Numerical Mathematics and AdvancedApplications 2009

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Gunilla Kreiss • Per Lötstedt • Axel MålqvistMaya NeytchevaEditors

Numerical Mathematicsand AdvancedApplications 2009

Proceedings of ENUMATH 2009,the 8th European Conference on NumericalMathematics and Advanced Applications,Uppsala, July 2009

123

EditorsGunilla KreissPer LötstedtAxel MålqvistMaya NeytchevaUppsala UniversityDepartment of Information Technology751 05 UppsalaSwedengunilla.kreiss@it.uu.seperl@it.uu.seaxel.malqvist@it.uu.semaya.neytcheva@it.uu.se

ISBN 978-3-642-11794-7 e-ISBN 978-3-642-11795-4DOI 10.1007/978-3-642-11795-4Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2010937319

Mathematics Subject Classification (2010): 65-06

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Preface

The European Conference on Numerical Mathematics and Advanced Applications(ENUMATH) was held from June 29–July 3, 2010, in Uppsala, Sweden. This wasthe eighth conference in a series of biannual meetings starting in Paris (1995). Sub-sequent conferences were organized in Heidelberg (1997), Jyväskylä (1999), Ischia(2001), Prague (2003), Santiago de Compostela (2005), and Graz (2007). ENU-MATH 2009 attracted over 330 attendees to the scientific programme, with teninvited speakers, one public lecture, 32 minisymposia, and more than 280 presenta-tions. This volume contains a selection of papers by the invited speakers and fromthe minisymposia and the contributed sessions.

The purpose of the conference was to create a forum for discussion and dis-semination of recent results in numerical mathematics and new applications ofcomputational methods. Many subjects were covered in the talks and a few of thetopics represented in these proceedings were discontinuous Galerkin methods, finiteelement methods in different applications, methods for fluid flow, electromagnetism,financial engineering, structural mechanics, optimal control, and biomechanics. Theminisymposia listed below with their organizers also give an impression of howbroad the scope of the conference was:

� Adaptivity for non-linear and non-smooth problems, part I & II, Ralf Kornhuber,Andreas Veeser

� Advanced techniques in radial basis function approximation for PDEs, part I& II, Natasha Flyer, Elisabeth Larsson

� Advances in numerical methods for non-Newtonian flows, part I & II, ErikBurman, Maxim Olshanskii, Stefan Turek

� Anisotropic adaptive meshes: error analysis and applications, part I & II, ThierryCoupez, Simona Perotto

� Asymptotic linear algebra, numerical methods, and applications, part I & II,Marco Donatelli, Stefano Serra-Capizzano

� Biomechanics, part I & II, Gerhard A. Holzapfel, Axel Klawonn� Embedded boundary methods for time-dependent problems, Daniel Appelö� Finite element software development, Anders Logg� Finite elements for convection-diffusion problems, part I, II & III, Miloslav

Feistauer, Petr Knobloch

v

vi Preface

� Finite element methods for flow problems, Johan Hoffman� Geometric aspects of the finite element modeling, part I & II, Sergey Korotov,

Tomas Vejchodsky� High frequency wave propagation, Olof Runborg� High order methods in CFD, Bernhard Müller� HPC-driven numerical methods and applications, part I & II, Svetozar Margenov,

Maya Neytcheva� Multiscale methods for differential equations, part I & II, Mats Larson, Axel

Målqvist� Numerical methods for multi-dimensional Lagrangian schemes, Pierre-Henri

Maire, Raphaël Loubere� Numerical methods for option pricing, Cornelis W. Oosterlee, Jari Toivanen� Numerical methods for stochastic partial differential equations, part I & II, Fabio

Nobile, Raul Tempone� Tensor numerical methods, Eugene Tyrtyshnikov, Boris Khoromskij� Theory and applications of non-conforming finite element methods, Emmanuil

Georgoulis, Max Jensen

The conference was organized by the Division of Scientific Computing of theDepartment of Information Technology at Uppsala University in collaboration withAkademikonferens in Uppsala. Uppsala University is not as old as the universities inParis, Heidelberg, and Prague, but it is the oldest university in the Nordic countries.It was founded in 1477 and the first professor in mathematics was appointed in1593. The first professor in numerical analysis, Heinz-Otto Kreiss, started his workin 1965.

The success of the conference was in a large part due to the invited speakersMartin Berggren, Daniele Boffi, Carsten Carstensen, Vit Dolejsi, Charlie Elliott,Claude Le Bris, Christian Lubich, Marco Picasso, Rob Stevenson, and Anna-KarinTornberg, as well as to Björn Engquist, who delivered the public lecture. The mem-bers of the program committee were Franco Brezzi, Miloslav Feistauer, RolandGlowinski, Rolf Jeltsch, Yuri Kuznetsov, Jacques Périaux, Rolf Rannacher, andEndre Süli. They selected the invited speakers and helped by sharing their knowl-edge of how are organized these conferences.

The scientific committee consisted of Christine Bernardi, Alfredo Bermudez deCastro, Albert Cohen, Claudio Canuto, Michael Griebel, Peter Hansbo, JaroslavHaslinger, Thomas Huckle, Karl Kunisch, Ulrich Langer, Stig Larsson, OlivierPironneau, Sergey Repin, Miro Rozloznik, J. J. Sanz-Serna, Stefan Sauter,Stefano Serra Capizzano, Valeria Simoncini, Olaf Steinbach, Rolf Stenberg, AndersSzepessy, Stefan Turek, Kees Vuik, Ragnar Winther, and Barbara Wohlmuth. Mem-bers of the committee, Martin Berggren and Bernhard Müller have served as refereesfor this volume.

