hybridizable discontinuous galerkin methods for time-harmonic … · 2012. 7. 4. · permanent...

Hybridizable discontinuous Galerkin methods fortime-harmonic Maxwell’s equations

Ting-Zhu Huang1 Stéphane Lanteri2, Liang Li1 and Ronan Perrussel3

1 : School of Mathematical Sciences, UESTC, Chengdu, China2 : NACHOS project-team, INRIA Sophia Antipolis - Méditerranée

3 : Laplace Laboratory, UMR CNRS 5213, INP/ENSEEIHT/UPS Toulouse

2012, July 3rd

Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 1 / 40

Introduction of the Team Led by Prof. Huang

Team Composition

Head: Prof. Dr. Ting-Zhu Huang

Permanent Employees (7 people): Dr. Yong Zhang, Dr. Hou-Biao Li, Dr. Guang-Hui Cheng,Dr. Jin-Liang Shao, Dr. Liang Li, Dr. Yan-Fei Jing, Dr. Chun-Wen

About 15 Ph.D. students

More than 20 master students

Many collaborators (domestic and overseas)

Research Topics

Our team focuses on numerical linear algebra with interdisciplinary studies of informationscience:

Numerical Linear Algebra

Iterative methods for large-scale linear algebraic systems

Preconditioning techniques

Eigenvalues and singular values

Interdisciplinary Studies

Electromagnetic computation

Digital image processing

Others - multiagent systems, Markov chains

0 50 100 150 200−30

θ (Degrees)

φ=90Y X

Research Topics

Our team focuses on numerical linear algebra with interdisciplinary studies of informationscience:

Numerical Linear Algebra

Iterative methods for large-scale linear algebraic systems

Preconditioning techniques

Eigenvalues and singular values

Interdisciplinary Studies

Electromagnetic computation

Digital image processing

Others - multiagent systems, Markov chains

0 50 100 150 200−30

θ (Degrees)

φ=90Y X

Iterative Methods

People

Dr. Yan-Fei Jing, Dr. Guang-Hui Cheng, Dr. Liang Li, Dr. Shi-Liang Wu and Dr. Jian-Lei Li

Subspace Methods

BiCOR/CORS/BiCORSTAB Lanczos Biconjugate A-Orthogonalization - Krylov subspacemethods

2D-DSPM (double successive projection method)

Restarted weighted full orthogonalization method

Other Methods

Modified SOR, AOR, SSOR, · · · for saddle point problem

Hermitian skew-Hermitian splitting methods

Modified Uzawa methods

Iterative Methods

People

Dr. Yan-Fei Jing, Dr. Guang-Hui Cheng, Dr. Liang Li, Dr. Shi-Liang Wu and Dr. Jian-Lei Li

Subspace Methods

BiCOR/CORS/BiCORSTAB Lanczos Biconjugate A-Orthogonalization - Krylov subspacemethods

2D-DSPM (double successive projection method)

Restarted weighted full orthogonalization method

Other Methods

Modified SOR, AOR, SSOR, · · · for saddle point problem

Hermitian skew-Hermitian splitting methods

Modified Uzawa methods

Preconditioning Techniques

People

Dr. Yong Zhang, Dr. Liang Li, Dr. Chun Wen, Mr. Liang-Jian Deng and Mr. Xian-Ming Gu

ILU FactorizationDynamic ordering schemes

Block ILU (algebraic recursive multilevel solver)

Other Techniques

Algebraic multigrid methods (aggregation-based) for Hermholtz equations

Schur complement free preconditiners for saddle point problems

Symmetric positive definite preconditioners and augmentation block triangularpreconditioners

Preconditioning Techniques

People

Dr. Yong Zhang, Dr. Liang Li, Dr. Chun Wen, Mr. Liang-Jian Deng and Mr. Xian-Ming Gu

ILU FactorizationDynamic ordering schemes

Block ILU (algebraic recursive multilevel solver)

Other Techniques

Algebraic multigrid methods (aggregation-based) for Hermholtz equations

Schur complement free preconditiners for saddle point problems

Symmetric positive definite preconditioners and augmentation block triangularpreconditioners

