Engrg., 209–212 (2012),184–196].

In this work we consider using lower and higher order conforming finite elementmethods with streamline upwind Petrov-Galerkin (SUPG) stabilization. In addition,discontinuity- or shock capturing stabilization as an additional consistent modifica-tion of the numerical scheme and stabilization in crosswind direction is applied. Tofurther improve the approximation quality and the efficiency of the calculations, wecombine the stabilized discretization with an a posteriori error control mechanism andan adaptive mesh generation algorithm based on a dual-weighted-residual approach.The dual weighted error estimator assesses the discretization error with respect to agiven quantity of physical interest.

We study different approaches for combining SUPG and shock-capturing stabiliza-tion with dual-weighted-residual error estimation. Various output functionals of thesolution are proposed. By numerical experiments we analyze and illustrate the effi-ciency and performance properties of the algorithms.

11-6


ROBUST ERROR ESTIMATES IN WEAK NORMS WITHAPPLICATION TO IMPLICIT LARGE EDDY SIMULATION

Erik Burman

Department of Mathematics, University College London, London, [email protected]

In this talk we will discuss a posteriori and a priori error estimates of filtered quantitiesfor solutions to some equations of fluid mechanics. For the computation of the solutionwe use low order finite element methods using either linear or nonlinear stabilization.To obtain estimates that are robust with respect to the diffusion/viscosity coefficientwe introduced a class of weak norms corresponding to taking a weighted H1-normof a filtered solution. For these weak norms we propose error estimates whose errorconstants depend only on the regularity of the initial data. In particular the estimatesare independent of the Reynolds number, the Sobolev norm of the exact solution attime t > 0, or nonlinear effects such as shock formation. It follows that we obtain acomplete assessment of the computability of the solution given the initial data. Aftera detailed description of the analysis in the case of the Burgers’ equation we widen thescope and discuss two dimensional incompressible turbulence and passive transportwith rough data within the same paradigm.

11-7


ANISOTROPIC LOCAL PROJECTION STABILIZATION INSTREAMLINE AND CROSSWIND DIRECTIONS

Helene Dallmanna and Gert Lubeb

Institute for Numerical and Applied Mathematics,Georg-August University of Gottingen, Germany


The local projection stabilization (LPS) method splits the discrete ansatz spaces intosmall and large scales and adds stabilization terms only on the small ones. In [2]an analysis for finite element discretizations of linearized incompressible flows usingthe local projection method is given. Here stabilizing terms for the streamline direc-tion of the velocity gradient, for the incompressibility constraint and the pressure areused. Furthermore, [1] also studies stabilizations in the crosswind direction for scalarequations.

We consider an anisotropic stabilization technique for the vector valued Oseen caseand present an analysis for the resulting problem; stabilizing terms in streamline andcrosswind direction are introduced into the equations. In addition, we look into thegeneralization to the nonlinear incompressible Navier Stokes model.

References

[1] Gabriel R Barrenechea, Volker John, and Petr Knobloch. A Local Projection Stabi-lization Finite Element Method with Nonlinear Crosswind Diffusion for Convection-diffusion-reaction Equations. WIAS, 2012.

[2] G. Lube, G. Rapin, and J. Lowe. Local projection stabilization for incompress-ible flows: Equal-order vs. inf-sup stable interpolation. Electronic Transactions onNumerical Analysis, 32:106–122, 2008.

11-8


ON SUPERCONVERGENCE FOR HIGHER-ORDER FEMIN CONVECTION-DIFFUSION PROBLEMS

Sebastian Franz

Institute for Numerical Mathematics, TU Dresden, [email protected]

For singularly perturbed convection-diffusion problems and many numerical methods asupercloseness property is known for bilinear elements. This means that the differencebetween the numerical solution uN , obtain by a Galerkin FEM or a stabilised FEM,and the bilinear interpolant of the exact solution u is convergent of order two in theenergy norm, although uN − u is only convergent of order one.

We will investigate similar properties for higher-order FEM and look especially atthe choice of suitable interpolation operators and results for stabilised methods. Havinga supercloseness property, it is cheap to obtain a better numerical solution by simplepostprocessing — a superconvergent solution. We will also address different possibilitiesfor postprocessing.

AN ADAPTIVE SUPG METHOD FOR EVOLUTIONARYCONVECTION-DIFFUSION EQUATIONS

Javier de Frutos1, Bosco Garcıa-Archilla2 and Julia Novo3

1IMUVA, Unversidad de Valladolid, [email protected]

Departamento de Matematica Aplicada II, Universidad de Sevilla, [email protected]

Departamento de Matematicas, Universidad Autonoma de Madrid, [email protected]

In [V. John & J. Novo, A robust SUPG norm a posteriori error estimator for stationaryconvection-diffusion equations, CMAME, 2013, 289-305] a robust residual-based a pos-teriori estimator for the SUPG finite element method applied to stationary convection-diffusion problems is proposed. In this work we extend this residual estimator toevolutionary convection-diffusion equations. The main idea is that the SUPG approx-imation to the evolutionary problem is also the SUPG approximation to a particularsteady convection-diffusion problem with right-hand side depending on the computedapproximation. Based on the a posteriori error estimator an adaptive algorithm is de-veloped. Some numerical experiments are shown in which the new adaptive procedurecompares favourably with the adaptive method based on the standard Galerkin finiteelement method proposed in [J. de Frutos, B. Garcıa-Archilla & J. Novo, CMAME,(2011), 3601-3612.]

11-9


SUPG FINITE ELEMENT METHOD FOR PDESIN TIME-DEPENDENT DOMAINS

Sashikumaar Ganesana and Shweta Srivastavab

Numerical Mathematics and Scientific Computing,SERC, Indian Institute of Science, Bangalore 560012, India.


The numerical solution of convection dominated convection–diffusion problems is one ofthe challenging and active research fields. It is well known that the standard Galerkinfinite element methods induce spurious oscillations in the numerical solution of theseproblems. Streamline–Upwind–Petrov–Galerkin (SUPG) is one of the popular sta-bilization methods proposed for steady-state convection dominated problems in [3].Recently, it has been analyzed for time-dependent scalar equations in [1,2,4].

In this talk, the SUPG finite element method for a time-dependent scalar equa-tion in a time-dependent domain will be presented. Apart form the other challengesassociates with the solution of convection dominated problems, the time-dependentdomain makes the problem more challenging. We handle the deformation of the do-main with the arbitrary Lagrangian–Eulerian (ALE) approach. The conservative andnon-conservative ALE form of the scalar problem will be discussed. Further, the anal-ysis of the SUPG applied to the conservative ALE form of the scalar problem will bepresented. Finally, the numerical results for an array of problems will be presented.

References

[1] E. Burman: Consistent SUPG-method for transient transport problems: Stabilityand convergence, Comput. Methods in Appl. Mech. and Engrg., 199, (2010) 1114–1123.

[2] S. Ganesan: An operator-splitting Galerkin/SUPG finite element method for pop-ulation balance equations: Stability and convergence, ESAIM: Mathematical Mod-elling and Numerical Analysis (M2AN), 46, (2012) 1447–1465.

[3] T.J.R. Hughes and A.N. Brooks: A multi-dimensional upwind scheme with no cross-wind diffusion, In Finite element methods for convection dominated flows, T.J.R.Hughes, ed., Vol AMD 34, ASME, New York (1979).

[4] V. John and J. Novo: Error Analysis of the SUPG Finite Element Discretizationof Evolutionary Convection-Diffusion-Reaction Equations, SIAM J. Numer. Anal.,49, (2011) 1149–1176.

11-10


STABILIZATION OF CONVECTION-DIFFUSION PROBLEMSBY SHISHKIN MESH SIMULATION

Bosco Garcıa-Archilla

Depto. de Matematica Aplicada II, Universidad de Sevilla, [email protected]

We present a new stabilization procedure for numerical methods for convection-diffusionproblems. It is based on a simulation of the interaction between the coarse and fineparts of a Shishkin grid, but it can be easily applied on coarse and irregular meshes andon domains with nontrivial geometries. The technique, which does not require adjust-ing any parameter, can be applied to different stabilized and non stabilized methods.Numerical experiments show it to obtain oscillation-free approximations on problemswith boundary and internal layers, on uniform and nonuniform meshes and on domainswith curved boundaries.

A ROBUST SUPG NORM A POSTERIORI ERROR ESTIMATORFOR STATIONARY CONVECTION-DIFFUSION EQUATIONS

Volker John1 and Julia Novo2

1Weierstrass Institute for Applied Analysis and Stochastics, WIAS, Berlin, [email protected]

2Departamento de Matematicas, Universidad Autonoma de Madrid, [email protected]

A robust residual-based a posteriori estimator is proposed for the SUPG finite ele-ment method applied to the stationary convection-diffusion-reaction equations. Theerror in the natural SUPG norm is estimated. The main concern of the paper is theconsideration of the convection-dominated regime. A global upper bound and a locallower bound for the error are derived, where the global upper estimate relies on somehypothesis. Numerical studies demonstrate the robustness of the estimator and thefulfillment of the hypothesis. A comparison to other residual-based estimators withrespect to the adaptive grid refinement is also provided.

11-11


VELOCITY-PRESSURE REDUCED ORDER MODELS FOR THEINCOMPRESSIBLE NAVIER–STOKES EQUATIONS

Volker John

Weierstrass Institute for Applied Analysis and Stochastics,Mohrenstr. 39, 10117 Berlin, Germany,

[email protected]

andFree University of Berlin, Department of Mathematics and Computer Science,

Arnimallee 6, 14195 Berlin, Germany

Reduced order modeling (ROM) uses global basis functions that were derived fromsimulations with standard discretizations or even from experimental data. In this way,only a small number of basis functions is necessary for performing simulations thatprovide the most important features of the solution. ROM is already often used in op-timization, where repeated simulations of a problem with (slightly) varying parametersare necessary.

In the case of the incompressible Navier–Stokes equations, mostly only a veloc-ity ROM is performed. However, the pressure is often of importance, e.g., for thecomputation of functionals of interest.

In this talk, three velocity-pressure ROMs from two classes will be introduced. Oneof the methods seems to be new. The methods will be compared in numerical studiesof a laminar flow around a cylinder, which consider, in particular, the drag and liftcoefficient at the cylinder.

A FINITE ELEMENT METHOD FOR A NONCOERCIVEELLIPTIC CONVECTION DIFFUSION PROBLEM

Klim Kavalioua and Lutz Tobiskab

Faculty of Mathematics, Otto von Guericke University, Magdeburg, [email protected], [email protected]

A combined finite-element finite-volume method is applied on a noncoercive ellipticboundary value problem. The method is based on triangulations of weakly acutetype and a secondary circumcentric subdivision. The properties of the continuousproblem, that the kernel is one-dimensional and spanned by a positive function, arepreserved in the discrete case. A priori error estimates of first order in the H1-norm areshown for sufficiently small mesh sizes. Numerical test examples confirm the theoreticalpredictions.

11-12


ON THE ROLE OF THE HELMHOLTZ DECOMPOSITION INMIXED METHODS FOR INCOMPRESSIBLE FLOWS

AND A NEW VARIATIONAL CRIME

Alexander Linke

Department of Mathematics and Computer Science,Free University Berlin, Arnimallee 3, 14195 Berlin, Germany,

[email protected]

In incompressible flows with vanishing normal velocities at the boundary, irrotationalforces in the momentum equations should be balanced completely by the pressuregradient. Unfortunately, nearly all available discretizations for incompressible flowsviolate this property. The origin of the problem is that discrete velocities are usually notdivergence-free. Hence, the use of divergence-free velocity reconstructions is proposedwherever an L2 scalar product appears in the discrete variational formulation, whichactually means committing a variational crime. The approach is illustrated and appliedto a nonconforming Crouzeix-Raviart finite element discretization. It will be provedand numerically demonstrated that a divergence-free velocity reconstruction based onthe lowest-order Raviart-Thomas element increases the robustness and accuracy of anexisting convergent discretization, when irrotational forces appear in the momentumequations.

11-13


A TWO-LEVEL LOCAL PROJECTION STABILISATIONON UNIFORMLY REFINED TRIANGULAR MESHES

Gunar Matthies1 and Lutz Tobiska2

1Fachbereich Mathematik und Naturwissenschaften, Institut fur Mathematik,Universitat Kassel, Heinrich-Plett-Straße 40, 34132 Kassel, Germany

[email protected]

2Institut fur Analysis und Numerik,Otto-von-Guericke-Universitat Magdeburg, PSF 4120, 39016 Magdeburg, Germany

[email protected]

The local projection stabilisation (LPS) has been successfully applied to scalar conv-ection-diffusion-reaction equations, the Stokes problem, and the Oseen problem.

A fundamental tool in its analysis is that the interpolation error of the approxima-tion space is orthogonal to the discontinuous projection space. It has been shown thata local inf-sup condition between approximation space and projection space is sufficientto construct modifications of standard interpolations which satisfy this additional or-thogonality.

There are different versions of the local projection stabilisation on the market; wewill consider the two-level approach based on standard finite element spaces Yh on amesh Th and on projection spaces Dh living on a macro mesh Mh. Hereby, the finermesh is generated from the macro mesh by a certain refinement rules. In the usual two-level local projection stabilisation on triangular meshes, each macro triangle M ∈Mh

is divided by connecting its barycentre with its vertices. Three triangles T ∈ Th areobtained. Then, the pairs (Pr,h, P

discr−1,2h), r ≥ 1, of spaces of continuous, piecewise

polynomials of degree r on Th and discontinuous, piecewise polynomials of degree r−1on Mh satisfy the local inf-sup condition and can be used within the LPS framework.

One disadvantage of this refinement technique is however that Th contains simpliceswith large inner angles even in the case of a uniform decomposition Mh into isoscelestriangles. Another drawback is that this refinement rule leads to non-nested meshesand spaces whereas the common refinement technique of one triangle into 4 similartriangles (called red refinement in adaptive finite elements) results into nested meshesand spaces.

We will show that in the two-dimensional case the pairs (Pr,h, Pdiscr−1,2h), r ≥ 2, satisfy

the local inf-sup condition with the refinement of one triangle into 4 triangles. Con-sequently, the LPS can be also applied on sequences of nested meshes and spaces andkeeping the same error estimates. Finally, we compare the properties of the two result-ing LPS methods based on the different refinement strategies by means of numericaltest examples for convection-diffusion problems with dominating convection.

11-14


A FLUX-CORRECTED TRANSPORT METHOD BASEDON LOCAL PROJECTION STABILIZATION FORNON-STATIONARY TRANSPORT PROBLEMS

Friedhelm Schieweck1 and Dmitri Kuzmin2

1Institut fur Analysis und Numerik, Otto-von-Guericke Universitat Magdeburg,Postfach 4120, D-39016 Magdeburg, Germany

[email protected]

2Applied Mathematics III, University Erlangen-Nuremberg,Cauerstr. 11, D-91058, Erlangen, Germana

[email protected]

The local projection stabilization (LPS) method is an attractive space discretizationtechnique for non-stationary convection-dominated transport problems. Combinedwith an implicit time discretization, it yields a stable and non-oscillatory high-orderfinite element approximation in regions where the exact solution is sufficiently smooth.However, the accuracy of the LPS solution deteriorates in the neighborhood of dis-continuities or steep gradients. Like any other linear high-order scheme, LPS tendsto produce spurious oscillations in these regions. The usual way to avoid such localoscillations is to add some shock-capturing terms acting as nonlinear artificial viscosity.In many cases, this fix leads to marked improvements but the results depend on thechoice of a free parameter, which undermines the practical utility of such schemes.

In contrast to traditional shock capturing, the flux-corrected transport (FCT)methodology makes it possible to prevent numerical oscillations and to enforce thediscrete maximum principle in a fail-safe manner. FCT can be regarded as a techniquefor blending a stable high-order discretization with a monotone low-order discretiza-tion which contains enough artificial diffusion to suppress undershoots and overshoots.To this end, the difference between the two approximations is decomposed into anti-diffusive fluxes or element contributions which are multiplied by solution-dependentcorrection factors. The FCT solution is guaranteed to be non-oscillatory everywhereand to revert to the underlying high-order approximation in smooth regions.

In the finite element literature, the FCT method has successfully been used tocombine an unstable high-order Galerkin scheme with a low-order upwind-biased dis-cretization. Since the unconstrained Galerkin approximation may exhibit global oscil-lations, the FCT limiter may need to be activated everywhere, which may destroy thehigh accuracy in regions of smoothness. This concern has led us to combine the stablehigh-order LPS discretization with a low-order artificial viscosity method. Since theLPS method is linearly stable, the FCT correction of antidiffusive fluxes is restrictedto small subdomains, whereas optimal accuracy is maintained elsewhere.

In this talk, we discuss the practical implementation of the proposed FCT methodin the case of one space dimension and present some numerical results.

11-15


TOWARDS ANISOTROPIC QUALITYTETRAHEDRAL MESH GENERATION

Hang Si

Weierstrass Institute for Applied Analysis and Stochastics, Berlin, [email protected]

Many physical problems exhibit anisotropic features, i.e., their solutions change moresignificantly in one direction than others. Examples include in particular convection-dominated problems whose solutions have, e.g., layers, shocks, or corner singularities.When numerical methods are used to approximate these problems, it is of great im-portance that the used meshes represent such features to achieve high accuracy at alow computational cost.

Tetrahedral meshes are very popular for discretizing three-dimensional domains.They can be easily adapted to arbitrary geometries, they can be refined locally, andthey can be generated automatically. However, the generation of anisotropic qualitytetrahedral meshes is a complex problem and it is an active research topic.

This talk addresses the state of the art in high quality tetrahedral mesh generation,presenting in particular a relative robust and efficient methodology implemented inthe program TetGen for generating isotropic meshes, and it discusses important openquestions which arise in generalizations to the anisotropic case.

11-16


A LOCAL PROJECTION STABILIZATION METHODFOR FINITE ELEMENT APPROXIMATIONOF A MAGNETOHYDRODYNAMIC MODEL

Benjamin Wackera and Gert Lubeb

Institute for Numerical and Applied Mathematics,University of Gottingen, Gottingen, Germany.


In this talk, we consider the equations of incompressible resistive magnetohydrody-namics. Based on a stabilized finite element formulation by S. Badia, R. Codina andR. Planas for the linearized equations [1], we propose a modification of this techniqueby a local projection stabilization finite element method for the approximation of thisproblem.

The introduced stabilization technique is then discussed by investigating the sta-bility and convergence analysis for the problem’s formulation thoroughly. We finallycompare our numerical analysis with other approximations presented in the literature.

Finally, we give an outlook to the application of the approach to the fully nonlinearMHD model [2].

References

[1] S. Badia, R. Codina and R. Planas. On an unconditionally convergent stabilizedfinite element approximation of resistive magnetohydrodynamics, Journal of Com-putational Physics, 234:399-416, 2013.

[2] D. Sondak, A. Oberai. Large eddy simulation models for incompressible magne-tohydrodynamics derived from the variational multiscale formulation. Physics ofPlasmas, 19, 102308, 2012.

11-17

12: Mini-Symposium: Finite Element Methods for Multiphysics Problems

12 Mini-Symposium: Finite Element Methods for

Multiphysics Problems

Organisers: Norbert Heuer and Salim Med-dahi

12-1


A STABILIZED FINITE VOLUME ELEMENT FORMULATIONFOR SEDIMENTATION-CONSOLIDATION PROCESSES

Raimund Burger1, Ricardo Ruiz-Baier2 and Hector Torres3

1 CI2MA and Departamento de Ingenierıa Matematica,Facultad de Ciencias Fısicas y Matematicas,

Universidad de Concepcion, Concepcion, [email protected]

2 CMCS-MATHICSE-SB, Ecole Polytechnique Federale de Lausanne,Lausanne, [email protected]

3 Departamento de Matematicas, Facultad de Ciencias,Universidad de La Serena, La Serena, Chile

[email protected]

A model of sedimentation-consolidation processes in so-called clarifier-thickener units isgiven by a parabolic equation describing the evolution of the local solids concentrationcoupled with a version of the Stokes system for an incompressible fluid describing themotion of the mixture. In cylindrical coordinates, and if an axially symmetric solutionis assumed, the original problem reduces to two space dimensions. This poses thedifficulty that the subspaces for the construction of a numerical scheme involve weightedSobolev spaces. A novel finite volume element method is introduced for the spatialdiscretization, where the velocity field and the solids concentration are discretizedon two different dual meshes. The method is based on a stabilized discontinuousGalerkin formulation for the concentration field, and a multiscale stabilized pair ofP1-P1 elements for velocity and pressure, respectively. In this presentation, numericalexperiments illustrate properties of the model and the satisfactory performance of theproposed method.

12-2


DOUBLE LAYER POTENTIAL BOUNDARY CONDITIONS FORTHE HYBRIDIZABLE DISCONTINUOUS GALERKIN METHOD

Zhixing Fu1,a, Norbert Heuer2 and Francisco-Javier Sayas1,c

1Department of Mathematical Sciences, University of Delaware, Newark DE, [email protected], [email protected]

2Facultad de Matematicas, Pontificia Universidad Catolica de Chile,Santiago, Chile.

[email protected]

In this talk we present a simple coupling strategy for the Discontinuous GalerkinMethod with Boundary Element Methods. The coupling is based on a Galerkin dis-cretization of the second (or hypersingular) boundary integral equation sharing nu-merical flows with the HDG scheme. We first show how the system can be hybridizedto be solved only on the skeleton of the triangulation and the coupling interface. Wenext give sufficient conditions on the diffusion parameter guaranteeing coercivity atthe discrete level. Finally, we will show some numerical experiments confirming thetheoretical finds and extending their applicability to situations where the coercivitythreshold is crossed.

12-3


A LINEAR FINITE ELEMENT SCHEME FOR THE STOCHASTICLANDAU–LIFSHITZ–GILBERT EQUATION

Beniamin Goldys1, Kim-Ngan Le2b and Thanh Tran2c

1School of Mathematics and Statistics,The University of Sydney, Sydney 2006, Australia

[email protected]

2School of Mathematics and Statistics,The University of New South Wales, Sydney 2052, Australia


The study of the theory of ferromagnetism involves the Landau–Lifshitz–Gilbert equa-tion. Since stationary solutions are in general not unique, it is important to describephase transitions between different equilibrium states induced by thermal fluctuationsof the effective field. This is done by adding noise to the effective field. The equationthen takes the form

dM =(λ1M ×∆M − λ2M × (M ×∆M )

)dt+ (M × g) dW (t),

where λ1 6= 0 and λ2 > 0 are constants, and g : D → R3 is a given bounded function,with D being a bounded domain in Rd, d = 2, 3, having smooth boundary ∂D. Here dW (t) stands for the Stratonovich differential of the Wiener process W (t). Theunknown function M : [0, T ]×D → R3 satisfies the following conditions

∂M (t, x)

∂n= 0 ∀t ∈ (0, T ) and x ∈ ∂D,

M(0, x) = M0(x) ∀x ∈ D,|M(t, x)| = 1 ∀t ∈ [0, T ] and x ∈ D.

In this talk we present a linear finite element scheme for the problem and showthat a weak martingale solution exists. Numerical experiments confirm our theoreticalresults.

12-4


A DECOUPLED PRECONDITIONING TECHNIQUE FORA MIXED STOKES-DARCY MODEL

Antonio Marquez1, Salim Meddahi2 and Francisco-Javier Sayas3

1Departamento de Construccion, Universidad de Oviedo, [email protected]

2Departamento de Matematicas, Universidad de Oviedo, [email protected]

3Department of Mathematical Sciences, University of Delaware, USA,[email protected]

Our aim is to provide an efficient iterative method to solve the mixed Stokes-Dracymodel for coupling fluid and porous media flow. We consider for this model a formu-lation relying on an H(div)-approach in the Darcy domain. The Stokes problem isexpressed in the usual velocity-pressure form. The resulting weak formulation leads toa coupled, indefinite, ill-conditioned and symmetric linear system of equations. Opti-mal iterative methods are then important for solving efficiently these linear equations.Ideally, the algorithm should uncouple the global model in such a way that, onlyindependent Stokes and Darcy subproblems are involved at each iteration step. We in-troduce a decoupled iterative process consisting in two nested MINRES methods whosepreconditioners only require the solution of several second-order H1-elliptic problemsin the Stokes and the Darcy domains. Theoretical analysis and numerical experimentsshow the optimality and efficiency of the proposed decoupled iterative solver.

CONFORMING AND DIVERGENCE-FREE STOKES ELEMENTS

Michael Neilan

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, [email protected]

A family of finite elements methods for the velocity-pressure formulation of the Stokesequations are developed. In our approach we enrich H(div; Ω)-conforming finite ele-ments locally with divergence-free rational bubble functions to enforce strong tangentalcontinuity while still preserving the inf-sup condition of the original H(div; Ω) finiteelement space. We show that the method converges optimally on general shape-regulartriangulations, and that the velocity error is completely decoupled from the pressureerror. Moreover, the pressure space is exactly the image of the divergence operator act-ing on the velocity space. Therefore, the discretely divergence-free velocity functionsare divergence-free pointwise. We also show how the proposed elements are related toa class of C1 Zienkiewicz-type finite elements through the use of a smooth discrete deRham (Stokes) complex.

12-5


AN EXACTLY DIVERGENCE-FREE FINITE ELEMENTMETHOD FOR A GENERALIZED BOUSSINESQ PROBLEM

Ricardo Oyarzua1 and Dominik Schotzau2

1Departmento de Matematica, Universidad del Bıo-Bıo, Concepcion, Chile,[email protected],

2Mathematics Department, University of British Columbia, [email protected]

In this talk we present a mixed finite element method with exactly divergence-free ve-locities for the numerical simulation of a generalized Boussinesq problem, describing themotion of a non-isothermal incompressible fluid subject to a heat source. The methodis based on using divergence-conforming elements of order k for the velocities, discon-tinuous elements of order k − 1 for the pressure, and standard continuous elements oforder k for the discretization of the temperature. The H1-conformity of the velocities isenforced by a discontinuous Galerkin approach. The resulting numerical scheme yieldsexactly divergence-free velocity approximations; thus, it is provably energy-stable with-out the need to modify the underlying differential equations. We prove the existenceand stability of discrete solutions, and derive optimal error estimates in the mesh sizefor small and smooth solutions.

HP-TIME-DISCONTINUOUS GALERKIN FORPRICING AMERICAN PUT OPTIONS

Ernst P. Stephan

Institute for Applied Mathematics,Leibniz Universitat Hannover, Hannover, Germany

[email protected]

The time-discontinuous Galerkin hp-finite element method is applied to the Black-Scholes partial differential equation for American put options. The first approachexploits a weak non-penetration condition and a Lagrange multiplier space which isspanned by biorthogonal basis functions. The arising problem is solved by a globalizedsemi-smooth Newton (SSN) method with a penalized Fischer-Burmeister non-linearcomplementarity function. It is shown that the reduced SSN method converges locallyQ-quadratic. The second, equivalent approach relaxes the non-penetration conditionto a discrete set of Gauss-Lobatto points in space and time which is incorporatedin the ansatz and test space yielding a non-symmetric linear variational inequality.Numerical examples confirm the superiority of the two hp-approaches in terms of errorreduction and computational time and of the mixed hp-method, i.e. the first approach,in particular.

12-6

13: Mini-Symposium: Finite Elements in Nonlinear Spaces

13 Mini-Symposium: Finite Elements in Nonlinear

Spaces

Organisers: Philipp Grohs and Oliver Sander

13-1


SUBDIVISION METHOD FOR THE CANHAN-HELFRICH MODEL

Jingmin Chen1, Sara Grundel2, Robert Kusner3, Thomas Yu1a and Andrew Zigerelli1

1Department of Mathematics, Drexel University, Phiadelphia, [email protected]

2Max Planck Institute for Dynamics of Complex Technical Systems,Sandtorstr. 1, 39106 Magdeburg, Germany

3Department of Mathematics,University of Massachusetts at Amherst, Amherst MA 01003, U.S.A.

Lipid bilayers are ubiquitious in biological systems, and their equilibrium shapes arewidely believed to be governed by the Canhan-Helfrich model. In the parametric FEMmethods by Bonito/Nochetto/Pauletti, Deckelnick/Dziuk/Elliott, etc. for numericallysolving this model, piecewise linear or piecewise quadratic elements are used for ap-proximating the membrane surface, and a technical weak formulation is derived. In thistalk, we develop a different numerical method based on a technique from computer-aided geometric design known as subdivision surface. Unlike piecewise polynomialsurfaces, subdivision surfaces have just enough regularity for us to directly and ac-curately compute their Willmore energy – the key functional in the curvature-basedCanham-Helfrich model; thus no weak formulation is needed and the equilibrium shapecan be computed based on a nonlinear optimization solver.

We also discuss, among other nontrivial mathematical properties, some curiousphase transition phenomena in the Canhan-Helfrich model observed based on our nu-merical method. Some of these were never addressed in either the biophysics or thegeometric analysis communities.

13-2


ON POTTS AND BLAKE-ZISSERMAN FUNCTIONALSFOR MANIFOLD-VALUED DATA

Laurent Demareta, Martin Storathb and Andreas Weinmannc

Institute of Biomathematics and Biometry,Helmholtz Zentrum Munchen and Department of Mathematics,

TU Munchen, [email protected],[email protected],[email protected]

For real-valued data Potts functionals are energy functionals of the form

Pγ(u) = γ · J(u) + ‖u− f‖22.

Here the regularizing term J(u) counts the number of jumps of u whereas the fidelityto univariate discrete data is measured in the `2 norm. The family of Blake-Zissermanfunctionals is obtained by replacing the jump penalty J by some cut-off quadraticvariation Js, i.e.,

Js(u) =∑

imin(s2, |ui − ui−1|2), s > 0.

Minimization of these functionals is usually used for denoising and for (multi-label)segmentation tasks.

If the data are not real-valued but take their values in a (connected) Riemannianmanifold (which includes the case of a vector space), we may still define (univariate)manifold-valued Potts functionals by

Pγ(u) = γ · J(u) +∑i

dist(ui, fi)2.

Here dist(·, ·) is the distance in the Riemannian manifold. Manifold-valued Blake-Zisserman functionals are obtained by replacing the data term likewise and by replacingthe absolute values |ui − ui−1| in Js(u) by the distances dist(ui, ui−1).

We obtain the existence of minimizers and an algorithm to compute minimizers forthe class of Cartan-Hadamard manifolds. Those are complete Riemannian manifolds ofnonpositive sectional curvature. Examples are the manifolds of positive matrices Posnwhich are the data space in diffusion tensor imaging, the hyperbolic and the euclideanspaces.

Even for real-valued data, the multivariate Potts and Blake-Zisserman problemsare NP-hard which means that we cannot expect to find a computationally feasiblealgorithm always yielding a minimizer. Accepting this fact we develop heuristic mini-mization strategies for multivariate manifold-valued data. We apply our algorithms todiffusion tensor data.

13-3


B-SPLINE QUASIINTERPOLATION OF MANIFOLD-VALUED DATA

Philipp Grohs

Seminar for Applied Mathematics, ETH Zurich, [email protected]

We consider the problem of approximating manifold-valued functions with approxima-tion spaces spanned by linear combinations of cardinal B-splines with control pointsconstrained to lie on the manifold, followed by a closest-point projection onto the man-ifold. Under certain conditions we can prove that these spaces realize the optimalapproximation rate. Applications for denoising of manifold-valued data and the com-putation of geometric PDEs will be discussed. This is joint work with Markus Sprecher(ETH Zurich).

INTRINSIC DISCRETIZATION ERROR BOUNDSFOR GEODESIC FINITE ELEMENT APPROXIMATIONS

OF ELLIPTIC MINIMIZATION PROBLEMS

Hanne Hardering

Freie Universitat Berlin, Institut fur Mathematik,Arnimallee 6, 14195 Berlin, Germany.

[email protected]

Geodesic finite elements (GFE) have been introduced recently for energy minimizationproblems over manifold-valued functions. They are based on the Karcher mean andform a generalization of Lagrangian finite elements. One of the main features of GFE isthat they do not rely on an embedding of the manifold into a linear space. We analyzethis approximation scheme by presenting a generalization of the standard discretizationerror bounds for elliptic problems. In particular, we introduce anH1-type error measureand a generalization of the well-known Cea-Lemma. Combined with the appropriategeneralization of interpolation error estimates this yields optimal a priori estimates forH1-elliptic problems, e.g. the approximation of harmonic maps into manifolds withnonpositive or small positive curvature. The results presented are joint work with O.Sander and P. Grohs.

13-4


SIMULATION OF Q-TENSOR FIELDS WITH CONSTANTORIENTATIONAL ORDER PARAMETER IN THE THEORY

OF UNIAXIAL NEMATIC LIQUID CRYSTALS

Alexander Raisch

Institute for Numerical Simulation, University of Bonn, Bonn, [email protected]

We propose a practical finite element method for the simulation of uniaxial nematicliquid crystals with a constant order parameter. In the simplest setting this reduces tothe task of computing harmonic maps with values in the nonorientable real projectiveplane. A monotonicity result for Q-tensor fields is derived under the assumption thatthe underlying triangulation is weakly acute. Using this monotonicity argument weshow the stability of a gradient flow type algorithm and prove the converge of outputsto discrete stable configurations as the stopping parameter of the algorithm tends tozero. We examine numerically the difference of orientable and non-orientable stableconfigurations of liquid crystals in a planar two-dimensional domain and on a curvedsurface. As an application, we examine tangential line fields on the torus and showthat there exist orientable and non-orientable stable states with comparing Landau-deGennes energy and regions with different tilts of the molecule.

13-5

14: Mini-Symposium: Finite elements for problems with singularities

14 Mini-Symposium: Finite elements for problems

with singularities

Organisers: Alexey Bespalov and Serge Nicaise

14-1


EIGENVALUE PROBLEMS IN A NON-LIPSCHITZ DOMAIN

Gabriel Acosta and Marıa Gabriela Armentanoa

Departamento de Matematica, Facultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires, IMAS-Conicet, 1428 Buenos Aires, Argentina.

[email protected]

In this work we analyze piecewise linear finite element approximations of the Laplaceeigenvalue problem in the plane domain Ω = (x; y) : 0 < x < 1; 0 < y < xα; whichgives, for α > 1, the simplest model of an external cusp. Since Ω is curved and non-Lipschitz, our problem is not covered by the known literature which, as far as we know,only deals with polygonal or smooth domains. Indeed, the classical spectral theory cannot be applied directly and in consequence we present the eigenvalue problem in aproper setting, and relying on known convergence results for the associated sourceproblem with 1 < α < 3 (see [1, 3, 4]), we obtain quasi optimal order of convergencefor the eigenpairs [2].

References

[1] G. Acosta and M. G. Armentano (2011), Finite element approximations in a non-Lipschitz domain: Part. II , Math. Comp. 80(276), pp. 1949-1978 .

[2] G. Acosta and M. G. Armentano (2013), Eigenvalue Problem in a non-Lipschitzdomain, to appear in IMA Journal of Numerical Analysis.

[3] G. Acosta, M. G. Armentano, R. G. Duran and A. L. Lombardi (2005), Nonhomo-geneous Neumann problem for the Poisson equation in domains with an externalcusp, Journal of Mathematical Analysis and Applications 310(2), pp. 397-411.

[4] G. Acosta, M. G. Armentano, R. G. Duran and A. L. Lombardi (2007), Finiteelement approximations in a non-Lipschitz domain, SIAM J. Numer. Anal. 45(1),pp. 277-295.

14-2


ANISOTROPIC MESH REFINEMENT IN POLYHEDRALDOMAINS: ERROR ESTIMATES WITH DATA IN L2(Ω)

Thomas Apel1a, Ariel Lombardi2, and Max Winkler1c

1Institut fur Mathematik und Bauinformatik,Universitat der Bundeswehr Munchen, Germany


2Departamento de Matematica, Universidad de Buenos Aires, and Instituto deCiencias, Universidad Nacional de General Sarmiento, Buenos Aires, Argentina

[email protected]

The presentation is concerned with the finite element solution of the Poisson equa-tion with homogeneous Dirichlet boundary condition in a three-dimensional domain.Anisotropic, graded meshes [2] are used for dealing with the singular behaviour of thesolution in the vicinity of the non-smooth parts of the boundary. The discretizationerror is analyzed for the piecewise linear approximation in the H1(Ω)- and L2(Ω)-norms by using a new quasi-interpolation operator. This new interpolant is introducedin order to prove the estimates for L2(Ω)-data in the differential equation which isnot possible for the standard nodal interpolant. These new estimates allow for theextension of certain error estimates for optimal control problems with elliptic partialdifferential equation and for a simpler proof of the discrete compactness property foredge elements of any order on this kind of finite element meshes, see [1].

References

[1] Th. Apel, A. L. Lombardi, and M. Winkler. Anisotropic mesh refinement in poly-hedral domains: error estimates with data in L2(Ω). Submitted, 2013.

[2] Th. Apel and S. Nicaise. The finite element method with anisotropic mesh gradingfor elliptic problems in domains with corners and edges. Math. Methods Appl. Sci.,21:519–549, 1998.

14-3


STRONG CONVERGENCE FOR GAUSS’ LAWWITH EDGE ELEMENTS

Patrick Ciarlet1, Haijun Wu2 and Jun Zou3

1 POEMS, ENSTA ParisTech, 828, bd des Marechaux,91762 Palaiseau Cedex, France,

[email protected]

2 Department of Mathematics, Nanjing University, Jiangsu, 210093, P.R. China,[email protected],

3 Department of Mathematics, The Chinese University of Hong Kong,Shatin, N.T., Hong Kong, P.R. China.

[email protected]

We propose and investigate edge element numerical schemes for the time-harmonicMaxwell equations and the stationary Maxwell equations in three dimensions. Theseapproximations have three novel features:

• First, the resulting discrete edge element linear systems can be solved itera-tively with the help of existing preconditioned solvers. Furthermore, numericalexperiments show an optimal rate of convergence: the number of iterations isindependent of the meshsize.

• Second, no saddle-point discrete systems are required to solve the stationaryMaxwell equations.

• Finally, these approximations ensure the strong convergence of the Gauss’ lawsin some appropriate norm, in addition to the standard convergence in energy-norm. These estimates hold under general regularity assumptions on the data in(non-convex) polyhedral domains, and for discontinuous coefficients.

In the presentation, we will focus on the norm estimates for the error on the divergenceof the fields, and on the new scheme for solving the stationary Maxwell equations. Inparticular, sharp estimates will be provided for intense fields that may occur in thepresence of reentrant corners and/or edges, the so-called geometrical singularities.

14-4


HP -ADAPTIVE FEM BASED ONCONTINUOUS SOBOLEV EMBEDDINGS

Thomas Fankhausera, Thomas P. Wihlerb and Marcel Wirzc

Mathematics Institute, University of Bern, [email protected], [email protected]

[email protected]

The aim of this talk is to present a new class of smoothness testing strategies in thecontext of hp-adaptive refinements based on continuous Sobolev embeddings. Herethe basic idea in deciding between h- and p-refinement is to monitor the continuityconstants of suitable Sobolev inequalities as the hp-FEM spaces are enriched. A fewnumerical experiments in the context of hp-adaptive FEM for linear elliptic PDE willbe performed.

14-5


MAPPING AND REGULARITY RESULTS FOR SCHROEDINGEROPERATORS WITH INVERSE SQUARE POTENTIALS

Eugenie Hunsicker1 Hengguang Li2, Victor Nistor3 and Vile Uski4

1Loughborough University, [email protected]

2Wayne State, USA

3Penn State, USA

4STFC Harwell Oxford, UK

Consider a Schroedinger operator H with periodic potential in R3. If the potential Vis smooth, then the spectrum of H can be studied through Bloch waves associated tovectors k in the first Brillouin zone. The Schroedinger operator Hk associated to such avector is a smooth elliptic operator on the torus R3/Γ, where Γ is the periodicity latticeof the potential, and thus has a unique self-adjoint extension to the second Sobolevspace, H2(R3/Γ). The eigenfunctions of this extension are smooth and form a Hilbertbasis of L2(R3/Γ). This means that they can be well approximated using either FEMwith arbitrary degree elements or by plane waves.

However, if the potential V is of inverse square type, that is V = W/|x − p|2 neara discrete set of points p for some continuous function W , and is elsewhere smooth,then we do not automatically get these results about the associated operators Hk.In particular, generally these do not have a unique self-adjoint extension, may nothave discrete spectrum and eigenfunctions are not generally smooth. In this talk, Iwill present some analytic tools which allow us to recover versions of mapping andregularity results for periodic Schroedinger operators with inverse square potentialssatisfying the condition that W (p) > −1/4 for all p.

V. Nistor will present applications of the results of this talk to FEM for Schrodingeroperators with inverse square potentials in his talk.

14-6


FINITE ELEMENT METHOD FOR SCHROEDINGER OPERATORSWITH INVERSE SQUARE POTENTIALS

Eugenie Hunsicker1a, Hengguang Li2, Victor Nistor3c, Jorge Sofo3d and Vile Uski1e

1Math. Dept, Loughborough U., Leicestershire, LE11 3TU, UK,[email protected], [email protected]

2Math. Dept, Wayne State U., Detroit, MI 48202, USA,[email protected]

3Math. Dept., Penn State U., University Park, PA 16802, USA,[email protected], [email protected]

We study the regularity and approximability of the eigenvalues and eigenfunctionsof periodic Schroedinger operators with inverse square potentials. This is part of abigger project together with E. Hunsicker, H. Li, J. Sofo, and V. Uski. In my talk, Iwill discuss the graded mesh approximation of the eigenfunctions, including numericaltests. The results are new even for the usual, Coulomb type potentials (of the form1/r). We consider, however, the stronger, inverse square potentials (of the form 1/r2)since the difficulties are more pronounced in this case, and thus our method is easierto justify. In particular, the solution can exhibit singularities of the form r−a, witha > 0. In spite of this rather strong singularity, we achieve higher order (hm) optimalrates of convergence.

14-7


LINEAR FINITE ELEMENTS MAY BE ONLY FIRST-ORDERPOINTWISE ACCURATE ON ANISOTROPIC TRIANGULATIONS

Natalia Kopteva

Mathematics and Statistics Department,University of Limerick, Limerick, Ireland.

[email protected]

It appears that there is a perception in the finite-element community that the computed-solution error in the maximum norm is closely related to the corresponding interpola-tion error. While an almost best approximation property of finite-element solutions inthe maximum norm has been rigourously proved (with a logarithmic factor in the caseof linear elements) for some equations on quasi-uniform meshes, there is no such resultfor strongly-anisotropic triangulations. Nevertheless, this perception is frequently con-sidered a reasonable heuristic conjecture to be used in the anisotropic mesh adaptation.

In this talk, we give a counterexample of an anisotropic triangulation on which

• the exact solution is in C∞(Ω) and has a second-order pointwise error of linearinterpolation O(N−2),

• the computed solution obtained using linear finite elements is only first-orderpointwise accurate, i.e. the pointwise error is as large as O(N−1).

Here the maximum side length of mesh elements is O(N−1) and the global number ofmesh nodes does not exceed O(N2).

Our example is given in the context of a singularly perturbed reaction-diffusionequation, whose exact solution exhibits a sharp boundary layer. Both standard andlumped-mass cases are addressed. A theoretical justification of the observed numer-ical phenomena is given by the following lemma (which is established using a finite-difference representation of the considered finite element methods).

Suppose Ω ⊃ Ω, where the subdomain Ω := (0, 2ε) × (−H,H) with the tensor-product mesh ωh := xi = ε i

N02N0i=0 ×−H, 0, H. The triangulation T in Ω is obtained

by drawing diagonals in each rectangle as shown below, using the mesh transition point(ε, 0).

T in Ω:

0−H

0

H

ε ε2

T0 in Ω0 ⊂ Ω:

Lemma. Let u = e−x/ε be the exact solution of the equation −ε24u + u = 0, subjectto a Dirichlet boundary condition, posed in a domain Ω ⊃ Ω, a triangulation T in Ω besuch that T ⊃ T , and U be the computed solution obtained using linear finite elements.For any positive constant C2, there exist sufficiently small constants C0 and C1 suchthat if N−1

0 ≤ C1 and ε ≤ C2H, then

maxΩ|U − u| ≥ C0N

−10 .

14-8


Similar phenomena will be also discussed in the context of singularly perturbedconvection-diffusion equations.

References

[1] N. Kopteva, Linear finite elements may be only first-order pointwise accurate onanisotropic triangulations, Math. Comp. (2013), accepted for publication; http:

//www.staff.ul.ie/natalia/pdf/kopteva_mc2012_R1.pdf.

HP FINITE ELEMENT METHODS FOR SINGULARLYPERTURBED TRANSMISSION PROBLEMS

Serge Nicaise1 and Christos Xenophontos2

1LAMAV, Universite de Valenciennes et du Hainaut Cambresis, [email protected]

2Department of Mathematics and Statistics, University of [email protected]

We consider singularly perturbed transmission problems with two different diffusioncoefficients, in one- and two-dimensions. Their solution will contain boundary layersonly in the part of the domain where the diffusion coefficient is high, interface layersalong the interface and, in the case of polygons, corner singularities at the vertices ofthe domain.

In each case, we are interested in regularity estimates for each solution component,that are explicit in the differentiation order and the singular perturbation parameter(i.e. the diffusion coefficient). These estimates will guide us in the construction (andanalysis) of robust hp finite element methods for the approximation of the solution.