The local committee was assisted by PhD students at our Division: Qaisar Abbas,Kenneth Duru, Magnus Gustafsson, Andreas Hellander, Stefan Hellander, KatharinaKormann, Martin Kronbichler, Erik Lehto, Anna Nissen, Elena Sundkvist, MartinTillenius, Salman Toor, and He Xin. The change between speakers in the sessionswould not have been so smooth without their presence in the lecture rooms. Specialthanks to Kenneth Duru for helping with the preparations of the proceedings.

Preface vii

The conference received financial support from Centre for InterdisciplinaryMathematics at Uppsala University, City of Uppsala, Comsol, Swedish Foundationfor Strategic Research, Swedish Research Council, Uppsala Multidisciplinary Cen-ter for Advanced Computational Science (UPPMAX), Wenner-Gren Foundations,and John Wiley & Sons. Their generous contributions helped to lower the fees forthe participants.

Last but not least, many thanks to Karin Hornay and Maria Bäckström fromAkademikonferens for sharing their invaluable experience in organizing confer-ences.

Uppsala Gunilla KreissMarch 2010 Per Lötstedt

Axel MålqvistMaya Neytcheva

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Contents

Part I Invited Papers

Discrete Differential Forms, Approximation of EigenvalueProblems, and Applicationto the p Version of Edge Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Daniele Boffi

Semi-Implicit DGFE Discretization of the CompressibleNavier–Stokes Equations: Efficient Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 15Vı́t Dolejšı́ and M. Holı́k

Some Numerical Approaches for Weakly RandomHomogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Claude Le Bris

Goal Oriented, Anisotropic, A Posteriori Error Estimatesfor the Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Frederic Alauzet, Wissam Hassan, and Marco Picasso

Part II Contributed Papers

Energy Stability of the MUSCL Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Qaisar Abbas, Edwin van der Weide, and Jan Nordström

Numerical Stabilization of the Melt Front for Laser BeamCutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Torsten Adolph, Willi Schönauer, Markus Niessen,and Wolfgang Schulz

Numerical Optimization of a Bioreactor for the Treatmentof Eutrophicated Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Lino J. Alvarez-Vázquez, Francisco J. Fernández,and Aurea Martı́nez

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Finite Element Approximation of a Quasi-3D Modelfor Estuarian River Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Mohamed Amara, Agnès Pétrau, and David Trujillo

Convergence of a Mixed Discontinuous Galerkinand Finite Volume Scheme for the 3 DimensionalVlasov–Poisson–Fokker–Planck System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Mohammad Asadzadeh and Piotr Kowalczyk

Infrastructure for the Coupling of Dune Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107Peter Bastian, Gerrit Buse, and Oliver Sander

FEM for Flow and Pollution Transport in a Street Canyon . . . . . . . . . . . . . . . . . . .115Petr Bauer, Atsushi Suzuki, and Zbyněk Jaňour

Stabilized Finite Element Methods with Shock-Capturingfor Nonlinear Convection–Diffusion-Reaction Models . . . . . . . . . . . . . . . . . . . . . . . . .125Markus Bause

Finite Element Discretization of the Giesekus Modelfor Polymer Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135Roland Becker and Daniela Capatina

A dG Method for the Strain-Rate Formulation of the StokesProblem Related with Nonconforming Finite Element Methods . . . . . . . . . . . . .145Roland Becker, Daniela Capatina, and Julie Joie

Numerical Simulation of the Stratified Flow Past a Body . . . . . . . . . . . . . . . . . . . . .155L. Beneš and J. Fürst

A Flexible Updating Framework for Preconditionersin PDE-Based Image Restoration Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163Daniele Bertaccini and Fiorella Sgallari

Stabilized Finite Element Methodfor Compressible–Incompressible Diphasic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . .171M. Billaud, G. Gallice, and B. Nkonga

An Immersed Interface Technique for the Numerical Solutionof the Heat Equation on a Moving Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .181François Bouchon and Gunther H. Peichl

Lid-Driven-Cavity Simulations of Oldroyd-B ModelsUsing Free-Energy-Dissipative Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191Sébastien Boyaval

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Adaptive Multiresolution Simulationof Waves in Electrocardiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199Raimund Bürger and Ricardo Ruiz-Baier

On the Numerical Approximation of the Laplace TransformFunction from Real Samples and Its Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .209R. Campagna, L. D’Amore, A. Galletti, A. Murli, and M. Rizzardi

A Motion-Aided Ultrasound Image Sequence Segmentation . . . . . . . . . . . . . . . . .217D. Casaburi, L. D’Amore, L. Marcellino, and A. Murli

A High Order Finite Volume Numerical Scheme for ShallowWater System: An Efficient Implementation on GPUs . . . . . . . . . . . . . . . . . . . . . . . .227M.J. Castro Dı́az, M. Lastra, J.M. Mantas, and S. Ortega

Spectral Analysis for Radial Basis Function CollocationMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .237R. Cavoretto, A. De Rossi, M. Donatelli, and S. Serra-Capizzano

Finite Element Solution of the Primitive Equationsof the Ocean by the Orthogonal Sub-Scales Method . . . . . . . . . . . . . . . . . . . . . . . . . . .245Tomás Chacón Rebollo, Macarena Gómez Mármol,and Isabel Sánchez MuQnoz

Solution of Incompressible Flow Equations by a High-OrderTerm-by-Term Stabilized Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253Tomás Chacón Rebollo, Macarena Gómez Mármol,and Isabel Sánchez MuQnoz

Solving Large Sparse Linear SystemsEfficiently on Grid Computers Using an AsynchronousIterative Method as a Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261T.P. Collignon and M.B. van Gijzen