Eigenvalues and Singular Values

People

Dr. Hou-Biao Li, Dr. Shu-Qian Shen, Dr. Guang-Hui Cheng

Spectral properties of preconditioned matrices for saddle point problems:Primal based penalty (PBP) preconditionerGeneralized SOR preconditionerHermitian and skew-Hermitian preconditioner

Estimation for eigenvalues together with singular values

Spectral properties for nonnegative matrices

Electromagnetic Computations

People

Dr. Liang Li, Dr. Yan-Fei Jing, Dr. Zhi-Gang Ren

Hybridizable Discontinuous Galerkin (HDG) Methods

Numerical performance of HDG methods for the solution of Maxwell’s equations

Locally well-posed HDG formulation

Domain decomposition methods (optimal Schwarz method)

Multigrid methods

HDG/BEM formulation

Efficient Solution of the Resulting Linear Systems

Application of Krylov subspace methods

Incomplete factorization preconditioner

Preconditioning techniques for FEM/BEM methods

Electromagnetic Computations

People

Dr. Liang Li, Dr. Yan-Fei Jing, Dr. Zhi-Gang Ren

Hybridizable Discontinuous Galerkin (HDG) Methods

Numerical performance of HDG methods for the solution of Maxwell’s equations

Locally well-posed HDG formulation

Domain decomposition methods (optimal Schwarz method)

Multigrid methods

HDG/BEM formulation

Efficient Solution of the Resulting Linear Systems

Application of Krylov subspace methods

Incomplete factorization preconditioner

Preconditioning techniques for FEM/BEM methods

Digital Image Processing

People

Dr. Xiao-Guang lv, Mr. Xi-Le Zhao, Mr. Jun Liu, · · ·

Efficient Solution to Special Matrices

Teoplitz, Hankel

AMG, Fourier/wavelet transformation based methods

Image restoration - New efficient boundary conditions

Deblurring and unmixing

Digital Image Processing

People

Dr. Xiao-Guang lv, Mr. Xi-Le Zhao, Mr. Jun Liu, · · ·

Efficient Solution to Special Matrices

Teoplitz, Hankel

AMG, Fourier/wavelet transformation based methods

Image restoration - New efficient boundary conditions

Deblurring and unmixing

Other Interdiscipline

Multiagent System

People: Dr. Jin-Liang Shao, Mr. Zhao-Jun Tang

Distributed coordinated control of multiagent systems

Neural networks - Stability

Markov Chains

High performance algorithms

People: Dr. Chun Wen

Multilevel aggregation

Triangular and skew-symmetric splitting method

Other Interdiscipline

Multiagent System

People: Dr. Jin-Liang Shao, Mr. Zhao-Jun Tang

Distributed coordinated control of multiagent systems

Neural networks - Stability

Markov Chains

High performance algorithms

People: Dr. Chun Wen

Multilevel aggregation

Triangular and skew-symmetric splitting method

Hybridizable discontinuous Galerkin methods fortime-harmonic Maxwell’s equations

Ting-Zhu Huang1 Stéphane Lanteri2, Liang Li1 and Ronan Perrussel3

1 : School of Mathematical Sciences, UESTC, Chengdu, China2 : NACHOS project-team, INRIA Sophia Antipolis - Méditerranée

3 : Laplace Laboratory, UMR CNRS 5213, INP/ENSEEIHT/UPS Toulouse

2012, July 3rd

Scientific context

Target applications:

interaction of EM fields with living tissues;

microwave imaging for the detection of buried objects.

Modeling context:

time-harmonic regime;

"high" frequency i.e. no quasi-static model.

Numerical ingredients:

unstructured meshes (triangles in 2D, tetrahedra in 3D);

high order polynomial interpolation of EM field components;

sparse direct solver and preconditioned iterative solvera.

aV. Dolean, S. Lanteri and R. Perrussel, Optimized Schwarz algorithms for solving time-harmonic Maxwell’s equationsdiscretized by a discontinuous Galerkin method, IEEE Trans. Magn., Vol. 44, No. 6, pp. 954-957 (2008)

Example - exposure of head tissues

Assessment:

complexity in modeling - discontinuous Galerkin FEM

large number of DOFs - hybridization to reduce the DOFs

Example - exposure of head tissues

Assessment:

complexity in modeling - discontinuous Galerkin FEM

large number of DOFs - hybridization to reduce the DOFs

Problem considered

2D time-harmonic Maxwell’s equationsiωεr E − curlH = 0, in Ω,

iωµr H+curlE = 0, in Ω,

with E = Ez and H =(Hx Hy

)Tand:curlE =

(∂y E −∂x E

curlH = ∂x Hy −∂y Hx ,

Boundary conditions: E = 0, on Γm,

E +(n×H) = E inc +(n×Hinc) = g inc, on Γa,

with Γm ∪Γa = ∂Ω.