Under the assumption of analytic input data, we show that the hp version of thefinite element method on so-called Spectral Boundary Layer Meshes yields exponen-tial rates of convergence in the energy norm, as the degree p of the approximatingpolynomials increases.

Numerical results illustrating our theoretical findings will also be presented.

14-9

http://www.staff.ul.ie/natalia/pdf/kopteva_mc2012_R1.pdf

http://www.staff.ul.ie/natalia/pdf/kopteva_mc2012_R1.pdf

15: Mini-Symposium: Foundations of isogeometric analysis

15 Mini-Symposium: Foundations of isogeometric

analysis

Organisers: Lourenco Beirao da Veiga andAnnalisa Buffa

15-1


TOWARDS ISOGEOMETRIC ANALYSIS FOR COMPRESSIBLEFLOW PROBLEMS AND UNSTRUCTURED MESHES

R. Abgrall

INRIA and Universite de Bordeaux, [email protected]

In this talk, we first show how the residual distribution formalism can be extended toBezier and Nurbs elements, using unstructured meshes. This leads to parameter freeschemes, able to compute possibly discontinous flows. Then we address the meshingissue. Given a CAD, we show how to compute automatically a ”curved” mesh, in twoand three dimensions, that is exactly compatible with the CAD representation of thegeometry. Using the combination of these tools (scheme and meshing), and degreeelevation, we show how to do mesh adaptation.

This work was done with C. Dobrzynski and A. Froehly, funding ERC advancedgrant ADDECCO # 226632.

15-2


ARBITRARY-DEGREE ANALYSIS-SUITABLE T-SPLINES

L. Beirao da Veiga, Annalisa Buffa, Giancarlo Sangallia and Rafael Vazquez

Dipartimento di Matematica – Universita di Pavia, [email protected]

IsoGeometric Analysis (IGA) is a numerical method for solving partial differentialequations (PDEs), introduced by Hughes et al. in [3]. In IGA, B-splines or Non-Uniform Rational B-Splines (NURBS), that typically represent the domain geometryin a Computer Aided Design (CAD) parametrization, become the basis for the solutionspace of variational formulations of PDEs. Local refinement strategies are possiblethanks to the non-tensor product extensions of B-splines, such as T-splines ([5, 6]).A T-spline space is spanned by a set of B-spline functions, named T-spline blendingfunctions, that are constructed from a T-mesh. The T-mesh breaks the global tensor-product structure by allowing so called T-junctions.

T-splines have been recognized as a promising tool for IGA in [1] and have been theobject of recent interest in literature. In particular, in the context of IGA, analysis-suitable (AS) T-splines have emerged: introduced in [4] in the bi-cubic case, they are asub-class of T-splines for which we have fundamental mathematical properties neededin a PDE solver. Linear independence of AS T-splines blending functions has beenfirst shown in [4]. In [2] it is shown that the condition of being AS, which is mainly acondition on the connectivity of the T-mesh, implies that the bi-cubic T-spline basisfunctions admit a dual basis that can be constructed as in the tensor-product setting.We present here generalizations of the results of [2] in a fundamental way. We definethe the class ASp,q of analysis suitable T-meshes of degrees p, q and show how theabstract theory results in properties of the T-spline space: existence of a dual basis,first of all, and then linear independence of the blending functions and existence of aprojector operator with optimal approximation properties.

References

[1] Y. Bazilevs, V.M. Calo, J.A. Cottrell, J.A. Evans, T.J.R. Hughes, S. Lipton, M.A.Scott, and T.W. Sederberg. Isogeometric analysis using T-splines. Comput. MethodsAppl. Mech. Engrg., 199(5-8):229 – 263, 2010.

[2] L. Beirao da Veiga, A. Buffa, D. Cho, and G. Sangalli. Analysis-Suitable T-splinesare Dual-Compatible. To appear in Comput. Methods Appl. Mech. Engrg., 2012.

[3] T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finiteelements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl.Mech. Engrg., 194(39-41):4135–4195, 2005.

[4] X. Li, J. Zheng, T.W. Sederberg, T.J.R. Hughes, and M.A. Scott. On linear inde-pendence of T-spline blending functions. Comput. Aided Geom. Design, 29(1):63 –76, 2012.

15-3


[5] T.W. Sederberg, D.L. Cardon, G.T. Finnigan, N.S. North, J. Zheng, and T. Lyche.T-spline simplication and local refinement. ACM Trans. Graph., 23(3):276–283,2004.

[6] T.W. Sederberg, J. Zheng, A. Bakenov, and A. Nasri. T-splines and T-NURCCSs.ACM Trans. Graph., 22(3):477–484, 2003.

15-4


ISOGEOMETRIC ANALYSIS AND NON-MATCHINGDOMAIN DECOMPOSITION METHODS

Michel Bercovier

Rachel and Selim Benin School of Comp. Sc. and Eng.,The Hebrew University of Jerusalem 91904, Jerusalem, Israel.?

[email protected]

Isogeometric analysis (IGA) is a rapidly developing paradigm for the discretizationof Partial Differential Equations (PDEs). The basic idea detailed in[J. A. Cottrell,T.. J. R.. Hugues, Y. Bazilevs , Isogeometric Analysis ,Wiley,UK, 2009 ] consists indefining the same global isoparametric transformation for the exact computationaldomain using B-Splines or NURBS and for the approximation functions for the PDEssolution. One of the aims is to avoid the costly steps of mesh generation and CADinterchange.

Domain decomposition methods are natural candidates for the solution of large IGAproblems and have been studied in [L. Beirao da Veiga, D. Cho, L. F. Pavarino, andS. Scacchi Overlapping Schwarz Methods for Isogeometric Analysis SIAM Journal onNumerical Analysis 2012, Vol. 50, No. 3, pp. 1394-1416], where overlapping domainscorrespond to matching grids.

In order to use some basic constructs of CAD (boolean operations such as unionor intersections), we introduced the simplest Schwarz Additive Domain DecompositionMethod (SADDM) [O. B. Widlund, A. Toselli, Domain Decomposition Method: Algo-rithms and Theory, Springer, 2004] in IGA. Ωi, i = 1, ..., n , are overlapping ( i.e. thereis always a pair (i, j) such that Ωi∩Ωj has a non void interior) and that the respectiveisoparametric transformations are non matching,: the pair of reference grid and knotsdefining each physical domain are not related.

The respective inverse isogeometric mapping from Ωi to Ωi defines in Ωj a trimmingline ( resp. a trimming surface in 3D), and the corresponding (partial) boundary Γj,i ofΩj In SADDM we need to compute at each iteration the trace of the solution ui obtainedin the sub-domain Ωi, on the boundary Γj,i of the sub-domain Ωj. It gives rise to anon homogeneous boundary condition (BC) problem on each sub domain. Applicationof this condition in IGA is not straightforward. We analyze different approaches suchas approximation, least square and others, and compare them.

We give several examples illustrating the power of this approach: direct use of CGSprimitives, local zooming instead of refinements, and parallelization for large problems.We show that there is no degradation of the powerful approximation properties of IGAwhen using non matching meshes.

The examples are computed by means of a standard open source IGA code for eachdomain, GeoPDEs [C. de Falco, A. Reali, and R. Vazquez. GeoPDEs: a research toolfor Isogeometric Analysis of PDEs. Advances in Software Engineering,40 (2011),1020-1034.]

This search has been done jointly with Ilya Soloveichik, who did most of the exam-ples implementations.

15-5


IMPLEMENTATION OF HIGH ORDER IMPEDANCE BOUNDARYCONDITIONS IN ISOGEOMETRIC METHODS

Annalisa Buffa1a, Luca Di Rienzo2 and Rafael Vazquez1c

1 Istituto di Matematica Applicata e Tecnologie Informatiche“Enrico Magenes” del CNR, via Ferrata 1, I-27100, Pavia, Italy


2 Dipartimento di Elettronica, Informazione e Bioingegneria,Politecnico di Milano, I-20133, Milan, Italy

[email protected]

The concept of surface impedance boundary conditions (SIBC) is now well-known incomputational electromagnetics. The idea is to replace the equation of the modelinside any conductor by approximate boundary conditions on its surface, restrictingthe computational domain to the exterior of the conductors.

The research on SIBC was started by Leontovich in the 40’s. He proposed an SIBCwhich only takes into account the local tangent plane at each point on the surface.Later, Rytov proposed an extension of Leontovich’s condition based on an asymptoticexpansion. The high order SIBCs proposed by Rytov, which are valid for smoothdomains, take into account the curvature of the surface and the second tangentialderivative of the field.

In this work we present the formulation of a scalar two-dimensional problem withhigh-order SIBCs, and its discretization with isogeometric methods. There are twomain advantages of isogeometric methods in this context: the exact computation ofthe curvature, and the possibility to compute the second tangential derivative whenneeded. The method is applied to the computation of the electromagnetic fields inmulticonductor transmission lines, to show its performance.

15-6


A COMPUTATIONAL COST ANALYSISOF ISOGEOMETRIC ANALYSIS

Nathan Collier1, Lisandro Dalcin2, David Pardo3, Maciej Paszynski4 and Victor Calo5

1King Abdullah University of Science and Technology, Saudi [email protected]

2Consejo Nacional de Investigaciones Cientıficas y Tecnicas, [email protected]

3The University of the Basque Country and Ikerbasque, [email protected]

4AGH University of Science and Technology, [email protected]

5King Abdullah University of Science and Technology, Saudi [email protected]

In this talk we discuss computational aspects of isogeometric analysis. We present somework estimates based on floating point operation counts for the assembly process as wellas review solver cost estimates for B-spline-based Galerkin and collocation methods.We show that the matrix assembly process for collocation methods is economical whencompared to both standard Galerkin methods as well as those using highly continuousB-splines. This suggests that despite their inferior convergence properties, collocationmethods can be an efficient alternative to Galerkin methods. We then conclude bypresenting a large amount of data in the form of a motion chart which highlights thetrends predicted by our work estimates.

15-7


MIXED ISOGEOMETRIC COLLOCATION METHODSFOR THE STOKES EQUATIONS

John A. Evansa, Dominik Schillingerb, Rene Hiemstrac and Thomas J.R. Hughesd

ICES, The University of Texas at Austin, Austin, TX, [email protected], [email protected],[email protected] [email protected]

Recently, isogeometric collocation has been introduced as a means of dramatically re-ducing the cost associated with isogeometric Galerkin methods [1, 2, 3]. Collocation isbased on the discretization of the strong form of a given partial differential equation at adiscrete set of collocation points and can be viewed as a specialized one-point quadra-ture scheme. In this talk, we present two classes of mixed isogeometric collocationmethods for incompressible fluid flow, focusing on the Stokes equations as a simpli-fied model problem. Our first class of methods is based on the use of Taylor-Hoodisogeometric elements while our second employs isogeometric divergence-conformingB-splines. We discuss stability and conservation properties of the two classes of collo-cation methods, details of implementation, and linear solution schemes, and we presentrelevant numerical results demonstrating the effectiveness of the methods. We finishby comparing the computational cost of the mixed collocation methods with the costassociated with mixed isogeometric Galerkin and finite element methods.

References

[1] F. Auricchio, L. Beirao da Veiga, T.J.R. Hughes, A. Reali, and G. Sangalli. Isogeo-metric collocation methods. Mathematical Models and Methods in Applied Sciences,20:2075-2107, 2010.

[2] F. Auricchio, L. Beirao da Veiga, T.J.R. Hughes, A. Reali, and G. Sangalli. Isoge-ometric collocation for elastostatics and explicit dynamics. Computer Methods inApplied Mechanics and Engineering, 249-252:2-14, 2012.

[3] D. Schillinger, J.A. Evans, A. Reali, M.A. Scott, and T.J.R. Hughes. Isogeometriccollocation: Cost comparison with Galerkin methods and extension to hierarchicalNURBS discretizations. Submitted for publication, 2013.

15-8


ALGEBRAIC MULTILEVEL PRECONDITIONINGIN ISOGEOMETRIC ANALYSIS

Krishan Gahalauta and Satyendra Tomarb

Radon Institute for Computational and Applied Mathematics (RICAM),Austrian Academy of Sciences, Altenbergerstrasse 69, A4040 Linz, Austria


The isogeometric analysis proposed by Hughes et al. in [1], has received great dealof attention in the computational mechanics community. In this talk we shall presentalgebraic multilevel iteration (AMLI) methods [2, 3] for solving linear system aris-ing from the isogeometric discretization of elliptic boundary value problems. AMLImethods are based upon the hierarchical splitting of the solution space. We presentthe multilevel structure of B-Splines and NURBS spaces and their corresponding hi-erarchical spaces. The matrix formulation of coarse grid operators and its hierarchicalcomplementary operators will be discussed for varying regularity of B-Spline basis func-tions. For NURBS, we generate these operators from B-Splines and the correspondingweights. We shall discuss the quality of splitting of spaces which is measured by theconstant γ in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality. For afixed p, the constant γ will be analyzed for different regularities of the B-Spline ba-sis functions. AMLI methods when applied in the framework of isogeometric analysisshows h-independent convergence rates. Supporting numerical results for CBS con-stant γ, and convergence factor and iterations count for linear AMLI V -cycles andW -cycle, and for nonlinear AMLI W -cycle are provided. Numerical results also showthat these methods exhibit almost p-independent convergence rates. Numerical testsare performed, in two-dimensions on square domain and quarter annulus, and in three-dimensions on thick ring. Moreover, for a uniform mesh on a unit interval, the explicitrepresentation of B-Spline basis functions for a fixed mesh size h is given for p = 2, 3, 4and for C0 and Cp−1 smoothness. In this work we present the construction of AMLImethods. A rigorous analysis, particularly for higher p and regularity, is still open andwill be the subject of our future research.

References

[1] T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs. Isogeometric analysis: CAD, finiteelements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl.Mech. Engrg. 194 (2005), 4135–4195.

[2] O. Axelsson, P.S. Vassilevski. Algebraic multilevel preconditioning methods I. Nu-mer. Math., 1989; 56:157–177.

[3] O. Axelsson, P.S. Vassilevski. Algebraic multilevel preconditioning methods II.SIAM J. Numer. Anal., 1990; 27:1569–1590.

15-9


GUARANTEED AND SHARP A-POSTERIORI ERRORESTIMATES IN ISOGEOMETRIC ANALYSIS

S.K. Kleissa and Satyendra Tomarb

RICAM, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040 Linz, [email protected], [email protected]

The potential and the performance of isogeometric analysis (IGA), introduced in [1],have been well-studied over the last years for applications from many fields, see themonograph [2]. Though not a pre-requisite, most of the studies of IGA are based onnon-uniform rational B-splines (NURBS). Since the straightforward implementation ofNURBS leads to a tensor-product structure, local mesh refinement methods are subjectof active current research. Despite the fact that adaptive mesh refinement is closelylinked to the question of reliable a posteriori error estimation, the latter is still in itsinfancy stage in isogeometric analysis.Functional-type a posteriori error estimates, see the recent monograph [3] and thereferences therein, which have also been studied for a wide range of problems, providereliable and efficient error bounds, which are fully computable and do not contain anygeneric, un-determined constants.In this talk functional-type a posteriori error estimates for isogeometric analysis will bediscussed. By exploiting the properties of NURBS, we will present efficient computationof these error estimates. The numerical realization and the quality of the computederror distribution will be addressed. The potential and the limitations of the proposedapproach will be illustrated using several computational examples.

References

[1] T.J.R. Hughes, J. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finiteelements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl.Mech. Engrg., 194(39-41):4135–4195, 2005.

[2] J. Cottrell, T.J.R. Hughes, and Y. Bazilevs. Isogeometric Analysis: Toward Inte-gration of CAD and FEA. Wiley, Chichester, 2009.

[3] S. Repin. A Posteriori Estimates for Partial Differential Equations. Walter deGruyter, Berlin, Germany, 2008.

15-10


LOCAL REFINEMENTS IN IGA BASED ON HIERARCHICALGENERALIZED B-SPLINES

Carla Manni1a, Francesca Pelosi1b and Hendrik Speleers2

1Dipartimento di Matematica, Universita di Roma “Tor Vergata”,Via della Ricerca Scientifica, 00133 Roma, Italy,


2Departement Computerwetenschappen, Katholieke Universiteit Leuven,Celestijnenlaan 200A, B-3001 Leuven, Belgium,

[email protected]

Tensor-product generalized B-splines can offer an interesting problem–oriented alter-native to NURBS (Non-Uniform Rational B-Splines) in IgA (Isogeometric Analysis) asinvestigated in [Manni, C., Pelosi, F., Sampoli, M.L.: Generalized B-splines as a tool inisogeometric analysis. Comput. Methods Appl. Mech. Engrg. 200 (2011) pp. 867–881]and [Manni, C., Pelosi, F., Sampoli, M.L.: Isogeometric analysis in advection-diffusionproblems: tension splines approximation. J. Comput. Appl. Math. 236 (2011) pp.511–528].

Generalized B-splines are piecewise functions with sections in more general spacesthan algebraic polynomial spaces (like classical B-splines). Suitable selections of suchspaces – typically including trigonometric or hyperbolic functions – allow an exact rep-resentation of polynomial curves, conic sections, helices and other profiles of salientinterest in applications. Moreover, generalized B-splines possess all fundamental prop-erties of algebraic B-splines (recurrence relation, compact minimum support, locallinear independence, . . . ) which are shared by NURBS as well. Finally, contrarily toNURBS, they behave completely similar to B-splines with respect to differentiationand integration.

Adaptive local refinement is a crucial ingredient in numerical treatment of partialdifferential equations. However, NURBS rely on a tensor-product structure so theydo not allow adequate local refinements. This motivates the interest in alternativestructures for IgA that permit local refinements. Among the others, hierarchical splineshave been profitably used in this context. The concept of hierarchical bases can beconsidered for more general spaces than tensor-product B-splines, see [Giannelli, C.,Juttler, B., Speleers, H.: Strongly stable bases for adaptively refined multilevel splinespaces. Preprint (2012)]. In particular, a hierarchical structure can be built for tensor-product generalized B-splines, with suitable section spaces, as they suffer from thesame drawbacks of NURBS with respect to local refinements.

In this talk we present a multilevel representation in terms of a hierarchy of tensor-product generalized B-splines, and we discuss its use in the context of IgA. In this way,we can combine the positive properties of a non-rational model with the possibility ofdealing with local refinements.

15-11


EFFICIENT ASSEMBLY METHOD FORISOGEOMETRIC DISCRETIZATIONS

Angelos Mantzaflaris1 and Bert Juttler2

1RICAM, Austrian Academy of Sciences, Linz, [email protected]

2AG, Johannes Kepler University, Linz, [email protected]

In a typical isogeometric analysis (IGA) pipeline we are given a parameterized geometry(physical domain) and a boundary value problem, whose unknown solution field isprojected onto a finite-dimensional sub-space, i.e. we restrict ourselves to finding asolution in that space. Then a linear system is generated, consisting of a computedmatrix with e.g. mass, stiffness terms, as well as a load vector containing the momentswith respect to the right-hand side. The solution of the resulting linear system yieldsthe coefficients of the unknown field in the chosen discretization space. At each ofthese steps, errors are introduced and accumulate in the final solution. In most casesthe principal error sources during the process are the discretization error coming fromprojection of the solution and the integration error made in the generation step.

Even though IGA has a clear advantage regarding the number of degrees of freedom,matrix generation (by means of numerical integration) constitutes a bottleneck in theoverall running times of isogeometric simulations. Similarly to finite element analysis,the standard choice for integral evaluation in IGA is Gaussian quadrature. A seriousproblem of the latter is that nodes-per-element needed increase rapidly with the degreeof the B-Spline basis and the dimension of the problem. Recent developments proposespecialized quadrature rules for B-spline bases, that reduce the number of quadraturepoints and weights used.

We propose a new, quadrature-free approach, based on interpolation of the “geome-try factor” and fast look-up operations for values of B-spline integrals for the assemblyof common matrix operators, such as the mass or stiffness matrices. The geometryfactor refers to the contributions of the (Jacobian of) the geometry mapping to theintegral transformation (and possibly to contributions of non-constant coefficients),which is the actual non-polynomial part of the integrand. A sufficiently accurate ap-proximation of this factor in terms of B-splines projects the integrand into a piecewisepolynomial space, where exact integration is possible, notably by the use of lookuptables. Theoretical error estimates support our experimental results regarding the ef-ficiency and overall convergence rate which are obtained by applying our method toelliptic problems.

15-12


COMPARISON OF BOUNDARY ELEMENT METHODDISCRETISATION TECHNOLOGIES

FOR ACOUSTIC ANALYSIS

Robert N. Simpson1a, Michael A. Scott2, Matthias Taus3,Derek C. Thomas4 and Haojie Lian1

1School of Engineering, Cardiff University,Queen’s Buildings, The Parade, Cardiff CF24 3AA, UK.

[email protected]

2Department of Civil and Environmental Engineering,Brigham Young University, Provo, Utah 84602, USA.

3Institute for Computational Engineering and Sciences,The University of Texas at Austin, Austin, Texas 78712, USA.

4Department of Physics & Astronomy,Brigham Young University, Provo, Utah 84602, USA.

A significant portion of boundary element method implementations make use of La-grangian discretisations to approximate both the geometry and unknown fields where,in the majority of cases, appropriate ’meshing’ software is required to generate suit-able analysis models. This methodology is well-established, but certain deficiencies areknown to exist. Perhaps the most significant of these is the disparity between ComputerAided Design (CAD) and analysis models which incurs significant overheads during theprototype design stages, further exaggerated by the iterative nature of design. In ad-dition, the geometrical error seen in Lagrangian discretisations can lead to significanterrors in the resulting solution unless extremely fine meshes are used.

A recent developing trend is the use of CAD discretisations to approximate both thegeometry and unknown fields for analysis. Termed ‘isogeometric analysis’ by Hugheset al.[1], the concept has received great attention, particular in the context of finiteelement methods where the use of CAD discretisations reveals many attractive prop-erties. In addition, recent attention has focussed on the use of CAD discretisationsin boundary element methods [2] which represents a particularly compelling approachwhere the use of the same model for both CAD and analysis completely overcomes themesh generation process.

In this work we illustrate the fundamental differences between conventional dis-cretisation technology and that used in the isogeometric concept for boundary elementacoustic analysis. The importance of geometrical accuracy is investigated where it isfound that the isogeometric approach offers significant improvements in accuracy overits Lagrangian counterpart. The ability of the isogeometric approach to provide a trulyintegrated design and analysis technology is also demonstrated, offering significant ben-efits for practical engineering design.

15-13


References

[1] T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD,finite elements, NURBS, exact geometry, and mesh refinement. Computer Methodsin Applied Mechanics and Engineering, 194:4135–4195, 2005.

[2] M. A. Scott, R. N. Simpson, J. A. Evans, S. Lipton, S. P. A. Bordas, T. J. R. Hughes,and T. W. Sederberg. Isogeometric boundary element analysis using unstructuredT-splines. Computer Methods in Applied Mechanics and Engineering, 254:197–221,2013.

SPLINES ON TRIANGULATIONS IN ISOGEOMETRIC ANALYSIS

Hendrik Speleers

Departement Computerwetenschappen, Katholieke Universiteit Leuven,Celestijnenlaan 200A, B-3001 Leuven, Belgium,

[email protected]

Isogeometric Analysis (IgA) is a novel paradigm for numerical simulation which com-bines Finite Element Analysis (FEA) with Computer Aided Design (CAD) methods.The CAD representations – usually in terms of tensor-product B-splines or Non-Uniform Rational B-Splines (NURBS) – are used both to describe the geometry andto approximate the unknown solutions of differential equations.

Adaptive local mesh refinement is an important ingredient for obtaining efficientlyan accurate solution of differential problems. In the context of classical FEA, localmesh refinement strategies are a well established procedure. Unfortunately, the tensor-product structure of NURBS spaces precludes strictly localized refinements. This mo-tivates the interest in alternative structures for IgA that permit local refinements.

In this talk we discuss the use of splines on triangulations for the numerical solutionof differential equations in the context of IgA. In particular, we focus on Powell-Sabin(PS) splines which are defined on triangulations with a particular macro-structure.These splines can be represented with basis functions possessing similar properties tothe classical (tensor-product) B-splines. The PS B-splines form a convex partitionof unity, and the coefficients of this representation have a clear geometric meaning.One can also easily define a rational extension of PS splines, so-called NURPS (Non-Uniform Rational PS). NURPS surfaces allow an exact representation of quadrics, andtheir shape can be locally controlled by control points and weights in a geometricallyintuitive way.

Thanks to their structure based on triangulations, PS/NURPS splines offer theflexibility of classical finite elements with respect to local mesh refinements. Moreover,they share with standard tensor-product NURBS the increased smoothness, the B-spline-like basis, and the ability to exactly represent profiles of interest in engineeringapplications as conic sections. Therefore, they constitute a natural bridge betweenclassical FEA and NURBS-based IgA. We will illustrate the use of PS/NURPS splinesin IgA with several numerical examples.

15-14


APPROXIMATION PROPERTIES OF SINGULARPARAMETRIZATIONS IN ISOGEOMETRIC ANALYSIS

Thomas Takacsa and Bert Juttlerb

Institute of Applied Geometry, Johannes Kepler University, Linz, [email protected], [email protected]

Isogeometric analysis is a numerical method based on the NURBS representation ofCAD models. The geometry mapping that parametrizes a 2-dimensional physical do-main posesses the tensor product structure of bivariate NURBS. Hence the domain isstructurally equivalent to a rectangle or to a hexahedron. The special case of singularlyparametrized NURBS surfaces is used to represent non-quadrangular domains withoutsplitting.

We analyze the approximation properties of the isogeometric test function spaceson singular parametrizations. We present local refinement strategies that lead to ge-ometrically regular splittings of singular patches. Using this we develop a generalframework to prove approximation results for singularly parametrized domains in iso-geometric analysis. We prove bounds for the L2 and H1 approximation error for twoclasses of singular parametrizations of two dimensional domains.

ADAPTIVE HIERARCHICAL B-SPLINES FOR LOCALREFINEMENT IN ISOGEOMETRIC ANALYSIS

Anh-Vu Vuonga and Bernd Simeonb

Felix-Klein-Centre for Mathematics, University of Kaiserslautern, [email protected] [email protected]

Adaptive simulation is one of the great challenges in Isogeometric Analysis, which com-bines finite elements with techniques from computer aided geometric design (CAGD).Especially, local refinement is a major obstacle because a straightforward approach byusing the standard CAGD routines is prevented by the tensor-product structure, whichcauses the insertion of various superfluous control points.

The main topic of this talk is a refinement technique based on hierarchical B-Splinesand its variants. It is very flexible and does not suffer under restrictions on degreeand continuity and has the advantage of offering local refinement by construction.Furthermore, properties like linear independence are ensured right from the beginningand the hierarchical spline spaces are fully integrated into the isogeometric setting byadopting well-established finite element techniques into this new context. For example,combined with an a posteriori multi-level error estimator this results in a promisingadaptive simulation.

The talk will introduce the fundamental idea of this approach, discuss its differentvariants and illustrate it by some numerical examples.

15-15

16: Mini-Symposium: Global and local error estimates for problems with singularities or low regularity

16 Mini-Symposium: Global and local error esti-

mates for problems with singularities or low reg-

ularity

Organisers: Alan Demlow and Dmitriy Leykekhman

16-1


ERROR ANALYSIS OF DISCONTINUOUS GALERKIN METHODSFOR THE STOKES PROBLEM UNDER MINIMAL REGULARITY

Santiago Badia1, Ramon Codina1, Thirupathi Gudi2 and Johnny Guzman3

1Universtitat Politecnica de Catalunya (UPC),Jordi Girona 1-3, 08034 Barcelona, Spain

2Department of Mathematics,Indian Institute of Science Bangalore, 560012 India

[email protected]

3Division of Applied Mathematics, Brown University, Providence RI 02912, USA

We analyze several discontinuous Galerkin methods (DG) for the Stokes problem underthe minimal regularity on the solution. We assume that the velocity u belongs to[H1

0 (Ω)]d and the pressure p ∈ L20(Ω). First, we analyze standard DG methods assuming

that the right hand side f belongs to [H−1(Ω) ∩ L1(Ω)]d. A DG method that is welldefined for f belonging to [H−1(Ω)]d is then investigated. The methods under studyinclude stabilized DG methods using equal order spaces and inf-sup stable ones wherethe pressure space is one polynomial degree less than the velocity space.

OPTIMALITY OF AN ADAPTIVE FEM FORCONTROLLING LOCAL ENERGY ERRORS

Alan Demlow

Department of Mathematics, University of Kentucky,715 Patterson Office Tower, Lexington, KY 40506–0027 USA

[email protected]

A posteriori error estimates and corresponding adaptive FEM for controlling finiteelement errors only on local subdomains of the overall computational domain haveappeared in the context of “two-grid” and parallel adaptive algorithms. In this talkwe will present optimality results for an AFEM designed to control local energy er-rors. Issues that arise include the necessity of controlling “pollution” effects of globalsolution properties on local solution quality and the effects of singularities on adaptiveconvergence rates.

16-2


OPTIMAL ERROR ESTIMATES FOR THE PARABOLICPROBLEM IN L∞(Ω;L2([0, T ])) NORM

Dmitriy Leykekhman1 and Boris Vexler2

1Department of Mathematics, University of Connecticut, [email protected],

2Technische Universitat Munchen, Faculty of Mathematics,Boltzmannstraße 3, 85748 Garching b. Munich, Germany

[email protected]

In this talk we discuss the local and global error estimates in L∞(Ω;L2([0, T ])) normfor the second order parabolic problem

ut(t, x)−∆u(t, x) = f(t, x), (t, x) ∈ I × Ω,

u(t, x) = 0, (t, x) ∈ I × ∂Ω,

u(0, x) = 0, x ∈ Ω,

where Ω is a bounded domain in RN , N ≥ 2. The norm is rather non-standard and notusually considered in the finite element literature, however such error estimates are im-portant for example for optimal control problems with point controls. For the N = 2,optimal error estimates were obtained in [D. Leykekhman and B. Vexler, Optimal apriori error estimates of parabolic optimal control problems with pointwise control,2013], however such optimal error estimates in higher dimensions pose significant dif-ficulties. In the talk I will discuss the difficulties and discuss a method to obtain suchestimates. The method is in the spirit of [A. H. Schatz and L. B. Wahlbin, Interiormaximum norm estimates for finite element methods, 1977] and requires new type oflocal energy estimates.

16-3


A POSTERIORI ESTIMATION OF HIERARCHICAL TYPE FOR ASCHRODINGER OPERATOR WITH INVERSE SQUARE POTENTIAL

Hengguang Li1 and Jeffrey S Ovall2

1Department of Mathematics, Wayne State University, Detroit, MI, [email protected]

2Department of Mathematics, University of Kentucky, Lexington, KY, [email protected]

We develop an a posteriori error estimate for mixed boundary value problems of theform (−∆ + δ2|x|−2)u = f , for some constant δ > 0, in Ω ⊂ R2, where Ω containsthe origin. Here r = |x|. Operators of this sort can arise in applications in quantummechanics, and require analysis in weighted Sobolev spaces for well-posedness andregularity results, as well as for the development of effective numerical algorithms. Wenote that the term |x|−2u is of the “same differential order” order as ∆u, so problemsof this sort cannot be view as lower-order perturbations of standard elliptic problems.If the origin is in the interior of Ω, u will generically have an |x|δ type singularity atthe origin, in addition to the usual boundary singularities at re-entrant corners in thedomain and points where the type of boundary condition changes. Therefore, sometype of adapted approximation is needed.

In two-dimensions a simple grading strategy may be chosen a priori which guar-antees optimal order convergence. Even in this case, however, a cheaply-computableerror estimate is desirable, if for no other reason than to establish a practical stop-ping criterion. We present an a posteriori error estimate of hierarchical-type, andargue that it is equivalent to the actual error in energy norm on a family of geometri-cally graded meshes appropriate for singular solutions of such problems. Experimentsdemonstrate the behavior of the method in practice. Because such meshes can havestrong anisotropy, negatively affecting the conditioning of the linear systems, we alsopresent comparisons with adaptively refined meshes driven by local indicators associ-ated with our approach.

16-4


LOCALIZED POINTWISE ESTIMATES FOR THE FULLYNONLINEAR MONGE-AMPERE EQUATION

Michael Neilan


In this talk, I will discuss localized pointwise error estimates for finite element meth-ods of the fully nonlinear Monge-Ampere equation. In our approach, we treat the fullynonlinear problem as a perturbed linear problem and resort to arguments given bySchatz who studied pointwise estimates of solutions satisfying a perturbed Galerkinorthogonality condition. Numerical examples will be presented which confirm the the-ory.

16-5


ROBUST LOCALIZATION OF THE BEST ERROR WITHFINITE ELEMENTS IN THE REACTION-DIFFUSION NORM

Francesca Tantardini1a, Andreas Veeser1b and Rudiger Verfurth2c

1Dipartimento di Matematica, Universita degli Studi di Milano, Milano, [email protected], [email protected]

2Fakultat fur Mathematik, Ruhr-Universitat, Bochum, [email protected]

We consider the problem of approximating a function in H10 in the reaction-diffusion

norm |||·|||2 := ‖·‖2L2 +ε‖∇·‖2

L2 with continuous finite elements on a given triangulation.We prove that the squared global best error is equivalent to the sum of the squares ofthe local best errors on pairs of elements:

infv∈S0,`

|||u− v|||2Ω ≤ C∑E

infP∈S0,`|ω(E)

|||u− P |||2ω(E), (1)

where the constant C is independent of ε. The sum on the right-hand side is overall the faces E of the triangulation and ω(E) is the pair of elements sharing the faceE. A result of this type has been proved for the H1-seminorm by A. Veeser. Therethe local errors are on the single elements, and do not involve the coupling betweenthe elements. We show that, for the reaction-diffusion norm, taking the local errorson single elements in place of the pairs entails that the constant in (1) grows withε−1. Robustness requires to enlarge the local domains, so that the local errors involvethe continuity constraint. Our result shows that it is enough to take pairs of adjacentelements.

Moreover we prove a variant of (1), where the local errors are defined on pairs thatare “minimal” in the context of bisection. This allows to define, for every element, alocal indicator that does not depend on the current triangulation, and so is suitablefor the tree approximation of P. Binev and R. DeVore. We can thus compute robustnear-best approximations, which may be used as benchmarks for other approximations,e.g. Galerkin solutions.

16-6

17: Mini-Symposium: High order finite element methods: A mini symposium celebrating LeszekDemkowicz’s contributions

17 Mini-Symposium: High order finite element meth-

ods: A mini symposium celebrating Leszek Demkow-

icz’s contributions

Organisers: Jay Gopalakrishnan and JoachimSchoberl

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FEM WITH DISCRETE TRANSPARENT BOUNDARY CONDITIONSFOR THE CAUCHY PROBLEM FOR THE SCHRODINGER

EQUATION ON THE WHOLE AXIS

Alexander Zlotnik

Department of Higher Mathematics at Faculty of Economics,National Research University Higher School of Economics University, Moscow, Russia.

[email protected]

The linear time-dependent Schrodinger equation is important in various fields of physics,and often it has to be solved in domains unbounded in space. To this end, a number ofapproaches were suggested, and the technique of discrete transparent boundary condi-tions (TBCs) is known among the best due to complete absence of spurious reflectionsfrom artificial boundaries, stable computations and clear mathematical backgroundleading to a rigorous stability theory [1, 2]. But constructing the discrete TBCs forhigher order methods is not a simple matter.

We consider the Cauchy problem for the 1D Schrodinger equation with variablecoefficients on the whole axis. To solve it, we study FEM of any order n ≥ 1 on aspace segment and of the Crank-Nicolson type in time coupled to the discrete TBCs.We prove stability bounds in L2 and in the energy-like norm uniformly in time, bothwith respect to initial data and a free term in the defining integral identity.

We also study the similar auxiliary method using an infinite mesh on the wholeaxis. Next, a model FEM for an auxiliary 2nd order ODE with a complex parameteron the half-axis is solved analytically. Studying the matrix pencil related to the stiffnessand mass matrices of the reference element and also applying the reproducing seriestechnique, we construct the discrete TBCs in the form of the Dirichlet-to-Neumann mapinvolving a discrete convolution in time. Its kernel is defined in turn as a n-multiplediscrete convolution of sequences expressed in terms of the Legendre polynomials. Thediscrete TBCs allows to restrict exactly the solution of the auxiliary FEM on the wholeaxis to a finite segment.

We also prove a collection of error bounds up to the orderO(τ 2+hn+1) in dependencewith smoothness of the initial function.

We present computational results for such typical problems as the free evolution ofthe Gaussian wave package and the tunnel effect for rectangular barriers. They clearlyshow that higher order finite elements coupled to the discrete TBCs are very effectiveeven in the case of strongly oscillating moving solutions and discontinuous potentials.

The results are got jointly with I. Zlotnik (MPEI, Russia) and given in part in [3].

References

[1] X. Antoine, A. Arnold, C. Besse et al., A review of transparent and artificialboundary conditions techniques for linear and nonlinear Schrodinger equations,Commun. Comp. Phys. (2008) 4 729-796.

[2] B. Ducomet and A. Zlotnik, On stability of the Crank-Nicolson scheme with ap-

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proximate transparent boundary conditions for the Schrodinger equation. Parts I,II, Commun. Math. Sci. (2006) 4 741-766; (2007) 5 267-298.

[3] A. Zlotnik and I. Zlotnik, Finite element method with discrete transparent bound-ary conditions for the time-dependent 1D Schrodinger equation, Kinetic and Re-lated Models (2012) 5 639-667.

HIGH ORDER FEM FOR WAVE PROPAGATION:LIKE IT OR LUMP IT

Mark Ainsworth

Division of Applied Mathematics, Brown University,182 George Street, Providence RI., USA


High order finite element methods have long been used for computational wave prop-agation for both first order and second order equations alike. A key issue with com-putational wave propagation is the phase accuracy of the methods: getting the wavesto propagate with the right speed is often at least as, if not more, important than theconvergence rate. The main variants are the finite element method (FEM) and spectralelement method (SEM) with each technique having its band of disciples. SEM can beviewed as a higher order mass-lumped FEM. Despite the predominance of these meth-ods, there is comparatively little by way of hard analysis on what each variant offersin terms of phase accuracy, with the result that there is considerable misinformationand confusion in the literature on this topic. In this presentation, we shall attempt toshed some light on the matter, and also briefly mention new methods that improve onboth FEM and SEM.

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COMMUTING QUASI INTERPOLANTS FOR T-SPLINE SPACES

Annalisa Buffa1a, Giancarlo Sangalli2 and Rafael Vazquez1

1 IMATI CNR “E. Magenes”, Via Ferrata 1, 27100 Pavia, Italya [email protected]

2 Universita degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy

In the papers [1] and [2], the spline discretization of differential forms is proposed andanalysed. In particular commuting projectors are constructed by exploiting the tensorproduct structure of the mesh and the basis functions. This construction can not inprinciple be extended to meshes with hanging nodes, i.e., to T-meshes and T-splines.

In this talk, I will present quasi-interpolant operators for Analysis Suitable T-splinesspaces which commute with the exterior derivative and which enjoy L2 stability proper-ties. Thanks to these commuting operators, we can use the theory of the finite elementexterior calculus to provide the mathematical analysis of the T-splines discretizationof several problems as e.g., Maxwell equations or Darcy flows.

References

[1] A. Buffa, J. Rivas, G. Sangalli, R. Vazquez, Isogeometric Discrete Differ-ential Forms in Three Dimensions, SIAM J. Numer. Anal. 49 (2011), pp. 818-842.

[2] A. Buffa, G. Sangalli, R. Vazquez, Isogeometric analysis in electromagnet-ics: B-splines approximation , Comput. Methods Appl. Mech. Engrg. 199 (2010),no. 17-20, pp. 1143–1152.

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A PDE-CONSTRAINED OPTIMIZATION APPROACH TO THEDISCONTINUOUS PETROV-GALERKIN METHOD WITH A

TRUST REGION INEXACT NEWTON-CG SOLVER

Tan Bui-Thanha and Omar Ghattasb

Institute for Computational Engineering and Sciences,the University of Texas at Austin, USA.


We introduce a PDE-constrained optimization approach to the discontinuous Petrov-Galerkin (DPG) method. This point of view allows us to use the full force of the state-of-the-art PDE-constrained optimization technique. The details of the our approachwill be presented. In particular, we will show how to compute the gradient and Hessian-vector product using efficient adjoint techniques. The gradient and Hessian-vectorproduct are in turn utilized in a trust region inexact Newton-CG to solve for the DPGsolution. The advantage of our approach is that it is valid for both linear and nonlinearPDEs. Moreover, our method is guaranteed to converge to the unique solution for well-posed linear PDEs and to a solution for nonlinear PDEs. Numerical results for severalPDEs including Laplace, Helmholtz, nonlinear viscous Burger, and compressible Eulerequations are presented to justify our new approach.

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A POSTERIORI ERROR CONTROL FOR DPG METHODS

Carsten Carstensen1, Leszek Demkowicz2 and Jay Gopalakrishnan3

1Institut fur Mathematik, Humboldt Universitat zu Berlin,Unter den Linden 6, 10099 Berlin, Germany.

[email protected]

2Institute for Computational Engineering and Sciences,The University of Texas at Austin, Austin, TX 78712, USA.

[email protected]

3Department of Mathematics & Statistics,PO Box 751, Portland State University, Portland, OR 97207-0751, USA.

[email protected]

Discontinuous Petrov Galerkin (DPG) methods was first presented to the communityby Leszek Demkowicz in the Babuska lecture of MAFELAP 2009. We give a briefoverview of this relatively new class of methods and report on the progress in ourresearch into its theoretical properties.

DPG methods minimize a residual in a nonstandard dual norm. The method com-bines least-square ideas with hybridization allowing certain dual norms of the residualto be locally computed. An interesting feature of the method is that it comes with abuilt-in error estimator thus making it attractive for practical use in scenarios requiringadaptivity.

The focus of this talk is on the proof of reliability and efficiency of the error estima-tor. A general a posteriori error analysis using the natural norms of the DPG schemesis now available. We show that a locally and inexactly computed residual norm is botha lower and an upper error bound, up to certain data approximation errors. This en-ables abstract and precise computable upper and lower error bounds. We apply theseideas to DPG discretizations of the equations of Laplace, Lame and Stokes.

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IMPROVED STABILITY ESTIMATES FOR THEHP -RAVIART-THOMAS PROJECTION OPERATOR

ON QUADRILATERALS

Alexey Chernov1 and Herbert Egger2

1Hausdorff Center for Mathematics and Institute for Numerical Simulation,University of Bonn, Germany,[email protected]

2Department of Mathematics, TU Darmstadt, [email protected]

Stability the hp-Raviart-Thomas projection operator has been addressed e.g. in [D.Schtzau, C. Schwab, and A. Toselli. Mixed hp-DGFEM for incompressible flows. SIAMJ. Numer. Anal., 40(6):2171–2194, 2002]. These results are suboptimal w.r.t. thepolynomial degree p. In this talk we present improved stability estimates for the hp-Raviart-Thomas projection operator on quadrilaterals. Such estimates may be usefulper-se, but also have important applications, e.g., in the inf-sup stability proofs anda-posteriori error estimation for DG methods.

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DPG METHOD FOR WAVE PROPAGATION PROBLEMS,A BETTER UNDERSTANDING

Leszek Demkowicz1, Jay Gopalakrishnan2, Jens Markus Melenk3,Ignacio Muga4 and David Pardo5

1ICES, U Texas at Austin, [email protected]

2Dept. of Math., Portland State U, [email protected]

3IASC, Vienna UT, Austria,[email protected]

4Dept. of Math., Pontifica U Catolica de Valparaiso, Chile,[email protected]

5Dept. of Applied Math., U Basque Country, Spain,[email protected]

Under appropriate assumptions on the domain and boundary conditions, the opera-tor corresponding to the linear acoustics equations is bounded below with a constantindependent of wave number k. This, in turn, implies the uniform stabilityresult forthe DPG ultraweak variational formulation and the uniform convergence result for theDPG method [1]: the Finite Element (FE) error is bounded by the Best Approxima-tion (BA) error times a k-independent stability constant. The result, unfortunately,does not explain the pollution-free behavior of the DPG method observed in com-putations. Indeed, the DPG solution consists of “fields” (pressure and velocity) and“traces” (trace of pressure and trace of normal velocity). Whereas fields are measuredin the pollution-free L2-norm, the traces are measured in a minimum-energy extensionnorm that hides derivatives and wave number k. As for every hybrid method, DPGrepresents a “team work”: the FE error (in fields and traces) is bounded by the (com-bined) BA error in fields and traces. And the BA error of traces scales the same wayas in the standard FEM: kp+1hp, one power of k too much...

We will present a new convergence analysis for the DPG method based on interpret-ing the DPG method as an implicit realization of a Petrov-Galerkin method with con-forming optimal test functions in the sense of Barret and Morton [2]. The globally op-timal test functions are implicitly approximated with weakly-conforming least squares,one might say that the least squares are working backstage for the DPG method. Theconvergence analys is hinges on Strang’s lemma and, besides the pollution-free L2 BAerror, includes a consistency error.