Hierarchical High Order Finite Element ApproximationSpaces for H(div) and H(curl) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .269Denise De Siqueira, Philippe R.B. Devloo, and Sônia M. Gomes

Some Theoretical Results About Stability for IMEX SchemesApplied to Hyperbolic Equations with Stiff Reaction Terms . . . . . . . . . . . . . . . . . .277Rosa Donat, Inmaculada Higueras, and Anna Martinez-Gavara

Stable Perfectly Matched Layers for the Schrödinger Equations . . . . . . . . . . . .287Kenneth Duru and Gunilla Kreiss

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Domain Decomposition Schemes for Frictionless MultibodyContact Problems of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .297Ivan I. Dyyak and Ihor I. Prokopyshyn

Analysis and Acceleration of a Fluid-Structure InteractionCoupling Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .307Michael R. Dörfel and Bernd Simeon

Second Order Numerical Operator Splitting for 3DAdvection–Diffusion-Reaction Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .317Riccardo Fazio and Alessandra Jannelli

Space-Time DG Method for NonstationaryConvection–Diffusion Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .325Miloslav Feistauer, Václav Kučera, Karel Najzar,and Jaroslava Prokopová

High Order Finite Volume Schemes for Numerical Solutionof Unsteady Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .335Petr Furmánek, Jiřı́ Fürst, and Karel Kozel

Multigrid Finite Element Method on Semi-Structured Gridsfor the Poroelasticity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .343F.J. Gaspar, F.J. Lisbona, and C. Rodrigo

A Posteriori Error Bounds for Discontinuous GalerkinMethods for Quasilinear Parabolic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .351Emmanuil H. Georgoulis and Omar Lakkis

An A Posteriori Analysis of Multiscale OperatorDecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .359Victor Ginting

Goal-Oriented Error Estimation for the DiscontinuousGalerkin Method Applied to the Biharmonic Equation . . . . . . . . . . . . . . . . . . . . . . .369João L. Gonçalves, Philippe R.B. Devloo, and Sônia M. Gomes

Solving Stochastic Collocation Systemswith Algebraic Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .377Andrew D. Gordon and Catherine E. Powell

Adaptive Two-Step Peer Methods for IncompressibleNavier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .387B. Gottermeier and J. Lang

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On Hierarchical Error Estimators for Time-Discretized PhaseField Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .397Carsten Gräser, Ralf Kornhuber, and Uli Sack

Nonlinear Decomposition Methods in Elastodynamics . . . . . . . . . . . . . . . . . . . . . . . .407Christian Groß, Rolf Krause, and Mirjam Walloth

An Implementation Framework for Solving High-DimensionalPDEs on Massively Parallel Computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .417Magnus Gustafsson and Sverker Holmgren

Benchmarking FE-Methods for the Brinkman Problem . . . . . . . . . . . . . . . . . . . . . .425Antti Hannukainen, Mika Juntunen, and Rolf Stenberg

Finite Element Based Second Moment Analysis for EllipticProblems in Stochastic Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .433H. Harbrecht

On Robust Parallel Preconditioning for Incompressible FlowProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .443Timo Heister, Gert Lube, and Gerd Rapin

Hybrid Modeling of Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .451Mats Holmström

A Priori Error Estimates for DGFEM Applied to NonstationaryNonlinear Convection–Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .459J. Hozman and V. Dolejšı́

Stable Crank–Nicolson Discretisation for IncompressibleMiscible Displacement Problems of Low Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . .469Max Jensen and Rüdiger Müller

Simulations of 3D/4D Precipitation Processes in a TurbulentFlow Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .479Volker John and Michael Roland

2D Finite Volume Lagrangian Scheme in Hyperelasticityand Finite Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .489Gilles Kluth and Bruno Després

Local Projection Method for Convection-Diffusion-ReactionProblems with Projection Spaces Defined on Overlapping Sets . . . . . . . . . . . . . .497Petr Knobloch

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Numerical Solution of Volterra Integral Equationswith Weak Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .507M. Kolk and A. Pedas

Non-Conforming Finite Element Method for the BrinkmanProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .515Juho Könnö and Rolf Stenberg

Error Control for Simulations of a Dissociative QuantumSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .523Katharina Kormann and Anna Nissen

A Comparison of Simplicial and Block Finite Elements . . . . . . . . . . . . . . . . . . . . . . .533Sergey Korotov and Tomáš Vejchodský

Five-Dimensional Euclidean Space Cannot be ConformlyPartitioned into Acute Simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .543Michal Křı́žek

The Discontinuous Galerkin Method for Convection-DiffusionProblems in Time-Dependent Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .551Václav Kučera, Miloslav Feistauer, and Jaroslava Prokopová

A Spectral Time-Domain Method for ComputationalElectrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .561James V. Lambers

Numerical Simulation of Fluid–Structure Interactionin Human Phonation: Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .571Martin Larsson and Bernhard Müller

Error Estimation and Anisotropic Mesh Refinementfor Aerodynamic Flow Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .579Tobias Leicht and Ralf Hartmann

A MHD Problem on Unbounded Domains: Coupling of FEMand BEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .589Wiebke Lemster and Gert Lube

A Stable and High Order Interface Procedure for ConjugateHeat Transfer Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .599Jens Lindström and Jan Nordström

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Local Time-Space Mesh Refinement for Finite DifferenceSimulation of Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .609Vadim Lisitsa, Galina Reshetova, and Vladimir Tcheverda

Formulation of a Staggered Two-Dimensional LagrangianScheme by Means of Cell-Centered Approximate RiemannSolver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .617R. Loubère, P.-H. Maire, and P. Váchal