Motivations for Discontinuous Galerkin Method

Advantages

Fexibility for the approximation inside each element: makes easier hp-adaptivity

Treatment of non-conforming finite element meshes is naturally included in theweak formulation

Efficiency for unsteady problems (time-domain Maxwell’s equations) with an explicittime-integration scheme

Naturally adapted to parallel computing

Drawback

the number of globally coupled degrees of freedom is huge comparedto conforming finite element methods for the same approximation order:

Ndof = 3Ne ·nde = 3Ne · (p+1)(p+2)2

Motivations for Discontinuous Galerkin Method

Advantages

Fexibility for the approximation inside each element: makes easier hp-adaptivity

Treatment of non-conforming finite element meshes is naturally included in theweak formulation

Efficiency for unsteady problems (time-domain Maxwell’s equations) with an explicittime-integration scheme

Naturally adapted to parallel computing

Drawback

the number of globally coupled degrees of freedom is huge comparedto conforming finite element methods for the same approximation order:

Ndof = 3Ne ·nde = 3Ne · (p+1)(p+2)2

Discontinuous Galerkin (DG) formulation

Discontinuous finite element spaces:V p

v ∈ L2(Ω) | v |K ∈ V ph (K ), ∀K ∈Th

v ∈ (L2(Ω))2 | v|K ∈ Vp

h(K ), ∀K ∈Th

with V ph (K )≡ Pp(K ) and Vp

h(K )≡ (Pp(K ))2.

Principles

Classical DG seeks an approximate solution (Eh,Hh) in the space V ph ×Vp

h satisfying for all K in Th:(iωεr Eh,v)K − (curlHh,v)K = 0, ∀v ∈ V p

h (K ),

(iωµr Hh,v)K +(curlEh,v)K = 0, ∀v ∈ Vph(K ),

with: (u,v)K =∫

Kuvdx and (u,v)K =

u.vdx.

Integration by parts:(iωεr Eh,v)K − (Hh,curlv)K−< n× Hh,v >∂K = 0, ∀v ∈ V p

h (K ),

(iωµr Hh,v)K +(Eh,curlv)K−< Eh,n×v >∂K = 0, ∀v ∈ Vph(K ),

with: < u,v >∂K =∫

∂Kuvds.

Numerical traces Eh and Hh ensure global consistency.

Discontinuous finite element spaces:V p

v ∈ L2(Ω) | v |K ∈ V ph (K ), ∀K ∈Th

v ∈ (L2(Ω))2 | v|K ∈ Vp

h(K ), ∀K ∈Th

with V ph (K )≡ Pp(K ) and Vp

h(K )≡ (Pp(K ))2.

Principles

Classical DG seeks an approximate solution (Eh,Hh) in the space V ph ×Vp

h satisfying for all K in Th:(iωεr Eh,v)K − (curlHh,v)K = 0, ∀v ∈ V p

h (K ),

(iωµr Hh,v)K +(curlEh,v)K = 0, ∀v ∈ Vph(K ),

with: (u,v)K =∫

Kuvdx and (u,v)K =

u.vdx.

Integration by parts:(iωεr Eh,v)K − (Hh,curlv)K−< n× Hh,v >∂K = 0, ∀v ∈ V p

h (K ),

(iωµr Hh,v)K +(Eh,curlv)K−< Eh,n×v >∂K = 0, ∀v ∈ Vph(K ),

with: < u,v >∂K =∫

∂Kuvds.

Numerical traces Eh and Hh ensure global consistency.

Notations

For an interface F = K+ ∩K−, let (v±,v±) be the traces of (v,v) on F from theinterior of K±.

Definition of averages and jumps:

vF=v+ +v−

vF=v+ + v−

JtvF K = tK+ v+ + tK−v−,

Jn×vKF = nK+ ×v+ +nK− ×v−,

with: t×n = 1.