The analysis does not prove that the DPG method is pollution-free but it explainsbetter what we observe numerically, and suggests a modification of the trial spacesimproving further the stability properties of the method.

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References

[1] L. Demkowicz, J. Gopalakrishnan, I. Muga, and J. Zitelli. “Wavenumber Ex-plicit Analysis for a DPG Method for the Multidimensional Helmholtz Equation”,CMAME, 213-216, pp.126-138, 2012.

[2] J. Chan, J. Gopalakrishnan and L. Demkowicz, “ Global properties of DPG testspaces for convection-diffusion problems”, ICES Report 2013/5.

A SPACE-TIME MULTIGRID METHOD FORHIGH ORDER TIME DISCRETIZATIONS

Martin Gander1, Martin Neumuller2a and Olaf Steinbach2b

1Section de Mathematiques, Universite Geneve, Geneve, [email protected]

2Institute of Computational Mathematics,Graz University of Technology, Graz, Austria.


For evolution equations we present a space-time method based on Discontinuous Galerkinfinite elements. Space-time methods have advantages when we have to deal with mov-ing domains and if we need to do local refinement in the space-time domain. For thismethod we present a multigrid approach based on space-time slabs. This method al-lows the use of parallel solution algorithms. In particular it is possible to solve parallelin time and space. Furthermore this multigrid approach leads to a robust method withrespect to the polynomial degree which is used for the DG time stepping scheme. Nu-merical examples will be given which show the performance of this space-time multigridapproach.

PARTIAL EXPANSION OF A LIPSCHITZ DOMAINAND SOME APPLICATIONS

Jay Gopalakrishnan1 and Weifeng Qiu2

1Department of Mathematics and Statistics, Portland State University

2Department of Mathematics, City University of Hong [email protected]

We show that a Lipschitz domain can be expanded solely near a part of its boundary,assuming that the part is enclosed by a piecewise C1 curve. The expanded domain aswell as the extended part are both Lipschitz. We apply this result to prove a regulardecomposition of standard vector Sobolev spaces with vanishing traces only on partof the boundary. Another application in the construction of low-regularity projectorsinto finite element spaces with partial boundary conditions is also indicated.

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DISPERSIVE AND DISSIPATIVE ERRORS IN THE DPGMETHOD WITH SCALED NORMS FOR HELMHOLTZ EQUATION

Jay Gopalakrishnan1a, Ignacio Muga2 and Nicole Olivares1b

1 Department of Mathematics and Statistics,Portland State University, Portland, OR 97207-0751, USA.


2 Instituto de Matematicas,Pontificia Universidad Catolica de Valparaıso, Valparaıso, Chile.

[email protected],

We consider the discontinuous Petrov-Galerkin (DPG) method, where the test spaceis normed by a modified graph norm. The modification scales one of the terms in thegraph norm by an arbitrary positive scaling parameter. Studying the application of themethod to the Helmholtz equation, we find that better results are obtained, under somecircumstances, as the scaling parameter approaches a limiting value. We perform adispersion analysis on the multiple interacting stencils that form the DPG method. Theanalysis shows that the discrete wavenumbers of the method are complex, explainingthe numerically observed artificial dissipation in the computed wave approximations.Since the DPG method is a nonstandard least-squares Galerkin method, we compareits performance with a standard least-squares method.

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ADAPTIVE AND HYBRIDIZED HERMITE METHODS FORINITIAL-BOUNDARY VALUE PROBLEMS

Thomas Hagstrom1, Daniel Appelo2 and Ronald Chen3

1Department of Mathematics, Southern Methodist University, Dallas TX [email protected]

2Department of Mathematics and Statistics,University of New Mexico, Albuquerque NM USA;

[email protected]

3College of Optical Sciences, Arizona Center for Mathematical Sciences,University of Arizona, Tucson AZ USA;

[email protected]

Hermite methods are general-purpose arbitrary-order volume discretizations with anumber of attractive properties. Foremost among these is the possibility to inde-pendently evolve large chunks of data within each computational cell over time stepsconstrained only by domain-of-dependence requirements. Thus high-order Hermitemethods maximize the computation-to-communication ratio, which is likely to be ofincreasing importance to exploit modern multicore architectures. In addition our ex-perience shows that the methods are quite robust - maintaining stability for problemswhere other methods fail. This talk will focus on two aspects of our ongoing efforts toenhance our Hermite solvers.

The first topic is the implementation and analysis of hp-adaptivity in space andtime. In space, the p-adaptive implementation allows the use of polynomials of differ-ing degrees in different cells. The only constraint is to maintain local dissipativity in theHermite interpolation process. We implement h-adaptivity with quadtree/octree spa-tial refinements, using local time-stepping to maintain efficiency. In time we considerthe use of adaptive Runge-Kutta methods independently within each cell.

The second development concerns the hybridization of Hermite methods with dis-continuous Galerkin (DG) methods to treat problems in complex geometry. The DGmethods are used on an unstructured grid layer near physical boundaries and utilize anindependent local time step. In practice we have taken this to be as much as an order ofmagnitude or more smaller than the time step in the Hermite cells. The latter are partof a Cartesian mesh which, ideally, will cover most of the computational volume. Datais interpolated from the DG cells to the Hermite cells as needed, while the Hermite cellsprovide fluxes to the DG cells. Our experiments show that the inerent dissipativity ofthe two discretization techniques is sufficient to render the hybrid method stable.

Acknowledgement: The work of the first two authors was supported in part by NSFGrant OCI-0905045. The first author was also supported in part by ARO ContractW911NF-09-1-0344. Any conclusions or recommendations expressed in this talk arethose of the authors and do not necessarily reflect the views of NSF or ARO.

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ON HP -BOUNDARY LAYER SEQUENCES

Harri Hakula

Department of Mathematics and Systems Analysis,Aalto University, Espoo, Finland

[email protected]

The focus of this talk is a practical implementation of Schwab’s concept of hp-approxi-mation of boundary layers [1] in two dimensions. Boundary layer functions are of theform

u(x) = exp(−a x/d), 0 < x < L,

where d ∈ (0, 1] is a small parameter, a > 0 is a constant. L is the diameter of thedomain, in other words, the longest length scale of the problem. Even though theclassical p-method, see e.g. [2], is capable of asymptotic superexponential convergence,judicious choice of a minimal number of elements using a priori knowledge of theboundary layers leads to far more efficient solution in the practical range of p. Moreover,in certain classes of problems, it is possible to choose a robust strategy leading toconvergence uniform in d. However, the distribution of the mesh nodes depends on p,and over a range of polynomial degrees p = 2, . . . , 8, say, the mesh is different for everyp! In 1D this is relatively simple, but in 2D much more difficult since we must allow forthe mesh topology to change over the range of polynomial degrees. For every boundarylayer in the problem, one should have an element of width O(p d) in the direction of thedecay of the layer. Notice, that if c p d → L as p increases (c constant), the standardp-method can be interpreted as the limiting method.

The algorithm discussed here is based on guiding the meshing process by trackingthe mesh lines via a geometric data-structure called an arrangement [3]. The geometricinformation required for moving the mesh lines is handled within the arrangement andthe actual meshes are realised using it. However, the meshes need not be topologicallyequivalent which distinguishes the proposed algorithm from the r-method, where themesh topology is fixed but the nodes can be moved.

The effectiveness of the algorithm is demonstrated using both source and eigenvalueproblems in computational mechanics.

References

[1] Ch. Schwab, p- and hp-Finite Element Methods, Oxford University Press, 1998.

[2] B. Szabo and I. Babuska, Finite Element Analysis, Wiley, 1991.

[3] H.Hakula, hp-Boundary Layer Mesh Sequences with Applicationsto Shell Problems, Computers and Mathematics with Applications,10.1016/j.camwa.2013.03.007, 2013.

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DISCONTINUOUS GALERKIN HP -BEMWITH QUASI-UNIFORM MESHES

Norbert Heuer1 and Salim Meddahi2

1Facultad de Matematicas, Pontificia Universidad Catolica de Chile, Santiago, [email protected]

2Departamento de Matematicas, Facultad de Ciencias,Universidad de Oviedo, Oviedo, Spain

[email protected]

We present and analyze a discontinuous variant of the hp-version of the boundaryelement Galerkin method with quasi-uniform meshes. The model problem is that of thehypersingular integral operator on an (open or closed) polyhedral surface. We prove aquasi-optimal error estimate and conclude convergence orders which are quasi-optimalfor the h-version with arbitrary degree and almost quasi-optimal for the p-version.Numerical results underline the theory.

We gratefully acknowledge support by FONDECYT-Chile through project 1110324and by Ministery of Education of Spain through project MTM2010-18427.

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TWO-GRID HP–ADAPTIVE DISCONTINUOUS GALERKINFINITE ELEMENT METHODS FOR SECOND–ORDER

QUASILINEAR ELLIPTIC PDES

Paul Houston

School of Mathematical Sciences, University of Nottingham,University Park, Nottingham, NG7 2RD, United Kingdom.

[email protected]

http://www.maths.nottingham.ac.uk/personal/ph/

In this talk we present an overview of some recent developments concerning the aposteriori error analysis and adaptive mesh design of h– and hp–version discontinu-ous Galerkin finite element methods for the numerical approximation of second–orderquasilinear elliptic boundary value problems. In particular, we consider the deriva-tion of computable bounds on the error measured in terms of an appropriate (mesh–dependent) energy norm, as well as for general target functionals of the solution, inthe case when a two-grid approximation is employed. In this setting, the fully non-linear problem is first computed on a coarse finite element space VH,P . The resulting‘coarse’ numerical solution is then exploited to provide the necessary data needed tolinearise the underlying discretization on the finer space Vh,p; thereby, only a linearsystem of equations is solved on the richer space Vh,p. Here, an adaptive hp–refinementalgorithm is proposed which automatically selects the local mesh size and local polyno-mial degrees on both the coarse and fine spaces VH,P and Vh,p, respectively. Numericalexperiments confirming the reliability and efficiency of the proposed mesh refinementalgorithm are presented.

This research has been carried out in collaboration with Scott Congreve (Universityof Nottingham) and Thomas Wihler (University of Bern).

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http://www.maths.nottingham.ac.uk/personal/ph/


NEW HYBRID DISCONTINUOUS GALERKIN METHODS

Youngmok Jeon1 and Eun-Jae Park2

1Department of Mathematics, Ajou University, Suwon, [email protected]

2Department of Mathematics and Department of Computational Scienceand Engineering, Yonsei University, Seoul 120-749, Korea

[email protected]

A new family of hybrid discontinuous Galerkin methods is studied for second-orderelliptic equations [3, 4]. Our proposed method is a generalization of CBE method [2]which allows high order polynomial approximations. Our approach is composed ofgenerating PDE-adapted local basis and solving a global matrix system arising froma flux continuity equation. Our method can be viewed a hybridizable discontinuousGalerkin method [1] using a Baumann-Oden type local solver.

First, optimal order error estimates measured in the energy norm are proved fornew triangular elements. Numerical examples are presented to show the performanceof the method. Next, quadratic and cubic rectangular elements are proposed andoptimal order error estimates measured in the energy norm are provided for ellipticequations [4]. Then, this approach is exploited to approximate Stokes equations andConvection-Diffusion equations. Numerical results are presented for various examples.

References

[1] B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization ofdiscontinuous Galerkin, mixed and continuous Galerkin methods for second orderelliptic problems, SIAM J. Numer. Anal. 47 (2009), pp. 1319–1365.

[2] Y. Jeon and E.-J. Park, Nonconforming cell boundary element methods forelliptic problems on triangular mesh, Appl. Numer. Math. 58 (2008), pp. 800–814.

[3] Y. Jeon and E.-J. Park, A hybrid discontinuous Galerkin method for elliptic prob-lems, SIAM J. Numer. Anal. 48 (2010), no. 5, 1968-1983.

[4] Y. Jeon and E.-J. Park, New locally conservative finite element methods on arectangular mesh, Numerische Mathematik 123 (2013), 97-119.

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GODUNOV SPH METHODS FOR SIMULATING COMPLEXFLOWS WITH FREE SURFACES OVER RAPIDLY

CHANGING NATURAL TERRAINS

Dinesh Kumar1, E. B. Pitman2b and A. K. Patra2c

1Department of Mech. and Aero. Eng.,University at Buffalo, Buffalo, NY 14260, USA

[email protected],

2Department of Mathematics,University at Buffalo, Buffalo, NY 14260, [email protected], [email protected]

The Lagrangian nature Godunov Smooth Particle Hydrodynamics is naturally suitablefor free-surface flows like those modeled as depth averaged (shallow water like model)granular avalanches. Yet enforcing boundary conditions on rapidly changing naturalsurfaces and high computational cost when compared to its grid-based Eulerian coun-terparts present major challenges. In this talk, we present a three-dimensional imple-mentation of Godunov-SPH method for such flows, on natural terrains. Godunov-SPHis based on the work of Inutsuka [J. Comp.Phys. 2002; 179:238267] that accuratelyresolves discontinuities without the need to use artificial viscosity. Our approach buildson the basic Inustsuka approach but introduces a modified ghost particle method tocorrectly enforce the essential boundary conditions and a Navier slip based boundarycondition for the natural boundary condition. The development here is motivated bythe need to improve upon depth averaged grid based models of large scale debris flowsand avalanches, often characterized as granular flows (see the Figure for a sample flowthat cannot be simulated using traditional depth average methods).

Sample simulations of a laboratory test of granular flows.

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APPLICATION OF HP FINITE ELEMENTS TO THEACCURATE COMPUTATION OF POLARISATION TENSORS

FOR THE EDDY CURRENT PROBLEM

P.D. Ledger1 and W.R.B. Lionheart2

1College of Engineering, Swansea University, [email protected]

2School of Mathematics, The University of Manchester, [email protected]

Engineers interested in the detection of unexplored ordnance have long since postulatedthat a formula exists that describes the change in a low frequency magnetic field causedby the presence of a conducting object e.g. [1]. Furthermore, it has been conjecturedthat the formula contains a polarisation tensor, which describes the shape and materialproperties of the object. Until very recently, however, it has not been clear how thispolarisation tensor can be computed in practice for different objects nor how the changein magnetic field depends on frequency. The key to answering these questions lies in thedevelopment of asymptotic expansions of the perturbed magnetic field in appropriatevariables.The leading order term in asymptotic expansions of perturbed electromagnetic fieldsdue to the presence of dielectric objects as δ → 0 and r → ∞, where δ is the objectsize and r the distance from the object to point of observation, has been obtained byAmmari and Volkov[2]. These expansions are expressed in terms of polarisation tensorsand hold great potential for the solution of inverse problems where the task is to deter-mine the shape, location and dielectric object properties from far field measurements.Recently, we have obtained the leading order terms in expansions that describe theperturbed fields as max(δ/r, kδ)→ 0. Our results are also written in terms of polarisa-tion tensors and describe low frequency perturbed fields at distances large compared tothe object size, where k is the free space wave number. It is expected that they will beuseful for finding dielectric objects from low frequency near field measurements. In thecase of highly conducting objects, at low frequencies, the eddy current approximationof Maxwell’s equations is often made. An important length scale for such problems isthe skin depth and, for the case where this is of same order as the object’s size, Am-mari, Chen, Chen et al. [4] have obtained an expansion for the perturbed field magneticfield as δ → 0. This result is expressed in terms of a new conductivity polarisationtensor and a new permeability tensor. The computation of these new tensors requiresthe solution of a Maxwell transmission problem. These developments offer possibilitiesfor finding conducting objects from low frequency field measurements.In this talk, we shall review these different asymptotic expansions and present anapproach for the efficient and accurate computation of the conductivity and polarisationtensors for the eddy current problem using hp edge finite elements.

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References

[1] Y. Das, J.E. McFee, J. Toews and G.C. Stuart, IEEE T. Geosci Remote, 28,278-288, 1990.

[2] H. Ammari and D. Volkov, Int. J. Multiscale Com., 3, 149-160, 2005.

[3] P.D. Ledger and W.R.B. Lionheart, Submitted 2012.

[4] H. Ammari, J. Chen, Z. Chen, J. Garnier and D. Volkov, Submitted 2012.

RECENT ADVANCES IN FINITE ELEMENT SIMULATION OFELECTROMAGNETIC WAVE PROPAGATION IN METAMATERIALS

Jichun Li

Department of Mathematical Sciences, University of Nevada, Las Vegas, [email protected]

Since 2000, there is a growing interest in the study of metamaterials due to theirpotential applications in areas such as design of invisibility cloak and sub-wavelengthimaging, etc. In this talk, I’ll first give a brief introduction to the short history ofmetamaterials. Then I’ll focus on mathematical modeling of metamaterials, and discusssome finite element schemes we developed in recent years. Finally, I’ll conclude thetalk with some interesting simulation results such as backward wave propagation andcloaking simulation. Some open issues will be mentioned too.

References

[1] J. Li and Y. Huang, “Time-Domain Finite Element Methods for Maxwell’s Equa-tions in Metamaterials”, Springer Series in Computational Mathematics, vol.43,Springer, Jan. 2013, 302pp.

[2] L. Demkowicz and J. Li, Numerical simulations of cloaking problems using a DPGmethod, Computational Mechanics. DOI 10.1007/s00466-012-0744-4

[3] Y. Huang, J. Li and Q. Lin, Superconvergence analysis for time-dependentMaxwell’s equations in metamaterials, Numerical Methods Partial Differ. Equ. 28(2012) 1794-1816.

[4] J. Li, Finite element study of the Lorentz model in metamaterials, Computer Meth-ods in Applied Mechanics and Engineering 200 (2011) 626-637.

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HP -FEM FOR SINGULAR PERTURBATIONSWITH MULTIPLE SCALES

Jens Markus Melenk and Christos Xenophontos

Institut fur Analysis und Scientific Computing,Technische Universitat Wien, AustriaWiedner Hauptstr. 8-10, A-1040 Wien

[email protected]

Department of Mathematics and Statistics, University of CyprusPO Box 20537, 1678 Nicosia, Cyprus

We consider the approximation of solutions of systems of elliptic equations that aresingularly perturbed. Based on suitably designed meshes, we show that the hp-versionof the FEM can achieve robust exponential convergence, i.e., the convergence is uniformwith respect to the singular perturbation parameters. The natural energy norm inwhich this convergence takes place is, however, rather weak and does not “see” thelayers. We will report on recent progress for convergence in a stronger norm. This normfeatures a different scaling of the parameters and is such that the layers are uniformlyO(1) as the singular perturbation parameters tend to zero. Thus, convergence in thisnorm yields relevant information about the convergence within the layer.

HP ADAPTIVE FINITE ELEMENT METHODS BASED ONDERIVATIVES RECOVERY AND SUPERCONVERGENCE

Hieu Nguyen1 and Randolph E. Bank2

1 Department of Mathematics, Heriot-Watt University, Edinburgh, [email protected]

2 Department of Mathematics, University of California, San Diego, USA.

In this talk, we present a hp-adaptive finite element method based on derivative recov-ery and superconvergence. In our approach, high-order derivatives of the solution areestimated by superconvergent approximations. These approximations are then used toformulate a posteriori error estimates for guiding adaptivity. The decision whether it isbeneficial to refine a given element in h (geometry) or in p (degree) is made via verifyinga superconvergence result on that element. A special set of transition basis functions isalso introduced to guarantee continuity across elements of different degrees. Numericalresults shows that our method achieves exponential rate of convergence predicted bytheory.

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COMPARISON OF DIFFERENT FINITE ELEMENT MODELSAND METHODS FOR THE GIRKMANN SHELL-RING PROBLEM

Antti H. Niemia and Julien Petitb

Department of Civil and Structural Engineering,Aalto University, Espoo, Finland


Finite element analysis of thin shell structures involves various explicit and implicitmodelling assumptions that extend way beyond the limits of mathematical conver-gence theory currently available. However, thanks to modern computation technology,such as the hp-adaptive finite element method as pioneered by Leszek Demkowicz,computations can also be based directly to elasticity theory. This approach rules outthe modelling errors arising from the use of dimensionally reduced structural modelsbut requires in general more degrees of freedom.

In this talk, we will present computational results for the so called Girkmann bench-mark problem involving a spherical shell stiffened by a foot ring. In particular, we willcompare the accuracy of finite element models based on elasticity theory to modelsbased on dimensionally reduced structural models.

PYRAMIDAL FINITE ELEMENTS

Nilima Nigama, Argyrios Petrasb and Joel Phillipsc

Department of Mathematics, Simon Fraser University, Burnaby, [email protected], [email protected], [email protected]

In this talk we present a construction of a family of high-order conforming finite ele-ments for pyramidal elements. This family satisfies the commuting diagram property,ensuring stability of mixed finite element discretizations. The analysis of errors due toquadrature is non-standard for these elements, and we describe the key ideas. Finally,we present some new developments towards a family of serendipity elements on thepyramid.

This work was motivated in large part by discussions with Prof. Leszek Demkowicz,whose many contributions to the study of FEM we would like to honour.

17-20


APPLICATION OF THE ADAPTIVE FINITE ELEMENT METHODTO NUMERICAL SIMULATIONS OF ARTERIES

Waldemar Rachowicz1 and Adam Zdunek2

1Institute of Computer Science, Cracow University of Technologyul. Warszawska 24, 31-155 Cracow, Poland

[email protected]

2Swedish Defence Research Agency, Stockholm, [email protected]

Computational biomechanics is a rapidly developing branch of computer simulationsbased on the Finite Element Method approximation. An area of special interest ismechanics of soft tissues, especially arteries with aneurysms, or with changes due toatherosclerosis. Both imply a high risk for rupture when pressurized. A commonlyaccepted approach to biomechanical simulation is to use low order very dense finiteelement (FE) meshes to discretize the biological objects of interest. Such an approach ismotivated by a necessity to accurately approximate a complex patient specific geometryof a piece of artery.

It is well known that one can use relatively coarse FE meshes and adapt themusing h-refinements (changing locally the size h of elements) and/or p-refinements(changing locally the order of approximation p). The mesh adaptations are intended toreduce the estimated discretisation error below a prescribed threshold at the minimumcomputational effort. The question arises if the techniques of error estimation andmesh adaptivity could be of any use for computational biomechanics.

In this paper we apply the machinery of error estimation and adaptivity for Fi-nite Element simulations of artery problems. Appropriate algorithms exist to someextent. Some adjustments are however needed. We developed the two-field formula-tion of displacement-pressure type [1] with no inter-element continuity enforced for thepressure variable.

We developed an error estimation technique which is applicable to finite elasticityproblems. It is a version of the residual method proposed by Bank and Weiser in thecontext of linear elliptic boundary-value problems. The distribution of error indicatorsmay be used to guide adaptivity of finite element meshes. We applied adaptive FEMto solve typical problems of mechanics of arteries.

We intend to use the adaptive version of the FEM to simulate the interactionbetween a medical device, namely a stent, and the artery. The stent is inserted into theartery and it is adequately deformed to widen the cross-section which has been narroweddue to the atherosclerosis. A precise control of this treatment, called angioplasty, isnecessary. This simulation involves a contact problem with large sliding and with finitedeformation of the bodies in contact.

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References

[1] Simo, J. C. and Taylor, R. L., Quasi-incompressible finite elasticity in principalstretches, Comput. Methods in Appl. Mech. Engng, 85, pp. 273−310, 1991.

DISCONTINUOUS PETROV-GALERKIN METHODS FORINCOMPRESSIBLE FLOW: STOKES AND NAVIER-STOKES

Nathan V. Robertsa, Leszek Demkowiczb and Robert Moserc

Institute of Computational Engineering and Sciences,University of Texas, Austin, TX, USA.


The discontinuous Petrov-Galerkin (DPG) finite element methodology first proposed in2009 by Demkowicz and Gopalakrishnan [1, 2]—and subsequently developed by manyothers—offers a fundamental framework for developing robust residual-minimizing fi-nite element methods, even for equations that usually cause problems for standardmethods, such as convection-dominated diffusion and the Stokes equations. For a verybroad class of well-posed problems, DPG offers provably optimal convergence rates witha modest convergence constant. In some of our experiments, DPG not only achievesthe optimal rates, but gets extremely close to the best approximation available in thediscrete space. Moreover, DPG provides a way to measure—not merely estimate—theerror in the approximate solution, which can then robustly drive adaptivity.

In this presentation, we apply DPG to Stokes and Navier-Stokes, discussing theo-retical convergence estimates [3] and illustrating these through numerical experimentsperformed with Camellia [4], a robust, flexible software framework for DPG researchand experimentation, built atop Trilinos.

References

[1] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous Petrov-Galerkinmethods. Part I: The transport equation. Computer Methods in Applied Mechanicsand Engineering, 199(23-24):1558 – 1572, 2010.

[2] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous Petrov–Galerkinmethods. Part II: Optimal test functions. Numerical Methods for Partial Differen-tial Equations, 27(1):70–105, 2011.

[3] Nathan V. Roberts, Tan Bui-Thanh, and Leszek F. Demkowicz. The DPG methodfor the Stokes problem. Technical Report 12-22, ICES, 2012.

[4] Nathan V. Roberts, Denis Ridzal, Pavel B. Bochev, and Leszek F. Demkowicz.A Toolbox for a Class of Discontinuous Petrov-Galerkin Methods Using Trilinos.Technical Report SAND2011-6678, Sandia National Laboratories, 2011.

17-22


PRECONDITIONING FOR HIGH ORDER HYBRID DG METHODS

Joachim Schoberl1 and Christoph Lehrenfeld2

1Institute for Analysis and Scientific Computing, Vienna UT, [email protected]

2 Institut fur Geometrie und Praktische Mathematik, RWTH Aachen, [email protected]

Discontinuous Galerkin methods provide a lot of freedom for the design of finite elementmethods, such as upwinding for convection dominated problems, or the choice of stablemixed finite elements. Hybridization of DG methods allows an efficient implementationin terms of matrix entries, and static condensation.

In this talk we focus on the analysis of preconditioners for the hybrid-DG methodfor elliptic problems. We prove a poly-logarithmic condition number in h and p for 2Das well as 3D. Techniques are p-version extension operators from vertices, edges, andfaces with respect to the norm induced by the stabilization term. We show that theBassi-Rebay method differs essentially from the interior penalty method, in 3D.

Numerical results for the model problem as well as more complex problems confirmthe theoretical results.

[J. Schoberl and C. Lehrenfeld: Domain Decomposition Preconditioning for HighOrder Hybrid Discontinuous Galerkin Methods on Tetrahedral Meshes, in AdvancedFinite Element Methods and Applications, Lecture Notes in Applied and Computa-tional Mechanics 66, Springer 2012, pages 27-56]

17-23


APPLICATION OF THE FULLY AUTOMATICHP -FEM TO ELASTIC-PLASTIC PROBLEMS

Marta Serafina and Witold Cecotb

Cracow University of Technology, Krakow, [email protected], [email protected]

The fully automatic (self automatic) hp-adaptive mesh refinement strategy [1] wasapplied to analysis of elastic-plastic problems. Since the solution is less regular atthe elastic-plastic interface, the finite element meshes should comply with elastic andplastic zones. However, the elastic and plastic zones are not known a-priori thus,appropriate adaptive mesh refinements are the way to construct meshes that, at least,approximately correspond to the shapes of the zones.

Generally, the self adaptive mesh refinement technique generates the aforemen-tioned meshes and, for the considered physically nonlinear problems, delivers the fastestconvergence of the error. We tested additional h-refinements and p-enrichment alongthe elastic-plastic interface, as well as exclusively h-adaptive or p-adaptive mesh refine-ments. Only in the case of cylinder additional h-refinements resulted in a speed up ofthe convergence. Presumably the reason for significant improvement only in this casewas the shape of elastic-plastic interface, which could be relatively easily captured inthe cylinder problem. In the other examples even anisotropic additional h-refinementsdid not result in meshes that exactly complied with elastic-plastic zones. Therefore, theoriginal hp-FEM delivered the fastest algebraic convergence for elastic-plastic problemsresulting in meshes with up to 8th order of approximation. In the future the fully au-tomatic hp mesh refinements will be supplemented with the r-adaptation, since it wassuccessfully used for p-refinements [2]. Also the algorithm of searching for the optimalnew hp meshes should undergo further testing. Currently, for the sake of efficiency, onlycertain selected from all possible refinements are considered. Such a strategy workscorrectly for linear problems but its validation for elastoplasticity is intended.

References

[1] L. Demkowicz, W. Rachowicz, and Ph. Devloo. A fully automatic hp-adaptivity.Journal of Scientific Computing, 17:127–155, 2002.

[2] V. Nubel and A. Duster and E. Rank An rp-adaptive finite element method for thedeformation theory of plasticity. Comput. Mech., 39:557–574, 2007.

17-24


A NEW ERROR ANALYSIS FOR CRANK-NICOLSONGALERKIN FEMS FOR A GENERALIZEDNONLINEAR SCHRODINGER EQUATION

Jilu Wang

Department of Mathematics, City University of Hong Kong, Hong [email protected]

This talk is concerned with unconditionally optimal L2 error estimates of linearizedCrank-Nicolson Galerkin FEMs for a generalized nonlinear Schrodinger equation, whilecertain conditions on time stepsize were always needed in previous works. A key to ouranalysis is an error splitting, in terms of the corresponding time-discrete system, withwhich the error is split into two parts, temporal error and spatial error. We can provethat the spatial error is τ -independent. By the inverse inequality, the boundednessof numerical solution in L∞-norm follows immediately. Then, the optimal L2 errorestimates are obtained by a routine method. Numerical results in both two and threedimensional spaces are given to confirm our theoretical analysis.

B-SPLINE FEM APPROXIMATION OF WAVE EQUATION

Hongrui Wanga and Mark Ainsworth

Department of Mathematics and Statistics, University of Strathclyde, [email protected]

The use of high order splines for the approximation of PDEs has recently attracteda lot of attention due to the work of Hughes [1]. However, it is as yet unclear howthese methods perform as the order of approximation is increased on a fixed mesh.We analyse this problem in the setting of the wave equation. We also show how themethods can be implemented efficiently and analyse its stability.

References

[1] John A. Evans, Yuri Bazilevs, Ivo Babuska, and Thomas J.R. Hughes. n-widths,sup-infs, and optimality ratios for the k-version of the isogeometric finite elementmethod. Computer Methods in Applied Mechanics and Engineering, 198:1726 –1741, 2009.

17-25


THE LOW-STORAGE CURVILINEAR DISCONTINUOUSGALERKIN METHOD

T. Warburton

Department of Computational and Applied Mathematics,Rice University, Houston, TX, USA.

[email protected]

Accurate modeling of wave scattering by curvilinear geometries can be achieved withcurvilinear finite element methods. In this talk we will describe a memory efficientdiscontinuous Galerkin method for scattering from curvilinear objects [1, 2, 3]. Wewill present an a priori convergence analysis and compare method specific sufficientconditions guaranteeing convergence for both the low-storage and traditional variants ofthe discontinuous Galerkin method. We will present computational results that confirmthe analysis and also demonstrate the potential for high performance computationsaccelerated by graphics processing units.

References

[1] T. Warburton. A low storage curvilinear discontinuous galerkin time-domainmethod for electromagnetics. In Electromagnetic Theory (EMTS), 2010 URSI In-ternational Symposium on, pages 996–999. IEEE, 2010.

[2] T. Warburton. Aspects of the a priori convergence analysis for the low storage curvi-linear discontinuous Galerkin method. In Theory and Applications of DiscontinuousGalerkin Methods, pages 44–46. Mathematisches Forschungsinstitut Oberwolfach,2012.

[3] T. Warburton. Analysis of the low-storage curvilinear discontinuous galerkinmethod for wave problems. SIAM Journal on Scientific Computing, Submitted.

17-26


A NOVEL FORMULATION FOR NEARLY INEXTENSIBLE ANDNEARLY INCOMPRESSIBLE FINITE HYPERELASTICITY

Adam Zdunek1, Waldemar Rachowicz2 and T. Eriksson3

1Swedish Defence Research Agency FOI, SE-164 90 Stockholm, Sweden,[email protected]

2Cracow University of Technology, Pl 31-155 Cracow, Poland,[email protected]

3School of Mathematics and Statistics, University of Glasgow, Glasgow, [email protected]

We present a novel formulation of the Hu-Washizu type suited for the modelling ofnearly incompressible materials soft in shear and reinforced by rapidly stiffening ornearly inextensible fibres or cords. Its strong form for a Spencer type [1] fibre-reinforcedfinite hyperelastic material is derived and illustrated with a couple of simple examples.For a single family of fibres it is a 5-field formulation in terms of displacement, an auxil-iary volume ratio, an auxiliary fibre stretch, and the corresponding Lagrange multipliersrepresenting mean pressure and deviatoric fibre stress, respectively. The formulationgeneralises the Simo-Taylor three-field formulation for near incompressibility [2], oftenused today in the field of soft tissue biomechanics [3, Sect. 4]. The right Cauchy-Greenstretch tensor in the novel formulation is unimodular and in addition it is stretch freein a fibre direction.

Furher, the novel formulation corrects a flaw in the class of very popular so-calledstandard reinforcing models [4], [3, Eqs. (33)-(36)]. The introduced uni-modular andstretch free right Cauchy-Green tensor corrects the modelling error, i.e. it removesthe erroneous strain energy contribution in the fibre direction from the ground sub-stance deformation. Simple analytical examples illustrate the modelling error causedusing the standard reinforcing model. The setting for the limiting incompressible andinextensible cases and for the fully coupled unconstrained case are also covered.

References

[1] A. J. M. Spencer, Continuum theory of the mechanics of fibre-reinforced compos-ites, Springer, New York, (1984).

[2] J. C. Simo , R. L. Taylor , K. S. Pister , Variational and projection methods forthe volume constraint in finite deformation elasto-plasticity., Computer Methodsin Applied Mechanics and Engineering 51 (1985) 177–208.

[3] G.A. Holzapfel Structural and numerical models for the (visco)elastic response ofarterial walls with residual stresses. In: G.A. Holzapfel and R.W. Ogden (eds.),”Biomechanics of Soft Tissue in Cardiovascular Systems”, CISM Courses and Lec-tures No. 441, International Centre for Mechanical Sciences, Springer: Wien, NewYork (2003), 109–184.

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[4] D. A. Poligone, C. O. Horgan, Cavitation for incompressibile anisotropic nolinearlyelastic spheres. J. Elasticity, 33 (1993) 27–65.

17-28

18: Mini-Symposium: Innovative compatible and mimetic discretizations for partial differential equa-tions

18 Mini-Symposium: Innovative compatible and mimetic

discretizations for partial differential equations

Organisers: Andrea Cangiani and GianmarcoManzini

18-1


BASIC PRINCIPLES OF VIRTUAL ELEMENT METHODS

L. Beirao da Veiga1, Franco Brezzi2, Andrea Cangiani3,Gianmarco Manzini4, L.D. Marini5 and Alessandro Russo6

1 Dipartimento di Matematica, Universita di Milano Statale, [email protected]

2 Department of Mathematics, University of Leicester, [email protected]

3 IUSS-Pavia and IMATI-CNR, Pavia, Italy and KAU, Jeddah, Saudi [email protected]

4 Group T-5, MS-B284 Los Alamos National Laboratory Los Alamos, NM 87545,USA

[email protected]

5 Dipartimento di Matematica, Universita di Pavia and IMATI-CNR, Pavia, [email protected]

6 Department of Mathematics and Applications, University of Milano-Bicocca, [email protected]

The Virtual Element Method (VEM) provides a framework for the extension of classicalfinite element methods to general meshes.

A virtual element space is a generalised finite element space consisting of bothpolynomials and non polynomials functions. The virtual space and degrees of freedomare carefully chosen so that the assembly of the discrete formulation can be basedsolely on the degrees of freedom, as in, say, (mimetic) finite difference and finite volumeapproaches, while the finite element setting permits us to analyse the method in thetypical finite element fashion.

This setting can easily deal with complicated element geometries and/or higher-order continuity conditions (like C1, C2, etc.).

18-2


MIMETIC DISCRETIZATIONS OF ELLIPTIC PROBLEMS

Gianmarco Manzini

Group T-5, MS-B284, Los Alamos National Laboratory,Los Alamos, NM 87545, USA

[email protected]

We present the family of mimetic finite difference method and discuss how these ap-proach can be used to design efficient schemes with arbitrary order of accuracy andarbitrary regularity to approximate elliptic problems. The diffusion tensor may beheterogeneous, full and anisotropic. These numerical techniques can be applied tocomputational meshes of polygonal or polyhedral cells with very general shape, alsonon-conforming as the ones of the Adaptive Mesh Refinement (AMR) method andnon-convex.

18-3


A VIRTUAL ELEMENT METHOD WITH HIGH REGULARITY

L. Beirao da Veigaa and Gianmarco Manzini

a Dipartimento di Matematica – Universita di Milano, [email protected]

The Virtual Element Method (VEM, introduced in [1]) is a generalization of the FiniteElement method that, by avoiding an explicit construction of discrete shape functions,achieves a higher degree of flexibility in terms of meshes and properties of the scheme.In order to neglect the explicit construction of basis functions, the method makes useof an approximated bilinear form ah(·, ·) that mimics the original bilinear form a(·, ·).For well-posedness and convergence to be guaranteed, the discrete bilinear form mustsatisfy precise conditions of stability and consistency. Nevertheless such conditionsleave a lot of additional freedom in the construction of the method.

In the present talk this feature allows us to design a family of numerical methods[2] that are associated with discrete spaces with arbitrary Cα regularity, polynomialdegree m ≥ α + 1 and are suitable to general unstructured polygonal meshes. Theparameter α determines the global smoothness of the underlying discrete space, whilethe parameter m determines the convergence rate for regular solutions.

After introducing and describing the method we will show both a-priori [2] anda-posteriori [3] error estimates for the scheme. In particular, the a-posteriori errorestimator is composed of various terms that are associated to the various sources oferror in the VEM discretization.

We finally show a selection of convergence tests, both for uniform and adaptivelygenerated mesh families.

References

[1] L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, A. Russo. Ba-sic principles of Virtual Element Methods. Math. Mod. Meth. Appl. Sci., 23(1):199–214, 2013.

[2] L. Beirao da Veiga, G. Manzini. A Virtual Element Method with arbitrary regu-larity. LANL technical report and in press on IMA J. Numer. Anal..

[3] L.Beirao da Veiga, G. Manzini. Residual a-posteriori error estimation of a VirtualElement Method. Submitted for publication.

18-4


THE VIRTUAL ELEMENT METHOD FOR GENERALSECOND-ORDER ELLIPTIC OPERATORS ONPOLYGONAL AND POLYHEDRAL MESHES

Franco Brezzi1, L. Donatella Marini2 and Alessandro Russo3

1IUSS-Pavia and IMATI-CNR, Pavia, Italyand KAU, Jeddah, Saudi Arabia

[email protected]

2Department of Mathematics, University of Pavia, Italyand IMATI-CNR, Pavia, [email protected]

3Department of Mathematics and Applications, University of Milano-Bicocca, Italyand IMATI-CNR, Pavia, [email protected]

General polygonal and polyhedral meshes naturally arise in the treatment of complexsolution domains and heterogeneous materials (e.g. reservoir models) and are particu-larly suited to moving meshes techniques as well as to adaptive mesh refinement andde-refinement.

In this talk we will present the Virtual Element Method and show how it can beused to approximate the solution of an elliptic second-order operator on a polygonalor polyhedral mesh with high-order accuracy.

18-5


NONSMOOTH INITIAL DATA ERROR ESTIMATES FORTHE FINITE VOLUME ELEMENT METHOD

FOR A PARABOLIC PROBLEM

Panagiotis Chatzipantelidis

Department of Mathematics, University of Crete, Heraklion, GR–71003, [email protected]

In this talk we present the results of [1], where we study spatially semidiscrete andfully discrete finite volume element methods for the homogeneous heat equation withhomogeneous Dirichlet boundary conditions and derive error estimates for smooth andnonsmooth initial data. We show that for smooth initial data, we obtain optimal resultsof second order as in the finite element method for piecewise linear functions. However,for initial data only in L2, a special condition is required, which is satisfied for sym-metric triangulations. Without such a condition, only first order convergence can beshown, which is illustrated by a counterexample. Improvements hold for less restrictivemeshes triangulations that are almost symmetric or piecewise almost symmetric.

References

[1] P. Chatzipantelidis, R. D. Lazarov, and V. Thomee, Some error estimates for thefinite volume element method for a parabolic problem, Computational Methods inApplied Mathematics, ( published online), DOI: [2]10.1515/cmam-2012-0006, Jan-uary 2013.

18-6


CONVECTION DOMINATED DISCONTINUOUSGALERKIN MULTISCALE METHOD

Daniel Elfversona and Axel Malqvistb

Division of Scientific Computing, Uppsala University, [email protected], [email protected]

In this work we study the solution of the convection dominated convection-diffusion-reaction problems, with heterogeneous and highly varying coefficients without anyassumptions on scale separation or periodicity. Problems o this type arise in manybranches of scientific computing. There are two reasons why standard continuousfinite element methods perform poorly for this kind of problems. That is, both thecoefficients describing the problem as well as boundary layers in the solution needsto be resolved. To this end, we propose a new Convection dominated discontinuousGalerkin multiscale method, based on a corrected basis calculated on localized patches,that takes the variations the fine variations into account without resolving it globallyon a single mesh. Let umsH be the solution obtained by the multiscale method where His the mesh size. Then the following result holds

|||u− umsH ||| ≤ |||u− uh|||+ CH,

under moderate assumptions on the magnitude of the convection and that the sizethe patches where the corrected basis is computed are chosen as O(H log(H−1)). Theconstant C is independent of the variation in the coefficients nor the mesh size, and uhis the (one scale) discontinuous Galerkin solution on the fine scale. This result holdsindependent of the regularity of the solution u.

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MHM METHOD FOR ADVECTIVE-REACTIVEDOMINATED MODELS

Christopher Hardera, Diego Paredesb and Frederic Valentinc

Laboratorio Nacional de Computacao Cientıfica - LNCC,Av. Getulio Vargas, 333 - 25651-075 Petropolis - RJ, Brazil


This work proposes a new family of Multiscale Hybrid-Mixed (MHM) finite elementmethods for advective-reactive dominated problems on coarse meshes. The MHMmethod is a consequence of a hybridization procedure. It results in a method that nat-urally incorporates multiple scales while providing solutions with high-order precisionfor the primal and dual (or flux) variables. The local problems are embedded in theupscaling procedure and are completely independent, meaning they can be naturallyobtained using parallel computation facilities. Also, the MHM method preserves localconservation properties from a simple post-processing of the primal variable.

The analysis results in a priori estimates showing optimal convergence in naturalnorms and provides a face-based a posteriori estimator. Regarding the latter, we provethat reliability and efficiency hold. Also, we introduce a new space adaptive strategywhich avoids any topological changes on the mesh. Numerical results verify theoreticalresults as well as a capacity to accurately incorporate heterogeneous and high-contrastcoefficients, and to approximate boundary layers. In particular, we show the greatperformance of the new a posteriori error estimator in driving space adaptivity.

We conclude that the MHM method, along with its associated a posteriori estima-tor, is naturally shaped to be used in parallel computing environments and appearsto be a highly competitive option to handle realistic multiscale singular perturbedboundary value problems with precision on coarse meshes.

References

[1] C. Harder, D. Paredes and F. Valentin A Family of Multiscale Hybrid-Mixed FiniteElement Methods for the Darcy Equation with Rough Coefficients. LNCC ResearchReport No. 02/2013. To appear in Journal of Computational Physics.

[2] R. Araya, C. Harder, D. Paredes and F. Valentin Multiscale Hybrid-Mixed Method.LNCC Research Report No. 03/2013.

[3] C. Harder, A.L. Madureira and F. Valentin New Finite Elements for Elasticity inTwo and Three-Dimensions. LNCC Research Report No. 04/2013.

18-8


TREFFTZ-DG METHODS FOR WAVE PROPAGATION:HP -VERSION AND EXPONENTIAL CONVERGENCE

Andrea Moiola1, Ralf Hiptmair2a, Ilaria Perugia3 and Christoph Schwab2b

1Department of Mathematics and Statistics, University of Reading, [email protected]

2Seminar for Applied Mathematics, ETH Zurich, [email protected], [email protected]

3Department of Mathematics, University of Pavia, [email protected]

The phenomena involving propagation and interaction of acoustic, electromagnetic andelastic linear waves in time-harmonic regime are often discretised using finite elementmethods. However, as soon as the wavelength becomes small compared to the diameterof the domain, simulations become excessively expensive. This is due to the highly os-cillatory structure of the solutions in the high frequency regime and to the accumulationof phase error, called numerical dispersion, which affects any local discretisation.

To cope with these fundamental difficulties, several recent methods incorporate in-formation about the equations in the design of the trial space. This can be achievedby using Trefftz methods, i.e. choosing test and trial functions that are piecewise solu-tions of the underlying PDE. Typical choices are plane, circular, spherical and angularwaves. Prominent examples of such methods are the ultra weak variational formulation(UWVF) of Cessenat and Despres; the discontinuous enrichment method (DEM/DGM)of Farhat and co-workers; the variational theory of complex rays (VTCR) of Ladeveze;and the wave based method (WBM) of Desmet.