Optimal Control for River Pollution Remediation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .627Aurea Martı́nez, Lino J. Alvarez-Vázquez,Miguel E. Vázquez-Méndez, and Miguel A. Vilar

An Anisotropic Micro-Sphere Approach Appliedto the Modelling of Soft Biological Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .637A. Menzel, T. Waffenschmidt, and V. Alastrué

Anisotropic Adaptation via a Zienkiewicz–Zhu ErrorEstimator for 2D Elliptic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .645S. Micheletti and S. Perotto

On a Sediment Transport Model in Shallow Water Equationswith Gravity Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .655T. Morales de Luna, M.J. Castro Dı́az, and C. Parés Madroñal

Adaptive SQP Method for Shape Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .663P. Morin, R.H. Nochetto, M.S. Pauletti, and M. Verani

Convergence of Path-Conservative Numerical Schemesfor Hyperbolic Systems of Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .675M.L. Muñoz-Ruiz, C. Parés, and M.J. Castro Dı́az

A Two-Level Newton–Krylov–Schwarz Methodfor the Bidomain Model of Electrocardiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .683M. Munteanu, L.F. Pavarino, and S. Scacchi

On a Shallow Water Model for Non-Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . .693G. Narbona-Reina and D. Bresch

On Stationary Viscous Incompressible FlowThrough a Cascade of Profiles with the ModifiedBoundary Condition on the Outflow and Large Inflow . . . . . . . . . . . . . . . . . . . . . . . .703Tomáš Neustupa

xvi Contents

Variational and Heterogeneous Multiscale Methods . . . . . . . . . . . . . . . . . . . . . . . . . . .713Jan Martin Nordbotten

Discrete Dislocation Dynamics and Mean Curvature Flow . . . . . . . . . . . . . . . . . . .721Petr Pauš, Michal Beneš, and Jan Kratochvı́l

Non-Symmetric Algebraic Multigrid Preconditionersfor the Bidomain Reaction–Diffusion system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .729Micol Pennacchio and Valeria Simoncini

Efficiency of Shock Capturing Schemes for Burgers’ Equationwith Boundary Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .737Per Pettersson, Qaisar Abbas, Gianluca Iaccarino,and Jan Nordström

FEM Techniques for the LCR Reformulation of ViscoelasticFlow Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .747A. Ouazzi, H. Damanik, J. Hron, and S. Turek

A Posteriori Estimates for Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . .755S. Repin

Review on Longest Edge Nested Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .763Maria-Cecilia Rivara

Simulation of Spray Painting in Automotive Industry . . . . . . . . . . . . . . . . . . . . . . . . .771Robert Rundqvist, Andreas Mark, Björn Andersson,Anders Ålund, Fredrik Edelvik, Sebastian Tafuri,and Johan S Carlson

Numerical Simulation of the Electrohydrodynamic Generationof Droplets by the Boundary Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .781P. Sarmah, A. Glière, and J.-L. Reboud

A General Pricing Technique Based on Theta-Calculusand Sparse Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .791Stefanie Schraufstetter and Janos Benk

A Posteriori Error Estimation in Mixed Finite ElementMethods for Signorini’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .801Andreas Schröder

Solution of an Inverse Problem for a 2-D Turbulent FlowAround an Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .809Jan Šimák and Jaroslav Pelant

Contents xvii

On Skew-Symmetric Splitting and Entropy ConservationSchemes for the Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .817Björn Sjögreen and H.C. Yee

Ideal Curved Elements and the Discontinuous GalerkinMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .829Veronika Sobotı́ková

Analysis of the Parallel Finite Volume Solverfor the Anisotropic Allen–Cahn Equation in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .839Pavel Strachota, Michal Beneš, Marco Grottadaurea,and Jaroslav Tintěra

Stabilized Finite Element Approximations of Flow Overa Self-Oscillating Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .847Petr Sváček and Jaromı́r Horáček

Multigrid Methods for Elliptic Optimal Control Problemswith Neumann Boundary Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .855Stefan Takacs and Walter Zulehner

Extension of the Complete Flux Scheme to Time-DependentConservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .865J.H.M. ten Thije Boonkkamp and M.J.H. Anthonissen

Solution of Navier–Stokes Equations Using FEMwith Stabilizing Subgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .875M. Tezer-Sezgin, S. Han Aydın, and A.I. Neslitürk

Multigrid Methods for Control-Constrained Elliptic OptimalControl Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .883Michelle Vallejos and Alfio Borzı̀

Modelling the New Soil Improvement Method Biogrout:Extension to 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .893W.K. van Wijngaarden, F.J. Vermolen, G.A.M. van Meurs,and C. Vuik

Angle Conditions for Discrete Maximum Principlesin Higher-Order FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .901Tomáš Vejchodský

Unsteady High Order Residual Distribution Schemeswith Applications to Linearised Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .911N. Villedieu, L. Koloszar, T. Quintino, and H. Deconinck

xviii Contents

Implicit–Explicit Backward Difference FormulaeDiscontinuous Galerkin Finite Element Methodsfor Convection–Diffusion Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .921Miloslav Vlasák and Vı́t Dolejšı́

A Cut-Cell Finite-Element Method for a Discontinuous SwitchModel for Wound Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .929S.V. Zemskov, F.J. Vermolen, E. Javierre, and C. Vuik

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .937

Part IInvited Papers

Discrete Differential Forms, Approximationof Eigenvalue Problems, and Applicationto the p Version of Edge Finite Elements

Daniele Boffi

Abstract We are interested in the approximation of the eigenvalues of Hodge–Laplace operator in the framework of de Rham complex by using exterior calculusand suitable equivalent formulations in mixed form. We discuss the role of discretecompactness property and show how it is related to the classic conditions for theconvergence of eigenvalues in mixed form. In this context, we review a recent resultconcerning the discrete compactness for the p version of discrete differential forms.One of the applications of the presented theory is the convergence analysis of the pversion of edge finite elements for the approximation of Maxwell’s eigenvalues.