We focus on a family of Trefftz-discontinuous Galerkin (TDG) schemes, which in-cludes the UWVF as a special case. In the case of the Helmholtz and Maxwell’sequations, a complete theory for the (a priori) convergence in h and p has been devel-oped, see e.g. [A. Moiola, Trefftz-discontinuous Galerkin methods for time-harmonicwave problems, PhD thesis, ETH Zurich, 2011] and the references therein. This anal-ysis is made possible by the special DG framework used, which ensures unconditionalstability and quasi-optimality (i.e., control of the error for any value of the wavenum-ber and the meshsize), and by the use of new approximation estimates for plane andspherical waves, which ensure high-order convergence. A special choice of the numeri-cal flux parameters allows to prove a priori error estimates in L2-norm for meshes thatare locally refined, for example near the corners of a scatterer [R. Hiptmair, A. Moiola,I. Perugia, Trefftz discontinuous Galerkin methods for acoustic scattering on locallyrefined meshes, Appl. Numer. Math., 2013]. However, the exponential convergence inthe number of DOFs of a full hp-version of the TDG is a highly desirable result.

To achieve this, new fully explicit error bounds for the approximation of har-monic functions by harmonic polynomials in star-shaped elements have been proved in[R. Hiptmair, A. Moiola, I. Perugia, Ch. Schwab, Approximation by harmonic polyno-mials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM, SAMreport 2012-38, ETH Zurich] relying on complex variable techniques and following theargument of M. Melenk. These bounds were used to prove exp(−b

√#dofs) orders of

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convergence for an hp-TDG method based on harmonic polynomials for Laplace BVPs(as opposed to exp(−b 3

√#dofs) of standard schemes). The extension, using Vekua’s

theory, to the Helmholtz case and plane/circular wave spaces is currently under way.

THE DISCRETE MAXIMUM PRINCIPLE IN THE FAMILY OFMIMETIC FINITE DIFFERENCE DISCRETIZATIONS

Daniil Svyatskiya, Konstantin Lipnikovb and Gianmarco Manzinic

Applied Mathematics and Plasma Physics, Theoretical Division,Los Alamos National Laboratory, Los Alamos, NM 87545, USA.

[email protected] [email protected]@lanl.gov

The Maximum Principle is one of the most important properties of solutions of ellipticpartial differential equations. Its numerical analog, the Discrete Maximum Principle(DMP), is one of the properties that are very difficult to incorporate into numericalmethods, especially when the computational mesh contains distorted or degenerate cellsor the problem coefficients are heterogeneous and anisotropic. To mimic this propertyin numerical simulations is very desirable in wide range of applications. Violation ofthe DMP leads to non-physical solutions with numerical artifacts, such as a heat flowfrom a cold material to a hot one. These oscillations can be significantly amplifiedby non-linearity of physics. Unfortunately, numerical schemes satisfying the DMPimpose severe limitations on mesh geometry and problem coefficients and often are notapplicable beyond simplicial meshes.

The family of the Mimetic Finite Difference (MFD) methods provides flexibility inthe choice of parameters which define a particular member of the family. For example,the MFD discretization scheme for quadrilateral meshes depends on three parameters.It is pertinent to note that these parameters are chosen locally and depend on geometryand material properties in a particular cells. The correct choice of these parametersmay guarantee that the resulting numerical scheme satisfy the DMP principle. Theanalysis of this strategy is based on the properties of M-matrices. The monotonicitylimits of MFD method are investigated in several practically important cases includingmeshes generated using the Adaptive Mesh Refinement (AMR) strategy.

18-10


A TWO-LEVEL METHOD FOR MIMETIC FINITE DIFFERENCEDISCRETIZATIONS OF ELLIPTIC PROBLEMS

Marco Verani

MOX-Department of Mathematics, Politecnico di Milano, Milano, [email protected]

Nowadays, the mimetic finite difference (MFD) method has become a very popularnumerical approach to successfully solve a wide range of problems. This is undoubtedlyconnected to its great flexibility in dealing with very general polygonal meshes and itscapability of preserving the fundamental properties of the underlying physical andmathematical models.

In this talk, we present and analyze a two level method for mimetic finite differencediscretizations of elliptic problems. The method, which is based on the introduction ofsuitable prolongation and restriction operators acting on polygonal meshes, is provedto be convergent. Several numerical experiments assess the efficacy of the proposedmethod and confirm the results of the theoretical analysis.

This is a joint work with Paola F. Antonietti (Politecnico di Milano) and LudmilZikatanov (Penn State University).

18-11

19: Mini-Symposium: Integrodifferential Relations in Direct and Inverse Problems of MathematicalPhysics

19 Mini-Symposium: Integrodifferential Relations

in Direct and Inverse Problems of Mathematical

Physics

Organisers: Georgy Kostin and Vasily Saurin

19-1


NORM-OPTIMAL ITERATIVE LEARNING CONTROL FOR AHEATING ROD BASED ON THE METHODOF INTEGRO-DIFFERENTIAL RELATIONS

Harald Aschemanna, Dominik Schindeleb and Andreas Rauhc

Chair of Mechatronics, University of Rostock,Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany.


In this contribution, a multi-variable norm-optimal iterative learning control (NOILC)is designed for a spatially one-dimensional heating rod. This application representsa distributed parameter system with two Peltier elements as control inputs. For sys-tem modelling, the Method of Integro-Differential Relations (MIDR) is applied withBernstein polynomials as ansatz functions [1]. It leads to an approximation in formof a linear state-space representation that fulfils the corresponding (energy) conser-vation law exactly. Moreover, the resulting approximation quality of the distributedparameter system can be assessed by evaluating an error measure for the constitutiverelations. Considering the heat transfer in the heating rod, the constitute relation isgiven by Fourier’s law for heat conduction in longitudinal direction, whereas the firstlaw of thermodynamics represents the conservation law.

The repetitive control task consists in tracking trajectories – which are repeatedperiodically – for the desired temperature at two selected points of the heating rod byactuating two Peltier elements as control inputs. The norm-optimal iterative learningcontrol law, cf. [2], makes use of both the tracking error of the previous iteration andtemperature measurements of the current iteration to reduce the tracking error fromiteration to iteration. Applying the MIDR approach, the system model results in aninth-order state-space model that is discretized w.r.t. time using the explicit Eulermethod.

Simulation results point out the benefits of the iterative learning control approach.The chosen NOILC design involves a combination of feedforward and feedback controland guarantees a fast convergence of the tracking errors despite model uncertaintiesand unknown disturbances due to convective heat losses.

References

[1] Rauh, Andreas; Senkel, Luise; Aschemann, Harald; Kostin, Georgy V.; Saurin,Vasily V.: Reliable Finite-Dimensional Control Procedures for Distributed Pa-rameter Systems with Guaranteed Approximation Quality, Proc. of IEEE Multi-Conference on Systems and Control, Dubrovnik, Croatia, 2012.

[2] Schindele, Dominik; Aschemann, Harald: Norm-Optimal Iterative Learning Controlfor a High-Speed Linear Axis with Pneumatic Muscles, Proc. of 8th IFAC Sympo-sium on Nonlinear Control Systems (NOLCOS), Bologna, Italy, pp. 463-468, 2010.

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VARIATIONAL FORMULATIONS OF INVERSEDYNAMICAL PROBLEMS IN LINEAR ELASTICITY

Georgy Kostina and Vasily Saurinb

Institute for Problems in Mechanics of the Russian Academy of Sciences,Moscow, Russia.


This contribution presents the method of integro-differential relations (MIDR) forinitial-boundary value problems in the linear theory of elasticity. The main idea ofMIDR is that the constitutive laws (stress-strain and momentum-velocity relations) arespecified by an integral equalities instead of their local forms [1]. The modified prob-lem is reduced to minimization of a nonnegative energy error functional over admissiblemomentum, displacement, and stress fields under equilibrium, kinematic, initial, andboundary constraints.

Based on the MIDR a parametric family of quadratic constitutive functionals areintroduced and corresponding variational formulations are presented. It is shown thatthe conventional dual Hamilton principles result from one of the variational statementsintroduced.

Numerical algorithms for direct and inverse dynamical problems are developedbased on the Ritz method and finite element technique with spline approximationsof the unknown functions in the space-time domain. The energy error functional isused to design integral criteria of the solution quality relying on the extremal proper-ties of finite-dimensional variational problems.

The efficiency of the estimates proposed are demonstrated on the example of con-trolled motions for an elastic body. The problem of its displacement from the initialdeformed state to the terminal one with the minimal total mechanical energy is consid-ered. The piecewise polynomial control of longitudinal displacements of a rectilinearthin rod is investigated. After FEM discretization, the original optimization problem isreduced to successive solving of two linear algebraic systems. The obtained numericalresults are analyzed and discussed.

References

[1] Kostin G.V., Saurin V.V. Integrodifferential relations in linear elasticity. DeGruyter Studies in Mathematical Physics 10. De Gruyter, Berlin, 2012.

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DESIGN AND EXPERIMENTAL VALIDATION OF CONTROLSTRATEGIES FOR A SPATIALLY TWO-DIMENSIONALHEAT TRANSFER PROCESS BASED ON THE METHOD

OF INTEGRO-DIFFERENTIAL RELATIONS

Andreas Rauha, Luise Senkelb and Harald Aschemannc

Chair of Mechatronics, University of Rostock,Justus-von-Liebig-Weg 6, D-18059 Rostock, Germany.


In previous work, different procedures have been developed for closed-loop control, ob-server design and optimization of distributed parameter systems on the basis of theMethod of Integro-Differential Relations (MIDR) [1,2]. Compared to other modelingapproaches for distributed parameter systems, the MIDR is characterized by its advan-tageous property of being able to quantify the approximation quality of a distributedparameter system, typically given by a set of partial differential equations, in termsof an error measure for the constitutive relations. In addition, the state variable ofthe distributed system is approximated in such a way that the corresponding (energy)conservation law is fulfilled exactly.

In the case of heat transfer processes, the constitute relations are given by Fourier’slaw in each relevant space direction. Moreover, the conservation law corresponds tothe first law of thermodynamics. Here, both the heat flux density and the temperaturedistribution have to be approximated in a suitable form such that the above-mentionedrequirements are fulfilled.

In this contribution, the modeling of a spatially two-dimensional heat transfer pro-cess is described for a test rig that is available at the Chair of Mechatronics at theUniversity of Rostock. The system consists of an aluminum plate that can be heatedand cooled from one side by an array of equally large Peltier elements. This array ofdistributed inputs, arranged in a chess board like structure, can be used for the imple-mentation of multi-input multi-output controllers and for the realization of distributeddisturbance inputs.

Simulations and experimental results highlight the main properties of the MIDRwhich exploits Bernstein polynomials for the parameterization of both the temperaturedistribution and the heat flux density with an efficient formulation of inter-elementconditions in a novel finite element scheme [2]. This presentation is concluded witha comparison of modeling, control and observer design that are based on either theMIDR or on a finite volume discretization of the two-dimensional heating system.

References

[1] Rauh, Andreas; Senkel, Luise; Aschemann, Harald; Kostin, Georgy V.; Saurin,Vasily V.: Reliable Finite-Dimensional Control Procedures for Distributed Pa-rameter Systems with Guaranteed Approximation Quality, Proc. of IEEE Multi-Conference on Systems and Control, Dubrovnik, Croatia, 2012.

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[2] Rauh, Andreas; Dittrich, Christina, Aschemann, Harald: The Method of Integro-Differential Relations for Control of Spatially Two-Dimensional Heat Transfer Pro-cesses, European Control Conference ECC’13, Zurich, Switzerland, 2013. Accepted.

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INTEGRO-DIFFERENTIAL RELATIONS INLINEAR ELASTICITY: STATIC CASE

Vasily Saurina and Georgy Kostinb

Institute for Problems in Mechanics of the Russian Academy of Sciences,Moscow, Russia.


Systems with distributed parameters are typically described by partial differential equa-tions (PDEs) and, in some cases, by integral or integro-differential relations. Thesemodels may also involve functionals of unknown variables. On an admissible set offunctions, such a functional attains its stationary value, i.e. desired solution of theproblem. This is usually associated with the problem statement based on a variationalprinciple.

One common characteristic feature inherent in variational methods is some ambigu-ity in the formulation of a finite approximation problem. It is not clear what relationsare best to be weakened. As an example, the equations of linear elasticity are consid-ered. In the original statement, there are 15 variables, namely, 12 components of thestress and strain tensors as well as three components of the displacement vector, whichcorrespond to 9 PDEs (equilibrium and kinematic equations) and 6 algebraic consti-tutive relations (Hooke’s law). If all the relations including the boundary conditionsare taken in integral (weak) form then this statement coincides with the Hu-Washizuprinciple, which contains 18 variables (three Lagrange multipliers are added) and noconstraints are imposed on them. By requiring the implementation of certain governingequations, the number of independent variables in the variational formulation can bereduced. For instance, it is possible to derive the Hellinger-Reissner principle in whichthere are 12 unknown functions. After successive elimination of variables, the classi-cal principle of minimum total potential energy is obtained, in which the only threevariables, components of displacement vector, remain. The equivalence of these princi-ples was theoretically justified, but from a practical point of view, it is a considerabledifference whether the problem is solved with respect to either 3 or 18 functions.

In the numerical simulations of linear elasticity problems discussed, approximatestress and displacement fields strictly obey the equilibrium equations, kinematic rela-tions, and boundary conditions, whereas the relations of Hooke’s law are weakened,i.e., satisfied in some integral sense [1]. It looks rather reasonable in numerical realiza-tion to present Hooke’s law as an integral over a function which is a quadratic formof the stress-strain relations. The functional of energy error gives one the possibilityto divide the problem originally formulated in terms of stresses and displacements intotwo independent subproblems: one in the displacements, the other in the stresses. Forvarious variational formulations following the method of integro-differential relations,the bilateral energy estimates of approximate solution quality are presented. Finiteelement algorithms were developed not only to check for model errors but also to refineadaptively FEM meshes in order to improve the solution quality.

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References

[1] Kostin G.V., Saurin V.V. Integrodifferential relations in linear elasticity. DeGruyter Studies in Mathematical Physics 10. De Gruyter, Berlin, 2012.

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20: Mini-Symposium: Large scale computing with applications

20 Mini-Symposium: Large scale computing with

applications

Organiser: Ulrich Rude

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ADAPTIVE ASYNCHRONOUS PARALLEL CALCULATIONSAT PETASCALE USING UINTAH

Martin Berzins

Scientific Computing and Imaging Institute72 S Central Campus Drive, Room 3750, Salt Lake City, UT 84112 USA

[email protected]

The past, present and future scalability of the Uintah Software framework is consideredwith the intention of describing a successful approach to large scale parallelism. Uintahallows the solution of large scale fluid-structure interaction problems through the use offluid flow solvers coupled with particle-finite element based solids methods. In additionUintah uses a combustion solver to tackle a broad and challenging class of turbulentcombustion problems. A unique feature of Uintah is that it uses an asynchronoustask-based approach with automatic load balancing to solve complex problems usingtechniques such as adaptive mesh refinement. At present, Uintah is able to makefull use of present-day massively parallel machines as the result of three phases ofdevelopment over the past dozen years. These development phases have led to anadaptive scalable run-time system that is capable of independently scheduling tasks tomultiple CPUs cores and GPUs on a node. In the case of solving incompressible low-mach number applications it is also necessary to use linear solvers and to consider thechallenges of radiation problems. The approaches adopted to achieve present scalabilityare described and their extensions to possible future architectures is considered.

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ACCELERATOR-FRIENDLY PARALLELADAPTIVE MESH REFINEMENT

Carsten Burstedde1, Lucas C. Wilcox2, Georg Stadler3 and Donna Calhoun4

1Institut fur Numerische Simulation, Universitat Bonn, [email protected]

2Department of Applied Mathematics, Naval Postgraduate School, USA

3Institute for Computational Engineering and Sciences,The University of Texas at Austin, USA

4Boise State University, USA

Recently, we have witnessed an impressive increase in total available compute power,which is commonly measured in flops per second. We have also seen applications thatsustain a rather high percentage of the theoretical peak performance. With regards tothe numerical solution of partial differential equations, achieving high peak percentageis a constant struggle for multiple reasons.

1. Sparse linear algebra is hard to vectorize.

2. The strong scaling behavior of multilevel solvers is suboptimal.

3. Finite element mesh traversal causes indirections which can be slow.

For the ostensibly simple case of explicit solvers, the first item still applies regardingmatrix-vector products, and the second still applies when local or hierarchical timestepping schemes are used. The third applies whenever the mesh is non-uniform ornon-structured.

With unstructured or tree-structured adaptive mesh refinement, the atomic meshprimitive is usually identified with a classical finite element, which entails all of theabove predicaments. One way out lies in the reinterpretation of the atomic meshprimitive as a macro-element, thus introducing a two-level discretization hierarchywhich is a better basis for writing optimized code. In fact, one of the earliest examplesof such an approach are unstructured conforming spectral elements demonstrated byTufo and Fischer in 1999.

We have developed h-adaptive non-conforming spectral elements for the numericalsolution of the elastic-acoustic wave equation in the past, which were subsequentlyported to multi-GPU clusters, applying MPI parallelization for the adaptive mesh andthread-level parallelization across the individual degrees of freedom of each high-orderspectral element. Currently, we are investigating a new approach to solving transportproblems, namely using uniform finite volume subgrids as the leafs of a tree-basedadaptively refined mesh. Work is underway to use thread-level parallelization on thesubgrids as well.

We will explain the details of our approach to cache- and accelerator-friendly scal-able adaptive mesh refinement and present numerical results for the solution of bothhyperbolic and parabolic partial differential equations.

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TOTAL EFFICIENCY OF CORE COMPONENTSIN FINITE ELEMENT FRAMEWORKS

Markus Geveler

Institute for Applied Mathematics and Numerics, TU Dortmund, [email protected]

Various techniques to exploit heterogeneous computational hardware in clusters areused for large scale Continuum Mechanics simulations. We restrict ourselves to FluidDynamics / Structure Mechanics solvers and concentrate on core components of Fi-nite Element frameworks like, e.g. the linear solvers therein. The target hardwareranges from clusters comprising Multicore-CPU / (multi-)GPU nodes to an experi-mental ARM-based cluster. Hardware-efficiency aspects (exploitation of all levels ofparallelism and accelerator hardware) as well as the numercial efficiency (using ad-justed Finite Element geometric Multigrid solvers over minor solvers) are consideredand finally, energy-efficiency is discussed briefly.

MASSIVE PARALLEL SIMULATION OF WATERAND SOLUTE TRANSPORT IN POROUS MEDIA

Olaf Ippischa, Markus Blatt and Jorrit Fahlke

Interdisciplinary Center for Scientific Computing, Heidelberg University, [email protected]

Water flow and solute transport are topics of high relevance for many problems withhigh importance for society. The multi-scale heterogeneity of natural porous requireslarge scale simulations with high spatial resolution.

Water flow in partially saturated porous media is described by Richards’ equation,a non-linear degenerate parabolic partial differential equation of second-order. Thenumerical solution is based on a cell-centred Finite-Volume discretisation, an implicitEuler-scheme in time, linearisation with an inexact Newton scheme with line searchand solution of the linear equation system with a BiCGStab solver preconditioned byalgebraic multi-grid.

A convection-diffusion equation is used for solute transport. As the flow field canbe quite heterogeneous and convection dominant, an explicit second order Godunovscheme with a minmod slope limiter is used for the convective part and a Finite-Volumediscretisation for the diffusive part.

The results of scalability tests with up to 150 billion unknowns on 294849 cores ofthe Bluegene/P type super computer JUGENE for the linear solver and the completemodel including I/O are presented as well as results from practical applications.

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SCALABLE PARALLEL MULTILEVEL SOLUTIONOF ELLIPTIC PROBLEMS

Peter K. Jimack1a, Mark A. Walkley1b and Jianfei Zhang2

1School of Computing, University of Leeds, Leeds LS2 9JT, [email protected], [email protected]

2Department of Engineering Mechanics, Hohai University,1 Xikang Road, Nanjing, China 210098

[email protected]

Multilevel solvers, such as geometric or algebraic multigrid, have been applied to theoptimal solution of algebraic systems arising from the discretization of elliptic partialdifferential equations (PDEs) for a number of decades. For standard finite differenceand finite element discretizations this optimality allows an accurate solution to beobtained at a cost of O(N) operations, where N is the number of degrees of freedom.The challenges of implementing such solvers on large scale parallel systems stem fromthe fact that the computations that must be undertaken at the coarsest levels do notgenerally contain sufficient work to hide or offset the parallelization overhead. In thiswork we consider the parallel solution of a scalar PDE and of systems of linear elasticityequations using a parallel algebraic multigrid preconditioner [A. Napov and Y. Notay,SIAM J. Sci. Comput., 34, 1079-1109, 2012]. In particular, we investigate the effectsof applying different approximations at the coarsest levels in order to obtain the bestparallel performance on large-scale systems. Numerical examples will be presented toshow that the best results are obtained when the accuracy of the coarse grid solve issacrificed for improved parallel performance.

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THOUGHTS ON GENERAL PURPOSE FINITE ELEMENTLIBRARIES AND HYBRID PROGRAMMING

Guido Kanschat

Interdisziplinares Zentrum fur Wissenschaftliches Rechnen (IWR),Universitat Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg Germany

[email protected]

Modern high performance architectures are characterized through a hierarchy of paral-lel computing units, from multicore processors, which share memory in a single physicalcomputing node equipped with GPU boards to clusters of thousands of such nodes andheterogeneous networks. The challenge originating from these hardware structures fordevelopers of a general purpose finite element library consists in the fact that the gapbetween achieved and peak performance widens more and more if software structuresdo not take these into account. Therefore, a balance has to be found between wideapplicability of the library and sufficient hardware adaptation. Accordingly, certainstructures and algorithms we have grown fond of will have to be replaced by more effi-cient or robust ones. We discuss the high performance computing support existing inthe deal.II library, its limits, and future developments to overcome current deficiencies.

PATCHING ADAPTIVITY FOR LARGE SCALE PROBLEMS- A NEW LIGHTWEIGHT ADAPTIVE SCHEME AND ITS

APPLICATION IN COMPUTATIONAL ELECTROCARDIOLOGY

Dorian Krause, Rolf Krausea, Thomas Dickopf and Mark Potse

Institute of Computational Science,Universita della Svizzera italiana, Lugano, Switzerland

[email protected]

Adaptive strategies are well known to reduce the number of degrees of freedom sub-stantially. However, in case of parallel large scale computations even a significantreduction in degrees of freedom does not necessarily lead to a similarly significant re-duction of overall computing time. This is mostly due to the associated overhead,which originates from error estimation, mesh adaptation, mesh redistribution, loadbalancing, and the use of the associated complicated data structures. Here, we presenta novel lightweight adaptive approach, which combines locally structured meshes witha non-conforming mortar discretisation. Our approach aims at combining the power ofadaptivity with computational efficiency, parallel scalability, and implementational sim-plicity. We present and analyse this new method in the framework of the monodomainequation from computational electrocardiology along with problems of different sizes.As it turns out, our lightweight adaptive scheme provides a very good balance betweenreduction in degrees of freedom and overall (parallel) efficiency.

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FAST AND SCALABLE ELLIPTIC SOLVERS FOR ANISOTROPICPROBLEMS IN GEOPHYSICAL MODELLING

Eike Mueller1a, Robert Scheichl1b and Eero Vainikko2

1 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, [email protected], [email protected].

2 Faculty of Mathematics and Computer Science, University of Tartu,Liivi 2, Tartu 50409, Estonia.

[email protected]

Semi-implicit time stepping is very popular and widely used in numerical weather- andclimate prediction models for various reasons, particularly since it allows for larger timesteps and thus for better efficiency. However, the bottleneck in semi-implicit schemesis the need for a three dimensional elliptic solve in each time step. With increasingmodel resolution this elliptic PDE can only be solved on operational timescales if highlyefficient algorithms are used and their scalability and performance on modern computerarchitectures can be guaranteed.

We have studied a typical model equation arising from semi-implicit semi-Lagrangiantime stepping; problems with a similar structure are encountered in other areas of geo-physical modelling, such as ocean circulation models and subsurface flow simulations.In particular, the vertical extent of the domain is significantly smaller than the hor-izontal size and one of the definining characteristics of elliptic PDEs encountered ingeophysical applications is a strong anisotropy in the vertical direction. To take thisinto account, these equations are usually discretised on grids with a tensor-productstructure with a unstructured (or semi-structured) horizontal mesh and a regular gridfor each vertical column. We implemented a bespoke, matrix-free geometric multigridsolver based on [Borm S., Hiptmair R., Numer. Algorithms 26: 200-1. (1999)] and[Buckeridge S, Scheichl R., Numerical Linear Algebra with Applications 17(2-3): 325–342 (2010)] which exploits the grid structure and strong vertical anisotropy and avoidsprecomputation of the matrix and coarse grid setup costs.

We demonstrated the superior performance of our solver on grids with a tensorproduct structure by comparing it to existing AMG solvers from the DUNE and Hyprelibraries and showed its scalability and robustness for systems with more than 1010

degrees of freedom on up to 65536 CPU cores of the HECToR supercomputer. Toinvestigate the performance on non-standard chip architectures, we ported the solverto a cluster of Graphics Processing Units (GPUs) using the CUDA-C programmingmodel. As the calculation is memory bound, the implementation was optimised toaccount for the GPU specific memory hierachy and the limited bandwidth of host-device data transfers.

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PARALLEL INCOMPRESSIBLE FLOW SIMULATIONSUSING DIVERGENCE-FREE FINITE ELEMENTS

Tobias Neckel

Department of Informatics, Technische Universitat MunchenBoltzmannstraße 3, 85748 Garching, Germany

[email protected]

Finite Elements have proven to be a valuable approach for numerical discretisation invarious fields of applications. In the case of incompressible flow simulations, differentvariants exist of how to combine a spatial discretisation (via FEM, e.g.) of the under-lying Navier-Stokes equations with integration in time: explicit methods, fully implicitdiscretisations, stabilised formulations, etc. The crucial point in all approaches is howto respect the continuity equation.

We developed a specific type of divergence-free elements in 2D (see [1, 4, 5]) whichexactly (i.e. continuously) fulfil the continuity equation. This enables a simultaneousconservation of momentum and energy, a feature that established element types typ-ically do not possess (cf. [3]) but that is advantageous for various applications suchas turbulent flow scenarios or coupled simulations. One important advantage of suchdivergence-free elements is the resulting skew-symmetric convection operator in the dis-cretisation. The divergence-free elements are similar to the well-known Q1Q0 elementsboth with respect to their low order and the corresponding computational costs.

In this contribution, we present the extension of the divergence-free finite elementsto the three-dimensional case. We use this spatial discretisation in a semi-implicitChorin projection scheme. The implementation has been realised in the PDE frame-work Peano [2] which uses (adaptive) Cartesian grids in combination with space-fillingcurves and stack data structures for efficient simulations. Due to the low order of theelements and the form of the grid cells, we precompute element matrices for all op-erators instead of performing numerical quadrature. Different benchmark simulationsare used to verify the code and show the validity of the approach. Furthermore, wediscuss certain aspects of the parallelisation of our discretisation. Applying a domaindecomposition for Cartesian grids, we simulated porous media-like scenarios (fracturenetworks in 2D and sphere packings in 3D) with several hundreds of CPUs on differentcomputing environments.

The focus of current work is on the implementation and verification of parallel flowsimulations using fully adaptive grids as well as on applying high numbers of processors.

References

[1] C. Blanke. Kontinuitatserhaltende Finite-Element-Diskretisierung der Navier-Stokes-Gleichungen. Diploma Thesis, Technische Universitat Munchen, 2004.

[2] H.-J. Bungartz, M. Mehl, T. Neckel, and T. Weinzierl. The PDE framework Peanoapplied to fluid dynamics: An efficient implementation of a parallel multiscale fluid

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dynamics solver on octree-like adaptive Cartesian grids. Computational Mechanics,46(1):103–114, June 2010.

[3] P. M. Gresho and R. L. Sani. Incompressible Flow and the Finite Element Method.John Wiley & Sons, 1998.

[4] T. Neckel. The PDE Framework Peano: An Environment for Efficient Flow Sim-ulations. Verlag Dr. Hut, 2009.

[5] T. Neckel, M. Mehl, and C. Zenger. Enhanced divergence-free elements for efficientincompressible flow simulations in the PDE framework Peano. In Proceedings of theFifth European Conference on Computational Fluid Dynamics, ECCOMAS CFD2010, 14th-17th June 2010, Lissabon, 2010.

ON LARGE-SCALE MECHANICS SIMULATIONSWITH THE PARALLEL TOOLBOX

Aurel Neic

University of Graz, Institute of Mathematics and Scientific Computing,Heinrichstrasse 36, 8010 Graz, Austria

[email protected]

The Cardiac Arrhythmias Research Package [G. Plank] simulates the electric and me-chanic behaviour of the heart tissue. Since the goal of this package is to resolve theanatomic properties of the heart in high detail, the FE discretisations usually consistof several millions of elements. This is a computationally challenging problem. Whileoriginally relying on PETSc for the data framework and solvers, CARP recently incor-porated the Parallel Toolbox for its GPU accelerated AMG-PCG solver. In this talkI will highlight the algorithmic and computational challenges in large-scale linear andnon-linear mechanics simulations on clusters of GPUs and CPUs. Additionally I willpresent new developments in the domain-decomposition parallelization techniques, suchas minimizing computational imbalances related to unbalanced subdomain boundariesand reducing the communication complexity of the Parallel Toolbox.

20-9


RECENT DEVELOPMENTS IN NGSOLVE FOR DISTRIBUTEDAND MANY-CORE PARALLEL COMPUTING

Joachim Schoberl

Institute for Analysis and Scientific Computing, Vienna UT, [email protected]

NGSolve is a general purpose high-order finite element package. Available applicationclasses are, among others, electromagnetics, wave propagation, and incompressibleflows. Numerical techniques include H(curl) and H(div) elements, hybrid-DG methods,and according preconditioning techniques.

One part of the talk reports on the distributed memory parallelization. We dis-cuss the rather small programming interface of the parallel layer to the numerical-components layer. We report on the performance of high-order domain decompositionmethods, in combination with direct and amg-based coarse grid preconditioners.

In the second part we discuss current software design for many core (GPGPU)hardware. Current hardware architecture providing coarse-grain and fine-grain paral-lelism is very attractive for high order finite element methods, which are more computeintensive than memory bandwidth consuming.

ANALYSIS OF ADAPTIVE SPACE-TIME FINITE ELEMENTSFOR PARABOLIC PROBLEMS

Kunibert G. Siebert

Institute for Applied Analysis and Numerical Simulation,Universitat Stuttgart, Germany

[email protected]

We present an adaptive space-time finite element method for parabolic problems. Thealgorithm is based on a classical adaptive time-stepping scheme supplemented by anadditional control of a potential energy increase of the discrete solution originatingfrom coarsening of the spatial meshes. We show that this algorithm is converging. Thismeans, given a positive tolerance, the algorithm reaches final time in a finite numberof steps and with an adaptive choice of the spatial meshes and the time-step-sizes suchthat the total space-time error is below the given tolerance.

This is joint research with Christian Kreuzer (Bochum), Christian A. Moller (Augs-burg), and Alfred Schmidt (Bremen).

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SCALABLE SOLVERS FOR ELLIPTIC PROBLEMS DISCRETIZEDBY ADAPTIVE HIGH-ORDER FINITE ELEMENTS

Georg Stadler1a, Tobin Isaac1b, Hari Sundar1c,Carsten Burstedde2 and Omar Ghattas1e

1Institute for Computational Engineering and SciencesThe University of Texas at Austin, Austin, TX USA

[email protected], [email protected],[email protected] [email protected]

2Institute for Numerical Simulation, University of Bonn, [email protected]

I will discuss challenges that arise in the solution of large-scale elliptic partial differ-ential equations discretized by higher-order adaptive finite elements. As examples, anequation with a positive definite operator as well as the Stokes equation will be con-sidered. The efficiency of iterative Krylov methods for the solution of these systemsrelies on the availability of scalable preconditioners, for which we consider algebraic andgeometric multigrid methods. We compare the scalability of these methods to tens ofthousands of CPU cores and discuss their ability to handle higher-order discretizations.

ALGEBRAIC MULTILEVEL PRECONDITIONING INH(CURL) AND H(DIV) SPACE

Satyendra Tomar

RICAM, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040 Linz, [email protected]

An algebraic multilevel iteration method for solving system of linear algebraic equa-tions arising in H(curl) and H(div) space will be presented. The algorithm is developedfor the discrete problem obtained by using the space of lowest order Raviart-Thomas-Nedelec elements of H(curl) and H(div). The theoretical analysis of the method isbased only on some algebraic sequences and generalized eigenvalues of local (element-wise) problems. Explicit recursion formulae are derived to compute the element ma-trices and the constant γ (which measures the quality of the space splitting) at anygiven level. It will be proved that the proposed method is robust with respect to theproblem parameters, and is of optimal order complexity. Supporting numerical results,including the case when the parameters have jumps, will also be presented.

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21: Mini-Symposium: Low Rank Tensor Based Numerical Methods

21 Mini-Symposium: Low Rank Tensor Based Nu-

merical Methods

Organisers: Lars Grasedyck, Boris Khorom-skij and Dmitry Savostyanov

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ADAPTIVE METHODS BASED ON TENSOR REPRESENTATIONSOF COEFFICIENT SEQUENCES AND THEIR

COMPLEXITY ANALYSIS

Markus Bachmayr and Wolfgang Dahmen

Institut fur Geometrie und Praktische Mathematik, RWTH Aachen, [email protected]

We consider a framework for the construction of iterative schemes for high-dimensionaloperator equations that combine adaptive approximation in a basis and low-rank ap-proximation in tensor formats. Our starting point is an operator equation Au = f ,where A is a bounded and elliptic linear operator mapping a separable Hilbert space H– for instance, a function space on a high-dimensional product domain – to its dual H ′.Assuming that a Riesz basis of H is available, the original problem can be rewritten asa linear system on `2, where the system matrix is bounded and continously invertible.

Under the given assumptions, a simple Richardson iteration on the infinite-dimensionalproblem converges, but of course cannot be realized in practice. This is the startingpoint for adaptive wavelet methods as introduced by Cohen, Dahmen and DeVore,which dynamically approximate such an ideal iteration by finite quantities, exploitingthe approximate sparsity of coefficient sequences.

The new aspect here is that, in order to significantly reduce computational com-plexity in a high dimensional context, we make use of an additional tensor productstructure of the problem. For this discussion, we assume H = H1 ⊗ · · · ⊗ Hd, i.e.,that H is a tensor product Hilbert space, and that we have a tensor product Rieszbasis of H. We now use a structured tensor format for the corresponding sequence ofbasis coefficients. Examples of suitable tensor structures are the Tucker format or theHierarchical Tucker format, where the latter can also be used for problems in very highdimensions. A crucial common feature of both formats is that there exist reliable pro-cedures for obtaining quasi-best approximations by lower-rank tensors with controllederror in `2-norm.

We are thus considering a highly nonlinear type of approximation: besides themultiplicative nonlinearity in the tensor representation, we aim to adaptively determinesimultaneously suitable finite approximation ranks, the active indices for the basisexpansions in the lower-dimensional spaces Hi, and corresponding coefficients. Weaccomplish this by a perturbed Richardson iteration, where approximation ranks andactive basis indices are adjusted implicitly in a sufficiently accurate approximationof the residual. The resulting growth in the complexity of iterates is kept in check bycombining a tensor recompression operation, which yields an approximation with lowerranks up to a specified error, with a coarsening operation that eliminates negligiblecoefficients in the lower-dimensional basis expansions. In the efficient realization ofthe latter, the special orthogonality properties of the considered tensor formats play acentral role.

Under the present quite general assumptions, we can then identify a choice of pa-rameters for the resulting iterative scheme that ensures its convergence and producesapproximations with near-minimal ranks. Under suitable further approximability con-

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ditions on the problem, we also obtain estimates for the total number of operationsrequired for reaching an approximate solution with a certain target accuracy. Further-more, we discuss the additional difficulties related to the preconditioning of problemsposed on Sobolev spaces in this setting. We consider some possible applications andillustrate our theory by numerical experiments.

BLACK BOX APPROXIMATION STRATEGIES INTHE HIERARCHICAL TENSOR FORMAT

Jonas Ballania and Lars Grasedyckb

Institut fur Geometrie und Praktische Mathematik, RWTH Aachen, [email protected], [email protected]

The hierarchical tensor format allows for the low-parametric representation of tensorseven in high dimensions d. The efficiency of this representation strongly relies onan appropriate hierarchical splitting of the different directions 1, . . . , d such that theassociated ranks remain sufficiently small. This splitting can be represented by abinary tree which is usually assumed to be given. In this talk, we address the questionof finding an appropriate tree from a subset of tensor entries without any a prioriknowledge on the tree structure. The proposed strategy can be combined with rank-adaptive cross approximation techniques such that tensors can be approximated in thehierarchical format in an entirely black box way. Numerical examples illustrate thepotential and the limitations of our approach.

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ALTERNATING MINIMAL ENERGY METHODS FOR LINEARSYSTEMS IN HIGHER DIMENSIONS.

PART II: FASTER ALGORITHM AND APPLICATIONTO NONSYMMETRIC SYSTEMS

Sergey V. Dolgov1 and Dmitry V. Savostyanov2

1Max Planck Institute for Mathematics in Sciences, Leipzig, Germany,[email protected]

2School of Chemistry, University of Southampton, UK,[email protected]

In this talk we further develop and investigate the rank-adaptive alternating methodsfor high-dimensional tensor-structured linear systems. The ALS method is reformu-lated in a recurrent variant, which performs a subsequent linear system reduction, andthe basis enrichment is derived in terms of the reduced system. This algorithm appearsto be more robust than the method based on a global steepest descent correction, andadditional heuristics allow to speedup the computations. Furthermore, the very samemethod is applied to nonsymmetric systems as well. Though its theoretical justificationis based on the FOM method, and is more difficult than in the SPD case, the practicalperformance is still very satisfactory, which is demonstrated on several examples of theFokker-Planck and chemical master equations.

Keywords: high–dimensional problems, tensor train format, ALS, DMRG, steepestdescent, convergence rate, superfast algorithms, NMR.

References: arXiv:1301.6068[math.NA], arXiv:1304.1222[math.NA]

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HP -DG TIME STEPPING FOR HIGH-DIMENSIONAL EVOLUTIONPROBLEMS WITH LOW-RANK TENSOR STRUCTURE

Vladimir Kazeev

Seminar for Applied Mathematics, ETH Zurich, CH-8092 Zurich, [email protected]

We consider linear evolution equations posed in spaces of possibly high dimensions. Im-portant examples are Kolmogorov equations: Fokker–Planck equations for stochasticODEs in finance and the Chemical Master Equation in chemistry and systems biol-ogy. While the latter often exhibit dynamics with pronounced transient phases, theformer may essentially require adaptivity to allow for time-inhomogeneous degeneratediffusion. Moreover, both tend to have high spatial dimension, which makes the useof classical adaptive or explicit full tensor-product discretizations in space unfeasible.For equations combining such features we propose an approach based on the hp-DGdiscretization in time and the low-rank tensor approximation in space.

In a suitable time-weighted Bochner space, a time-inhomogeneous degenerate evo-lution problem is shown to be well-posed and the analytic regularity of the time-dependence of the solution is studied. The hp-discontinuous Galerkin time discretiza-tion is shown to converge exponentially with respect to the number of temporal degreesof freedom.

To overcome the “curse of dimensionality” in space, the Tensor Train (TT) repre-sentation of high-dimensional arrays, based on the separation of variables, is employed.The low-rank approximation of a tensor in the TT format can be performed with theuse of standard, well-established matrix algorithms. On the other hand, for many ap-plications, the complexity and memory requirements are linear or almost linear withrespect to the number of dimensions. The use of quantization, leading to the QTT de-composition, allows to resolve even more structure in the matrices and vectors involvedand to further reduce the complexity and memory requirements.

Numerical experiments with equations of the two types mentioned above demon-strate the efficientcy of the proposed approach.

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HARTREE-FOCK EIGENVALUE SOLVER USINGTENSOR-STRUCTURED TWO-ELECTRON INTEGRALS

Venera Khoromskaia

Max-Planck Institute for Mathematics in the Sciences, Leipzig, [email protected]

The Hartree-Fock eigenvalue problem with the 3D integro-differential operator repre-sents the basic model in ab-initio electronic structure calculations. Due to presence ofmultiple strong cusps in electron density of molecules, the traditional approach to thesolution of the Hartree-Fock equation is based on accurate analytical precomputationof the arising 6D convolution type integrals with the Newton kernel, the so-called two-electron integrals (TEI), using the naturally separable Gaussian-type basis functions.We present an alternative, fast “black-box“ Hartee-Fock solver by the tensor numer-ical methods based on the rank-structured calculation of the core Hamiltonian andTEI using a general, well separable basis discretized on n × n × n 3D Cartesian grid[3, 4, 5, 7]. The arising 6D convolution integrals are approximated by 1D algebraicoperations in O(n log n) complexity on large spatial grids up to the size n3 ≈ 1014,thus providing high resolution of molecular cusps at low cost [1, 2]. The truncatedCholesky decomposition of TEI matrix is based on multiple factorizations, includingthe purely algebraic directional “1D density fitting“ depending on a threshold ε > 0,yielding an almost irreducible number of product basis functions for building the TEItensor [5]. The factorized TEI matrix is applied in tensor-based post-Hartree-Fock cal-culations [6]. We present on-line (in Matlab) ab initio ground state energy calculationsfor compact molecules, including glycine and alanine amino acids [7].

References

[1] B. N. Khoromskij and V. Khoromskaia. Multigrid Tensor Approximation of Func-tion Related Arrays. SIAM J Sci. Comp., 31(4), 3002-3026, 2009.

[2] V. Khoromskaia. Computation of the Hartree-Fock Exchange in the Tensor-structured Format. CMAM, Vol. 10, No 2, pp.204-218, 2010.

[3] B. N. Khoromskij, V. Khoromskaia and H.-J. Flad. Numerical solution of theHartree-Fock equation in Multilevel Tensor-structured Format. SIAM J Sci. Comp.,33(1), 45-65, 2011.

[4] V. Khoromskaia, D Andrae and B.N. Khoromskij. Fast and Accurate Tensor Cal-culation of the Fock Operator in a General Basis. CPC, 183, 2392-2404, 2012.

[5] V. Khoromskaia, B.N. Khoromskij and R. Schneider. Tensor-structured Calcula-tion of the Two-electron Integrals in a General Basis. Preprint 29/2012 MIS MPI,Leipzig, 2012. SIAM J. Sci. Comp., 2013, to appear.

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[6] V. Khoromskaia, and B.N. Khoromskij. Møller-Plesset Energy Correction UsingTensor Factorizations of the Grid-based Two-electron Integrals. Preprint 26/2013MIS MPI Leipzig, 2013.

[7] V. Khoromskaia. Fast 3D grid-based Hartree-Fock solver by tensor methods. Inprogress, 2013.

SUPER-FAST SOLVERS FOR PDES DISCRETIZEDIN THE QUANTIZED TENSOR SPACES

Boris Khoromskij

Max-Planck Institute for Mathematics in the Sciences, Leipzig, [email protected]

Tensor numerical approximation provides the efficient separable representation of mul-tivariate functions and operators on large n⊗d-grids, that allows the solution of d-dimensional PDEs with linear complexity scaling in the dimension, O(dn). Modernmethods of separable approximation combine the canonical, Tucker, as well as the ma-trix product state (MPS) and tensor train (TT) formats [2]. The recent quantizedTT (QTT) approximation [1] is proven to provide the logarithmic data-compressionon a wide class of functions and operators. It makes possible to solve high-dimensionalsteady-state and dynamical problems in quantized tensor spaces, with the log-volumecomplexity scaling in the full-grid size, O(d log n), instead of O(nd).

In this talk I show how the grid-based QTT tensor approximation applies to hardproblems arising in electronic structure calculations, such as many-electron integrals[3]. The QTT approximation method provides the efficient solvers for parametric PDEs[4] as well as for high-dimensional time-dependent models, in particular, the molecularSchrodinger, the Fokker-Planck [5] and chemical master equations [6]. We presentnumerical tests indicating the efficiency of the QTT tensor approximation in somesteady-state and dynamical calculations.

References

[1] B.N. Khoromskij. O(d logN)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling. J. Constr. Approx. v. 34(2), 257-289 (2011).

[2] B.N. Khoromskij. Tensors-structured Numerical Methods in Scientific Computing:Survey on Recent Advances. Chemometr. Intell. Lab. Syst. 110 (2012), 1-19. DOI:10.1016/j.chemolab.2011.09.001.

[3] V. Khoromskaia, B.N. Khoromskij, and R. Schneider. Tensor-structured calcula-tion of two-electron integrals in a general basis. SIAM J. Sci. Comput., 2013 (toappear). Preprint 29/2012, MPI MiS, Leipzig 2012.

[4] B.N. Khoromskij, and Ch. Schwab, Tensor-Structured Galerkin Approximation ofParametric and Stochastic Elliptic PDEs. SIAM J. Sci. Comp., 33(1), 2011, 1-25.

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[5] S.V. Dolgov, B.N. Khoromskij, and I. Oseledets. Fast solution of multi-dimensionalparabolic problems in the TT/QTT formats with initial application to the Fokker-Planck equation. SIAM J. Sci. Comput., 34(6), 2012, A3016-A3038.

[6] S. Dolgov, and B.N. Khoromskij. Tensor-product approach to global time-space-parametric discretization of chemical master equation. Preprint 68/2012, MPI MiS,Leipzig 2012 (submitted).

ALTERNATING MINIMAL ENERGY METHODS FORLINEAR SYSTEMS IN HIGHER DIMENSIONS.

PART I: SPD SYSTEMS

Dmitry V. Savostyanov1 and Sergey V. Dolgov2

1School of Chemistry, University of Southampton, UK,[email protected]

2Max Planck Institute for Mathematics in Sciences, Leipzig, Germany,[email protected]

We propose a new algorithm for the approximate solution of large–scale high–dimensio-nal tensor–structured linear systems. It can be applied to high-dimensional differentialequations, which allow a low–parametric approximation of the multilevel matrix, righthand side and solution in the tensor train format. We combine the Alternating LinearScheme approach with the basis enrichment idea using Krylov–type vectors. We obtainthe rank–adaptive algorithm with the theoretical convergence estimate not worse thanthe one of the steepest descent. The practically observed convergence is significantlyfaster, comparable or even better than the convergence of the DMRG–type algorithm.The complexity of the method is at the ALS level. The method is successfully appliedfor a high–dimensional problem of quantum chemistry, namely the NMR simulation ofa large peptide.

Keywords: high–dimensional problems, tensor train format, ALS, DMRG, steepestdescent, convergence rate, superfast algorithms, NMR.

References: arXiv:1301.6068[math.NA], arXiv:1304.1222[math.NA]

21-8

22: Mini-Symposium: Mathematical and statistical modeling in biology

22 Mini-Symposium: Mathematical and statistical

modeling in biology

Organiser: H.T. Banks

22-1


ACOUSTIC LOCALISATION OF CORONARY ARTERY STENOSIS:WAVE PROPAGATION IN SOFT TISSUE MIMICKING GEL

H. Thomas Banks4, Malcolm J. Birch2, Mark P. Brewin1,2, Steve E. Greenwald1a,Shuhua Hu4, Zackary Kenz4, Carola Kruse3,

Dwij Mehta1b, Simon Shaw3 and John R. Whiteman3

1Blizard Institute, Barts and The London School of Medicine and Dentistry,Queen Mary, University of London, UK


2Clinical Physics, Barts Health NHS Trust, London, UK

3BICOM, Institute of Computational Mathematics, Brunel University, UK

4CRSC, Department of Mathematics,North Carolina State University, Raleigh NC, USA

Plaque developing in a coronary artery produces turbulent flow downstream and wallshear stresses varying at a frequency around 1kHz. These give rise to low amplitudeacoustic shear waves which propagate through the chest and can be measured by skinsensors. This acoustic surface signature may provide a cheap non-invasive means ofdiagnosing arterial disease. We will discuss measurements of the propagation of 1-D freeoscillations induced in tissue mimicking gel specimens following the sudden release ofshear or compressive stresses and the results of subsequent measurements of the surfacestrain field, using a novel optical technique, due to 2-D forced oscillations, induced inthe gel by an electro mechanical vibrator. We will also describe measurements ofthe strain field on the surface of cuboidal and cylindrical gel specimens containingunobstructed and stenosed (partially obstructed) tubes through which laminar andturbulent flow is passed. A companion presentation will describe the results of directand inverse solver software to simulate the response of the gel to the shear waves. Thismathematical loop makes it possible to characterise the source given the signal and tocompare material data with predicted values.

22-2


EFFICIENT NUMERICAL METHODS FOR COUPLED PDE-ODESYSTEMS: AN APPLICATION IN INTERCELLULAR SIGNALING

Thomas Carraroa, Elfriede Friedmannb and Daniel Gerechtc

Institute for Applied Mathematics, Heidelberg University, Heidelberg, [email protected],

[email protected]@iwr.uni-heidelberg.de

A combined experimental and theoretical study [Busse et al, PNAS 2010] has shownthat intercellular diffusion of the cytokine Interleukin-2 (IL-2) is a key regulatory stepin the induction of immune responses in the lymph node. Showing the importance ofspatial distribution, it implies that a realistic 3D configuration of the cells is necessaryfor understanding the observed processes. We developed efficient numerical methodsto solve the nonlinear system of equations consisting of a PDE coupled with ODEs.Our principal components are a Galerkin space discretization by finite elements, a fullycoupled multilevel algorithm as solver and an adaptive time scheme. 3D simulationsshow how the competition of T helper cells and regulatory T cells for IL-2 influencethe activation of the T cells. We compare results of a homogenized model to the fullsystem.

MODELING AND INVERSE PROBLEM CONSIDERATIONSFOR A VISCOELASTIC TISSUE MODEL

Zackary Kenz

Center for Research in Scientific Computation,North Carolina State University, Raleigh, NC 27695-8205, U.S.A.

[email protected],

Existing methods used to detect an arterial stenosis are laborious, expensive, and ofteninvasive. A method has been proposed whereby one places sensors on the surface ofthe chest and attempts to detect shear waves generated by turbulent flow (generatedby a blockage) impacting the vessel wall. We focus here on the propagation of shearand pressure waves through a viscoelastic medium. We will discuss the development ofone-dimensional models for wave propagation in a tissue-mimicking agar gel cylinder.Corresponding experimental data from the gel has been obtained by our collaborators.We present a sample of our inverse problem model parameter results using this data,and also examine the robustness of the estimated parameters.

This is work performed in collaboration with H.T. Banks and Shuhua Hu, NCSU;with Carola Kruse, Simon Shaw, and John R. Whiteman at BICOM, Brunel University,London; with Stephen E. Greenwald and Mark P. Brewin at Blizard Institute, Bartsand the London School of Medicine and Dentistry, Queen Mary, University of London;and with Malcom J. Birch at Clinical Physics, Barts Health NHS Trust, London.

22-3


HIGH ORDER SPACE-TIME FINITE ELEMENT SCHEMESFOR THE DYNAMICS OF VISCOELASTIC SOFT TISSUE

Carola Kruse1a, Simon Shaw1, John R. Whiteman1, H. Thomas Banks2,Zackary Kenz2,

Shuhua Hu2, Steve E. Greenwald3, Mark P. Brewin3 and Malcolm J. Birch4

1BICOM, Department of Mathematical Sciences, Brunel University, Uxbridge, [email protected]

2Center for Research in Scientific Computation,North Carolina State University, Raleigh, NC 27695-8212, USA

3Blizard Institute, Barts and The London School of Medicine and Dentistry,Queen Mary, University of London, UK

4Clinical Physics, Barts and the London National Health Service Trust, UK

As plaque builds up in a coronary artery, blood flow past the stenosed region becomesturbulent and creates abnormal variations in wall shear stresses in the wake. Theseshears drive low amplitude acoustic shear waves at around 1 kHz through the soft tis-sue in the thorax which appear at the chest wall and can be measured non-invasivelyby placing sensors on the skin. This acoustic surface signature (bruit) has thus the po-tential to provide a cheap non-invasive means of diagnosing Coronary Artery Disease[Banks and Pinter, Multiscale Model. Simul., 3: 395 - 412, 2005]. An efficient andaccurate solver with the ability to resolve these low energy surface fluxes will be anessential ingredient.

With this as our motivation we will describe the development and formulation of ahigh order solver for a space-time elasto- and visco-dynamic problem formed by merg-ing Hookes law with the Zener and Kelvin-Voigt models for viscoelasticity. We employa spectral finite element method to discretize in space and a high order discontinuousGalerkin finite element discretization in time using normalized Legendre polynomialsof arbitrary degree, r, say. This choice allows the linear system to be decoupled byfollowing Werder et al.s technique [Comp. Meth. Appl. Mech. Engrg., 190: 6685 -6708, 2001] and results in a set of (r+1) complex symmetric systems for each time slab.We illustrate the effect of the decoupling with respect to accuracy and computationtime.

22-4

23: Mini-Symposium: New advances in a posteriori error estimation

23 Mini-Symposium: New advances in a posteriori

error estimation

Organisers: Mark Ainsworth, Alexandre Ernand Martin Vohralik

23-1


COMPUTABLE ERROR BOUNDS FOR FINITE ELEMENTAPPROXIMATION ON NON-POLYGONAL DOMAINS

Mark Ainsworth1 and Richard Rankin2

1Division of Applied Mathematics, Brown University,182 George Street, Providence, Rhode Island, 02912, USA.


2Department of Computational and Applied Mathematics, Rice University,6100 Main Street, Houston, Texas, 77005, USA.

[email protected]

We consider the case of piecewise affine finite element approximation of the solutionto the Poisson problem with pure Neumann boundary conditions on domains whichare non-polygonal. We obtain an a posteriori error estimator which takes the effect ofthe boundary approximation into account. The estimator provides a guaranteed upperbound on the energy norm of the error and, up to a constant and oscillation terms,local lower bounds on the energy norm of the error.

23-2


GUARANTEED AND ROBUST ERROR BOUNDS FORSINGULARLY PERTURBED PROBLEMS

IN ARBITRARY DIMENSION

Mark Ainsworth1 and Tomas Vejchodsky2

1Division of Applied Mathematics, Brown University, Providence, USA,mark [email protected]

2Mathematical Institute, University of Oxford, UKand Institute of Mathematics, Academy of Sciences, Prague, Czech Republic,

[email protected]

We present guaranteed, robust, and fully computable error bounds for finite elementsolutions of singularly perturbed reaction-diffusion problem

−∆u+ κ2u = f in Ω, u = 0 in ∂Ω.

The bounds are proved to be robust for the entire range of values of the reaction coeffi-cient κ including the singularly perturbed case. The method is based on equilibration ofinterelement fluxes and on a subsequent reconstruction of the complementary flux. Theconstruction and equilibration of interelement fluxes that yields robust error boundswas proposed already in [1]. However, error bounds from [1] cannot be computed ex-actly and if they are approximated then they do not guarantee the upper bound on theerror. In order to overcome this issue, we applied the complementarity technique in [2]to and we obtained a robust, fully computable and guaranteed upper bounds on theerror. Nevertheless, the reconstruction of the complementary flux in [2] is technicallydemanding and applicable to two-dimensional problems only. In the talk we present areconstruction of the complementary flux that is more elegant and provides guaranteed,fully computable, and robust error bounds in arbitrary dimension.

References

[1] Ainsworth, M., Babuska, I., Reliable and robust a posteriori error estimating forsingularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. 36 (1999)331–353.

[2] Ainsworth, M., Vejchodsky, T., Fully computable robust a posteriori error boundsfor singularly perturbed reaction-diffusion problems. Numer. Math. 119 (2011) 219–243.

23-3


INSTANCE OPTIMALITY FOR THE MAXIMUM STRATEGY

Lars Diening

LMU Munich, Institute of Mathematics, Theresienstr. 39, 80333 Munich, [email protected]

We study the adaptive finite element approximation of the Dirichlet problem −∆u = fwith zero boundary values using linear Ansatz functions and newest vertex bisection.Our approach is based on the minimization of the corresponding Dirichlet energy. Weshow that the maximums strategy attains every energy level with a number of degreesof freedom, which is proportional to the optimal number. As a consequence we achieveinstance optimality of the error. This is a joint work with Christian Kreuzer (Bochum)and Rob Stevenson (Amsterdam).

23-4


A FRAMEWORK FOR ROBUST A POSTERIORI ERROR CONTROLIN UNSTEADY NONLINEAR ADVECTION-DIFFUSION PROBLEMS

Vıt Dolejsı1, Alexandre Ern2 and Martin Vohralık3

1Charles University in Prague, Faculty of Mathematics and Physics,Sokolovska 83, 186 75 Praha 8, Czech Republic.

[email protected]

2Universite Paris-Est, CERMICS, Ecole des Ponts ParisTech,77455 Marne-la-Vallee, France.

[email protected]


We derive a framework [1] for a posteriori error estimates in unsteady, nonlinear, possi-bly degenerate, advection-diffusion problems. Our estimators are based on a space-timeequilibrated flux reconstruction and are locally computable. They are derived for theerror measured in a space-time mesh-dependent dual norm stemming from the problemand meshes at hand augmented by a jump seminorm measuring possible nonconformi-ties in space. Owing to this choice, a guaranteed and globally efficient upper boundis achieved, as well as robustness with respect to nonlinearities, advection dominance,domain size, final time, and absolute and relative size of space and time steps. Local-in-time and in-space efficiency is also shown for a localized upper bound of the errormeasure. In order to apply the framework to a given numerical method, two simpleconditions, local space-time mass conservation and an approximation property of thereconstructed fluxes, need to be verified. We show how to do this for the interior-penalty discontinuous Galerkin method in space and the Crank–Nicolson scheme intime. Numerical experiments illustrate the theory.

References

[1] Dolejsı V., Ern A., Vohralık, M., A framework for robust a posteriori error controlin unsteady nonlinear advection-diffusion problems. SIAM J. Numer. Anal. (2013),DOI 10.1137/110859282.

23-5


QUASI-OPTIMAL AFEM FOR NON-SYMMETRIC OPERATORS

Michael Feischla Thomas Fuhrerb and Dirk Praetoriusc



In our talk, we present our recent preprint [arXiv:1210.8369], where adaptive mesh-refinement for conforming FEM of general linear, elliptic, second-order PDEs is ana-lyzed. For a bounded Lipschitz domain Ω ⊂ Rd, our model problem thus reads

Lu(x) := −divA(x)∇u(x) + b(x) · ∇u(x) + c(x)u(x) = f(x) x ∈ Ω,

u(x) = 0 x ∈ ∂Ω.

For a given conforming simplicial mesh T`, we allow continuous T`-piecewise polyno-mials of arbitrary, but fixed polynomial order with homogeneous boundary conditionsSp0 (T`) as ansatz functions. As e.g. in [Cascon-Kreuzer-Nochetto-Siebert, SINUM 2008],adaptivity is driven by the residual error estimator ρ`, and we prove convergence evenwith quasi-optimal algebraic convergence rates.

The advantages over the state of the art read as follows: Unlike prior works forlinear non-symmetric operators, e.g. [Cascon-Nochetto, IMA JNA 2012], our analysisavoids the artificial quasi-symmetry assumptions ∇ · b = 0 and c ≥ 0 as well as theinterior node property for the refinement. Moreover, the differential operator L has tosatisfy a Garding inequality only. If L is uniformly elliptic, no additional assumptionon the initial mesh is posed. Finally, our analysis also covers certain nonlinear problemsin the frame of strongly monotone operators.

On a technical level, the heart of the matter is a novel quasi-orthogonality estimatewhich builds on the observation that estimator reduction already implies convergenceof the adaptive scheme. Moreover and unlike e.g. [Cascon-Kreuzer-Nochetto-Siebert,SINUM 2008] and [Cascon-Nochetto, IMA JNA 2012], our analysis avoids the use ofthe so-called total error and quasi error and directly works with the estimator. As aconsequence, the bound for the range of optimal marking parameters 0 < θ < θ? doesnot depend on lower bounds for the error (so-called efficiency estimates).

23-6


A POSTERIORI ERROR ESTIMATES FOR THE WAVE EQUATION

Omar Lakkis1, Emmanuil H. Georgoulis2 and Charalambos Makridakis3

1Department of Mathematics, University of Sussex,Falmer near Brighton, England UK

[email protected]

2Department of Mathematics, University of Leicester,University Road, Leicester LE1 7RH, United Kingdom

3Department of Applied Mathematics,University of Crete, L. Knosou GR 71409 Heraklion, Greece

We present a posteriori error bounds in L∞(0, T, L2(Ω)) norm for an implicit time-stepping method in combination with a finite element method in space to solve theinitial-boundary value problem for the wave equation on the space-time domain Ω ×(0, T ). The technique relies on appropriately constructed time-reconstruction in com-bination with Baker’s function method from Baker (SINUM, 1976). I will mentionrelated results concerning the leapfrog method as well. In the latter case, the time-reconstruction has to be also carefully designed as to obtain optimal-order estimators.

23-7


ON MATHEMATICAL METHODS GENERATING FULLYRELIABLE A POSTERIORI ESTIMATES FOR

NONLINEAR BOUNDARY VALUE PROBLEMS

Sergey Repin

V. A. Steklov Institute of Mathematics at St. Petersburg,Fontanka 27, 191023, St. Petersburg, Russian Federation

[email protected]

Modern theory of partial differential equations has elaborated several ways of derivingfully guaranteed estimates, which can be used as a posteriori estimates for numerical(e.g., FEM) solutions and as computable bounds of modeling errors. For problemsgenerated by linear differential operators, the corresponding theory is well developed.It is difficult to say the same about many classes of nonlinear problems, which aremuch less studied in the context of a posteriori analysis. One of the most importantquestions is how to define the right (suitable) measure of the error for a stronglynonlinear problem. We discuss possible answers to this and other questions with theparadigm of a class of convex variational problems and motivate the selection of acertain error measure. It is supplied with bounds, which are directly computable(i.e., they do not require exact satisfaction of some additional conditions). Also, wediscuss closely related mathematical questions, which are principally important forquantitative analysis of nonlinear problems and present generalized forms of the Prager-Synge estimate, Mikhlin’s variational identity, Helmgholtz decomposition theorem, andderive a general estimate of the distance to the set equilibrated fields.

ADAPTIVE FINITE ELEMENTS FOR PDE CONSTRAINEDOPTIMAL CONTROL PROBLEMS

Kunibert G. Siebert

Institute for Applied Analysis and Numerical Simulation,Universitat Stuttgart, Germany

[email protected]

Many optimization processes in science and engineering lead to optimal control prob-lems where the sought state is a solution of a partial differential equation (PDE).Control and state may be subject to further constraints. The complexity of such prob-lems requires sophisticated techniques for an efficient numerical approximation of thetrue solution. One particular method are adaptive finite element discretizations.

We report on ongoing research about control constrained optimal control problems.We give a summary about recent findings concerning sensitivity analysis, a posteriorierror control, and convergence of adaptive finite elements.

This is joint work with Fernando D. Gaspoz (Stuttgart).

23-8

24: Mini-Symposium: Non-Standard Finite Elements and Solvers in Solid Mechanics

24 Mini-Symposium: Non-Standard Finite Elements

and Solvers in Solid Mechanics

Organisers: Axel Klawonn and Gerhard Starke

24-1


A NEW COARSE SPACE FOR FETI-DP IN THE CONTEXTOF ALMOST INCOMPRESSIBLE ELASTICITY

Sabrina Gippert1a. Axel Klawonn 1b and Oliver Rheinbach2

1Mathematisches Institut, Universitat zu Koln, [email protected], [email protected]

2Fakultat fur Mathematik und Informatik,Technische Universitat Bergakademie Freiberg, Germany

[email protected]

Domain decomposition methods such as FETI-DP have been considered successfullyfor almost incompressible elasticity problems. To ensure a good condition numberit is known, that for mixed finite element discretizations with discontinuous pressureelements a zero net flux condition on each subdomain is needed. This is usually done en-forcing the constraint for each edge and each face separately. Here, the edge constraintsare implemented using a transformation of basis approach with partial assembly, whilethe face terms are enforced using a deflation framework. For this new approach it issufficient to implement the zero net flux condition in this deflation method using oneconstraint for each subdomain instead of one constraint for each face, which allows amuch smaller coarse space. This new approach will be discussed and numerical resultswill be presented.

NONLINEAR FETI-DP AND BDDC METHODS

Axel Klawonn1a, Martin Lanser1b, and Oliver Rheinbach2

1Mathematisches Institut, Universitat zu Koln, [email protected], [email protected]

c Fakultat fur Mathematik und Informatik,Technische Universiat Bergakademie Freiberg,[email protected]

New nonlinear FETI-DP (Dual-Primal Finite Element Tearing and Interconnecting)and BDDC (Balancing Domain Decomposition by Constraints) domain decompositionmethods are introduced. In all of these methods, in each iteration, local nonlinearproblems are solved on the subdomains. The new approaches have a potential to reducecommunication and show a significantly improved performance. Numerical results forthe p-Laplace operator are presented.

24-2


LSFEM FOR GEOMETRICALLY AND PHYSICALLYNONLINEAR ELASTICITY PROBLEMS

Benjamin Muller1a, Gerhard Starke1b, Jorg Schroder2c,

Alexander Schwarz2d and Karl Steeger2e

1Faculty of Mathematics, University Duisburg - Essen, Essen, [email protected], [email protected]

2Faculty of Engineering, University Duisburg - Essen, Essen, [email protected], [email protected]

[email protected]

Deformation processes of solid materials are omnipresent and can be described by sys-tems of partial differential equations in continuum mechanics. In this talk we present aleast squares finite element method based on the momentum balance and the constitu-tive equation for hyperelastic materials. Our approach is motivated by a well - studiedleast squares formulation for linear elasticity. This method is generalized to an ap-proach which takes physical as well as geometrical nonlinearities into account. Thenovelty of our approach is that, in addition to the displacement u, we consider the fullfirst Piola - Kirchhoff stress tensor P and approximate both simultaneously.In the discrete formulation we use quadratic Raviart - Thomas elements for the stresstensor and continuous quadratic elements for the displacement vector. For the min-imization of the nonlinear least squares functional, the Gauss - Newton method withbacktracking line search is used.We will emphasize the advantages of our approach in comparison to other solutionmethods. The talk will end with an illustration of the performance for some two di-mensional problems in plane strain configuration and some three dimensional problems.

AN APPROACH TO ADAPTIVE COARSESPACES IN FETI-DP METHODS

Oliver Rheinbach1, Axel Klawonn2b and Patrick Radtke2c

1 Fakultat fur Mathematik und Informatik,Technische Universitat Bergakademie Freiberg, Germany

[email protected]

2 Mathematisches Institut, Universitat zu Koln, [email protected], [email protected]

Adaptive coarse spaces for domain decomposition methods can be used to obtain inde-pendence on coefficient jumps for highly heterogeneous problems, even when coefficientjumps inside subdomains are present. In this talk, for FETI-DP methods, we present anew approach to obtain independence of the coefficient jumps by solving certain localeigenvalue problems and enriching the FETI-DP coarse space with eigenvectors.

24-3


GEODESIC FINITE ELEMENTS

Oliver Sander

Institut fur Geometrie und Praktische Mathematik,RWTH Aachen University, Germany

[email protected]

Geodesic finite elements are a novel way to discretize problems involving functions withvalues on a Riemannian manifold. Examples for such problems include Cosserat mate-rials and liquid crystals. Geodesic finite elements of any order can be constructed, andare conforming in the sense that they are first-order Sobolev functions. The construc-tion is equivariant under isometries of the value manifold, which means that frame-indifference in mechanics is preserved. Optimal discretization error bounds have beenshown analytically, and can be observed in numerical experiments. We present thetheory of geodesic finite elements and give a few example applications.

24-4


ASPECTS ON MIXED LEAST-SQUARES FINITE ELEMENTSFOR HYPERELASTIC PROBLEMS

Alexander Schwarz1a, Karl Steeger1ba, Jorg Schroder1c,Gerhard Starke2d and Benjamin Muller2e

1 Faculty of Engineering, Universitat Duisburg-Essen, [email protected], [email protected],

[email protected]

2 Faculty of Mathematics, Universitat Duisburg-Essen, [email protected], [email protected]

The main goal of this contribution is the solution of geometrically nonlinear prob-lems using the mixed least-squares finite element method (LSFEM). The basis for theproposed element formulations are div-grad first-order systems consisting of the equi-librium condition and the constitutive equation both written in residual forms, see e.g.[2] and [1]. Generally, L2-norms are adopted on these residuals leading to function-als depending on displacements and stresses, which are the basis for the associatedminimization problems. In particular we consider different hyperelastic free energyfunctions in order to define the stress response of the material. Besides some numer-ical advantages, as e.g. an inherent symmetric structure of the system of equationsand an error estimator, it is known that least-squares methods have also a drawbackconcerning accuracy, especially when lower-order elements are used. Therefore, a focusof the presentation is on performance and implementation aspects of triangular mixedfinite elements with different interpolation order. In order to approximate the stresses,shape functions related to the edges are chosen. These vector-valued functions are usedfor the interpolation of the rows of the stress tensor and belong to a Raviart-Thomasspace, which guarantees a conforming discretization of the Sobolev space H(div). Fur-thermore, standard polynomials associated to the vertices of the triangle are usedfor the continuous approximation of the displacements in W 1,p with p > 2. Finally,the proposed formulations will be compared considering various benchmark problems,computational costs and accuracy.

References

[1] A. Schwarz, J. Schroder, G. Starke, and K. Steeger. Least-squares mixed finiteelements for hyperelastic material models. In Report of the Workshop 1207 atthe “Mathematisches Forschungsinstitut Oberwolfach” entitled “Advanced Compu-tational Engineering”, organized by O. Allix, C. Carstensen, J. Schroder, P. Wrig-gers, pages 14–16, 2012.

[2] G. Starke, B. Muller, A. Schwarz, and J. Schroder. Stress-displacement least squaresmixed finite element approximation for hyperelastic materials. In Report of theWorkshop 1207 at the “Mathematisches Forschungsinstitut Oberwolfach” entitled“Advanced Computational Engineering”, organized by O. Allix, C. Carstensen, J.Schroder, P. Wriggers, pages 11–13, 2012.

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ON ISOGEOMETRIC FINITE ELEMENTS IN SOLIDMECHANICS AND VIBRATIONAL ANALYSIS

Bernd Simeona and Oliver Weegerb

Department of Mathematics, Felix-Klein-Zentrum, TU Kaiserslautern, [email protected], [email protected]

Over the last years, the new paradigm of Isogeometric Analysis has demonstratedits potential to bridge the gap between Computer Aided Design (CAD) and the FiniteElement Method (FEM). The distinguished feature of isogeometric finite elements is theusage of one common geometry representation for creating CAD models, for meshing,and for numerical simulation. In this way, a seamless integration of all computationaltools within a single design loop comes into reach. Moreover, increased smoothness ofthe basis functions and an exact representation of the boundary are properties whichare also attractive from a numerical viewpoint.

The presentation is aimed at the application of isogeometric methods in the fieldof solid mechanics, in particular vibrational analysis. We start with a short overviewon the methodology, point out the common features and differences when comparedto the classical finite element method, and concentrate then on static and vibrationalanalysis of linear and nonlinear elasticity problems.

By studying typical geometries and problem setups, we analyze the pros and consof employing NURBS-based shape functions. In this context, we also summarize ourwork on implementing a 3D isogeometric solver with multi-patch and large deformationcapabilities. Finally, we explore the benefits of higher smoothness in the field of non-linear vibrational analysis. The method of harmonic balance is well-established hereand allows the tracing of resonance phenomena. Using a well-understood nonlinearEuler-Bernoulli beam model as benchmark problem, we demonstrate that spline-baseddiscretizations with higher global smoothness lead to an improved accuracy in theharmonic balance analysis and are thus a promising approach for this particular appli-cation.

This work is supported by the European Union within the FP7-project TERRIFIC:Towards Enhanced Integration of Design and Production in the Factory of the Futurethrough Isogeometric Technologies.

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MOMENTUM BALANCE IN FIRST-ORDER SYSTEMFINITE ELEMENT METHODS FOR ELASTICITY

Gerhard Starke

University Duisburg-Essen, [email protected]

The purpose of this talk is to demonstrate the importance of accurate momentum bal-ance approximation for stress-displacement first-order system formulations in elasticity.Our study includes first-order system least squares methods as well as mixed methodsof saddle point structure and establishes close relations between these different ap-proaches. Since the latter class of methods treat momentum balance as a constraint,this approach appears to be favourable. With a slight modification, however, first-ordersystem least squares formulations also lead to surprisingly high momentum balance ac-curacy. The advantage of this approach is that Raviart-Thomas elements (of degree k)for the stress approximation can be combined with standard conforming elements (ofdegree k+1) for the displacements without stability problems. For bending-dominatedproblems it is observed that improved momentum balance accuracy goes hand in handwith much better convergence of pointwise values. While the theoretical connectionsare mainly derived for linear elasticity models we also investigate geometrically non-linear elasticity using a hyperelastic material of Neo-Hookean type.

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25: Mini-Symposium: Novel Methods for Time-Harmonic Wave Equations

25 Mini-Symposium: Novel Methods for Time-Harmonic

Wave Equations

Organisers: Antti Hannukainen and LotharNannen

25-1


ANALYSIS OF A CARTESIAN PML APPROXIMATION TOACOUSTIC SCATTERING PROBLEMS IN RN

James H. Bramblea and Joseph E. Pasciakb

Department of Mathematics, Texas A&M University, College Station, TX, [email protected], [email protected]

We consider the application of a perfectly matched layer (PML) technique in Cartesiangeometry to approximate solutions of the acoustic scattering problem in the frequencydomain. This work provides new stability estimates for both the infinite domain PMLapproximation as well as the truncated (bounded) domain PML approximation. Thestability estimates are new in the sense that they demonstrate the interplay betweenthe size of the computational domain M and the strength of the PML stretchingfunction σ0. In particular, we show that the stability and exponential convergencedepends on the product σ0M . This means that one can obtain the desired accuracywithout changing the domain size by simply increasing the PML parameter σ0. Weillustrate these results with computations showing that even infinitesimal PML layersare adequate provided that the PML strength is suitably adjusted. These stabilityestimates allow for the case of piecewise C1 stretching as well as piecwise C0 stretching.The second case gives rise to PML variational equations with jumping coefficients.

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A HIGH FREQUENCY BEM FOR SCATTERINGBY A CLASS OF NONCONVEX OBSTACLES

Simon Chandler-Wilde1, David Hewett1a,2, Stephen Langdon1 and Ashley Twigger1


2Current address: Mathematical Institute, University of Oxford, UK.

There is considerable current interest in the development of numerical methods fortime-harmonic acoustic and electromagnetic scattering problems that are able to effi-ciently resolve the scattered field at high frequencies. Conventional finite or boundaryelement methods, with piecewise polynomial approximation spaces, suffer from the re-striction that a fixed number of degrees of freedom is required per wavelength in orderto represent the oscillatory solution, leading to excessive computational cost when thescatterer is large compared to the wavelength.

The ‘hybrid numerical-asymptotic’ (HNA) approach aims to reduce the computa-tional cost at high frequencies by using a numerical approximation space incorporatingoscillatory functions, chosen based on partial knowledge of the high frequency asymp-totic behaviour of the solution. This is a fast-evolving field - for a recent review of theHNA approach in the boundary element context, see [1]. However, to date, the vastmajority of algorithms, and all the numerical analysis, have been restricted to problemsof scattering by single convex obstacles (e.g. [2]).

In this talk we describe a HNA hp-BEM for acoustic scattering by a class of sound-soft nonconvex polygons. We demonstrate via a rigorous, frequency-explicit error anal-ysis, supported by numerical examples, that to achieve any desired accuracy it is suffi-cient for the number of degrees of freedom to grow only in proportion to the logarithmof the frequency as the frequency increases, in contrast to the at least linear growthrequired by conventional methods. This appears to be the first such numerical analysisresult for any problem of scattering by a nonconvex obstacle.

The main difficulty in moving from the convex case to the nonconvex case is that thehigh frequency asymptotic behaviour of the solution, knowledge of which is required forthe design of the HNA approximation space, becomes significantly more complicated.In particular, there are two new complexities to consider: first, multiply-reflected anddiffracted-reflected rays may be present in the asymptotic solution; second, one mustdeal with the rapid variation of the field across the shadow boundaries between theilluminated and shadow regions of previously diffracted ray fields. Our full hp-erroranalysis is based on new results concerning the high frequency asymptotic solution,and its analytic continuation into the complex plane.

Further details are available in [3].

References

[1] S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and E. A. Spence,Numerical-asymptotic boundary integral methods in high-frequency acoustic scat-

25-3


tering, Acta Numer., 21 (2012), pp. 89–305.

[2] D. P. Hewett, S. Langdon and J. M. Melenk, A high frequency hp boundaryelement method for scattering by convex polygons, SIAM J. Numer. Anal., 51(1)(2013), pp. 629–653.

[3] S. N. Chandler-Wilde, D. P. Hewett, S. Langdon, and A. Twigger,A high frequency boundary element method for scattering by a class of noncon-vex obstacles , University of Reading Department of Mathematics and Statisticspreprint MPS-2012-04, submitted for publication.

A HIGH FREQUENCY BOUNDARY ELEMENT METHODFOR SCATTERING BY TWO-DIMENSIONAL SCREENS

Simon Chandler-Wilde1, David Hewett1,2, Stephen Langdon1c and Ashley Twigger1


2Current address: Mathematical Institute, University of Oxford, UK.

We propose a numerical-asymptotic boundary element method for problems of time-harmonic acoustic scattering of an incident plane wave by a sound-soft two-dimensionalscreen. Standard numerical schemes have a computational cost that grows at leastlinearly with respect to the frequency of the incident wave. Here, we enrich our ap-proximation space with oscillatory basis functions carefully designed to capture thehigh frequency behaviour of the solution. We show that in order to achieve any desiredaccuracy it is sufficient to increase the number of degrees of freedom only in proportionto the logarithm of the frequency, as the frequency increases, and for fixed frequency wedemonstrate exponential convergence with respect to the number of degrees of freedom.

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SOLVING THE STEADY-STATE AB-INITIOLASER THEORY WITH FEM

Sofi Esterhazy1, Matthias Liertzer2a, Jens Markus Melenk1 and Stefan Rotter2

1Institute for Analysis and Scientific Computing,Vienna University of Technology, A-1040 Vienna, Austria, EU

2Institute for Theoretical Physics,Vienna University of Technology, A-1040 Vienna, Austria, EU

[email protected]

The key equations of semi-classical laser theory are the Maxwell-Bloch equations. Thistime-dependent set of nonlinearly coupled partial differential equations describes awealth of phenomena in laser physics ranging from chaotic or pulsed lasing to the mostcommonly studied scenario of steady-state lasing. In the latter case the laser outputconsists of several harmonically oscillating modes which can be described by a reducedset of time-independent nonlinearly coupled Helmholtz equations with open boundaryconditions [1]. In this only very recently proposed approach, which is known as thesteady-state ab-initio laser theory (SALT), each laser mode Ψµ is described as follows,[

∆ + k2µ

(εc(x) +

γ⊥kµ − ka + iγ⊥

D0(x)

1 +∑N

ν=1 Γν |ψν(x)|2

)]Ψµ(x) = 0, (1)

where all of these N equations (i.e., µ = 1 . . . N) are nonlinearly coupled to each otherby the denominator

∑Nν=1 Γν |ψν(x)|2. Note that both the number of modes N , as

well as the modes Ψµ and the corresponding frequencies kµ are a priori unknown andneed to be found within the constraint that the frequencies kµ are real. The givenparameters are the dielectric function εc(x), the Lorentzian gain curve with a peakfrequency of ka and width 2γ⊥, and the pump D0(x).

Up to now, the above SALT equations have only been solved by expanding themodes Ψµ on a special set of biorthogonal basis vectors, also known as constant-fluxstates [1]. In this talk I will present a novel strategy for efficiently solving these equa-tions on top of a finite element discretization without the requirement of such a basisset. For this purpose we employ a nonlinear eigensolver for treating the frequencieskµ, as well as an iterative Newton-Raphson scheme for dealing with the nonlinearityin Ψµ.

References

[1] H.E. Tureci, A.D. Stone, B. Collier, Phys. Rev. A 74, 043822; H.E. Tureci,A.D. Stone, L. Ge, S. Rotter, R.J. Tandy, Nonlinearity 22, C1 (2009), L. Ge,Y. Chong, A.D. Stone, Phys. Rev. A 82, 063824 (2010)

[2] S. Esterhazy, M. Liertzer, J.M. Melenk, S. Rotter, in preparation

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HOW SHOULD ONE CHOOSE THE SHIFT FOR THE SHIFTEDLAPLACIAN TO BE A GOOD PRECONDITIONER

FOR THE HELMHOLTZ EQUATION?

Martin Gander1, I. G. Graham2b and E. A. Spence2c

1 Section de Mathematiques, Universite de Geneve, CH-1211 Geneve, Switzerland,[email protected]

2 Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK,[email protected], [email protected]

There has been much recent research on preconditioning the Helmholtz equation ∆u+k2u = 0 with the inverse of the operator ∆+(k2+iε) (i.e. the Helmholtz operator with acomplex shift), with this method often known as “the shifted Laplacian preconditioner”.Despite many numerical investigations there has been relatively little analysis of howone should chose the shift, ε, for the type of Helmholtz problems arising in applications.

In this talk we present sufficient conditions for the shifted problem to be a goodpreconditioner for the original Helmholtz problem for finite element discretisations ofthe following Helmholtz boundary value problems: (i) the interior impedance problem,and (ii) the sound-soft scattering problem (with the radiation condition imposed as animpedance boundary condition).

For example, let Ω be a bounded Lipschitz domain in Rd with boundary Γ. Let Adenote the matrix arising when the standard variational formulation of the problem

∆u+ k2u = −f in Ω,∂u

∂n− iku = g on Γ, (1)

is solved using the Galerkin method, and let Aε denote the Galerkin matrix arisingfrom the standard variational formulation of the problem with ε, i.e.

∆u+ (k2 + iε)u = −f in Ω,∂u

∂n− iku = g on Γ. (2)

Theorem (Sufficient conditions for A−1ε to be a good preconditioner for A)

Suppose that either Ω is a C1,1 domain in 2- or 3-d that is star-shaped in the sensethat in the sense that

infx∈Γ

(x · n(x)) > 0,

or Ω is a convex polygon or polyhedron, and suppose that the Galerkin discretisationsof both the Helmholtz problem (1) and the problem with a shift (2) are formed usingconforming, piecewise-linear finite elements on a quasi-uniform mesh.

Assume that there exists a c > 0 such that ε ≤ ck2 for all k. Then, there exists ak0 > 0 and C1, C2 > 0 (independent of h, k, and ε) such that if hk

√|k2 − ε| ≤ C1 then∥∥I−A−1

ε A∥∥

2≤ C2

ε

k

for all k ≥ k0.Thus, if ε/k is sufficiently small, A−1

ε is a good preconditioner for Aε.

25-6


HYBRID NUMERICAL-ASYMPTOTIC APPROXIMATIONFOR HIGH FREQUENCY SCATTERING BY

PENETRABLE CONVEX POLYGONS

Samuel Grotha, David Hewett and Stephen Langdon

Department of Mathematics and Statistics, University of Reading, [email protected]

In this talk we consider the two-dimensional problem of scattering of a time-harmonicwave by a penetrable convex polygon Ω with boundary ∂Ω. Standard numerical meth-ods for scattering problems, using piecewise polynomial approximation spaces, requirea fixed number of degrees of freedom per wavelength in order to represent the os-cillatory solution. This leads to prohibitive computational expense in the high fre-quency regime. For problems of scattering by impenetrable scatterers, where thereis just one wavenumber k, much work has been done on developing and analysinghybrid numerical-asymptotic (HNA) methods (see [1]) which overcome this limitation.These HNA methods approximate the unknown boundary data v in a boundary elementmethod framework using an ansatz of the form

v(x, k) ≈ v0(x, k) +M∑m=1

vm(x, k) exp(ikψm(x)), x ∈ ∂Ω, (1)

where the phases ψm are chosen using knowledge of the high frequency asymptotics.The expectation is that if v0 (the geometric optics) and ψm are chosen well, then vmwill be much less oscillatory than v and so can be more efficiently approximated bypiecewise polynomials than v itself.

This talk discusses the challenging task of generalising the HNA methodology toso-called “transmission problems” involving penetrable scatterers. The main difficultyin this generalisation is that the high frequency asymptotic behaviour is significantlymore complicated than for the impenetrable case. In particular, the boundary of thescatterer represents an interface between two media with different wavenumbers, andso we expect to need to modify the ansatz (1) to include terms oscillating at bothwavenumbers.

We discuss how appropriate phases are chosen in the penetrable case using highfrequency asymptotics and hence show how effective HNA approximation spaces canbe constructed for this problem. Moreover, we demonstrate, via comparison with areference solution, that these HNA approximation spaces can approximate the highlyoscillatory solution of the transmission problem accurately and efficiently at all fre-quencies. Full details can be found in [2].

References

[1] S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and E. A. Spence,Numerical-asymptotic boundary integral methods in high-frequency acoustic scat-tering, Acta Numer., 21 (2012), pp. 89–305.

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[2] S. P. Groth, D. P. Hewett, S. Langdon, Hybrid numerical-asymptotic ap-proximation for high frequency scattering by penetrable convex polygons , Univer-sity of Reading Department of Mathematics and Statistics preprint MPS-2013-02,submitted for publication.

25-8


ANALYSIS OF PRECONDITONERS FOR HELMHOLTZEQUATION USING PESUDOSPECTRUM

Antti Hannukainen

Department of Mathematics and Systems Analysis, Aalto University, [email protected]

Finite element simulation of time-harmonic wave propagation problems leads to so-lution of very large indefinite linear systems. When losses, absorbing boundary orimpedance boundary conditions are present, as often in realistic engineering applica-tions, these linear systems are complex valued, non-Hermitian and non-normal. Solvingsuch systems is very challenging. The large size of the system restricts the use of di-rect solvers making preconditioned iterative solvers the method of choice, especially inhigh-frequency domain.

Analyzing the convergence properties of such preconditioned iterative methods forwave-propagation problems is difficult. This is mainly due to indefiniteness and non-normality. Because of the non-normality, the eigenvalues alone do not give informationof the convergence. This problem can be handled by using a suitable convergencecriterion.

In this talk, we focus on convergence analysis of the preconditioned GMRES methodfor the Helmholtz equation with first order absorbing boundary conditions. For GM-RES, convergence criteria suitable for analyzing non-normal systems are based on es-timating the field of values (FOV) or the pseudospectrum.

The FOV based convergence criterion has been used to study two-level and Laplacepreconditioners for Helmholtz equation in media with losses, [1, 2]. The major short-coming of this approach is in handing indefiniteness. This is due to the fact that theFOV is always a convex set containing all eigenvalues. The FOV delivers GMRES con-vergence estimates only when the origin does not belong to the FOV, thus it cannotbe successfully applied when the eigenvalues are clustered around the origin, althoughthey would be far from it. This is the case for time-harmonic Helmholtz equation withabsorbing boundary conditions. Due to this difficulty, we consider a pseudospectrumbased convergence criteria. In this talk, we demonstrate how it can be applied toanalyze preconditioners when the FOV based criteria fails.

References

[1] A. Hannukainen. Field of values analysis of a two-level preconditioner for theHelmholtz equation. SIAM Journal on Numerical Analysis, (accepted for publica-tion), 2013.

[2] M. B. van Gijzen, Y. A. Erlangga, and C. Vuik. Spectral analysis of the discreteHelmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput.,29(5):1942–1958, 2007.

25-9


A DOMAIN DECOMPOSITION PRECONDITIONERFOR MIXED HYBRID INFINITE ELEMENTS

Martin Hubera, Lothar Nannenb and Joachim Schoberlc

Institute for Analysis and Scientific Computing,Vienna University of Technology, Vienna, Austria


In a direct numerical simulation of scalar time-harmonic scattering problems the num-ber of unknowns typically increase like O(κ3) with increasing wavenumber κ. Due tothe pollution effect the situation is even worse. Since direct solvers like PARDISO orMUMPS are due to the memory requirements only useful for small and medium scaleproblems, iterative solvers are of interest for large scale problems. Because usual finiteelement methods lead to indefinite discretization matrices, a suitable preconditioneris needed to ensure convergence of the iterative solver. An additional difficulty arisesfrom the fact, that scattering problems are non-local and therefore often a transpar-ent boundary condition is needed to restrict the problem to a bounded computationaldomain.

In a standard variational formulation of the Helmholtz equation −∆u − κ2u = fthe solution u is assumed to be in H1(Ω). In this talk we are using the mixed hybridformulation of [1] for the solution u ∈ L2(Ω), the discontinuous gradient field σ :=1iκ∇u ∈ Hdiscont(div,Ω), the solution uF ∈ L2(F) on the skeleton F of a triangulationT and the normal components of the gradient field σF ∈ L2(F) on the skeleton. Sincethe volume terms are local, they can be eliminated with static condensation and adiscrete problem for the skeleton variables uF and σF remains.

We use a Robin-type domain decomposition preconditioner with additional penaltyterms in the mixed hybrid formulation, which was presented by Huber and Schoberlon the international conference on domain decomposition methods 2012 in Rennes(see [2]). This preconditioner does not need a coarse grid correction and is thereforewell suited for large scale problems. To realize the transparent boundary conditionthe method is completed by a mixed hybrid discontinuous version of the Hardy spaceinfinite elements presented in [3].

References

[1] P. Monk, J. Schoberl, A. Sinwel. Hybridizing Raviart-Thomas elements for theHelmholtz equation. Electromagnetics, 30(1): 149–176, 2010.

[2] M. Huber. Hybrid discontinuous Galerkin methods for the wave equation. PhDthesis, Vienna University of Technology, Austria, 2013.

[3] L. Nannen, T. Hohage, A. Schadle, and J. Schoberl. Exact sequences of highorder Hardy space infinite elements for exterior Maxwell problems. SIAM J. Sci.Comput., 2013, published online: http://dx.doi.org/10.1137/110860148.