1 Introduction

The use of homological techniques for the analysis of finite element approxima-tions of partial differential equations has become a very popular and effective tool(see [3, 4]). In the framework of de Rham complex it is natural to consider theeigenvalue problem associated with Hodge–Laplace operator. There are severaleigenvalue problems of interest for the applications, which can be related to theHodge–Laplace eigenvalue problem: for instance the standard Laplace eigenvalueproblem fits within this framework (0-forms), as well as the Maxwell eigenvalueproblem (1-forms in two or three space dimensions), or the eigenvalue problemassociated with grad div operator (2-forms in three dimensions).

The main object of this paper is to extend the results of [11] to differential forms.First of all, we consider two mixed variational formulations which give the same

solutions as the standard formulation originally designed for the Hodge–Laplaceeigenvalue problem. The theory developed in [13] can be used for the analysis ofthe mixed formulations in order to show the convergence of the eigenpairs; classicresults ([6, 32]) give the order of convergence for eigenvalues/eigenfunctions.

D. BoffiDipartimento di Matematica “F. Casorati”, Università di Pavia, via Ferrata 1 Pavia, Italye-mail: daniele.boffi@unipv.it

G. Kreiss et al. (eds.), Numerical Mathematics and Advanced Applications 2009,DOI 10.1007/978-3-642-11795-4 1, © Springer-Verlag Berlin Heidelberg 2010

3

daniele.boffi@unipv.it

4 D. Boffi

Then, we recall the discrete compactness property that can be naturally written inthe context of differential forms. The notion of discrete compactness has been usedsince long time in the literature: we recall, in particular, the works by Stummel [37],Vaı̆nikko [38], Anselone [1], and the more recent book by Chatelin [19]. In theapproximation of Maxwell’s eigenvalues, it has been used firstly by Kikuchi [30]and then reinterpreted and rephrased by several authors [7, 8, 10, 11, 18, 21, 31, 33].We refer the interested reader also to [28, 35] and to the references therein for areview on this topic.

Following [11], we show that the discrete compactness property and standardapproximation properties are equivalent to the natural convergence conditions forthe two equivalent mixed formulation we have introduced.

One of the consequences of the presented theory is that we can show that adiscretization that satisfies the discrete compactness property provides convergenteigenpairs and that such convergence can be analyzed by means of the standardBabuška–Osborn theory. A similar result has been obtained in [9] as a consequenceof the theory developed by [18] which makes use of the results of [26].

In [5] it has been introduced a comprehensive theory for the convergence of theeigenmodes of the Hodge–Laplace operator. The theory has been used (togetherwith a suitably defined projection operator) for the analysis of the convergence ofthe h version of finite elements applied to k-forms in any space dimensions (whensuitable discrete differential forms are used). The abstract hypotheses of [5] imply,in particular, our discrete compactness property.

When discussing the p version of finite elements for the approximation of theeigenmodes of the Hodge–Laplace operator, it is still an open problem to seewhether the assumptions of [5] are satisfied for discrete differential forms. On theother hand, in [9] it has been shown that the discrete compactness in p is valid as aconsequence of a recent result on the Poincaré operator (see [20]). In particular, thisimplies that two- and three-dimensional edge elements provide a good convergencein p for the eigenvalues/eigenfunctions of the Maxwell cavity problem.

The results of the present paper and, in particular, the relationships between theeingevalue problem associated with the Hodge–Laplace operator for differentialforms and suitable mixed formulations are discussed in more detail in [12].

2 Short Introduction to de Rham Complexand Differential Forms

Given a domain ˝ � Rn and k with 0 � k � n, we denote by C1.˝;�k/ thespace of smooth differential forms of order k on ˝ . For the sake of simplicity, weassume that˝ is simply connected, but the results of this paper might be generalizedto non-trivial cohomologies with natural modifications.

We suppose that we are given an exterior derivative

dk W C1.˝;�k/! C1.˝;�kC1/

Discrete Differential Forms and the p Version of Edge Finite Elements 5

for any k. The space L2.˝;�k/ denotes the space of differential k-forms on ˝with square integrable coefficients in their canonical basis representation; its innerproduct is given by

.u; v/ DZ˝

u ^ ? v;

where ? denotes the Hodge star operator mapping k-forms to .n � k/-forms.We shall make use of the Hilbert spaces

H.dk;˝/ D fv 2 L2.˝;�k/ W dkv 2 L2.˝;�kC1/g

andH0.dk;˝/ D fv 2 H.dk ;˝/ W tr@˝v D 0g:

We refer the interested reader to [4] for a canonical definition of the trace operatortr@˝ . In particular, we are interested in the following complex

R ����! H0.d0;˝/ d0����! H0.d1;˝/ d1����! � � � dn�1����! H0.dn;˝/ ����! 0:We recall in particular that dkC1 ı dk D 0 and that the range of dk coincides withthe kernel of dk C 1.