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IMPROVING THE SHIFTED LAPLACE PRECONDITIONERBY MULTIGRID DEFLATION

A. H. Sheikha, D. Lahayeb and C. Vuikc

Delft Institute of Applied Mathematics, TU Delft, Delft, [email protected], [email protected], [email protected]

How does deflation by multigrid vectors affect the performance of the shifted Lapla-cian preconditioned for the Helmholtz equation? We investigate two deflation variantsthat differ in the choice of the coarse grid operator. A rigorous Fourier mode analysisfor the one-dimensional problem with Dirichlet boundary conditions shows that theuse of deflation results in tighter clustering of the spectrum at low wavenumber, andthat undesirable small eigenvalues reappear at high wavenumber. Numerical resultsfor two-dimensional problems show an iteration count that remains constant for lowwavenumber and that increases linearly after a certain threshold value. This thresh-old value is larger in the deflation variant with the coarse grid operator that is moreexpensive to compute. Numerical results with the multilevel extension of the defla-tion algorithm on three-dimensional problems shows a speed-up for sufficiently largeproblems size.

Reference A. H. Sheikh, D. Lahaye and C. Vuik, On the convergence of shiftedLaplace preconditioner combined with multilevel deflation, NLAA, 2013(DOI: 10.1002/nla.1882).

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26: Mini-Symposium: Numerical Methods for Parabolic Equations

26 Mini-Symposium: Numerical Methods for Parabolic

Equations

Organiser: Thomas Wihler

26-1


ENERGY CONSERVATIVE/DISSIPATIVE APPROXIMATIONSOF NONLINEAR EVOLUTION PROBLEMS

Charalambos Makridakis

School of Mathematical and Physical Sciences, University of Sussex, UK.and

Department of Applied Mathematics, University of Crete, [email protected]

We discuss DG methods for nonlinear evolution PDEs including and higher-order termsdescribing diffusion and dispersion/capillarity effects. In particular we shall considerthe isothermal Navier-Stokes Korteweg system for which we present thermodynami-cally consistent DG schemes. We discuss issues related to the error analysis of theapproximations by utilizing appropriate local reconstructions.

26-2


A POSTERIORI ERROR ANALYSISFOR DG IN TIME ALE FORMULATIONS

Andrea Bonito1, Irene Kyza2 and Ricardo H. Nochetto3

1Department of Mathematics, Texas A&M University,College Station, TX 77843-3368, USA,

[email protected],

2Division of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland, UK& IACM-FORTH,

Nikolaou Plastira 100, Vassilika Vouton, GR 700 13 Heraklion-Crete, Greece,[email protected],

3Department of Mathematics, University of Maryland,College Park, MD 20742-4015, USA,

[email protected]

Arbitrary Lagrangian Eulerian (ALE) formulations are useful when approximating so-lutions of problems defined in time-dependent domains, such as fluid-structure inter-actions. For realistic simulations involving fluids in 3d, it is important that the ALEmethod is at least of second order of accuracy. For finite element (FE) schemes, higherorder in time ALE formulations, without any constraints on the time-steps, were notavailable in the literature before [1, 2]. In [1], we propose unconditionally stable discon-tinuous Galerkin (dG) methods in time of any order, for a time-dependent diffusion-dominated problem defined on deformable domains, by enforcing a discrete Reynolds’identity. In [2], we provide optimal order a priori error estimates.

As a continuation to [1, 2], in this talk, we discuss a posteriori error analysis for thedG methods introduced in [1]. The analysis is based on the reconstruction technique,and in particular, in the proposition of a novel dG reconstruction in the ALE frame-work. This reconstruction generalises the notion of the dG reconstruction for problemson time-independent domains, introduced earlier by Makridakis & Nochetto in [3]. Us-ing the properties of the reconstruction and pde techniques for the problem writtenon the ALE framework, we prove optimal order a posteriori error bounds without anyrestrictions on the time-steps. Our analysis allows variable time-steps and gives im-portant information on the behaviour of the error with respect to the movement ofthe domain. More precisely, adaptivity is proven to be beneficial in cases of highlyoscillatory ALE maps. The analysis is illustrated by insightful numerical experiments.

References

[1] A. Bonito, I. Kyza, R.H. Nochetto, Time discrete higher order ALE formulations:Stability, SIAM J. Numer. Anal. 51 (2013) 577–604.

[2] A. Bonito, I. Kyza, R.H. Nochetto, Time discrete higher order ALE formulations:A priori error analysis, to appear in Numer. Math., DOI: 10.1007/s00211-013-0539-3.

26-3


[3] Ch. Makridakis, R.H. Nochetto, A posteriori error analysis for higher order dis-sipative methods for evolution problems,Numer. Math. 104 (2006) 489–514.

DISCONTINUOUS GALERKIN APPROXIMATION OF POROUSFISHER-KOLMOGOROV EQUATIONS

Fausto Cavalli1a, Giovanni Naldi1b and Ilaria Perugia2

1Dipartimento di Matematica, Universita di Milano,Via Saldini 50, 20133 Milano, Italy


2Dipartimento di Matematica, Universita di Pavia, Via Ferrata 1, 27100 Pavia, [email protected]

The reaction-diffusion equation plays an important role in dissipative dynamical sys-tems for physical, chemical and biological phenomena. For example, in chemicalphysics, in order to describe concentration and temperature distributions, the heatand mass transfer are modeled by a non linear diffusion term, while the rate of heatand mass production are described by a non linear reaction term. These terms are oftenconsidered in the form of mass action law. In population dynamics, where the focusis on the evolution of a population density, diffusion terms correspond to a randommotion of individuals, and reaction terms describe their reproduction and interaction,as in a predator-prey model.

Originally, continuum models of population spreading were based on linear diffusion.Then, several authors have pointed out that these diffusion mechanisms can be morerealistically described by (degenerate) non linear diffusion models. The typical nonlinear reaction-diffusion model is as follows:

∂u

∂t−∆p(u) = r(u) in Ω× (0,+∞),

∇p(u) · nΩ = 0 on ∂Ω× (0,+∞),u|t=0 = u0 in Ω.

where Ω ⊂ Rd, with d = 1, 2, is a bounded domain, u is a density, p(u) is a suitablediffusivity, r(u) a reaction function, and u0 the initial datum. If p(u) ' uγ, with γ > 0,the previous equation is known as the porous media equation, which is degeneratein the sense that p′(0) = 0. In this case the solution can develop interfaces betweenthe regions with zero and non zero population density. These sharp fronted solutionsrepresent population density profiles. The reaction is modeled by a generalized Fisher-Kolmogorov term, namely r(u) = u(1− uβ).

From a numerical point of view, the discretization of this problem is challengingbecause the numerical scheme has to reproduce shock waves or fronts of the analyticalsolutions, and preserve stability and invariance properties. We present a new high-ordernumerical methods for the approximation of the porous Fischer-Kolmogorov equation,based on a discontinuous Galerkin space discretization and Runge-Kutta time stepping.These methods are capable to reproduce the main properties of the analytical solutions.We present some preliminary theoretical results and provide several numerical tests.

26-4


ON ADAPTIVE DISCONTINUOUS GALERKIN METHODSFOR PARABOLIC PROBLEMS

Emmanuil H. Georgoulis

Department of Mathematics, University of Leicester,University Road, Leicester LE1 7RH, United Kingdom

[email protected]

Spatial discretisation via discontinuous Galerkin (dG) methods is particularly usefulfor various classes of parabolic problems, such as time-dependent convection-diffusion-reaction and/or spatially fourth order initial/boundary value problems. The suitabilityof spatial discretisation via dG methods for the aforementioned classes of PDE problemslies in their ability to accommodate discontinuous numerical fluxes across element faces:for the former the ability to introduce (discontinuous) upwind fluxes is crucial, whereasfor the latter the lack of H2-conformity is practically desirable. In this talk, some recentexperience on the use of adaptive algorithms based on a posteriori error estimation forfully discrete dG methods of interior penalty type for parabolic convection-diffusion andfourth order parabolic problems will be reviewed. The non-conformity of the spatialdiscretisation introduces some new challenges both in terms of a posteriori error controland in the design of adaptive algorithms, which will be discussed. The talk is basedon joint work with Andrea Cangiani, Stephen Metcalfe, and Juha Virtanen from theUniversity of Leicester, UK.

THE HP -ADAPTIVE GALERKIN TIME STEPPINGMETHOD FOR NONLINEAR DIFFERENTIAL EQUATIONS

WITH FINITE TIME BLOW UP

Barbel Janssen and Thomas P. Wihler

Mathematisches Institut, Universitat Bern,Sidlerstrasse 5, CH-3012 Bern, Switzerland

[email protected]

We consider hp-adaptive Galerkin time stepping methods for nonlinear ordinary dif-ferential equations. The occuring nonlinearity is assumed to be bounded by a constanttimes the solution to a power β which is larger than one. We prove dual based aposteriori error estimates. Existence of discrete solutions is shown using reconstruc-tion techniques. By means of numerical examples we show that the blow up-is wellpreserved.

26-5


MAXIMUM-NORM STRONG APPROXIMATION RATESFOR NOISY REACTION-DIFFUSION EQUATIONS

Omar Lakkisa, G.T. Kossioris and M. Romito

Department of Mathematics, University of Sussex,Falmer near Brighton, England UK

[email protected]

Pointwise error and maximum norm estimates are important in estimating the risk ofrare events happening. I will present new convergence results for the approximation viafinite element in space-time and Monte Carlo (or accelerated versions thereof) in theprobability to the exact solution process of the stochastic Allen-Cahn equation’s. Con-vergence rates are established for the expected maximum norm in space-time, and are”strong” in this sense. This improves previous results by Katsoulakis et al (2007) and,although we focus on specific case, our results can be applied to more general reaction–diffusion equations. The results are based on joint work with Georgios Kossioris andMarco Romito.

26-6


A NEW APPROACH TO ERROR ANALYSIS OFFULLY DISCRETE FINITE ELEMENT METHODS

FOR NONLINEAR PARABOLIC EQUATIONS

Buyang Li1 and Weiwei Sun2

1Department of Mathematics, Nanjing University, Nanjing, [email protected]

2Department of Mathematics, City University of Hong Kong, Hong [email protected]

Error analysis of fully discrete finite element methods for nonlinear parabolic equa-tions often requires certain time-step size conditions (or stability conditions) such as∆t = O(hk), which are widely used in the analysis of the nonlinear PDEs from mathe-matical physics, such as the Navier–Stokes equations, viscoelastic flow, the thermistorproblem, the time-dependent Ginzburg–Landau equations. Such stability conditionsare necessary for explicit schemes, however, due to the various nonlinearities and thecoupling of the equations, it is not known whether these time-step size restrictions arenecessary for linearized semi-implicit schemes or implicit schemes. In numerical anal-ysis people often assume such stability conditions to derive optimal error estimates,however, whether these conditions are necessary is seldom answered. In this talk, weaddress this question by exploring a new approach to analyse the error of fully discretefinite element methods for nonlinear parabolic equations. By this new approach, wefound that the time-step size restrictions are not necessary for most problems. Toillustrate our idea, we analyse two models — the thermistor problem and incompress-ible miscible flow in porous media, with linearized backward Euler scheme for the timediscretization. Previous works on the two models all required certain time-step size con-ditions. Our idea is to introduce a system of elliptic PDEs (the time-discrete parabolicPDEs) and split the error into two parts: ‖un−Un

h ‖ ≤ ‖un−Un‖+ ‖Un−Unh ‖, where

un is the exact solution, Unh is the fully discrete finite element solution, and Un is the

solution of the elliptic PDEs. By our choice, the fully discrete solution Unh is also the

finite element solution of the elliptic PDEs, i.e. finite element approximations of Un.Therefore, the first part of the error is due to time discretization and depends only on∆t, the second part of the error is due to the finite element discretization of the ellipticPDEs and depends only on h. With such an error splitting, the time-discretization andthe spatial discretization are completely decoupled. Analysis of the second part of theerror relies on rigorous analysis of the regularity of Un.

26-7

27: Mini-Symposium: Numerical Methods for Reaction-Transport Equations with Applications inMedicine

27 Mini-Symposium: Numerical Methods for Reaction-

Transport Equations with Applications in Medicine

Organisers: Jennifer Ryan and Fred Ver-molen

27-1


FINITE ELEMENT ANALYSIS OF THE MECHANO-CHEMICALREGULATION OF WOUND CONTRACTION

IN SURGICAL WOUNDS

Etelvina Javierre1, Clara Valero2, Maria Jose Gomez-Benito2

and Jose Manuel Garcia-Aznar2

1 Centro Universitario de la Defensa de Zaragoza, Zaragoza, [email protected]

2 Multiscale in Mechanical and Biological Engineering (M2BE)Aragon Institute of Engineering Research (I3A), Universidad de Zaragoza, Spain

Wound contraction is a highly orchestrated process in which biological and mechanicalsignals regulate collective cell migration and extracellular matrix synthesis to restorethe integrity of a damaged tissue. The aim of this work is to elucidate the effect ofwound depth on the contraction kinetics. Hence, only the primary elements in woundcontraction are considered (i.e. fibroblasts as the main cell species in the dermis, myofi-broblasts as the contractile phe- notype of fibroblasts, a generic inflammatory growthfactor signaling and enhancing cell function, and collagen as the main component ofthe extracellular matrix). Fur- thermore, we consider a direct coupling between cellfunction and ECM deformation, including different cell mechanosensing and mechan-otransduction mechanisms. The governing equations are obtained from conservationlaws for the cellular and chemical species and the ECM momentum. These conservationlaws give rise to a set of nonlinearly coupled convection-diffusion-reaction equationsthat are solved applying the finite element method. Wound morphology and boundaryconditions are discussed as well. We limit our analysis to deep and elongated wounds,for which plane strain hypotheses can be assumed reducing the model to two spatialdimensions. Thus, the upper part of the domain (representing the interface betweenthe tissue and the external environment) behaves as a free boundary with no displace-ment constrains. The results of this model are compared with earlier works that neglectwound depth (treating the wound as a planar surface). Both approaches give a similaroverall contraction kinetics regarding the contracted area vs time plots. However,taking into account wound depth shows significant changes on the migration patternand the contraction evolution.

Acknowledgements: The authors gratefully acknowledge the support of the SpanishMinistry of Economy and Competitiveness through the project DPI2012-32880.

27-2


PRESENTATION OF RESULTS OF FINITE-ELEMENT ANALYSESON A TWO-DIMENSIONAL MECHANOCHEMICAL MODEL

FOR DERMAL WOUND HEALING

D.C. Koppenola and Fred J. Vermolenb

DIAM, Delft University of Technology, Delft, The [email protected], [email protected]

The healing of full-thickness dermal wounds involves a complicated sequence of spa-tially and temporally coordinated processes that can be classified roughly into fourconsecutive, partly overlapping phases: haemostasis, inflammation, proliferation, andremodeling. One of the key processes that takes place during the proliferative phaseis the contraction of the wound. During this process the wound boundaries are drawninwards by biomechanical mechanisms so that the size of the injury is reduced. Thisphenomenon is an important and intrinsic feature in the healing process and it is usuallybeneficial when it is well-balanced. If this balance is disrupted however, then this maycause delayed and / or impaired healing in case of insufficient contraction, while exces-sive contraction may induce low quality repair with substantial scarring. Even thoughintensive research over the last few decades has produced much knowledge about thebiomechanical mechanisms underlying wound contraction and its associated patholo-gies, there is much that remains understood incompletely about these mechanisms andthe etiology of the pathologies that may develop.

In order to gain more insight into the mechanisms underlying wound contraction, a newdeterministic model, consisting of a system of nonlinearly coupled parabolic-hyperbolicpartial differential equations, has been developed recently by Murphy et al. [1]. Thismodel allows a detailed evaluation of the effects of strong interactions between me-chanical changes and some of the most important biological entities involved in thecontraction of full-thickness dermal wounds. This evaluation is accomplished by de-scribing the interactions between (myo-)fibroblasts, two growth factors, collagen, theenzyme collagenase and a mechanical force balance as a result of the physico-chemicalproperties of the extracellular matrix and the cell-generated traction forces. Murphy etal. assume that the wound is long, thin and much longer than it is deep and therefore itis appropriate for them to consider a one-dimensional representation. After obtainingsome results from a numerical analysis of this one-dimensional model, they made acomparison between the model predictions and experimental data on human dermalwound healing which shows that all the essential features are well matched.

Given this nice match between the one-dimensional model predictions and the experi-mental data, and the substantial impact the geometry of the wound has on the healingprocess, we investigate the influence of geometry. To this end, we extended the model ofMurphy et al. to two spatial dimensions and we used various initial conditions to simu-late several wound geometries. We will present some of the results from finite-elementanalyses.

27-3


References

[1] Murphy, K.E., Hall, C.L., Maini, P.K., McCue, S.W., McElwain, D.L.S., A Fi-brocontractive Mechanochemical Model of Dermal Wound Closure IncorporatingRealistic Growth Factor Kinetics. Bulletin of Mathematical Biology 74, pp. 1143-1170, 2012.

MATHEMATICAL MODELLING AND NUMERICAL SIMULATIONSOF ACTIN DYNAMICS IN THE EUKARYOTIC CELL

Anotida Madzvamuse1a, Uduak George1 and Angelique Stephanou2

1University of Sussex, School of Mathematical and Physical Sciences,Pevensey III, 5C15. BN1 9QH. Brighton, UK.

[email protected]

2UJF-Grenoble 1, CNRS, Laboratoire TIMC-IMAG UMR 5525,DyCTiM research team, Grenoble, F-38041, France

In this talk I will present a model for cell deformation and cell movement that couplesthe mechanical and biochemical properties of the cortical network of actin filamentswith its concentration. Actin is a polymer that can exist either in filamentous form (F-actin) or in monometric form (G-actin) (Chen et al., in Trends Biochem Sci 25:1923,2000) and the filamentous form is arranged in a paired helix of two protofilaments(Ananthakrishnan et al., in Recent Res Devel Biophys 5:3969, 2006). By assumingthat cell deformations are a result of the cortical actin dynamics in the cell cytoskele-ton, we consider a continuum mathematical model that couples the mechanics of thenetwork of actin filaments with its biochemical dynamics. Numerical treatment of themodel is carried out using the moving grid finite element method (Madzvamuse et al.,in J Comput Phys 190:478500, 2003). Furthermore, by assuming slow deformations ofthe cell, we use linear stability theory to validate the numerical simulation results closeto bifurcation points. Far from bifurcation points, we show that the mathematicalmodel is able to describe the complex cell deformations typically observed in exper-imental results. Our numerical results illustrate cell expansion, cell contraction, celltranslation and cell relocation as well as cell protrusions in agreement with experimen-tal observations. In all these results, the contractile tonicity formed by the associationof actin filaments to the myosin II motor proteins is identified as a key bifurcationparameter. Cell migration plays a critical and pivotal role in a variety of biologicaland biomedical disease processes and is important for emerging areas of biotechnologywhich focus on cellular transplantation and the manufacture of artificial tissues andsurfaces, as well as for the development of new therapeutic strategies for controllinginvasive tumor cells.

27-4


ANALYZING THE TREATMENT OF A BACTERIAL INFECTIONIN A WOUND USING OXYGEN THERAPY

Richard Schugart

Department of Mathematics, West Kentucky University, [email protected]

In this work, a reduced mathematical model is to be presented for the treatmentof a bacterial infection in a wound using oxygen therapy. The model considers theinteractions of neutrophils and bacteria in a wound combined with the use of hyperbaricand topical oxygen therapies to aide in the removal of bacteria from the wound. Botha traveling wave solution and numerical simulations of the model will be presented.The application of optimal control theory for the model will be discussed.

27-5


A SEMI–STOCHASTIC MODEL FORTHE IMMUNE RESPONSE SYSTEM

Fred J. Vermolen

Delft Institute of Applied Mathematics,Delft University of Technology, The Netherlands

[email protected]

The immune response system is a vital defense mechanism for human beings. Defectscan be caused by a shortage of white blood cells (such as lymphocytes, monocytes,neutrophils, or macrophages), or by an impairment of the transmittivity of the venule(small blood vessel) wall. In case of a (bacterial) infection, bacterial secretion alertsmacrophages that secrete chemokines that are detected by the endothelial cells thatconstitute the venule walls. From there, cytokines are released and detected by thewhite blood cells that are advected in the blood stream. The white blood cells movetowards the venule walls and leave the venule by transmigration through the venulewall.

In this presentation, we will discuss a semi-stochastic model for the immune re-sponse system. The model is based on simple principles, such as Poisseuille flow throughvenules, Brownian motion of bacteria, fundamental solutions for bacterially secreted,and migration and deformation of white blood cells. The gradient of a chemical stim-ulus acts as the driving force of white blood cell migration and deformation. Thisgradient is obtained by a superposition of point sources that mimic bacterial secre-tion. The white blood cells are modeled to be elastic and hence spring forces definethe driving force of the white blood cell to get reshaped in its original geometry. Thecellular migration and deformation is modeled through straightforward formalisms forcurve shape evolution, and not by the solution of partial differential equations. Hencethe model is based on systems of ordinary (stochastic) differential equations and finiteelement strategies are not used.

We show the mathematical formulation of the model, as well as some implicationsof the model, such as a parameter variation based on statistical principles.

27-6


MULTISCALE MODELS OF TUMOR CELLS: FROM IN-VITROAGGREGATES TO IN-VIVO VASCULARIZED TUMORS

Irene Vignon-Clementela, Nick Jagiella and Dirk Drasdo

INRIA, Paris Rocquencourt, Francea [email protected]

This work aims at better understanding the dynamic interplay between tumor cells andtheir environment, and its assessment by medical imaging, to in fine improve treatment.A multiscale model was built that represents the generic features of such an interplay.Tumor cells are modeled individually, interacting and evolving by rules that depend ontheir environment. In turn, the cells modify their environment (nutrients, etc), whichis represented by nonlinear continuum reaction diffusion equations. Additional brickswere added to model the vascularization that provides nutrients to the tumor cellswhen studying the in-vivo situation. Angiogenesis & vessel remodeling induced by theover proliferating and hypoxic tumor cells, have been included in this generic modelto study this interplay via the molecular scale. A second part of the work has beento get closer to real settings. To study the in-vitro multicellular growth of a specificlung cancer cell type, the model has been refined based on experimental data. Selectedmechanisms have been proposed to explain the spatio-temporal information of theseexperimental data. Finally, we will present some challenges and novel research axes togo to the in-vivo scale. Numerical methods and results will be presented throughoutthe talk.

27-7

28: Mini-Symposium: Numerical methods for contact and other geometrically non-linear problems

28 Mini-Symposium: Numerical methods for con-

tact and other geometrically non-linear prob-

lems

Organisers: Alexey Chernov and MatthiasMaischak

28-1


PARALLEL SOLUTION OF CONTACT SHAPE OPTIMIZATIONPROBLEMS WITH COULOMB FRICTION BASED

ON DOMAIN DECOMPOSITION

P. Beremlijskia Tomas Brzobohatyb Tomas Kozubekc and Alexandros Markopoulosd

VSB-Technical University of Ostrava,17. listopadu 15, 708 33 Ostrava-Poruba, Czech Republic.

[email protected], [email protected],[email protected], [email protected]

We shall first briefly introduce the FETI based domain decomposition methodologyadapted to the solution of multibody contact problems with Coulomb friction. Theseproblems play a role of the state problem in contact shape optimization problems withCoulomb friction. We use a modification of FETI that we call Total FETI, whichimposes not only the compatibility of a solution across the subdomain interfaces, butalso the prescribed displacements. For solving a state problem we use the methodof successive approximations. Each iterative step of the method requires us to solvethe contact problem with given friction. As a result, we obtain a convex quadraticprogramming problem with a convex separable nonlinear inequality and linear equal-ity constraints. For the solution of such problems we use a combination of inexactaugmented Lagrangians in combination with active set based algorithms.

The discretized problem with Coulomb friction has a unique solution for small coef-ficients of friction. The uniqueness of the equilibria for fixed controls enables us to applythe so-called implicit programming approach. Its main idea consists in minimizationof a nonsmooth composite function generated by the objective and the control-statemapping. The implicit programming approach combined with the differential calculusof Clarke was used for a discretized problem of 2D shape optimization. There is nopossibility to extend the same approach to the 3D case. To get subgradient informationneeded in the used numerical method we use the differential calculus of Mordukhovich.Application of Total FETI method to the solution of the state problem and sensitivityanalysis allows massively parallel solution of these problems. The effectiveness of ourapproach is demonstrated by numerical experiments.

References

[1] P. Beremlijski, J. Haslinger, M. Kocvara, R. Kucera and J. Outrata: Shape Op-timization in Three-Dimensional Contact Problems with Coulomb Friction. In:SIAM Journal on Optimization 20/1, 2009, pp. 416-444.

[2] P. Beremlijski, T. Brzobohaty, T. Kozubek, A. Markopoulos and J. Outrata: Par-allel solution of contact shape optimization problems with Coulomb friction basedon domain decomposition. In: WIT Transactions on the Built Environment 124,2012, pp. 285-295.

28-2


PARALLEL SOLUTION OF ELASTO-PLASTIC PROBLEMS

Martin Cermaka and Michal Mertab

Centre of Excellence IT4Innovations, VSB-TU Ostrava, Czech [email protected], [email protected]

In this work we present the parallel solution of elasto-plastic problems. We assumethe von Mises plastic criterion with kinematic hardening and the associated plasticflow rule. For the time discretization we use the implicit Euler method and the corre-sponding one-time-step problem is formulated with respect to unknown displacement.For the space discretization we use the finite element method and we parallelize theresulting problem using the Total-FETI method.

Our parallel implementations are based on the Trilinos and the PETSc softwareframeworks. Their performance and scalability are compared on 2D and 3D bench-marks. The scalability tests were carried out using the HECToR supercomputer atEPCC, UK.

28-3


CONVERGENCE ANALYSIS FOR MULTILEVEL VARIANCEESTIMATORS IN MULTILEVEL MONTE CARLO METHODSAND APPLICATION FOR RANDOM OBSTACLE PROBLEMS

Alexey Chernova and Claudio Bierigb

Hausdorff Center for Mathematics and Institute for Numerical Simulation,University of Bonn, Germany.


The Multilevel Monte Carlo Method (MLMC) is a recently established sampling ap-proach for uncertainty propagation for problems with random parameters. Under cer-tain assumptions, MLMC allows to estimate e.g. the mean solution and k-point corre-lation functions at essentially the same overall computational cost as the cost requiredfor solution of one forward problem for a fixed deterministic set of parameters.

However, in many practical applications estimation of the variance (along with themean) is the main goal of the computations. In this case the variance can be potentiallycomputed from correlation functions in the post-processing step. This approach hastwo drawbacks:

1. Optimal complexity approximation of correlation functions involves quite cum-bersome sparse tensor product constructions. It is desirable to avoid it if thevariance is the aim of the computation.

2. Computation of the variance from the 2-point correlation function is prone tonumerical instability, specially in the case of small variances.

Much less is known about direct estimation of the variance, potentially overcomingthese difficulties. In this talk we present new convergence theorems for the multilevelvariance estimators. As a result, we prove that under certain assumptions on theparameters, the variance can be estimated at essentially the same cost as the mean,and consequently as the cost required for solution of one forward problem for a fixeddeterministic set of parameters. We comment on fast and stable evaluation of theestimators suitable for parallel large scale computations.

The suggested approach is applied to a class of scalar random obstacle problems,a prototype of contact between deformable bodies. In particular, we are interested inrough random obstacles modelling contact between car tires and variable road surfaces.Numerical experiments support and complete the theoretical analysis.

28-4


SCALABLE ALGORITHMS AND CONDITIONING OFCONSTRAINTS ARISING FROM VARIATIONALLY CONSISTENT

DISCRETIZATION OF CONTACT PROBLEMS

Zdenek Dostal, Tomas Kozubek and Oldrich Vlach

VSB–Technical University of Ostrava, CZ-70833 Ostrava, Czech [email protected]

The results related to the development of theoretically supported scalable algorithmsfor the solution of large scale transient contact problems of elasticity will be brieflyreviewed [1]. The algorithms that were originally developed for matching discretiza-tion combine the Total FETI/BETI based domain decomposition methods adaptedto the solution of 2D and 3D multibody contact problems of elasticity with optionalpreconditioning by conjugate projector or dual scaling with our in a sense optimal al-gorithms for the solution of resulting quadratic programming or QPQC problems. Thetheoretical results are qualitatively the same as the classical results on scalability ofFETI/BETI for linear elliptic problems, i.e., the inequality constraints are treated ina sense for free. In this presentation we discuss generalization of these results to theproblems discretized by non-matching grids. We consider implementation of the non-penetration condition by the variationally consistent discretization introduced recentlyby B. I. Wohlmuth [2] and study performance of related algorithms. We give boundson the spectrum of the related matrices for some mortar discretizations and comparethem with the numerical values obtained for some special cases [3]. We also provideresults of numerical experiments showing that variationally consistent discretizationpreserve efficiency of the scalable TFETI/TBETI solvers.

References

[1] Z. Dostal, T. Kozubek, T. Brzobohaty, A. Markopoulos, and O. Vlach, ScalableTFETI with optional preconditioning by conjugate projector for transient contactproblems of elasticity, Computer Methods in Applied Mechanics and Engineering247–248 (2012) 37–50.

[2] B. I. Wohlmuth Variationally consistent discretization schemes and numericalalgorithms for contact problems, Acta Numerica (2012) 569–734.

[3] Z. Dostal, T. Kozubek, O. Vlach, and T. Brzobohaty, On scalable algorithms andconditioning of constraints arising from variationally consistent discretization ofcontact problems.

28-5


PARALLEL SOLUTION OF CONTACTPROBLEMS BASED ON TFETI

Zdenek Dostala, Tomas Brzobohatyb, Tomas Kozubekc,Alexandros Markopoulosd and Oldrich Vlache

VSB-Technical University of Ostrava,17. listopadu 15, 708 33 Ostrava-Poruba, Czech [email protected], [email protected],


In our contribution we briefly review the TFETI based domain decomposition method-ology adapted to the solution of 2D and 3D multibody contact problems. For thesolution of the resulting quadratic programming problems our in a sense optimal al-gorithms are used and we will present them together with their powerful ingredients.Our results obtained for elastic contact problems are extended to the contact problemswith non-matching grids which necessarily arise, e.g., in the solution of transient orshape optimization problems. We consider both standard engineering approaches suchas node to segment, or mortar elements. The aim is to get the constraint matrix B withnearly orthogonal rows, which is required assumption of our algorithms. The simplebounds on the singular values of the resulting matrix B as well as the results of numeri-cal experiments, including both the academic examples and some problems of practicalinterest will be presented. We conclude that the normalized orthogonal mortars pro-posed by Wohlmuth can be used to approximate the non-penetration conditions in away that complies with the requirements of the FETI methods.

References

[1] Z. Dostal, D. Horak, R. Kucera: Total FETI - an easier implementable variantof the FETI method for numerical solution of elliptic PDE, Communications inNumerical Methods in Engineering 22 (2006) 1155–1162.

[2] Z. Dostal: Optimal quadratic programming algorithms: with applications to vari-ational inequalities, Springer, New York, 2009.

[3] Z. Dostal, T. Kozubek, A. Markopoulos, M. Mensık: Cholesky decdompositionand a generalized inverse of the stiffness matrix of a floating structure with knownnull space, Applied Mathematics and Computation 217 (2011) 6067-6077.

[4] R. Kucera, T. Kozubek, A. Markopoulos: On large-scale generalized inverses insolving two-by-two block linear systems, submitted to Linear Algebra and Its Ap-plications (2012).

[5] T. Kozubek, Z. Dostal, T. Brzobohaty, O. Vlach, A. Markopoulos: ScalableTFETI with optional preconditioning by conjugate projector for transient Com-puter, Methods in Applied Mechanics Engineering, 2012, 247-248, 37-50

28-6


[6] B. I. Wohlmuth: Variationally consistent discretization schemes and numericalalgorithms for contact problems. Acta Numerica, 20 (2011), 569734.

28-7


LOCAL AVERAGING OF CONTACT WITHNON MATCHING MESHES

Guillaume Drouet1 and Patrick Hild2

1 LaMSID - Laboratoire de Mecanique des Structures Industrielles Durables,UMR 8193, EDF, CNRS, CEA, France,

[email protected]

2 Institut de Mathematiques de Toulouse,UMR 5219, Universite Toulouse 3, CNRS, France,

[email protected]

Finite element methods are currently used to approximate the unilateral contact prob-lem. Such a problem shows a nonlinear boundary condition, which roughly speakingrequires that (a component of) the solution u is nonpositive (or equivalently nonnega-tive) on (a part of) the boundary of the domain Ω. This nonlinearity leads to a weakformulation written as a variational inequality which admits a unique solution) andthe regularity of the solution shows limitations whatever the regularity of the data is.A consequence is that only finite element methods of order one and of order two are ofinterest.

In this talk we limit ourselves to the finite element method of order one in two andthree space dimensions and we consider a discrete contact condition which requires,in the case of a sole body in contact with a rigid foundation, that the approximatesolution uh is nonpositive in average on some local patches comprising several contactelements that form a partition of the contact zone. The corresponding discrete convexcone of admissible functions is given by

Kh =

vh ∈ V h :

∫Tm

vhNdΓ ≤ 0 ∀Tm ∈ TM

,

where TM is a macro-mesh. The discrete convex cone of admissible solutions is not asubset of the continuous convex cone of admissible solutions. So, in order to achievethe error analysis, we have to bound two terms, the approximation error and theconsistency error.

First, we show that if the macro-mesh Tm satisfies a reasonnable technical assump-tion, we are able to construct an average preserving operator wich leads us to anoptimal approximation error so that the convergence error of the method will be givenby the consistency error. In dimension 2, this choice gives us slightly better theoreticalresults than the existing ones.

Then, we show that this approach can be easily extended to multi body contactwith non matching meshes in two and three space dimensions which is the goal of thismethod.

Finally, we show that the assumption made on TM could be linked to the inf-supconditions when considering the mixed method associated to the variational inequality.

28-8


BE/FE APPROXIMATION OF HIGHER ORDER FOR NONSMOOTHPROBLEMS, EFFECTIVE QUADRATURE, AND TIME

DISCRETIZATION BY IMPLICIT RUNGE-KUTTA METHODS

Joachim Gwinner

Institute of Mathematics, Department of Aerospace Engineering,University of the Federal Army Munich, D - 85577 Neubiberg, Germany

[email protected]

In this talk we address higher order approximation in the FEM , BEM or in FEM-BEMcoupling for unilateral problems including nonsmooth friction-type functionals. Suchan approximation leads to a nonconforming discretization scheme.To fix ideas considerthe friction-type functional

j(v) =

∫ΓC

g|v| ds .

Then in contrast to previous related work we approximate such a functional usingGauss-Lobatto quadrature by its e.g. p-approximation

jp(v) =∑

e∈Sh,e⊂ΓC

p∑j=0

ωp+1j

∣∣∣v Fe(ξp+1j )

∣∣∣and take the quadrature error of the friction functional into account of the error anal-ysis.

Moreover we are concerned with full space time discretization of related nons-mooth parabolic and evolutionary inequality problems employing implicit Runge-Kuttametods for time discretization. This talk is based on the recent work [1, 2, 3].

References

[1] J. Gwinner, On the p-version approximation in the boundary element method fora variational inequality of the second kind modelling unilateral contact and givenfriction, Appl. Numer. Math. 60 (2010) 689 – 704.

[2] J. Gwinner, hp-FEM convergence for unilateral contact problems with Tresca fric-tion in plane linear elastostatics, J. Comput. Appl. Math. (2013, in press).

[3] J. Gwinner & M. Thalhammer, Full discretisations for nonlinear evolutionary in-equalities based on stiffly accurate Runge-Kutta and hp-finite element methods, toappear.

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A DISCRETIZATION FOR DYNAMIC LARGEDEFORMATION CONTACT PROBLEMS OFNONLINEAR HYPERELASTIC CONTINUA

Ralf Kornhuber1a, Oliver Sander2 and Jonathan Youett1c

1Department of Mathematics and Computer Science,Freie Universitat Berlin, Germany.

[email protected], [email protected],

2Department of Mathematics, IPGM, RWTH Aachen, [email protected]

In this talk we present a discretization for dynamic large deformation frictionless con-tact problems of nonlinear hyperelastic continua. The equations of motion are derivedby a nonsmooth Hamilton principle which results in a differential inclusion in theframework of generalized gradients [F.H. Clarke, Optimization and Nonsmooth Anal-ysis, 1983]. We use a mortar method with dual basis functions for the discretizationof the contact constraints which are known to be very robust. For the time discretiza-tion of the inclusion we apply a contact-stabilized midpoint rule which leads to spatialproblems that can be reformulated as minimization problems. These non-convex min-imization problems are then solved using a quasi Newton SQP method. The efficiencyof SQP methods mainly depends on the quality of the solver for the quadratic sub-problems. In our proposed method we use an approximation of the linearized contactconstraints for the subproblems, which allows us to apply a special basis transformationthat leads to a decoupling of the constraints [B. Wohlmuth and R. Krause, MonotoneMethods on Non-Matching Grids for Nonlinear Contact Problems,2003]. The result-ing quadratic problem is then solved fast and efficiently using a monotone multigridmethod. Numerical results illustrate the energy and convergence behaviour of theproposed scheme.

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PARALLEL LEVEL SET METHODS FOR LARGEDEFORMATION CONTACT PROBLEMS

Rolf Krause1a, Valentina Poletti1b, Roberto Croce1c and Petr Kotas2

1Institute of Computational Science, University of Lugano, [email protected], [email protected]

[email protected]

2Department of Applied Mathematics,Technical University of Ostrava, Czech Republic

In this talk, we present and discuss a new approach for the efficient and highly parallelcomputation of the distance between a deformable body and an obstacle using level-setmethods.

The numerical solution of contact problems involving large deformations requiresan acurate treatment of the non-linear contact constraints. Since these constraintsdepend on the current configuration, during any iterative solution process they haveto be updated repeatedly. Unfortunately, straight-forward approaches as, e.g. thedirect computation of the closest point projection, are computationally intensive andpossibly error-prone. In view of the potential ambiguity of closest points we thereforefollow an alternative approach where the contact condition is described by a signeddistance function to the obstacles boundary. This approach allows for a fast androbust computation of the distance to the obstacle for different configurations by simplyevaluating the precomputed signed-distance function.

However, in view of the ever increasing parallelism on modern computers and super-computers, the signed-distance function should be computable in parallel. In this talk,we first discuss the difficulties when aiming at a parallel computation of the signed-distance function. We then present our new and fully parallel fast marching method,which allows for the efficient and parallel computation of the signed-distance function.Numerical examples illustrating the performance of our approach will be given.

28-11


ERROR ESTIMATORS FOR A PARTIALLY CLAMPEDPLATE PROBLEM WITH BOUNDARY ELEMENTS

Matthias Maischak

BICOM, Department of Mathematical Sciences, Brunel University, Uxbridge, [email protected]

The biharmonic equation models the bending of a thin elastic plate. Restricting thecorresponding minimization problems on a convex subset of possible boundary condi-tions, cf. [2], describing restrictions on the clamping of the plate boundary, we obtaina variational inequality. Using the fundamental solution we obtain a symmetric inte-gral operator representation [5]. The higher regularity requirements of the biharmonicoperator lead to the usage of C1 basis functions, as well as to a C1 regular representa-tion of the boundary. We will first present a high-order numerical quadrature scheme,suitable for a hp-method on curved boundaries and second, we will derive a-posteriorierror estimates based on a hierarchical decomposition, cf. [8, 9]. Several numericalexamples underline the theoretical results.

References

[1] L. Caffarelli, The obstacle problem, (1998).

[2] L. A. Caffarelli, A. Friedman, and A. Torelli, The two-obstacle problemfor the biharmonic operator, Pacific Journal of Mathematics, 103 (1982), pp. 325–335.

[3] F. Cuccu, B. Emamizadeh, and G. Porru, Optimization problems for anelastic plate, Journal of Mathematical Physics, 47 (2006).

[4] W. Han, D. Hua, and L. Wang, Nonconfoming finite element methods for aclamped plate with elastic unlateral obstacle, Journal of Integral Equations andApplications, 18 (2006), pp. 267–284.

[5] G. Hsiao and W. L. Wendland, Boundary Integral Equations, vol. 164 ofApplied Mathematical Sciences, Springer-Verlag, Berlin, 2008.

[6] Y. Jeon and W. McLean, A new boundary element method for the biharmonicequations with dirichlet boundary conditions, Advances in Computational Mathe-matics, 19 (2003), pp. 339–354.

[7] M. Maischak, P. Mund, and E. P. Stephan, Adaptive multilevel bem foracoustic scattering, Comput.Methods Appl.Mech.Engrg., 150 (1997), pp. 351–367.

[8] M. Maischak and E. P. Stephan, Adaptive hp versions of bem for Signoriniproblems, Applied Numerical Methods, 54 (2005), pp. 425–449.

[9] , Adaptive hp-versions of boundary element methods for elastic contact prob-lems, Comput. Mech., 39 (2007), pp. 597–607.

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[10] C. Tosone and A. Maceri, The clamped plate with elastic unilateral obstacles:a finite element approach, Math. Models Methods Appl. Sci., 13 (2003), pp. 1231–1243.

OPTIMAL ACTIVE-SET AND SPECTRAL ALGORITHMSFOR THE SOLUTION OF 3D CONTACT PROBLEMS

WITH ANISOTROPIC FRICTION

Lukas Pospısila, Zdenek Dostalb and Tomas Kozubekc

FEI VSB-Technical University Ostrava,Tr 17 listopadu, CZ-70833 Ostrava, Czech Republic


The formulation of a contact problem with anisotropic friction in terms of contactstresses leads to the minimization of a strictly convex quadratic function subject toseparable inequality ellipsoidal constraints and non-penetration inequality linear con-straints.For solving this optimizing problem, we present a modification of our recently developedin a sense optimal MPGP algorithms [1]. These active-set based algorithms explorethe faces by the conjugate gradients and change the active sets and active variables bythe gradient projection with the constant steplength, which guarantees the monotonedescend of the cost function. We present also a modification, which replaces the con-stant steplength by the projected Barzilai-Borwein (spectral) method [2], that rapidlydecreases the number of projection steps.

The efficiency of our algorithms is illustrated on the solution of a 3D contact problemof one cantilever beam and rigid obstacle in mutual contact with the Tresca friction.

F

ΓD

ΓCx

y

z

rigid obstacle

ΓFFg

For solving such a large-scaled problem, we use a Total FETI (TFETI) based domaindecomposition, which involves additional equality linear constraints. Here MPGP isused in the inner loop of the Semi-monotonic augmented Lagrangian method.

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References

[1] Dostal, Z., Kozubek, T.: An optimal algorithm and superrelaxation for minimiza-tion of a quadratic function subject to separable convex constraints with applications,Mathematical Programming, vol. 135, pp. 195–220, 2012.

[2] Birgin, E.G., Martınez, J.M., and Raydan, M.: Nonmonotone spectral projectedgradient methods on convex sets, SIAM Journal on Optimization 10, pp. 1196–1211, 2000.

HP -ADAPTIVE FEM WITH BIORTHOGONAL BASISFUNCTIONS FOR ELLIPTIC OBSTACLE PROBLEMS

Andreas Schroder1 and Lothar Banz2

1 Department of Mathematics, University of Salzburg, Austria,[email protected]

2 Institute of Applied Mathematics, Leibniz University Hanover, Germany,[email protected]

In this talk, the discretization of a non-symmetric elliptic obstacle problem with hp-adaptive conforming finite elements is discussed. For this purpose, a higher-ordermixed finite element discretiziation is introduced where the dual space is discretized viabiorthogonal basis functions. hp-adaptivity is realized via automatic adaptive mesh re-finement based on a posteriori error estimates. The use of biorthogonal basis functionsleads to an algebraic system including box constraints and componentwise complemen-tarity conditions. This structure is exploited to apply efficient semismooth Newtonmethods using a penalized Fischer-Burmeister NCP-function in each component. Toinclude meshes with hanging nodes and varying polynomial degrees resulting from hp-adaptive mesh refinements the use of appropriate connectivity matrices is proposed.Several numerical experiments confirm the applicability of the hp-adaptive scheme.

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HIGH ORDER BEM FOR FRICTIONAL CONTACT PROBLEMS

Ernst P. Stephan

Institute for Applied Mathematics,Leibniz Universitat Hannover, Hannover, Germany.

[email protected]

We consider frictional contact for non-linear elasticity in R2. We use the Poincare-Steklov operator, which realizes the Dirichlet-to-Neumann map, and represent thenegative of the unknown normal traction on the contact boundary by a Lagrange mul-tiplier. Herewith we derive a mixed formulation which is equivalent to a variationalinequality on the contact boundary, where the non-penetration condition is incorpo-rated in the convex set of admissible ansatz and test functions. Both formulations areuniquely solvable. We use Gauss-Lobatto-Lagrange basis functions on a regular meshon the contact boundary for the primal variable and biorthogonal basis functions ofthe same degree on the same mesh for the Lagrange multiplier. We present a reliableand efficient a posteriori error estimate of residual type for the Galerkin solution ofthe mixed formulation. The discrete mixed system is solved by the semi-smooth New-ton algorithm in combination with a penalized Fischer-Burmeister complementarityfunction taking care of the contact condition. Numerical experiments are given whichsupport our theoretical results.