Given spaces of discrete differential forms V kp � H0.dk;˝/, a typical set-ting involves appropriate projection operators �kp WH0.dk;˝/ ! V kp such that thefollowing full de Rham complex commutes

R ����! H0.d0;˝/ d0����! H0.d1;˝/ d1����! � � � dn�1����! H0.dn;˝/ ����! 0�0p

??y �1p??y �np

??yR ����! V 0p

d0����! V 1pd1����! � � � dn�1����! V np ����! 0:

(1)

Remark 1. We use the index p for discrete spaces, so that it is explicit that we areinterested in the p version of finite elements; nevertheless, the abstract theory weare going to present is valid for general Galerkin discretizations where V kp are finitedimensional subspaces ofH0.dk;˝/.

Remark 2. In general, we are not going to assume that the full diagram (1) com-mutes. When we are interested in differential forms of degree k, it will be enoughto consider a small portion of (1) in the vicinity of k-forms.

The coderivative operator ık D ? dn�k ? maps C1.˝;�k/ to C1.˝;�k�1/and leads to the definition of the Hilbert space

H.ık;˝/ D fv 2 L2.˝;�k/ W ıkv 2 L2.˝;�k�1/g:

The spaces of differential forms when n D 2; 3 have been studied intensively.Table 1 recalls the representation of the involved quantities in terms of vectorproxies.

6 D. Boffi

Table 1 Identification between differential forms and vector proxies in R2 and R3

Differential form Proxy representationd D 2 d D 3

k D 0 d0 grad gradtr@˝� �j@˝ �j@˝H0.d0;˝/ H

10 .˝/ H

10 .˝/

ı1 � div � divk D 1 d1 curl curl

tr@˝u .u � t/j@˝ n� .u � n/j@˝H0.d1;˝/ H0.curl/ H0.curl/

ı2�!curl curl

k D 2 d2 0 divtr@˝q 0 .q � n/j@˝H0.d2;˝/ L

20.˝/ H0.div/

ı3 � grad

3 The Hodge–Laplace Eigenvalue Problem

Given k with 0 � k � n, we are interested in the following symmetric eigenvalueproblem: find � 2 R and u 2 H0.dk;˝/ with u 6D 0 such that

.dku; dkv/ D �.u; v/ 8v 2 H0.dk;˝/: (2)

The interest for the eigenvalue problem (2) arose in [9]: the case k D 1 (both forn D 2 and n D 3) corresponds to the Maxwell eigenvalue problem, since d1 can beidentified to the curl operator (see Table 1).

Problem (2) is strictly related to the so called Hodge–Laplace elliptic eigenvalueproblem (see [4, 5]): find ! 2 R and u 2 H0.dk;˝/ \ H.ık ;˝/ with u 6D 0such that

.dku; dkv/C .ıku; ıkv/ D !.u; v/ 8v 2 H0.dk;˝/\H.ık;˝/: (3)

It is well-known that problem (3) is associated with a compact solution opera-tor; this is consequence of the compact embedding of H0.dk;˝/ \H.ık;˝/ intoL2.˝;�k/ (see [36]). Moreover, the eigensolutions of (3) split into two separatefamilies: the first one consists of eigenvalues corresponding to eigenfunctions uwith dku D 0 and the second one of eigenvalues corresponding to eigenfunctionsu with ıku D 0. The second family corresponds to all the solutions to our originaleigenvalue problem (2) with positive frequencies. In addition, the zero frequencysolves problem (2) with the infinite dimensional eigenspace dk�1.H0.dk�1;˝//.

Given a finite dimensional discretization V kp of H0.dk;˝/, the discrete version

of (2) is: find �p 2 R and up 2 V kp with uh 6D 0 such that

.dkup; dkv/ D �p.uh; v/ 8v 2 V kp : (4)

Discrete Differential Forms and the p Version of Edge Finite Elements 7

One of the main issues for the convergence of the solutions of (4) towards thoseof (2) is the infinite dimensional kernel of (2). In general, we would like that thepositive frequencies of (4) provide a good approximation of the positive frequen-cies of (2). From the above discussion, it follows that adding the condition � > 0to problem (2) is equivalent to adding the condition ıku D 0 to the solution ofproblem (2). This last property can also be proved by taking v D dk�1t in (2) withan arbitrary t 2 H0.dk�1;˝/. The variational equation with � 6D 0 implies then.u; dk�1t/ D 0, that is ıku D 0.

In order to isolate the positive frequencies of problems (2) and (4) it is veryconvenient to consider equivalent mixed formulations.

3.1 First Mixed Formulation

A first mixed formulation of problem (2) can be obtained as a generalization of theso-called Kikuchi formulation for Maxwell’s eigenvalue problem (see [29]). It uses.k � 1/- and k-forms as follows: find � 2 R and u 2 H0.dk;˝/ with u 6D 0 suchthat for s 2 H0.dk�1;˝/ it holds

(.dku; dkv/C .dk�1s; v/ D �.u; v/ 8v 2 H0.dk;˝/.dk�1t;u/ D 0 8t 2 H0.dk�1;˝/: (5)

It can be easily shown that all eigensolutions of problem (5) have � > 0and solve the original problem (2), that dk�1s is always equal to zero (take v Ddk�1s in the first equation of (5)), and that all eigensolutions of (2) with positiveeigenvalue solve (5) as well. When k D 1 (which is the case for Maxwell’s eigen-value problem), we additionally have that s D 0 from d0s D 0 and the boundaryconditions.

A discretization of (5) involves the discrete spaces V k�1p � H0.dk�1;˝/ andV kp � H0.dk;˝/ as follows: find �p 2 R and up 2 V kp with up 6D 0 such that forsp 2 V k�1p it holds

(.dkup ; dkv/C .dk�1sp ; v/ D �p.up ; v/ 8v 2 V kp.dk�1t;up/ D 0 8t 2 V k�1p :

(6)

We assume the fundamental inclusion

dk�1.V k�1p / � V kp (7)

which is a compatibility condition valid whenever diagram (1) is satisfied. Underhypothesis (7) the discrete problem (6) is equivalent to (4) in the sense that all solu-tions corresponding to positive frequencies are the same. Hence the convergenceanalysis of the solutions of (6) towards those of (5) can be used in order to pursue

8 D. Boffi

our goal of studying the convergence of the positive solutions of (4) to the positivesolutions of (2).

It follows from the theory of [13, Sect. 3] that the conditions we are goingto present, ensure (and, in a sense, are necessary for) the convergence of theeigensolutions of (6) towards those of (5).

We need the discrete kernel of the ık operator (or, better, the kernel of the discreteık operator), defined as follows:

K1p D fv 2 V kp W .v; dkt/ D 0; 8t 2 V k�1p g:

Moreover, we introduce the solution spaces of the source problem associatedwith (5): V k0 and V

k�10 are the subspaces of H0.dk;˝/ and H0.dk�1;˝/, respec-

tively, containing all the first and second components u 2 H0.dk;˝/ and s 2 H0.dk�1;˝/, respectively, of the solution of the source problem

(.dku; dkv/C .dk�1s; v/ D .f; v/ 8v 2 H0.dk;˝/.dk�1t;u/ D 0 8t 2 H0.dk�1;˝/;

when f varies in L2.˝;�k/. Spaces V k0 and Vk�10 will be endowed with their

natural norms.

Definition 1. The ellipticity in the kernel is satisfied if there exists a positiveconstant ˛, independent of p, such that

.dkv; dkv/ � ˛.v; v/ 8v 2 K1p :

Definition 2. The weak approximability of V k�10 is satisfied if there exists �1.p/,tending to zero, such that for every s 2 V k�10

supv2K1p

.v; dk�1s/kvkH.dk ;˝/

� �1.p/kskV k�10

:

Definition 3. The strong approximability of V k0 is satisfied if there exists �2.p/,tending to zero, such that for every u 2 V k0 there exists uI 2 K1p such that

ku � uIkH.dk ;˝/ � �2.p/kukV k0:

3.2 Second Mixed Formulation

We are now going to present an alternative mixed formulation of problem (2)which is a generalization of the one introduced in [10] and which makes use of thespaces H0.dk;˝/ and W kC1 D dk.H.dk ;˝// � H0.dkC1;˝/: find � 2 R and

Discrete Differential Forms and the p Version of Edge Finite Elements 9

u 2 H0.dk;˝/, with u 6D 0 such that for 2 W kC1 it holds(.u; v/C .dkv; / D 0 8v 2 H0.dk;˝/.dku;'/ D ��. ;'/ 8' 2 W kC1:

(8)

It can be shown (see [10]) that all solutions of (8) have positive frequencies andcorrespond to the solutions of (2) with positive frequencies.

A discretization of (8) involves the space V kp � H0.dk;˝/ and a suitable dis-cretization of W kC1 � H0.dkC1;˝/. Since we are not going to approximate (8)numerically, but we only use the mixed formulations for the numerical analysisof (2) and (4), the most natural choice for the approximation of W kC1 consistsin taking

W kC1p D dk.V kp /: (9)In particular, equation (9) is the analogue of (7) for this mixed formulation. The dis-crete problem is: find �p 2 R and up 2 V kp , with up 6D 0 such that for p 2 W kC1pit holds

(.up; v/C .dkv; p/ D 0 8v 2 V kp.dkup;'/ D ��p. p ;'/ 8' 2 W kC1p :

(10)

Thanks to (9) is follows that all solutions of (10) correspond to the solutionsof (4) with positive frequencies. As for the first mixed formulation, we can thenanalyze the convergence of problem (10) to (8) in order to study the convergence ofthe positive solutions of (4) towards the positive solutions of (2).

We now describe the conditions presented in [13, Sect. 4] which are sufficient(and in a sense necessary) for the convergence of (10) to (8). We consider thediscrete kernel of the operator dk , that is

K2p D fv 2 V kp W .dkv;'/ D 0 8' 2 W kC1p g:

Moreover, we need the solutions spaces W k0 and WkC10 which contain all the first

and second components u 2 H.dk ;˝/ and 2 W kC1, respectively, of the solutionof the source problem

(.u; v/C .dkv; / D 0 8v 2 H0.dk;˝/.dku;'/ D �.g;'/ 8' 2 W kC1:

when g varies in L2.˝;�kC1/. The spaces W k0 and WkC10 are endowed with their

natural norms.

Definition 4. The weak approximability of W kC10 is satisfied if there exists �3.p/,tending to zero, such that

.dkv;'/ � �3.p/kvkL2.˝;�k/k'kW kC10

8v 2 K2p 8' 2 W kC10 :

10 D. Boffi

Definition 5. The strong approximability ofW kC10 is satisfied if there exists �4.p/,tending to zero, such that for every 2 W kC10 there is I 2 W kC1p with

k � IkL2.˝;�kC1/ � �4.p/k kW kC10

:

An operator˘p W W k0 ! V kh is called Fortin operator if it satisfies(.dk.u �˘pu/;'/ D 0 8u 2 W k0 8' 2 W kC1pk˘pukH.dk ;˝/ � CkukW k

08u 2 W k0 :

Definition 6. The Fortid property is satisfied if there exists a Fortin operator whichconverges to the identity in norm, that is, there exists �5.p/, tending to zero, suchthat

ku �˘pukL2.˝;�k/ � �5.p/kukW k0:

3.3 Discrete Compactness Property

The eigenfunctions u of problem (2) corresponding to nonzero frequencies are char-acterized by the constraint ıku D 0. The discrete compactness property mimicks,at discrete level, the compactness of the subspace of H0.dk;˝/ consisting of func-tions with vanishing ık , into L2.˝;�k/. It makes use of discrete differential formsof order k � 1 and k: V k�1p and V kp are finite dimensional internal approximationsof H0.dk�1;˝/ and H0.dk;˝/, respectively.