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29: Mini-Symposium: Numerical methods for fully nonlinear elliptic equations

29 Mini-Symposium: Numerical methods for fully

nonlinear elliptic equations

Organisers: Susanne Brenner, Klaus Bohmerand Michael Neilan

29-1


PSEUDO TRANSIENT CONTINUATION AND TIME MARCHINGMETHODS FOR MONGE-AMPERE TYPE EQUATIONS

Gerard Awanou

Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois, Chicago

322 Science and Engineering Offices (SEO) M/C 249, 851 South Morgan Street,Chicago, IL 60607-7045, USA

[email protected]

We present two numerical methods for the fully nonlinear elliptic Monge- Ampereequation. The first is a pseudo transient continuation method and the second is a purepseudo time marching method. The methods perform well across a wide range of dis-cretizations: C1 conforming approximations, standard Lagrange finite element method,standard finite difference method, etc. We will present recent results on the conver-gence of the iterative methods for solving the nonlinear system of equations resultingfrom the discretizations and prove the convergence of the discretizations for smoothsolutions. We give numerical evidence that the methods are also able to capture theviscosity solution of the Monge-Ampere equation. Even in the case of the degener-ate Monge-Ampere equation, the time marching method appears also to compute theviscosity solution.

GENERAL FULL DISCRETIZATIONS FOR CENTER MANIFOLDS,HERE FOR FULLY NONLINEAR EQUATIONS AND FEMS

Klaus Bohmer

Fachbereich Mathematik und Informatik, Philipps University, Marburg, [email protected]

Dynamical systems are often studied for parabolic PDEs and their discretization. Thenonlinear elliptic parts are either equations or system of order 2 or 2m,m > 1. Spaceand time discretization methods, so called full discretizations, are necessary to deter-mine the dynamics on stable and center manifolds for these problems.

My general theory essentially starts with the standard space discretization methodsused for nonlinear elliptic PDEs. The coefficients of the Taylor expansion of a spacediscretized center manifold and its normal form converge to those of the original centermanifold. Then standard, e.g., Runge–Kutta, or geometric time discretization methodscan be applied to the discrete center manifold, a small dimensional system of ordinarydifferential equations. These results are applicable to any of the recent FEMs for fullynonlinear elliptic/parabolic PDEs, some of them presented in this mini symposium.

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NUMERICAL SOLUTION OF MONGE-AMPERE EQUATIONON DOMAINS BOUNDED BY PIECEWISE CONICS

Oleg Davydov1 and Abid Saeed2

1Department of Mathematics and Statistics, University of Strathclyde, Scotland,[email protected],

2Department of Mathematics, Kohat University of Science & Technology, Kohat,Pakistan,

[email protected]

We introduce new C1 polynomial finite element spaces for curved domains bounded bypiecewise conics using Bernstein-Bezier techniques. These spaces are employed to solvefully nonlinear elliptic equations. Numerical results for several test problems for theMonge-Ampere equation on domains of various smoothness orders endorse theoreticalerror bounds given previously by K. Bohmer.

DISCONTINUOUS GALERKIN FINITE ELEMENTDIFFERENTIAL CALCULUS AND APPLICATIONS

Xiaobing Feng

Department of Mathematics, The University of Tennessee,Knoxville, TN 37996, U.S.A.

[email protected]

In this talk I shall first present a newly developed discontinuous Galerkin finite ele-ment differential calculus theory for approximating weak (or distributional) derivativesof broken Sobolev functions. After the definition is introduced, various properties andcalculus rules (such as product and chain rule, integration by parts formula and di-vergence theorem) for the numerical derivatives will be outlined. I shall then discusshow the proposed discontinuous Galerkin finite element differential calculus can be(conveniently) used to build discretization methods for linear and nonlinear (includingfully nonlinear) PDEs. This is a jointly work with Michael Neilan of University ofPittsburgh and Tom Lewis of the University of Tennessee at Knoxville.

29-3


A FINITE ELEMENT METHOD FORHAMILTON-JACOBI-BELLMAN EQUATIONS

Max Jensen1 and Iain Smears2

1Department of Mathematics, University of Sussex, [email protected]

2Mathematical Institute, University of Oxford, [email protected]

Hamilton-Jacobi-Bellman equations describe how the cost of an optimal control prob-lem changes as problem parameters vary.

This talk will address how Galerkin methods can be adapted to solve these equationsefficiently. In particular, it is discussed how the convergence argument by Barles andSouganidis for finite difference schemes can be extended to Galerkin finite elementmethods to ensure convergence to viscosity solutions. A key question in this regard isthe formulation of the consistency condition. Due to the Galerkin approach, coercivityproperties of the HJB operator may also be satisfied by the numerical scheme. In thiscase one achieves besides uniform also strong H1 convergence of numerical solutionson unstructured meshes.

ADAPTIVITY AND FULLY NONLINEAR PROBLEMS

Omar Lakkis1 and Tristan Pryer2

1 Department of Mathematics, University of Sussex, Brighton, [email protected]

2 School of Mathematics, Statistics and Actuarial Sciences,University of Kent, Canterbury, England.

[email protected]

In this talk we discuss the development of residual based h–adaptivity for a class offinite element discretisation of fully nonlinear problems introduced in [LP:2011]. Wepay particular attention to the Monge Ampere equation, the Infinity Laplacian andPucci’s equation, looking at the efficiency of the resulting estimator for these problems.

References

[LP:2011] Lakkis, Omar and Pryer, Tristan. A nonvariational finite element methodfor nonlinear elliptic problems. Submitted - tech report available on ArXiVhttp://arxiv.org/abs/1103.2970, 2011.

29-4


FINITE ELEMENT METHODS FOR THEMONGE-AMPERE EQUATION

Michael Neilan


In this talk, I will discuss finite element methods for the fully nonlinear Monge-Ampereequation. The main feature of our discretizations is that their linearizations are co-ercive over the finite element space. I will describe a simple procedure to constructsuch schemes and briefly discuss the difficulties in the convergence analysis. Finally,numerical examples for a series of benchmark problems will be presented.

29-5

30: Mini-Symposium: Numerical modeling of flow in subsurface reservoirs

30 Mini-Symposium: Numerical modeling of flow

in subsurface reservoirs

Organisers: Ahmed H. Elsheikh, Ben Ganisand Mary Wheeler

30-1


MULTIPHASE FLOW THROUGH POROUS MEDIA: AN ADAPTIVECONTROL VOLUME FINITE ELEMENT METHOD

Peyman Mostaghimia, James R. Percival, Brendan S. Tollit,Stephen J. Neethling, Gerard J. Gorman, Matthew D. Jackson,

Christopher C. Pain and Jefferson L.M.A. Gomes

Department of Earth Science and Engineering, Imperial College London,South Kensington Campus, London SW7 2AZ, UK

[email protected]

Numerical simulation of multiphase flow in porous media is of importance in a widerange of applications in science and engineering. We present the formulation for anincompressible adaptive control volume finite element method for accurate and efficientmodelling of flow in porous media. The saturation equation is spatially discretized us-ing a node centred control volume method on an unstructured finite element mesh. Thepressure equation is spatially discretized using a continuous control volume finite ele-ment method (CVFEM) to achieve consistency with the discrete saturation equation.The proposed scheme is CV-wise locally mass conservative while geometric flexibility ofthe finite element method is retained. The numerical simulation is implemented in theframework of Fluidity, an open source and general purpose fluid flow simulator capableof solving a number of different governing equations on arbitrary unstructured meshes.The scheme is equipped with dynamic anisotropic mesh adaptivity to update the meshresolution to capture the evolving features of flow through the porous medium. It usesmetric advection between adaptive meshes in order to predict the future density ofmesh. We demonstrate the advantage of using dynamic mesh adaptivity for flow inporous media. In addition, we compare the obtained results for simulation of fluid flowin some challenging benchmark geometries against conventional finite-difference simu-lations. We also apply the method for large-scale simulation of heap leaching processfor mineral recovery applications. The obtained results demonstrate the capability ofthe scheme for flow simulation in complex geometries with high spatial accuracy at lowcomputational cost through the use of anisotropic mesh adaptivity.

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MULTIPOINT FLUX DOMAIN DECOMPOSITIONTIME-SPLITTING METHODS ON GENERAL GRIDS

Andres Arraras1a, Laura Portero1b and Ivan Yotov2

1Departamento de Ingenierıa Matematica e Informatica,Universidad Publica de Navarra, Pamplona, Spain


2Department of Mathematics, University of Pittsburgh, Pittsburgh, [email protected]

In this work, we propose and analyze efficient discretizations for mixed formulations ofevolutionary diffusion problems on general grids. The spatial approximation is based onthe multipoint flux mixed finite element method, which allows for local flux eliminationby using suitable finite element spaces and special quadrature rules. As a result, weobtain a cell-centered pressure system on triangular, quadrilateral, tetrahedral andhexahedral meshes. Such a system is subsequently partitioned via an overlappingdomain decomposition splitting technique. A proper combination of this techniquewith multiterm fractional step diagonally implicit Runge–Kutta methods reduces theglobal system to a collection of uncoupled subdomain problems that can be solvedin parallel. The fully discrete scheme is unconditionally stable and computationallyefficient, since it avoids the need for Schwarz-type iteration procedures. We derive apriori error estimates for both the semidiscrete and fully discrete formulations, andfurther illustrate the theoretical results with numerical experiments.

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AN OPTIMIZATION APPROACH TO LARGE SCALESIMULATIONS OF FLUID FLOWS IN FRACTURED MEDIAWITH FINITE ELEMENTS ON NONCONFORMING GRIDS

Stefano Berronea, Sandra Pieraccinib and Stefano Scialoc

Dipartimento di Scienze Matematiche, Politecnico di Torino,Corso Duca degli Abruzzi 24, 10129 Torino, Italy.

[email protected], [email protected]@polito.it

Numerical simulation of subsurface flows is a challenging problem since geometricalcomplexity and huge dimensions of underground geological reservoirs require a largeamount of computational power, and uncertainty in the data on rock properties canonly be handled with a stochastic approach and performing a huge number of simula-tions. As a consequence efficiency and large applicability of numerical algorithms is ofparamount importance.A novel optimization method to discrete fracture network (DFN) approach for simula-tions of subsurface flows is presented here. DFN models are complex 3D networks ofplanar polygons resembling underground fractures and, as a first approximation, onlyflow in the fractures is evaluated, neglecting the percolation through the surroundingrock matrix which occurs on a different temporal scale. A classical approach to theproblem consists in the generation of a finite element conforming triangulation of thenetwork and the resolution of the resulting large algebraic system of equations. Theoptimization approach described in [1, 2, 3], instead, seeks the solution as the mini-mum of a PDE constrained functional. The large scale DFN problem is thus split in anumber of quasi-independent smaller problems on the fractures that can be solved inparallel on multi core or GPU based computer architectures. The algorithm does notrely on a conforming triangulation of the fracture system, so that the meshing processcan be performed independently on each fracture. This is of paramount importance forDFN simulations, given the difficulties in the generation of a good quality conformingmesh arising from the intricate nature of fracture intersections. The non conformingmeshes are handled by means of modified finite element methods. The application ofextended finite element methods (XFEM) is fully analysed [4], but also virtual elements(VEM) are tested.Good efficiency for the optimization algorithm is achieved in conjunction with precon-ditioning techniques and multi-grid approaches appear to be very promising to speedup the convergence of the method. Numerical results on complex configurations areprovided.

References

[1] , A PDE-constrained optimization formulation for discrete fracture networkflows, SIAM Journal on Scientific Computing, 35 (2013), pp. A908–A935.

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[2] S. Berrone, S. Pieraccini, and S. Scialo, On simulations of discrete fracturenetwork flows with an optimization-based extended finite element method, SIAMJournal on Scientific Computing, 35 (2013), pp. B487–B510.

[3] , An xfem optimization approach for large scale simulations of discrete fracturenetwork flows. Submitted for pubblication, 2013.

[4] , The extended finite element method for discrete fracture network simulations.In preparation.

PRESSURE JUMP INTERFACE LAW FOR THESTOKES-DARCY COUPLING: CONFIRMATION

BY DIRECT NUMERICAL SIMULATIONS

Thomas Carraroa and Christian Gollb

Institute for Applied Mathematics, Heidelberg University,69120 Heidelberg, Germany.

[email protected]@iwr.uni-heidelberg.de

We consider slow incompressible viscous flow over a porous bed which is made upof a periodic repetition of a so called ‘unit cell’. The flow is modeled by the steadyStokes equation on the complete domain including the porous part. For computationalpurposes we upscale the Stokes equation in the porous bed and replace it by the Darcylaw. After upscaling, an interface Γ appears between the two domains and relevantboundary conditions at the interface have to be defined.

It is well known that on the interface Γ the Beavers-Joseph-Saffmann conditionholds true for the effective fluid-velocity ueff :

ueff1 + εCbl

1

∂ueff1

∂x2

= 0. (BJS)

Additionally, after [1, 2], the Darcy pressure in the porous domain pD and the effectivepressure in the fluid region peff fulfill a pressure jump law:

pD = peff + Cblω

∂ueff1

∂x2

. (PJL)

The subject of this talk is the numerical confirmation of the pressure jump (PJL) bya direct finite element simulation of the flow on the microscopic level. Additionally, weshow the numerical results for the confirmation of (BJS). Therefore, we compute theconstants Cbl

1 and Cblω , which are defined as averages of an appropriate boundary layer

problem solved by a finite element approximation. A goal oriented adaptive scheme isapplied for the grid refinement allowing a reduction of the computational costs withoutloss of accuracy.

To verify the interface law, we compute the solution of the microscopic problem fordifferent values of ε (the ratio between the length of the unit cell and the height of the

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computational domain) and show several convergence results. All the computationsare performed for two different inclusions, namely circles and ellipses, since the shapeof the pores strongly influences the behavior of the pressure at the interface.

References

[1] Marciniak-Czochra, Anna and Mikelic, Andro, Effective Pressure Inter-face Law for Transport Phenomena between an Unconfined Fluid and a PorousMedium Using Homogenization, Multiscale Modeling & Simulation, 10, 285–305,(2012)

[2] Jager, Willi and Mikelic, Andro, On the Interface Boundary Condition ofBeavers, Joseph, and Saffman, SIAM Journal on Applied Mathematics, 60, 1111–1127, (2000)

EFFICIENT BAYESIAN UNCERTAINTY QUANTIFICATION OFSUBSURFACE FLOW MODELS USING NESTED SAMPLING AND

SPARSE POLYNOMIAL CHAOS SURROGATES

Ahmed H. Elsheikh1a,2, Mary F. Wheeler1 and Ibrahim Hoteit2

1 Center for Subsurface Modeling (CSM),Institute for Computational Engineering and Sciences (ICES),

University of Texas at Austin, TX, [email protected]

2 Dept. of Earth Sciences and Engineering,King Abdullah University of Science and Technology (KAUST),

Thuwal, Saudi Arabia

An accelerated Bayesian uncertainty quantification method based on the nested sam-pling (NS) algorithm and non-intrusive stochastic collocation method is presented.Nested sampling is an efficient Bayesian sampling algorithm that builds a discrete rep-resentation of the posterior distributions by iteratively re-focusing a set of samplesto high likelihood regions. NS allows representing the posterior probability distri-bution function (PDF) with a smaller number of samples and reduces the curse ofdimensionality effects. The main difficulty of the nested sampling algorithm is in aconstrained sampling step which is commonly performed using a random walk Markovchain Monte-Carlo (MCMC) algorithm. In the current manuscript, we perform a two–stage sampling using a polynomial chaos response surface to filter out rejected samplesin the Markov chain Monte-Carlo method. The combined use of nested sampling andthe two–stage MCMC based on approximate response surfaces provides significate com-putational gains in terms of the number of simulation runs. The proposed algorithmis applied for calibration and model selection of subsurface flow models.

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HIGH-ORDER CUT-CELL TECHNIQUES FORNUMERICAL UPSCALING IN POROUS MEDIA

Christian Engwer

Institute for Computational und Applied Mathematics,University of Munster, Germany

[email protected]

For simulations of processes in porous media usually a homogenized macroscopic scaleis considered. On the macro scale empirical laws are used, which require effectiveparameters. The models, as well a the parameters are directly linked to pore scalephysics and the geometry of the pore space. Pore scale simulations can help whereexperiments are not possible, or hard to conduct. They give an opportunity for differentapplication, e.g. parameter estimation or model verification.

We discuss practical challenges when trying to employ direct pore-scale simulationsfor numerical upscaling. Mesh-generation problems can be overcome using cut-celltechniques. We present the Unfitted Discontinuous Galerkin method, a higher ordercut-cell method. This method allows to solve PDEs on domains with a complicatedgeometric shape. It uses finite element meshes which are significantly coarser then thoserequired by standard conforming finite element approaches and is flexible enough to beused for elliptic, hyperbolic and parabolic problems. Essential boundary conditions areincorporated weakly. Recent extensions allow to solve coupled bulk-surface problems.Further challenges arise for the linear solver and for parallel computations.

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ADJOINTS OF FINITE ELEMENT MODELS

Patrick E. Farrell1a,2 and Simon W. Funke1,3

1Department of Earth Science and Engineering,Imperial College London, London, [email protected]

2Center for Biomedical Computing, Simula Research Laboratory, Oslo, Norway

3Grantham Institute for Climate Change, Imperial College London, London, UK.

The derivatives of PDE models are key ingredients in many important algorithms ofcomputational mathematics. They find applications in diverse areas such as sensitivityanalysis, PDE-constrained optimisation, continuation and bifurcation analysis, errorestimation, and generalised stability theory.

These derivatives, computed using the so-called tangent linear and adjoint models,have made an enormous impact in certain scientific fields (such as aeronautics, me-teorology, and oceanography). However, their use in other areas has been hamperedby the great practical difficulty of the derivation and implementation of tangent lin-ear and adjoint models. In his recent book [U. Naumann, The Art of DifferentiatingComputer Programs, SIAM, 2011], Naumann describes the problem of the robust au-tomated derivation of parallel tangent linear and adjoint models as “one of the greatopen problems in the field of high-performance scientific computing”.

In this talk, we present an elegant solution to this problem for the common casewhere the forward model may be written in variational form, and discuss some of itsapplications.

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A GLOBAL JACOBIAN METHOD FOR MORTARDISCRETIZATIONS OF NONLINEAR POROUS MEDIA FLOWS

Benjamin Ganis1a, Mika Juntunen1, Gergina Pencheva1,Mary F. Wheeler1 and Ivan Yotov2

1Center for Subsurface Modeling (CSM),Institute for Computational Engineering and Sciences (ICES),


2Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

We describe a non-overlapping domain decomposition algorithm for nonlinear porousmedia flows discretized with the multiscale mortar mixed finite element method. Thenew methods are designed to be less complex than a previously developed nonlinearmortar algorithm, which required nested Newton iterations and a forward differenceapproximation. Furthermore, efficient linear preconditioners can be applied to speedup the iteration. Consequently, we are able to demonstrate greatly improved parallelscalability for large nonlinear systems.

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DIRECT NUMERICAL SIMULATION OFTWO-PHASE FLOW AT THE PORE SCALE

Ali Q Raeinia, Branko Bijeljicb and Martin J Bluntc

Department of Earth Science and Engineering, Imperial College London,South Kensington Campus, London SW7 2AZ, UK


Direct numerical simulations of two-phase flow at the micron scale help understand andprovide vital information on the pore-scale mechanisms controlling two-phase flow inporous media. In this study, we discuss a new sharp surface force model developed forcomputing capillary forces at the micron scale, in a volume off fluid based finite volumeframework. We discuss the difficulties encountered in modelling the high capillaryforces at such small scales; and how this new formulation, along with a new filteringscheme to remove non-physical forces from the capillary forces, helps to overcome thesedifficulties and predict the flow by solving the Navier-Stokes equations. We presentexemplar applications of this new method in modelling two-phase flow through simplepure geometries. Particularly, we present simulation results on snap-off and layer flow,in drainage and imbibition, and a methodology to upscale the results to obtain therelevant information required to describe the flow at larger scales. Finally, we presentsample simulations for modelling two-phase flow directly on micro-CT images of porousmedia, and study the effect of capillary number on the macroscopic properties of porousmedia, such as relative permeability curves and residual non-wetting phase saturations.

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MODELING FLOW WITH NONPLANAR FRACTURES

Gurpreet Singha, Omar Al-Hinaib, Gergina Penchevac and Mary F. Wheelerd

Center for Subsurface Modeling,Institute for Computational Engineering and Sciences,

University of Texas at Austin, Austin, TX, [email protected], [email protected],


Modeling flow with fractures can be challenging due to complex fracture geometries,strong variations in length and time scales as well as the need to combine multipleflow models. In this talk we propose a general methodology for coupling fracture andreservoir flow which is capable of handling multiple physics under the same frame-work while accounting for reservoir complexities. This is accomplished by separaterepresentations of the fracture and reservoir models, followed by a coupling of the twoproblems using appropriate boundary conditions and forcing functions. A multi-pointflux mixed finite element method and mimetic finite difference method are used as thediscretization schemes for the reservoir and the fracture, respectively. The methodsare locally conservative and allow for accurate flux approximation as well as generalhexahedral grids and non-planar fractures. An appropriate choice of convergence cri-teria and numerical stabilization consistent with the physical problem is then utilizedto iteratively couple the resulting two flow systems.

COMPUTATIONAL ENVIRONMENTS FOR ENERGY ANDENVIRONMENTAL MODELING IN POROUS MEDIA

Mary F. Wheeler

Center for Subsurface Modeling (CSM),Institute for Computational Engineering and Sciences (ICES),


We discuss a framework for modeling flow, mechanics and chemistry in porous media.Applications include carbon sequestration in saline aquifers, polymer flooding, andgeomechanics. Issues include accuracy of discretizations on general grids includingdistorted hexahedra meshes, solvers and incorporating uncertainty quantification andparameter estimation in the framework.

30-11


MULTISCALE DOMAIN DECOMPOSITION METHODS FORPOROUS MEDIA FLOW COUPLED WITH GEOMECHANICS

Ivan Yotov


We consider numerical modeling of the system of poroelasticity, which describes fluidflow in deformable porous media. The focus is on locally mass conservative flow dis-cretizations that provide efficient and accurate multiscale approximations on roughgrids and for highly heterogeneous media. We employ a multiscale mortar finite el-ement method, where the equations in the coarse elements (or subdomains) are dis-cretized on a fine grid scale, while continuity of normal velocity and stress betweencoarse elements is imposed via a mortar finite element space on a coarse grid scale.With an appropriate choice of polynomial degree of the mortar space, optimal orderconvergence is obtained for the method on the fine scale. The algebraic system is re-duced via a non-overlapping domain decomposition to a coarse scale mortar interfaceproblem that is solved efficiently using a multiscale flux basis.

This is joint work with Ben Ganis, Bin Wang, and Mary Wheeler (UT Austin),Ruijie Liu (BP), and Gary Xue (Shell).

30-12

31: Mini-Symposium: PDEs on Surfaces

31 Mini-Symposium: PDEs on Surfaces

Organiser: Charlie Elliott

31-1


UNFITTED FINITE ELEMENT METHODS USING BULK MESHESFOR SURFACE PARTIAL DIFFERENTIAL EQUATIONS

Klaus Deckelnick1, Charles M. Elliott2a and Tom Ranner2b

1Institut fur Analysis und Numerik, Otto–von–Guericke–Universitat Magdeburg,Universitatsplatz 2, 39106 Magdeburg, Germany;

[email protected]

2Mathematics Institute, Zeeman Building,University of Warwick, Coventry CV4 7AL, UK;


In this talk we propose two different unfitted finite element methods for solving anelliptic partial differential equation on a given hypersurface. These methods use aregular triangulation of an ambient domain and perform calculations on an inducedtriangulation of either the surface or a narrow band around the surface. We presentan error analysis for both approaches and show how these methods can be combinedin order to solve a parabolic equation on a family of evolving hypersurfaces.

31-2


DISCONTINUOUS GALERKIN METHODS FOR SURFACE PDES

Andreas Dednera, Pravin Madhavanb and Bjorn Stinner

Mathematics Institute and Centre for Scientific Computing,University of Warwick, Coventry, UK.


Partial differential equations (PDEs) on manifolds have become an active area of re-search in recent years due to the fact that, in many applications, models have to beformulated not on a flat Euclidean domain but on a curved surface. For example, theyarise naturally in fluid dynamics and material science but have also emerged in areasas diverse as image processing and cell biology.

Finite element methods (FEM) for elliptic problems and their error analysis havebeen successfully applied to problems on surfaces via the intrinsic approach based oninterpolating the surface by a triangulated one. This approach has subsequently beenextended to parabolic problems as well as evolving surfaces. However, as in the planarcase, there are a number of situations where FEM may not be the appropriate numer-ical method, for instance, advection dominated problems which lead to steep gradientsor even discontinuities in the solution. Discontinuous Galerkin (DG) methods are aclass of numerical methods that have been successfully applied to hyperbolic, ellipticand parabolic PDEs arising from a wide range of applications. Some of its main advan-tages compared to ‘standard’ finite element methods include the ability of capturingdiscontinuities as arising in advection dominated problems, and less restriction on gridstructure and refinement as well as on the choice of basis functions which makes themideal for a posteriori error estimation and adaptive refinement.

In this talk we will extend the discontinuous Galerkin (DG) framework for a linearsecond-order elliptic problem on a compact smooth connected and oriented surface.An interior penalty (IP) method is introduced on a discrete surface and we derive apriori error estimates by relating the latter to the original surface via a surface liftingoperator. The estimates suggest that the geometric error terms arising from the surfacediscretisation do not affect the overall convergence rate of the IP method. This is thenverified numerically for a number of test problems.

31-3


PATTERN FORMATION IN MORPHOGENESISON EVOLVING BIOLOGICAL SURFACES:

THEORY, NUMERICS AND APPLICATIONS

Anotida Madzvamuse1, Raquel Barreira2, Charles M. Elliott3,Ammon J. Meir4 and Necibe Tuncer5

1University of Sussex, School of Mathematical and Physical Sciences,Pevensey III, 5C15. BN1 9QH. Brighton, UK.

[email protected]

2Escola Superior de Tecnologia do Barreiro/IPS,Rua Americo da Silva Marinho-Lavradio, 2839-001 Barreiro, Portugal

3Mathematics Institute and Centre for Scientific Computing,University of Warwick, Coventry CV4 7AL, UK

4Department of Mathematics and Statistics, Auburn University, USA

5Department of Mathematics, University of Tulsa, USA

In this talk, I will present our most recent results based on two finite element formula-tions: (i) the surface finite element and (ii) radially projected finite element methodsapplied to solving partial differential equations of reaction-diffusion type on arbitrarystationary and evolving surfaces. Reaction-diffusion equations on evolving surfaces areformulated using the material transport formula, surface gradients and diffusive conser-vation laws. The evolution of the surface is defined by a material surface velocity. Theradially projected finite element method differs from the surface finite element methodin that it provides a conforming finite element discretization which is ”logically” rect-angular. However, this property restricts the general applicability of the numericalmethod to arbitrarily evolving surfaces, a key advantage for the evolving surface finiteelement method. To demonstrate the capability, flexibility, versatility and generalityof the numerical methodologies proposed, I will present various numerical results. Thismethodology provides a framework for solving partial differential systems on contin-uously evolving domains and surfaces with numerous applications in developmentalbiology, cancer research, wound healing, tissue regeneration, and cell motility amongmany others , where reaction-diffusion systems are routinely applied.

31-4


AN ALE ESFEM FOR SOLVING PDES ON EVOLVING SURFACES

Vanessa Styles

Department of Mathematics, University of Sussex, Brighton, [email protected]

Numerical methods for approximating the solution of partial differential equations onevolving hypersurfaces using surface finite elements on evolving triangulated surfacesare presented. In the ALE ESFEM the vertices of the triangles evolve with a velocitywhich is normal to the hypersurface whilst having a tangential velocity which is ar-bitrary. This is in contrast to the original evolving surface finite element method inwhich the nodes move with a material velocity. Numerical experiments are presentedwhich illustrate the value of choosing the arbitrary tangential velocity to improve meshquality.

31-5


NUMERICAL SIMULATIONS OF CHEMOTAXIS-DRIVENPDES ON SURFACES

Stefan Tureka and Andriy Sokolovb

Institute of Applied Mathematics, TU Dortmund, [email protected]

[email protected]

In the last twenty years one can observe a rapid and consistent growth of interest for bio-mathematical applications. Among them are modeling of tumor invasion and metasta-sis (Chaplain et al.), modeling of vascular network assembly (Preziosi et al.), patternformations due to the Turing-type instability (Murray et al.) or chemotaxis-drivenprocesses (Horstmann et al.), protein-protein interaction on the membrane (Gory-achev, Bastiaens) and others. These mathematical models, presented as systems ofadvection-reaction-diffusion equations, can take into consideration various biologicalprocesses (e.g. transport, random walk, reaction, chemotaxis, growth and decay, etc.).In our recent research we combine the system of the generalized Keller-Segel modelfor multi-species with reaction-diffusion equations on (evolving-in-time) surfaces whichcan be mathematically written in the following form

∂ui∂t

=

random walk/diffusion︷︸︸︷Dui ∆ui +∇ ·

species-species interaction︷︸︸︷(

n∑k=1,k 6=i

κi,kui∇uk

)−

species-agents interaction︷︸︸︷(m∑k=1

χi,kui∇ck

) +

+

kinetics︷︸︸︷fi(c,ρ), in Ω, (1)

∂cj∂t

= Dcj∆cj −

decay︷︸︸︷m∑k=1

αk,jck +

production︷︸︸︷n∑k=1

βk,juk +gj(u,ρ), in Ω (2)

∂ρl∂t

=

diffusion on surface︷︸︸︷Dρl ∆Γρl +

source︷︸︸︷sl(u, c), on Γ(t) (3)

where ui(x, t) (i = 1, n) and cj(x, t) (j = 1,m) are some solutions (for example, speciesand chemical agents) defined in a domain Ω ⊂ Rd (d = 1, 2, 3), and ρl (l = 1, p) aresolutions defined on a surface Γ(t) ⊂ Ω. In order to get an accurate numerical solutionin a reasonably finite time one has to construct an efficient, fast and robust numericalscheme for a sufficiently large class of partial differential equations. In our talk weinvestigate FCT and level-set techniques which help to overcome numerical challengeswhile preforming numerical simulation for the chemotaxis-driven PDEs on evolving-in-time surfaces (1)–(3).

31-6

32: Mini-Symposium: Sensitivity analysis and optimization for fluid-structure interaction problems

32 Mini-Symposium: Sensitivity analysis and op-

timization for fluid-structure interaction prob-

lems

Organisers: Thomas Richter and ThomasWick

32-1


PARAMETER ESTIMATION IN FLUID-STRUCTUREINTERACTION AND SUBSURFACE FLOWS

Ahmed H. Elsheikh1, Thomas Richter2, Mary F. Wheeler1 and Thomas Wick1,2a

1 Center for Subsurface Modeling,Institute for Computational Engineering and Sciences,

The University of Texas at Austin, USA

2 Institute of Applied Mathematics, Heidelberg University, Germanya [email protected]

We present forward and inverse modeling for coupled problems such as fluid-structureinteraction and the coupled Biot system with elasticity. Our aim is to estimate materialcoefficients such as permeability coefficients or Lame parameters. In particular, suchestimation problems are quite important in subsurface modeling because this is a hottopic in reservoir engineering.

Two optimization algorithms are presented. First, standard gradient-based opti-mization (for a stationary setting) showing the robustness when all derivatives arecorrectly assembled. This is exemplified with the help of a prototypical setting wherean elastic structure interacts with a surrounding fluid. Second, we use non-intrusiveBayesian inverse modeling for the coupled, nonstationary Biot-Lame-Navier problem.Here, some reservoir (the pay-zone) is modeled as a poroelastic medium with the helpof Biot’s equations. A surrounding medium (the non-pay zone) is modeled as a staticelastic structure. In both cases, we prefer a monolithic setting of the forward problem.In fact, for gradient-based optimization this a indispensable requirement.

We present different examples in two and three dimensions in order to show theperformance of our algorithms.

32-2


TOWARDS OPTIMAL CONTROL OF LARGE DEFORMATION FSIPROBLEMS INCLUDING CONTACT AND TOPOLOGY CHANGE

Thomas Richtera and Thomas Wick

Institute of Applied Mathematics, Heidelberg University, [email protected]

Some fluid-structure interaction (FSI) problems involve very large deformation of theelastic structures. A prominent example is the flow of blood in the heart. If the movingheart valves are included in the simulation, the structure will even undergo self-contactwhich can lead to topology change (if the heart chamber is closing).

This contribution has two parts. First, we will introduce the fully Eulerian formula-tion as a monolithic model for FSI problems. This formulation can be regarded as theantipode to the Arbitrary Lagrangian Eulerian (ALE) formulation for FSI problems.While the ALE method uses fixed reference domains (Lagrangian in the solid) for mod-eling both subproblems, the Eulerian formulation uses a natural Eulerian frameworkfor both parts. This construction allows for the simulation of problems with very largedeformation and contact.

Second, we discuss gradient based optimization techniques for FSI problems. Thebasic concepts will be introduced using the ALE formulation. It will be easy to see,that this ALE formulation is not able to handle problems with large deformation oreven contact. Finally, we will describe first steps for an extension of gradient basedoptimization techniques to the Eulerian formulation.

CALCULATION OF SENSITIVITIES FORFLUID-STRUCTURE INTERACTIONS

Thomas Wick1 and Winnifried Wollner2

1 The Institute for Computational Engineering and Sciences (ICES),The University of Texas at Austin, USA

[email protected]

2University of Hamburg, Department of Mathematics,Bundesstr 55, 20146 Hamburg, Germany

[email protected]

In this talk we will consider a stationary model for fluid structure interaction of anincompressible fluid with an elastic structure using the ALE-framework. In particularwe will discuss differentiability of the solution operator of the problem.

32-3

33: Mini-Symposium: Stochastic finite elements and PDEs

33 Mini-Symposium: Stochastic finite elements and

PDEs

Organiser: Max Gunzburger

33-1


A POSTERIORI ERROR ESTIMATIONFOR STOCHASTIC GALERKIN FEMS

Alex Bespalov1,2a, Catherine Powell2b and David Silvester2c

1 School of Mathematics, University of Birmingham, Birmingham, UK,

2 School of Mathematics, University of Manchester, Manchester, [email protected] [email protected],

[email protected]

Stochastic Galerkin finite element approximation is an increasingly popular approachfor the solution of elliptic PDE problems with correlated random data. Given aparametrisation of the data in terms of a large, possibly infinite, number of randomvariables, this approach allows the original PDE problem to be reformulated as a para-metric, deterministic PDE on a parameter space of high, possibly infinite, dimension.A typical strategy is to combine conventional (h-) finite element approximation on thespatial domain with spectral (p-) approximation on a finite-dimensional manifold inthe (stochastic) parameter space.

For approximations relying on low-dimensional manifolds in the parameter space,stochastic Galerkin finite element methods have superior convergence properties tostandard sampling techniques. On the other hand, the desire to incorporate more andmore parameters (random variables) together with the need to use high-order polyno-mial approximations in these parameters inevitably generates very high-dimensionaldiscretised systems. This in turn means that adaptive algorithms are needed to ef-ficiently construct approximations, and fast and robust linear algebra techniques areessential for solving the discretised problems.

Both strands will be discussed in the talk. We outline the issues involved in a poste-riori error analysis of computed solutions and present a practical a posteriori estimatorfor the approximation error. We introduce a novel energy error estimator that uses aparameter-free part of the underlying differential operator—this effectively exploits thetensor product structure of the approximation space and simplifies the linear algebra.We prove that our error estimator is reliable and efficient. We also discuss differentstrategies for enriching the discrete space and establish two-sided estimates of the er-ror reduction for the corresponding enhanced approximations. These give computableestimates of the error reduction that depend only on the problem data and the originalapproximation. We show numerically that these estimates can be used to choose theenrichment strategy that reduces the error most efficiently.

This work is supported by the EPSRC grant EP/H021205/1.

33-2


WEAK TRUNCATION ERROR ESTIMATES FOR ELLIPTICPDES WITH LOGNORMAL COEFFICIENTS

Julia Charrier1 and Arnaud Debussche2

1 LATP, Aix Marseille universite, 13453 Marseille, [email protected]

2 ENS Cachan Bretagne, 35170 Bruz, [email protected]

In the context of uncertainty quantification, uncertainties on some physical propertiesof a medium can be modeled by using random fields, leading to partial differentialequations with random coefficients. We are here concerned with the case of ellipticpartial differential equations with lognormal coefficients. More precisely, we considerthe following equation: for almost all ω

−div(a(ω, x)∇u(ω, x)) = f(x) on Du(ω, x) = 0 on ∂D,

where a(ω, x) = eg(ω,x) and g is a gaussian field. We include in particular in ourstudy cases where the lognormal coefficient a may have realizations with low regularity(typically only Holder continuous). The aim is then to compute the law of the solutionu. Such models are in particular widely used in hydrogeology to model flow in porousmedia with uncertainty. Severeral numerical methods for such equations (such asstochastic galerkin or stochastic collocation methods among others) are based on theapproximation of the random coefficient a by a finite number N of random variables.The numerical cost of these methods tends generally to increase drastically with thenumber N of random variables used, hence it is important to get sharp estimates ofthe error resulting on the solution.

In this talk we present weak error bounds, in the case where the approximation ofthe coefficient a is performed through a truncation of the Karhunen-Loeve expansionof the gaussian field g. More precisely we give bounds in C1 norm of the error on thelaw of the solution u resulting from the approximation of the coefficient a. We get aweak order of convergence equal to twice the strong order of convergence and illustratenumerically the optimality of the order of convergence obtained. These results (whichcan be found in [2]) complete and improve the results of [1]. Moroever we complementthese results by providing the first term of the asymptotic expansion of the weaktruncation error in a particular case, the aim of studying such asymptotic expansionbeing to justify the use of Richardson extrapolation with respect to the truncationorder N .

References

[1] Charrier J. - Strong and weak error estimates for the solutions of elliptic partial differentialequations with random coefficients, SIAM Journal on Numerical Analysis, 2012.

33-3


[2] Charrier J., Debussche A. - Weak truncation error estimates for elliptic PDEs with lognor-

mal coefficients, Stochastic Partial Differential Equations : Analysis and Computations, 2013

ADAPTIVE ANISOTROPIC SPECTRAL STOCHASTIC METHODSFOR UNCERTAIN SCALAR CONSERVATION LAWS

Alexandre Ern1, Olivier Le Maıtre2 and Julie Tryoen1,3

1Universite Paris-Est, CERMICS, Ecole des Ponts ParisTech,77455 Marne la Vallee cedex 2, France

[email protected]

2LIMSI-CNRS, UPR-3251, Orsay, [email protected]

3INRIA Bordeaux Sud-Ouest, Bacchus Team, 33405 Talence cedex 1, [email protected]

We develop adaptive anisotropic discretization schemes for conservation laws withstochastic parameters. A Finite Volume scheme is used for the deterministic discretiza-tion, while a piecewise polynomial representation is used at the stochastic level. Themethodology is designed in the context of intrusive Galerkin projection methods withRoe-type solver, see [1]. The adaptation aims at selecting the stochastic resolution levelbased on the local smoothness of the solution in the stochastic domain. In addition,the stochastic features of the solution greatly vary in the space and time so that theconstructed stochastic approximation space depends on space and time. The dynami-cally evolving stochastic discretization uses a tree-structure representation that allowsfor the efficient implementation of the various operators needed to perform anisotropicmultiresolution analysis. Efficiency of the overall adaptive scheme is assessed on thestochastic traffic equation with uncertain initial conditions and velocity leading to ex-pansion waves and shocks that propagate with random velocities. Numerical testshighlight the computational savings achieved as well as the benefit of using anisotropicdiscretizations in view of dealing with problems involving a larger number of stochasticparameters. More details on the methodology and the numerical results can be foundin [2].

References

[1] J. Tryoen, O. Le Maıtre, M. Ndjinga and A. Ern, Intrusive Galerkinmethods with upwinding for uncertain nonlinear hyperbolic systems, J. Comput.Phys., 229, 6485–6511 (2010).

[2] J. Tryoen, O. Le Maıtre, and A. Ern, Adaptive anisotropic spectral stochasticmethods for uncertain scalar conservation laws, SIAM J. Sci. Comput., 34(5),A2459–A2481 (2012).

33-4


SPARSE ADAPTIVE TENSOR GALERKIN APPROXIMATIONSOF STOCHASTIC PDE-CONSTRAINED CONTROL PROBLEMS

Angela Kunoth

Institut fur Mathematik, Universitat Paderborn,Warburger Str. 100, 33098 Paderborn, Germany

[email protected]

For control problems constrained by linear elliptic or parabolic PDEs depending oncountably many parameters, i.e., on σj with j ∈ N, we prove analytic parameterdependence of the state, the co-state and the control. Moreover, we establish that thesefunctions allow expansions in terms of sparse tensorized generalized polynomial chaos(gpc) bases. Their sparsity is quantified in terms of p-summability of the coefficientsequences for some 0 < p ≤ 1. Resulting a-priori estimates establish the existence ofan index set Λ, allowing for concurrent approximations of state, co-state and controlfor which the gpc approximations attain rates of best N -term approximation.

These results serve as the analytical foundation for the development of correspond-ing sparse realizations in terms of deterministic adaptive Galerkin approximations ofstate, co-state and control on the entire, possibly infinite-dimensional parameter space.

The results were obtained jointly with Christoph Schwab (ETH Zurich).

FINITE ELEMENT APPROXIMATION OFTHE CAHN-HILLIARD-COOK EQUATION

Stig Larsson

Department of Mathematical Sciences,Chalmers University of Technology, SE–412 96 Gothenburg, Sweden

[email protected]

We study the Cahn-Hilliard-Cook equation, i.e., the Cahn-Hilliard equation perturbedby additive colored noise. We show almost sure existence and regularity of solutions.We introduce spatial approximation by a standard finite element method and proveerror estimates of optimal order on sets of probability arbitrarily close to 1. We alsoprove strong convergence without known rate.

This is joint work with M. Kovacs and A. Mesforush.

33-5


EXPLORING EMERGING MANYCORE ARCHITECTURESFOR UNCERTAINTY QUANTIFICATION THROUGHEMBEDDED STOCHASTIC GALERKIN METHODS

Eric Phipps1, H. Carter Edwards2, Jonathan Hu3 and Jakob T. Ostien4

1Sandia National Laboratories,Optimization and Uncertainty Quantification Department, Albuquerque, NM, USA

, [email protected],

2Sandia National Laboratories,Multiphysics Simulation Technologies Department, Albuquerque, NM, USA,

[email protected],

3Sandia National Laboratories,Scalable Algorithms Department, Livermore, CA, USA,

[email protected],

4Sandia National Laboratories,Mechanics of Materials Department, Livermore, CA, USA,

[email protected]

We explore approaches for improving the performance of intrusive or embedded stochas-tic Galerkin uncertainty quantification methods on emerging computational architec-tures. Our work is motivated by the trend of increasing disparity between floating-point throughput and memory access speed on emerging architectures, thus requiringthe design of new algorithms with memory access patterns more commensurate withcomputer architecture capabilities. We first compare the traditional approach for im-plementing stochastic Galerkin methods to non-intrusive spectral projection methodsemploying high-dimensional sparse quadratures on relevant problems from computa-tional mechanics, and demonstrate the performance of stochastic Galerkin is reason-able. Several reorganizations of the algorithm with improved memory access patternsare described and their performance measured on contemporary manycore architec-tures. We demonstrate these reorganizations can lead to improved performance formatrix-vector products needed by iterative linear system solvers, and highlight furtheralgorithm research that might lead to even greater performance.

Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia

Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of

Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

33-6


ADAPTIVE, SPARSE QUADRATURES FORBAYESIAN INVERSE PROBLEMS

Claudia Schillingsa and Christoph Schwabb

Seminar for Applied Mathematics, ETH Zurich, [email protected],[email protected]

Based on sparsity results of a parametric, deterministic formulation for Bayesian in-verse problems, we will propose a class of adaptive, deterministic sparse tensor Smolyakquadrature schemes for the efficient approximate numerical evaluation of expectationsunder the posterior, given data. Convergence rates for the quadrature approximationare shown, both theoretically and computationally, to depend only on the sparsity classof the unknown and in particular, are provably higher than those of Monte-Carlo (MC)and Markov-Chain Monte-Carlo methods. Problem classes considered are PDEs andlarge ODE systems with uncertain coefficients and parameters.This work is supported by the European Research Council under FP7 Grant AdG247277.

MULTILEVEL MARKOV CHAIN MONTE CARLO ALGORITHMSFOR UNCERTAINTY QUANTIFICATION IN SUBSURFACE FLOW

Aretha Teckentrup

Department of Mathematical Sciences, University of Bath, Bath, BA2 3JU, [email protected]

The quantification of uncertainty in groundwater flow plays a central role in the safetyassessment of radioactive waste disposal and of CO2 capture and storage underground.Stochastic modelling of data uncertainties in the rock permeabilities lead to ellipticPDEs with random coefficients. Typical models used for the random coefficients, suchas log-normal random fields with exponential covariance, are unbounded and haveonly limited spatial regularity, making practical computations very expensive and therigorous numerical analysis challenging.

To overcome the problem of the prohibitively large computational cost of existingMarkov chain Monte Carlo (MCMC) methods, we develop and analyse a new multilevelMCMC algorithm, based on a hierarchy of spatial levels/grids. We will demonstrateon a typical model problem the significant gains with respect to conventional MCMCthat are possible with this new approach, and provide a full convergence analysis ofthe new algorithm.

33-7

34: Mini-Symposium: Superconvergence in DG: analysis and recovery

34 Mini-Symposium: Superconvergence in DG: anal-

ysis and recovery

Organisers: Lilia Krivodonova and JenniferRyan

34-1


A NEW LAX-WENDROFF DISCONTINUOUSGALERKIN METHOD WITH SUPERCONVERGENCE

Wei Guo1, Jianxian Qiu2 and Jing-Mei Qiu3

1 Department of Mathematics, University of Houston, Houston, [email protected]

2 School of Mathematical Sciences, Xiamen University, [email protected]

3 Department of Mathematics, University of Houston, Houston, [email protected]

Various superconvergence properties of discontinuous Galerkin (DG) and local DG(LDG) methods for linear hyperbolic and parabolic equations have been investigatedin the past. Due to these superconvergence properties, DG and LDG methods havebeen known to provide good wave resolution properties, especially for long time inte-grations. In this work, we investigate the super convergence property of DG coupledwith Lax-Wendroff time discretization. We develop a new Lax-Wendroff (LW) DGmethod for hyperbolic conservation laws for its super convergence in terms of negativenorm, post-processed solutions and extra long time behavior. Especially, we proposeto use the local DG method in approximating higher order spatial derivatives in theLW approach. As a results, 2k + 1-th order of convergence is observed for the postprocessed LWDG solution. The eigen-structure of the new Lax-Wendroff DG schemeis analyzed symbolically; and the 2k + 1-th order of convergence is observed for thephysically relevant eigenvalues. A collection of numerical examples for linear equationsare presented to verify our observations.

ENERGY NORM ERROR ESTIMATION FOR AVERAGEDDISCONTINUOUS GALERKIN METHODS

IN ONE SPACE DIMENSION

Ferenc Izsak

Institute of Mathematics, Eotvos Lorand University, Budapest, [email protected]

In the numerical solution of elliptic boundary value problems, a natural measure ofthe error is the H1-(semi)norm. This does not make sense in case of discontinuousapproximations. In the talk, we present an error estimate for the local average of theinterior penalty method in the H1-seminorm in one space dimension. Instead of thewell-known post-processing we immediately compute the averaged approximation. Italso gives a new viewpoint: the interior penalty method can be derived as a lower ordermodification of a conforming method.

34-2


SMOOTHNESS-INCREASING ACCURACY-CONSERVING (SIAC)FILTERING: PRACTICAL CONSIDERATIONS

WHEN APPLIED TO VISUALIZATION

Robert M. Kirby

School of Computing, University of Utah, [email protected]

The discontinuous Galerkin (DG) method continues to maintain heightened levels ofinterest within the simulation community because of the discretization flexibility itprovides. Although one of the fundamental properties of the DG methodology andarguably its most powerful property is the ability to combine high-order discretizationson an inter-element level while allowing discontinuities between elements, this flexibilitygenerates a plethora of difficulties when one attempts to post-process DG fields foranalysis and evaluation of scientific results. Smoothness-increasing accuracy-conserving(SIAC) filtering enhances the smoothness of the field by eliminating the discontinuitybetween elements in a way that is consistent with the DG methodology; in particular,high-order accuracy is preserved and in many cases increased.

In this talk, we will present our efforts as pertains to attempting to apply SIACfiltering to DG fields for the purposes of visualization. Included in the topics to bediscussed will be comparisons between exact and approximate quadrature algorithms[1], extensions of SIAC filtering to triangular meshes [2, 3] and implementation andparallelization details [4]. This is joint work with the PhD work of Dr. Hanieh Mirzaee(School of Computing, University of Utah).

References

[1] Hanieh Mirzaee, Jennifer K. Ryan and Robert M. Kirby,“Quantification of ErrorsIntroduced in the Numerical Approximation and Implementation of Smoothness-Increasing Accuracy Conserving (SIAC) Filtering of Discontinuous Galerkin (DG)Fields”, Journal of Scientific Computing, 45, 447–470, (2010).

[2] Hanieh Mirzaee, Liang Yue, Jennifer K. Ryan and Robert M. Kirby,“Smoothness-Increasing Accuracy-Conserving Post-Processing (SIAC) Postprocessing for Dis-continuous Galerkin Solutions Over Structured Triangular Meshes”, SIAM Journalof Numerical Analysis, Vol. 49, No. 5, pages 1899-1920, 2011.

[3] Hanieh Mirzaee, James King, Jennifer K. Ryan and Robert M. Kirby,“Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for Discontinuous Galerkin Solu-tions Over Unstructured Triangular Meshes”, SIAM Journal of Scientific Com-puting, Vol. 35, No. 1, pages 212-230, 2013.

[4] Hanieh Mirzaee, Jennifer K. Ryan and Robert M. Kirby, “Efficient Implementationof Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for DiscontinuousGalerkin Solutions”, Journal of Scientific Computing, Vol. 52, No. 1, pages 85-112,2012.

34-3


ERROR ESTIMATION FOR THE DISCONTINOUS GALERKINMETHOD APPLIED TO HYPERBOLIC CONSERVATION LAWS

Lilia Krivodonovaa and Noel Chalmersb

Department of Applied Mathematics, University of Waterloo,Waterloo, Ontario, Canada.


Error estimation for hyperbolic problems presents theoretical and computational dif-ficulties due to discontinuities that such equations develop. Additionally, the popularadjoint problem approach to computing an estimate of the error is not very suitablefor transient hyperbolic equations.

We derive a partial differential equation that describes propagation of the exacterror for one-dimensional hyperbolic problems. We solve this equation numericallyto obtain accurate error estimates even in the presence of discontinuities. We dis-cuss the connection of the error governing equation with the superconvergence at theRadau points. We then show that the structure of the error is more complex thansuperconvergence theory would allow and that the proposed estimator works wheresuperconvergence based estimators fail.

34-4


COMPUTATIONALLY EFFICIENT BOUNDARY FILTERINGUSING SMOOTHNESS-INCREASING

ACCURACY-CONSERVING (SIAC) METHODS

Xiaozhou Li

Delft Institute of Applied Mathematics,Delft University of Technology 2628 CD Delft, The Netherlands.

[email protected]

The discontinuous Galerkin (DG) method continues to be a popular method for manyscientific areas such as fluid-structure interaction, two phase flow, etc. By investigatingthe superconvergence properties of a DG approximation, it is possible to obtain a higherorder accurate solution through Smoothness-Increasing Accuracy-Conserving Filteringby post-processing as outlined in [1, 2]. This post-processing technique relies on thenegative-order norm estimates of DG approximation and can filter out oscillations inthe error and maintain or improve the accuracy. However, extending this technique tonon-periodic boundary conditions is challenging.

In this talk, we discuss creating a boundary filter that both improves accuracy andsmoothness and is furthermore computationally efficient. We address improving theone-sided problem by introducing general B-splines near the boundary region which ismore efficient. For the discontinuous Galerkin method, we show that this one-sidedpost-processing technique increases the smoothness and conserves the accuracy notonly for uniform mesh but also for non-uniform mesh. This is joint work with JenniferRyan (University of East Anglia), Mike Kirby (University of Utah) and Kees Vuik(Delft University of Technology).

References

[1] B. Cockburn, M. Luskin, C.-W. Shu, E. Suli, Enhanced accuracy by post-processing for finite element methods for hyperbolic equations, Mathematics ofComputation, 72 (2003), pp.577-606.

[2] M. Steffan, S. Curtis, R.M. Kirby, and J.K. Ryan, Investigation of smoothnessenhancing accuracy-conserving filters for improving streamline integration throughdiscontinuous fields, IEEE-TVCG, 14 (2008), pp. 680-692.

[3] X. Li, J.K. Ryan, R.M. Kirby and C. Vuik, Computationally efficient position-dependent smoothness-increasing accuracy-conserving (SIAC) filtering: the uni-form mesh case, Submitted.

34-5


SUPERCONVERGENCE OF A HDG METHODFOR FRACTIONAL DIFFUSION PROBLEMS

Kassem Mustapha1 and Bernardo Cockburn2

1Department of Mathematics and Statistics,King Fahd University of Petroleum and Minerals, Saudi Arabia

[email protected]

2School of Mathematics, University of Minnesota, [email protected]

In this talk, we propose a spatial hybridizable discontinuous Galerkin (HDG) methodfor the numerical solution of fractional sub-diffusion problems of the form ∂tu −∂−αt ∇2u = f on (0, T ) × Ω with −1 < α < 0. Noting that, as α → 0, our modelproblem becomes u′ − ∇2u = f , which is just the classical heat equation. For exacttime-marching, we derive optimal algebraic error estimates assuming that the exactsolution is sufficiently regular. Thus, if the HDG approximations are taken to be piece-wise polynomials of degree r ≥ 0, the approximations to u in the L∞

(0, T ;L2(Ω)

)-norm

and to −∇u in the L∞(0, T ; L2(Ω)

)-norm are proven to converge with the rate hr+1,

where h is the maximum diameter of the elements of the spatial mesh. Moreover, forr ≥ 1 and by using quasi-uniform meshes, we obtain a superconvergence result whichallows us to compute, in an elementwise manner, a new approximation for u convergingwith a rate faster than

√log(Th−2/(α+1)) hr+2. Some numerical examples will be given

at the end of the presentation.

34-6


POST-PROCESSING DISCONTINUOUS GALERKIN SOLUTIONS TOVOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS:

ANALYSIS AND SIMULATIONS

Jennifer K. Ryan1 and Kassem Mustapha2

1School of Mathematics, University of East Anglia, Norwich NR4 7TJ, [email protected]

2Department of Mathematics and Statistics,King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

[email protected]

In this talk we discuss superconvergence extraction for Volterra integro-differentialequations with smooth and non-smooth kernels. Specifically, we discuss viable superconvergence extraction techniques via a post-processed discontinuous Galerkin (DG)method. Using nodal interpolation, a global superconvergence error bound (in the L∞-norm) is established. For a non-smooth kernel, a family of non-uniform time meshesis used to compensate for the singular behaviour of the exact solution near t = 0.The derived theoretical results are numerically validated in a sample of test problems,demonstrating higher-than-expected convergence rates.

34-7

35: Mini-Symposium: Time-Domain Boundary Integral Equations

35 Mini-Symposium: Time-Domain Boundary In-

tegral Equations

Organisers: Lehel Banjai and Stefan Sauter

35-1


A FULLY DISCRETE KIRCHHOFF FORMULA BASEDON CQ AND GALERKIN BEM

Lehel Banjai1, Antonio Laliena2 and Francisco-Javier Sayas3

1 School of Mathematical and Computer Sciences,Heriot-Watt University, Edinburgh, UK

[email protected]

2Dep. Matematicas, EUPLA, Universidad de Zaragoza, La Almunia, [email protected]

3Department of Mathematical Sciences, University of Delaware, Newark DE, [email protected]

In this talk we present some novel analytical techniques to study all discretization errorsfor a fully discrete approximation of the Kirchhoff Boundary Integral Equation using:Galerkin BEM, data interpolation, and Convolution Quadrature in the time domain.The most delicate part of this analysis is related to the Galerkin semidiscretization-in-space. This analysis is carried out by studying a dynamical system that is equivalentto the set of semidiscrete equations. This entire process is developed without any re-course to the Laplace transform of the discrete operator, thus providing tight estimateswith well controled constants for long time simulations. Finally, time discretization isstudied in a similar way, transfering well known tools for analysis of discretizations ofthe wave equation on bounded sets to exotic transmission problems.

35-2


TIME-DOMAIN FEM/BEM COUPLING

Lehel Banjai1, Volker Gruhne2, Christian Lubich3 and Francisco-Javier Sayas4

1 Heriot-Watt University, Edinburgh, [email protected]

2 Max Planck Institute for Mathetics in the Sciences, Leipzig, Germany.

3 University of Tubingen, Germany.

4 University of Delaware, USA.

We will discuss the numerical simulation of acoustic wave propagation with localizedinhomogeneities. To do this we will apply a standard finite element method (FEM) inspace and explicit time-stepping in time on a finite spatial domain containing the in-homogeneities. The equations in the exterior computational domain will be dealt witha time-domain boundary integral formulation discretized by the Galerkin boundaryelement method (BEM) in space and convolution quadrature in time. We will give theanalysis of the proposed method, starting with the proof of a positivity preservationproperty of convolution quadrature as a consequence of a variant of the Herglotz the-orem. Combining this result with standard energy analysis of leap-frog discretizationof the interior equations will give us both stability and convergence of the method.Numerical results will also be given.

35-3


CONVOLUTION-IN-TIME APPROXIMATIONS OF TDBIES

Penny J Davies1 and Dugald B Duncan2

1Department of Mathematics and Statistics, University of Strathclyde, Glasgow, [email protected]

2Maxwell Institute for Mathematical Sciences, Department of Mathematics,Heriot-Watt University, Edinburgh, UK

[email protected]

We present a new framework for the temporal approximation of TDBIEs which can becombined with either collocation or Galerkin approximation in space. It shares someproperties of convolution quadrature, but instead of being based on an underlying ODEsolver the approximation is explicitly constructed in terms of basis functions which havecompact support, and hence has sparse system matrices.

The time-stepping method is derived as an approximation for convolution Volterraintegral equations (VIEs): at time step tn = nh the VIE solution is approximated in a“backward time” manner in terms of basis functions φj by u(tn−t) ≈

∑nj=0 un−jφj(t/h)

for t ∈ [0, tn]. When the basis functions are cubic B-splines with the “parabolic runout”conditions at t = 0 the method is fourth order accurate, and numerical test resultsindicate that it gives a very stable approximation scheme for acoustic scattering TDBIEproblems.

35-4


QUADRATURE SCHEMES AND ADAPTIVITY FOR 2D TIMEDOMAIN BOUNDARY ELEMENT METHODS (TD-BEM)

Matthias Glafkea and Matthias Maischakb

BICOM, Department of Mathematical Sciences, Brunel University, Uxbridge, [email protected], [email protected]

In this talk we study the transient scattering of acoustic waves by an obstacle in aninfinite two dimensional domain, where the scattered wave is represented in terms oftime domain boundary layer potentials. The problem of finding the unknown solutionof the scattering problem is thus reduced to finding the unknown density of the timedomain boundary layer operators on the obstacle’s boundary, subject to the boundarydata of the known incident wave.

Using a Galerkin approach, the unknown density is approximated by a piecewisepolynomial function, the coefficients of which can be found by solving a linear system.The entries of the system matrix of this linear system involve, for the case of the twodimensional scattering problem under consideration, integrals over four dimensionalspace-time manifolds. An accurate computation of these integrals is crucial for thestability of this method. Using piecewise polynomials of low order, the two tempo-ral integrals can be evaluated analytically, leading to kernel functions for the spatialintegrals with complicated domains of piecewise support.

These spatial kernel functions can be generalised into a class of admissible kernelfunctions which, as we prove, belong to countably normed spaces [1]. Therefore, aquadrature scheme for the approximation of the two dimensional spatial integrals withadmissible kernel functions converges exponentially [3]. Similar results for the threedimensional case can be found in [2, 4].

We also present a fully adaptive scheme that modifies both the spatial and thetemporal mesh according to an error estimator, and show numerical experiments un-derlining the theoretical results [1].

References

[1] M. Glafke. Adaptive Methods for Time Domain Boundary Integral Equations. PhDThesis, Brunel University, 2013.

[2] E. Ostermann. Numerical Methods for Space-Time Variational Formulations ofRetarded Potential Boundary Integral Equations. PhD Thesis, Institut fur Ange-wandte Mathematik, Leibniz Universitat Hannover, 2010.

[3] C. Schwab. Variable order composite quadrature of singular and nearly singularintegrals. Computing 53, 2 (1994), 173–194.

[4] E. P. Stephan, M. Maischak, E. Ostermann. Transient boundary element methodand numerical evaluation of retarded potentials. In Computational Science – ICCS2008, Vol. 5102 of Lecture Notes in Computational Science , Springer, 2008, 321–330.

35-5


SOLVING THE HEAT EQUATION WITH A FASTMULTIPOLE GALERKIN BOUNDARY ELEMENT METHOD

Michael Messner1,a, Johannes Tausch2, Martin Schanz1,c and Wolfgang Weiss3

1Institute of Applied Mechanics, Graz University of Technology, Graz, [email protected], [email protected],

2Mathematics Department, Southern Methodist University, Dallas, Texas, [email protected],

3Institute of Tools and Forming, Graz University of Technology, Graz, [email protected]

We are solving boundary integral equations related to the transient heat equation [1].A straight forward space-time Galerkin discretization with equidistant time steps leadsto a lower Toeplitz structure in time with dense spatial contributions. The cost ofsolving such a system by forward substitution is O(N2M2) with N being the numberof time steps and M the number of spatial unknowns. We accelerate the spatio-temporal farfield by the parabolic fast multipole method [2, 3] to reduce the overallcomplexity to almost O(NM). On some benchmark examples we show that we achievethe correct convergence rates for a variety of pure and mixed initial boundary valueproblems. Moreover, we present some results from the thermal simulation of industrialapplications.

References

[1] M. Costabel, Boundary Integral Operators for the Heat Equation. Integral Equa-tions and Operator Theory 13:498-552, 1990.

[2] J. Tausch, A fast method for solving the heat equation by layer potentials. Journalof Computational Physics 224(2):956-969, 2007.

[3] M. Messner and J. Tausch and M. Schanz, Fast Galerkin Mehtod for ParabolicSpace-Time Boundary Integral Equations. submitted to: Journal of ComputationalPhysics

35-6


A GENERALIZED CONVOLUTION QUADRATUREWITH VARIABLE TIME STEPPING

Stefan A. Sauter

Institut fur Mathematik, Universitat Zurich,Winterthurerstr 190, CH-8057 Zurich, Switzerland

[email protected]

We will present a generalized convolution quadrature for solving linear parabolic andhyperbolic evolution equations. The original convolution quadrature method by Lubichworks very nicely for equidistant time steps while the generalization of the method andits analysis to non-uniform time stepping is by no means obvious. We will introducethe generalized convolution quadrature allowing for variable time steps and developa theory for its error analysis. This method opens the door for further developmenttowards adaptive time stepping for evolution equations. As the main application of ournew theory we will consider the wave equation in exterior domains, which is formulatedas a retarded boundary integral equation.

This work is in collaboration with Maria Lopez-Fernandez.

ADAPTIVE METHODS FOR RETARDEDBOUNDARY INTEGRAL EQUATIONS

Stefan A. Sautera and Alexander Veitb

Institute of Mathematics, University of Zurich, [email protected], [email protected]

We consider retarded boundary integral formulations of the three-dimensional waveequation in unbounded domains. Our goal is to apply a Galerkin method in spaceand time in order to solve these problems numerically. In this approach the accuratecomputation of the system matrix entries is the major bottleneck. In order to simplifythe arising quadrature problem we use globally smooth and compactly supported basisfunctions for the time discretization. This furthermore easily allows the use of a variabletime-stepping and a variable order of the approximation in time.

In order to obtain a scheme that automatically adapts the time grid to local irregu-larities in the solution we use suitable a posteriori error estimators. Various numericalexperiments show the behavior of the adaptive algorithm.

35-7


BEM FOR PARABOLIC PHASE PHANGEPROBLEMS WITH MOVING INTERFACES

Johannes Tauscha and Elizabeth Caseb

Department of Mathematics, Southern Methodist University, Dallas, TX, [email protected], [email protected]

Many phase change problems are governed by the heat equation in the liquid and soliddomain and the Stefan condition

vn = ks∂us∂n− kl

∂ul∂n

,

on the boundary between the two. Here, vn is the normal velocity of the interface, uis the temperature, k is a non-dimensionalized diffusion constant, and subscripts in-dicate solid and liquid phase. The goal is to determine the evolution of the unknowninterface. We consider a boundary integral formulation based on the Green’s repre-sentation formula of the heat equation Unlike the more standard approaches to thistype of problem there are only unknowns on the boundary. However, every time stepinvolves a convolution over the entire history of the problem.

The time-dependent integral equation is discretized with the Nystrom method of[J. Tausch, Applied Num. Math.59, pp. 2843-2856 (2009)]. Here, special attentionmust be given to extra terms in the Green’s representation formula and the discreteoperators due to the moving interfaces. We obtain a simple time-stepping methodand illustrate the stability and convergence on a number of integral formulations ofsolidification problems.

35-8


A HYBRID APPROACH TO THE TIME MARCHINGSOLUTION OF MAXWELL’S EQUATIONS

Daniel S. Weile

Department of Electrical and Computer Engineering, Newark, DE, [email protected]

The solution of the transient boundary integral formulation of electromagnetics gen-erally proceeds by one of two methods based on the temporal discretization of theequation. In the convolute quadrature (CQ) approach, the equation is discretized by amapping from the continuous Laplace domain into the discrete frequency domain andthen inverse transformed. On the other hand, the temporal Galerkin (TG) approachworks by approximating the temporal dependence of the unknown current as a linearcombination of basis functions, and then forcing the projection of the approximationon a given testing space to vanish.

The great advantage of CQ relative to TG arises from its predictable convergenceproperties and the ease of computing the convolution coefficients it requires. Unfortu-nately, another predictable outcome of the CQ procedure is an unphysical numericaldispersion, which can alter the appearance of important time domain effects such ascoherence and pulse shaping. Because time delays are computed explicitly in TG,numerical dispersion is avoided. The exact computation of the time delays, however,complicates the evaluation of the convolution coefficients because of the bizarre regionshapes that arise in the electromagnetic interaction of patches.

In this talk, we introduce a hybrid CQ/TG approach that combines the stabilityof CQ with the dispersion properties of TG. Specifically, CQ is used to compute theinteractions between proximal patches, and TG is used for laches at a greater dis-tance. Numerical results will demonstrate that the stability of CQ is preserved and itsdispersion is mitigated.

35-9


USING SPACE-TIME GALERKIN STABILITY THEORY TO DEFINEA ROBUST COLLOCATION METHOD FOR TIME-DOMAIN

BOUNDARY INTEGRAL EQUATIONS IN ELECTROMAGNETICS

Elwin van ’t Wout1a, Duncan R. van der Heul2b,Harmen van der Ven1c and Kees Vuik2d

1National Aerospace Laboratory NLR, Amsterdam, Netherlands,[email protected], [email protected]

2Delft Institute of Applied Mathematics,Delft University of Technology, Delft, Netherlands,


The time domain boundary integral equation (TDIE) method is an effective compu-tational method for modelling electromagnetic scattering phenomena. Because of theformulation in time domain, scatterers with a nonlinear response can be simulatedfor incident fields with a large frequency band. As a boundary element method, thenumber of spatial degrees of freedom scales quadratically with the electrical size of thescatterer. These advantages make it a promising method for industrial use in stealthtechnology and target recognition of electrically large structures.

Two of the most widely used discretisation schemes for TDIE methods are space-time Galerkin and collocation in time, also called Marching-on-in-Time (MoT). Bothhave distinct advantages. The design of space-time Galerkin schemes has been basedon a functional analysis of the variational formulation. Coerciveness and stability havebeen proven for specific Sobolev spaces. The efficiency of MoT schemes has beenimproved tremendously with the use of accelerators based on plane-wave expansionsand fast Fourier transforms. This has led to the successful application of the MoTscheme to electrically large and penetrable structures.

Aim of this paper is to combine the two numerical schemes into one method thatpossesses the advantages of both. That is, the stability theory of the space-timeGalerkin method will be used to define a robust collocation scheme. To this end,the functional analytic framework of the space-time Galerkin method will be adaptedfor different versions of the Electric Field Integral Equation (EFIE). This results ininfinite dimensional Sobolev spaces for which the variational formulations are stable.On a discrete level, the two methods will be shown to be equivalent for specific choicesof test and basis functions. So for a given space-time Galerkin scheme, a discretelyequivalent MoT scheme can be derived. The test and basis functions of the space-time Galerkin scheme will be chosen as elements of specific Sobolev spaces. Then, theequivalencies advocate the use of quadratic spline basis functions in the MoT scheme.

Computational experiments confirm the robustness of the MoT scheme when thebasis functions in time are chosen according to the mathematical framework of space-time Galerkin schemes. It is anticipated that with the combined experiences fromspace-time Galerkin and collocation schemes, a TDIE method can be formulated thatis sufficiently robust for industrial applications.

35-10

Author index

Abdul-Rahman Razi, 2-3Abert Claas, 6-2Abgrall R., 15-2Acosta Gabriel, 14-2Ahmed Naveed, 2-4Ainsworth Mark, 11-2, 17-3, 17-25, 23-2,

23-3Al-Hinai Omar, 30-11Allendes Alejandro, 11-2Antonietti Paola F., 7-2Apel Thomas, 3-2–3-4, 14-3Appelo Daniel, 17-11Armentano Marıa Gabriela, 14-2Arndt Daniel, 11-3Arraras Andres, 30-3Arridge S.R., 4-2Artina Marco, 2-5Aschemann Harald, 19-2, 19-4Awanou Gerard, 29-2Ayuso de Dios Blanca, 7-2

Bachmayr Markus, 21-2Badia Santiago, 16-2Bai Yun, 10-2Baiges Joan, 11-4Ballani Jonas, 21-3Bandara Kosala, 4-17Banjai Lehel, 35-2, 35-3Bank Randolph E., 17-19Banks H. Thomas, 22-2, 22-4Bansch Eberhard, 2-6Banz Lothar, 28-14Barreira Raquel, 31-4Barrenechea Gabriel R., 11-2, 11-5Bauman Paul T., 10-3Bause Markus, 11-6Beirao da Veiga L., 18-2Beirao da Veiga L., 15-3, 18-4Bercovier Michel, 15-5Beremlijski P., 28-2Berrone Stefano, 30-4Berzins Martin, 20-2Bespalov Alex, 33-2Betcke T., 4-2Beuchler Sven, 3-5Bierig Claudio, 28-4

Bijeljic Branko, 30-10Birch Malcolm J., 22-2, 22-4Blatt Markus, 20-4Blunt Martin J, 30-10Bohmer Klaus, 29-2Boffi Daniele, 2-11Bokhove Onno, 8-5Bonito Andrea, 26-3Bordas Stephane P.A., 1-2Bramble James H., 25-2Brenner Susanne C., 1-1Brewin Mark P., 22-2, 22-4Brezzi Franco, 18-2, 18-5Bruckner Florian, 6-3Brzobohaty Tomas, 28-2, 28-6Buffa Annalisa, 1-4, 15-3, 15-6, 17-4Bugert Beatrice, 4-3Bui-Thanh Tan, 17-5Burger Raimund, 12-2Burman Erik, 11-7Burstedde Carsten, 20-3, 20-11

Calhoun Donna, 20-3Calo Victor, 15-7Cangiani Andrea, 18-2Carey Varis, 10-3Carraro Thomas, 22-3, 30-5Carstensen Carsten, 2-7, 9-2, 9-3, 17-6Carter Edwards H., 33-6Case Elizabeth, 35-8Cavalli Fausto, 26-4Cavallini Nicola, 2-11Cecılio Diogo Lira, 2-8Cecot Witold, 17-24Cermak Martin, 28-3Cesmelioglu Aycil, 8-2Chalmers Noel, 34-4Chandler-Wilde Simon, 2-12, 25-3, 25-4Charrier Julia, 33-3Chatzipantelidis Panagiotis, 18-6Chen Jingmin, 13-2Chen Ronald, 17-11Chernov Alexey, 17-7, 28-4Childs. P.N., 2-12Chkadua Otar, 5-2Chrysafinos Konstantinos, 3-5

35-11

Chung Eric, 7-3Ciarlet Patrick, 14-4Cirak Fehmi, 4-17Cockburn Bernardo, 8-2, 8-5, 34-6Codina Ramon, 11-4, 16-2Collier Nathan, 15-7Costabel Martin, 1-5Croce Roberto, 28-11

Dahmen Wolfgang, 21-2Dalcin Lisandro, 15-7Dallmann Helene, 11-8Dauge Monique, 9-4Davies Penny J, 35-4Davydov Oleg, 29-3Dawson Clint N, 1-1Debussche Arnaud, 33-3Deckelnick Klaus, 3-6, 31-2Dedner Andreas, 7-4, 31-3Demaret Laurent, 13-3Demkowicz Leszek, 17-6, 17-8, 17-22Demlow Alan, 16-2Devloo Philippe R B, 2-8Dickopf Thomas, 20-6Diening Lars, 23-4Di Rienzo Luca, 15-6Dolejsı Vıt, 2-9, 23-5Dolgov Sergey V., 21-4, 21-8Dostal Zdenek, 28-5, 28-6, 28-13Drasdo Dirk, 27-7Drouet Guillaume, 28-8Duncan Dugald B, 2-10, 35-4

Egger Herbert, 17-7El-Kacimi A., 2-16Elfverson Daniel, 18-7Elliott Charles M., 1-6, 31-2, 31-4Elsheikh Ahmed H., 30-6, 32-2Engstrom Christian, 9-5Engwer Christian, 30-7Eriksson T., 17-27Ern Alexandre, 10-4, 23-5, 33-4Esterhazy Sofi, 25-5Evans John A., 15-8

Fahlke Jorrit, 20-4Fankhauser Thomas, 14-5Farrell Patrick E., 30-8Faustmann Markus, 4-4Feischl Michael, 4-5, 4-7, 23-6

Feng Xiaobing, 7-5, 29-3Fierro Francesca, 10-5Flaig Thomas G., 3-2, 3-3Fonrasier Massimo, 2-5Franz Sebastian, 11-9Frei Stefan, 3-7Friedmann Elfriede, 22-3Frutos Javier de, 11-9Fu Zhixing, 12-3Fuhrer Thomas, 4-5, 23-6Funke Simon W., 30-8

Gahalaut Krishan, 15-9Gallistl Dietmar, 9-3Gander Martin, 17-9, 25-6Ganesan Sashikumaar, 11-10Ganesh M., 4-8Ganis Benjamin, 30-9Garcıa-Archilla Bosco, 11-9, 11-11Garcia-Aznar Jose Manuel, 27-2Gardini Francesca, 2-11Gastaldi Lucia, 2-11Gedicke Joscha, 9-2, 9-3George Uduak, 27-4Georgoulis Emmanuil H., 7-5, 23-7, 26-5Gerds Peter, 9-6Gerecht Daniel, 22-3Geveler Markus, 20-4Ghattas Omar, 17-5, 20-11Giani Stefano, 9-7, 9-8Gimperlein Heiko, 4-9Gippert Sabrina, 24-2Glafke Matthias, 35-5Goldys Beniamin, 12-4Goll Christian, 30-5Gomes Jefferson L.M.A., 30-2Gomez-Benito Maria Jose, 27-2Gopalakrishnan Jay, 17-6, 17-8–17-10Gorman Gerard J., 30-2Graham I. G., 25-6Grasedyck Lars, 9-6, 21-3Greenwald Steve E., 22-2, 22-4Grohs Philipp, 13-4Groth Samuel, 25-7Grubisic Luka, 9-7, 9-8Gruhne Volker, 35-3Grundel Sara, 13-2Grzhibovskis Richards, 4-14, 5-3Gudi Thirupathi, 3-7, 7-6, 16-2

35-12

Gunzburger Max, 1-7Guo Wei, 34-2Guzman Johnny, 7-6, 16-2Gwinner Joachim, 28-9

Hagstrom Thomas, 17-11Hakula Harri, 9-7, 17-12Hannukainen Antti, 25-9Harasim Petr, 2-2Harder Christopher, 18-8Hardering Hanne, 13-4Hesthaven J. S., 4-8Heuer Norbert, 4-9, 4-10, 12-3, 17-13Hewett David, 25-3, 25-4, 25-7Hiemstra Rene, 15-8Hild Patrick, 28-8Hinze Michael, 3-6Hiptmair Ralf, 18-9Hoteit Ibrahim, 30-6Houston Paul, 17-14Howarth C. J., 2-12Hrkac Gino, 6-4Hu Jonathan, 33-6Hu Shuhua, 22-2, 22-4Huber Martin, 25-10Hughes Thomas J.R., 15-8Hunsicker Eugenie, 14-6, 14-7

Ippisch Olaf, 20-4Isaac Tobin, 20-11Izsak Ferenc, 34-2

Jackson Matthew D., 30-2Jagiella Nick, 27-7Janssen Barbel, 26-5Jansson Johan, 2-6Javierre Etelvina, 27-2Jensen Max, 29-4Jeon Youngmok, 17-15Jimack Peter K., 20-5John Volker, 11-5, 11-11, 11-12Juntunen Mika, 30-9Juttler Bert, 15-12, 15-15

Kamaludin Sarah, 2-3Kanschat Guido, 7-7, 20-6Karakatsani Fotini, 2-6Karkulik Michael, 4-5, 4-10Kavaliou Klim, 11-12Kazeev Vladimir, 21-5

Kenz Zackary, 22-2–22-4Khoromskaia Venera, 21-6Khoromskij Boris, 21-7Kirby Robert M., 34-3Klawonn Axel, 24-2, 24-3Kleiss S.K., 15-10Klofkorn Robert, 7-4Knobloch Petr, 11-5Kohler Karoline, 2-7Koppenol D.C., 27-3Kopteva Natalia, 14-8Kornhuber Ralf, 28-10Kossioris G.T., 26-6Kostin Georgy, 19-3, 19-6Kotas Petr, 28-11Kovar Petr, 4-11Kovarova Tereza, 4-11Kozubek Tomas, 28-2, 28-5, 28-6, 28-13Krause Dorian, 20-6Krause Rolf, 20-6, 28-11Krivodonova Lilia, 34-4Kruse Carola, 22-2, 22-4Kubatko Ethan, 8-3Kucera Vaclav, 2-14Kuerten J.G.M., 8-6Kumar B. V. Rathish, 2-22Kumar Dinesh, 17-16Kunoth Angela, 33-5Kusner Robert, 13-2Kuzmin Dmitri, 11-15Kyza Irene, 26-3

Laghrouche O., 2-16Lahaye D., 25-11Lakkis Omar, 23-7, 26-6, 29-4Laliena Antonio, 35-2Langdon Stephen, 2-12, 25-3, 25-4, 25-7Lanser Martin, 24-2Larsson Stig, 33-5Le Kim-Ngan, 12-4Ledger P.D., 17-17Lee Chang-Ock, 2-15Lehrenfeld Christoph, 17-23Le Maıtre Olivier, 33-4Leykekhman Dmitriy, 3-8, 3-9, 16-3Li Buyang, 26-7Li Hengguang, 14-6, 14-7, 16-4Li Jichun, 17-18Li Xiaozhou, 34-5

35-13

Lian Haojie, 15-13Liertzer Matthias, 25-5Linke Alexander, 11-13Lionheart W.R.B., 17-17Lipnikov Konstantin, 8-4, 18-10Liu Xuefeng, 9-9Lombardi Ariel, 14-3Lube Gert, 11-8, 11-17Lubich Christian, 35-3Lukas Dalibor, 4-11

Madhavan Pravin, 31-3Madzvamuse Anotida, 27-4, 31-4Mahmood M., 2-16Maischak Matthias, 28-12, 35-5Makridakis Charalambos, 2-6, 23-7, 26-2Malqvist Axel, 18-7Maly Lukas, 4-11Manni Carla, 15-11Mantzaflaris Angelos, 15-12Manzini Gianmarco, 18-2–18-4, 18-10Marini L. Donatella, 18-5Marini L.D., 18-2Markopoulos Alexandros, 28-2, 28-6Marquez Antonio, 12-5Mateos Mariano, 3-4Matthies Gunar, 2-4, 11-14Mazzieri Ilario, 7-2Meddahi Salim, 9-10, 12-5, 17-13Medvedeva Tatyana, 8-5Mehrmann V., 9-3Mehta Dwij, 22-2Meidner Dominik, 3-9Meir Ammon J., 31-4Melenk Jens Markus, 4-4, 4-12, 7-7, 17-8,

17-19, 25-5Merta Michal, 4-11, 28-3Messner Michael, 35-6Micheletti Stefano, 2-5Miedlar Agnieszka, 9-3, 9-8Mikhailov Sergey E., 5-3, 5-4Moller Matthias, 2-18Moiola Andrea, 18-9Monk Peter, 1-8Mora David, 9-10Moser Robert, 17-22Mostaghimi Peyman, 30-2Muller Benjamin, 24-5Mueller Eike, 20-7

Muller Benjamin, 24-3Muga Ignacio, 17-8, 17-10Mustapha Kassem, 34-6, 34-7

Naldi Giovanni, 26-4Nannen Lothar, 25-10Nappi Angela, 8-3Naß Martin, 3-10Nataraj Neela, 3-7Natroshvili David, 5-6Neckel Tobias, 20-8Neethling Stephen J., 30-2Neic Aurel, 20-9Neilan Michael, 12-5, 16-5, 29-5Neitzel Ira, 3-10Neumuller Martin, 17-9Nguyen Hieu, 17-19Nguyen Ngoc Cuong, 8-2Nicaise Serge, 3-2, 14-9Niemeyer Julia, 2-19Niemi Antti H., 17-20Nigam Nilima, 17-20Nistor Victor, 14-6, 14-7Nochetto Ricardo H., 10-6, 26-3Novo Julia, 11-9, 11-11

Oden J. Tinsley, 1-9Of Gunther, 4-13, 4-17Olivares Nicole, 17-10Ostien Jakob T., 33-6Ovall Jeffrey S, 9-7, 9-8, 16-4Oyarzua Ricardo, 12-6

Page Marcus, 6-4, 6-5Pain Christopher C., 30-2Paredes Diego, 18-8Pardo David, 15-7, 17-8Park Eun-Hee, 2-15Park Eun-Jae, 17-15Parsania Asieh, 7-7Pasciak Joseph E., 25-2Paszynski Maciej, 15-7Patra A. K., 17-16Payne Mark, 2-10Pelosi Francesca, 15-11Pencheva Gergina, 30-9, 30-11Peraire Jaime, 8-2Percival James R., 30-2Perotto Simona, 2-5Perugia Ilaria, 18-9, 26-4

35-14

Petit Julien, 17-20Petras Argyrios, 17-20Pfefferer Johannes, 3-4, 3-10Phillips Joel, 17-20Phipps Eric, 33-6Pieper Konstantin, 3-11Pieraccini Sandra, 30-4Pitman E. B., 17-16Poletti Valentina, 28-11Portero Laura, 30-3Pospısil Lukas, 28-13Potse Mark, 20-6Powell Catherine, 33-2Praetorius Dirk, 4-4, 4-5, 4-7, 4-12, 6-5,

23-6Pryer Tristan, 29-4

Qiu Jianxian, 34-2Qiu Jing-Mei, 34-2Qiu Weifeng, 8-2, 17-9Quarteroni Alfio, 7-2

Rachowicz Waldemar, 17-21, 17-27Radtke Patrick, 24-3Raeini Ali Q, 30-10Raisch Alexander, 13-5Rankin Richard, 11-2, 23-2Rannacher Rolf, 3-7Ranner Tom, 31-2Rauh Andreas, 19-2, 19-4Repin Sergey, 23-8Rhebergen Sander, 7-9, 8-5Rheinbach Oliver, 24-2, 24-3Richter Thomas, 32-2, 32-3Riviere Beatrice, 7-8Rjasanow Sergej, 4-14Roberts Nathan V., 17-22Rodrıguez Rodolfo, 9-10Rosch Arnd, 3-4, 3-10Romito M., 26-6Rotter Stefan, 25-5Ruede Ulrich, 1-10Ruiz-Baier Ricardo, 12-2Russo Alessandro, 18-2, 18-5Ryan Jennifer K., 34-7

Sadhanala Veeranjaneyulu, 3-7Saeed Abid, 29-3Salgado A.J., 10-6Salgado Pilar, 2-21

Sander Oliver, 24-4, 28-10Sangalli Giancarlo, 1-4, 15-3, 17-4Sangwan Vivek, 2-22Santos Erick Raggio Slis, 2-8Saurin Vasily, 19-3, 19-6Sauter Stefan A., 7-7, 9-10, 35-7Savostyanov Dmitry V., 21-4, 21-8Sayas Francisco-Javier, 4-9, 12-3, 12-5, 35-

2, 35-3Schanz Martin, 35-6Schedensack Mira, 9-3Scheichl Robert, 20-7Schieweck Friedhelm, 11-15Schillinger Dominik, 15-8Schillings Claudia, 33-7Schindele Dominik, 19-2Schmidt Alfred, 10-5Schmidt Gunther, 4-3Schneider Rene, 2-23Schoberl Joachim, 17-23, 20-10, 25-10Schotzau Dominik, 12-6Schroder Jorg, 24-3, 24-5Schroder Andreas, 28-14Schugart Richard, 27-5Schwab Christoph, 18-9, 33-7Schwarz Alexander, 24-3, 24-5Schwegler Kristina, 11-6Schweiger M., 4-2Scialo Stefano, 30-4Scott Michael A., 15-13Sebestova Ivana, 2-9Segeth Karel, 2-23Selgas Virginia, 2-21Sellier A., 4-15Senkel Luise, 19-4Serafin Marta, 17-24Sharma Natasha, 7-7Shaw Simon, 22-2, 22-4Sheikh A. H., 25-11Si Hang, 11-16Siebert Kunibert G., 20-10, 23-8Silvester David, 33-2Simeon Bernd, 2-19, 15-15, 24-6Simpson Robert N., 15-13Singh Gurpreet, 30-11Slodicka Marian, 6-6Smears Iain, 29-4Smigaj W., 4-2Sofo Jorge, 14-7

35-15

Sokolov Andriy, 31-6Speleers Hendrik, 15-11, 15-14Spence E. A., 25-6Srivastava Shweta, 11-10Stadler Georg, 20-3, 20-11Stamm B., 4-8Starke Gerhard, 24-3, 24-5, 24-7Steeger Karl, 24-3, 24-5Steinbach Olaf, 3-11, 4-13, 4-17, 17-9Stephan Ernst P., 12-6, 28-15Stephanou Angelique, 27-4Stinner Bjorn, 31-3Storath Martin, 13-3Styles Vanessa, 31-5Suess Dieter, 6-4Sun Weiwei, 26-7Sundar Hari, 20-11Sung Li-yeng, 3-12Svyatskiy Daniil, 18-10

Takacs Thomas, 15-15Tantardini Francesca, 16-6Taus Matthias, 15-13Tausch Johannes, 35-6, 35-8Teckentrup Aretha, 33-7Thomas Derek C., 15-13Tian Lulu, 8-6Tobiska Lutz, 11-12, 11-14Tollit Brendan S., 30-2Tomar Satyendra, 15-9, 15-10, 20-11Torres Hector, 12-2Tran Thanh, 12-4Trevelyan J., 2-16Tryoen Julie, 33-4Tuncer Necibe, 31-4Turek Stefan, 31-6Twigger Ashley, 25-3, 25-4

Unger Gerhard, 4-16Uski Vile, 14-6, 14-7

Vainikko Eero, 20-7Valdman Jan, 2-2Valentin Frederic, 18-8Valero Clara, 27-2Van Bockstal Karel, 6-6van der Heul Duncan R., 35-10van der Vegt Jaap, 7-9, 8-5, 8-6van der Ven Harmen, 35-10van der Zee K.G., 10-6

van ’t Wout Elwin, 35-10Vassilev Danail, 8-4Vazquez Rafael, 1-4, 15-3, 15-6, 17-4Veeser Andreas, 10-5, 16-6Veit Alexander, 35-7Vejchodsky Tomas, 23-3Verani Marco, 18-11Verfurth Rudiger, 16-6Vermolen Fred J., 27-3, 27-6Vexler Boris, 3-8, 3-9, 3-11, 16-3Vignon-Clementel Irene, 27-7Vlach Oldrich, 28-5, 28-6Vohralık Martin, 2-9, 10-4, 23-5Vuik C., 25-11Vuik Kees, 35-10Vuong Anh-Vu, 15-15

Wacker Benjamin, 11-17Walkley Mark A., 20-5Wang Hongrui, 17-25Wang Jilu, 17-25Warburton T., 17-26Weeger Oliver, 24-6Weile Daniel S., 35-9Weinmann Andreas, 13-3Weiss Wolfgang, 35-6Wheeler Mary F., 30-6, 30-9, 30-11, 32-2Whiteman John R., 22-2, 22-4Wick Thomas, 32-2, 32-3Wihler Thomas P., 7-10, 14-5, 26-5Wilcox Lucas C., 20-3Winkler Max, 14-3Winther Ragnar, 1-10Wirz Marcel, 7-10, 14-5Wohlmuth B., 4-12Wollner Winnifried, 3-7, 32-3Wu Haijun, 14-4

Xenophontos Christos, 14-9, 17-19Xu Yan, 8-6

Ye Qiang, 9-11Yotov Ivan, 8-4, 30-3, 30-9, 30-12Youett Jonathan, 28-10Yu Thomas, 13-2

Zapletal Jan, 4-17Zdunek Adam, 17-21, 17-27Zhang Jianfei, 20-5Zigerelli Andrew, 13-2

35-16

Zlotnik Alexander, 17-2Zou Jun, 14-4

35-17

mafelap 2013 - brunel university...

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