Definition 7. The discrete compactness property is satisfied if every sequence fupgin V kp , bounded in H0.dk;˝/ and with

.up; dk�1t/ D 0 8t 2 V k�1p ;

contains a subsequence which converges in L2.˝;�k/.

If the space V k�1p is a good approximation ofH0.dk�1;˝/ then it is not difficultto see that the limit u in Definition 7 satisfies .u; d t/ D 0 for all t 2 H0.dk�1;˝/,that is ıku D 0.Definition 8. The strong discrete compactness property is satisfied if the limit u ofthe subsequence in Definition 7 satisfies ıku D 0.

The strong discrete compactness property is strictly related to the the standarddiscrete compactness property and the (CDK) property (completeness of discretekernels) as it has been defined in [18] and used, for instance, in [9].

Discrete Differential Forms and the p Version of Edge Finite Elements 11

Before stating our main theorem, we need to make explicit a standard approxi-mation property for the discrete space V kp :

limp

infup2V kp

ku � upkH.dk;˝/ D 0 8u 2 H0.dk ;˝/ with ıku D 0: (11)

Theorem 1. Let us suppose that (7) and (9) are satisfied, so that the setting ofSects. 3.1 and 3.2 can be adopted. Then the following three sets of conditions areequivalent:

1. strong discrete compactness property (Definition 8) and approximation prop-erty (11);

2. ellipticity in the kernel (Definition 1), weak approximability of V k�10 (Defini-tion 2, and strong approximability of V k0 (Definition 3);

3. weak approximability of W kC10 (Definition 4), strong approximability of WkC10

(Definition 5), and Fortid property (Definition 6).

Proof. Due to the page restriction of the present paper, we cannot reproduce the fullproof of this result. Nevertheless, the reader is referred to [11, Theorem 3] where theanalogous result has been proved in a particular case (k D 1, n D 3 and the usualproxy representation where d1 corresponds to the curl operator). The proofs of [11,Propositions 3–6] leading to [11, Theorem 3] can be repeated practically identicalin our more general setting.

We take this opportunity to remark that [11, Proposition 3] which has beenproved in [33, Corollary 4.2] should assume the strong discrete compactness prop-erty and not only the standard discrete compactness property. This change has noconsequences for the final result.

The meaning of Theorem 1 is that each of the three equivalent conditions isa sufficient condition for the good approximation of the positive eigenvalues (andcorresponding eigenspaces) of (4) to the positive eigenvalues (and correspondingeigenspaces) of (2) in the spirit of [6]. The task of evaluating the order of conver-gence is usually less difficult since it is possible to take advantage of the smoothnessof the eigenfunctions. We refer the interested reader to [6] for the general case andto [32] for the case of mixed approximations. A review of the abstract theory willappear in [12].

4 The p Version of Edge Finite Elements

While the h version of edge finite elements for the approximation of Maxwell’seigenvalue problem has been the object of a rich literature, few results are availableabout the p and hp versions. Numerical results showed that the pure spectral method(one cubic element and p going to infinity) provides good results if suitable Nédélecfinite elements are used (see [34]), on the other hand the first theoretical results about

12 D. Boffi

the p version of edge finite elements are presented in [15], where the two dimen-sional triangular case is studied for the hp version. The analysis, however, relies ona conjectured estimate which has only been demonstrated numerically. In [14] thefirst proof of the discrete compactness in the case of the rectangular hp version ofedge elements (allowing for 1-irregular hanging nodes) has been proposed.

A significant step forward for the analysis of the p version of edge finite elementcomes from the results of [20], where a regularized Poincaré lifting is introduced.This is one of the main ingredients for the analysis reported in [9], where the discretecompactness in p for a wide class of edge finite elements is proved.

We refer the interested reader to the details in [9] for the technical assump-tions and the abstract setting. The main conclusion, related to edge finite elementsfor the approximation in p of Maxwell’s eigenvalue problem, is that edge finiteelements satisfy the discrete compactness in p in two dimensions (triangles andparallelograms) and in three dimensions (tetrahedra and parallelepipeds). The anal-ysis relies on the already mentioned Poincaré lifting and on recently introducedprojection-based interpolation operators (see, in particular, [17, 22–25]).

It is clear that after the results of [9] an important part of the analysis has beencompleted; nevertheless, there are still open problems that we hope can be solvedin a near future. First of all, the technique of [9] does not apply to the general hpversion in a straightforward way. There, we used a fixed mesh and the estimatesdepend on the mesh. Then, other finite element geometries might be used (prisms,pyramids, etc.) and the case of general quadrilaterals or hexehedra can be consid-ered for which, so far, only relatively negative results concerning the approximationproperties on distorted meshes are available (see [2, 16, 27]).

The approximation theory for eigenvalue problem in the framework of differ-ential forms has been studied recently also in [5]. There, the discrete compactnessproperty is studied by means of suitably constructed projection-based interpolationoperators which satisfy the strong property of being bounded in L2 and are con-structed by a means of an extension-regularization procedure. It is not clear whetherthis assumption is met by the interpolation operators used for the present analysis; itwould be interesting to further investigate this point and to see whether the assump-tions in [5] are stronger than the discrete compactness property discussed in thispaper.

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