Classical DGMNumerical traces define the couplings between neighboring elements:

Eh = Eh +αHJn×HhK and Hh = Hh +αEJtEhK.

Centered DG scheme: αE = αH = 0

Upwind DG scheme: αE = αH =12

Hybridizable discontinuous Galerkin (HDG) methods

B. Cockburn, J. Gopalakrishnan and R. LazarovUnified hybridization of discontinuous Galerkin, mixed, and continuousGalerkin methods for second order elliptic problemsSIAM J. Numer. Anal., Vol. 47, No. 2 (2009)

N.C. Nguyen, J. Peraire and B. CockburnAn implicit high-order hybridizable discontinuous Galerkin method for linearconvection-diffusion equationsJ. Comput. Phys., Vol. 228, No. 9 (2009)

N.C. Nguyen, J. Peraire and B. CockburnAn implicit high-order hybridizable discontinuous Galerkin method for nonlinearconvection-diffusion equationsJ. Comput. Phys., Vol. 228, No. 23 (2009)

S.C. Soon, B. Cockburn and H.K. StolarskiA hybridizable discontinuous Galerkin method for linear elasticityInt. J. Numer. Meth. Engng., Vol. 80, No. 8 (2009)

N.C. Nguyen, J. Peraire and B. CockburnA hybridizable discontinuous Galerkin method for Stokes flowComput. Meth. App. Mech. Engng., Vol. 199, No. 9-12 (2010)

N.C. Nguyen, J. Peraire and B. CockburnHybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equationsJ. Comput. Phys., Vol. 230, No. 19 (2011)

B. Cockburn, J. Gopalakrishnan and R. LazarovUnified hybridization of discontinuous Galerkin, mixed, and continuousGalerkin methods for second order elliptic problemsSIAM J. Numer. Anal., Vol. 47, No. 2 (2009)

N.C. Nguyen, J. Peraire and B. CockburnAn implicit high-order hybridizable discontinuous Galerkin method for linearconvection-diffusion equationsJ. Comput. Phys., Vol. 228, No. 9 (2009)

N.C. Nguyen, J. Peraire and B. CockburnAn implicit high-order hybridizable discontinuous Galerkin method for nonlinearconvection-diffusion equationsJ. Comput. Phys., Vol. 228, No. 23 (2009)

S.C. Soon, B. Cockburn and H.K. StolarskiA hybridizable discontinuous Galerkin method for linear elasticityInt. J. Numer. Meth. Engng., Vol. 80, No. 8 (2009)

N.C. Nguyen, J. Peraire and B. CockburnA hybridizable discontinuous Galerkin method for Stokes flowComput. Meth. App. Mech. Engng., Vol. 199, No. 9-12 (2010)

N.C. Nguyen, J. Peraire and B. CockburnHybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equationsJ. Comput. Phys., Vol. 230, No. 19 (2011)

Main ideasNumerical traces will not directly couple neighboring elements

These traces will depend on a hybrid variable living on the interfaces of the elementsof the mesh

A conservativity condition has to be enforced to make the problem solvable

The new hybrid variable λh is chosen as an element of the traced finite element space:

η ∈ L2(Fh) | η |F ∈ Pp(F), ∀F ∈Fh and η |Γm

Note that Mph consists of functions which are continuous on an edge, but discontinuous

at its ends. Fh: set of all the faces of Th

Definition of numerical tracesEh = λh,

Hh = Hh + τK (Eh −λh)t,∀F ∈Fh,

with τK > 0 being a stabilization parameter.

Main ideasNumerical traces will not directly couple neighboring elements

These traces will depend on a hybrid variable living on the interfaces of the elementsof the mesh

A conservativity condition has to be enforced to make the problem solvable

The new hybrid variable λh is chosen as an element of the traced finite element space:

η ∈ L2(Fh) | η |F ∈ Pp(F), ∀F ∈Fh and η |Γm

Note that Mph consists of functions which are continuous on an edge, but discontinuous

at its ends. Fh: set of all the faces of Th

Definition of numerical tracesEh = λh,

Hh = Hh + τK (Eh −λh)t,∀F ∈Fh,

with τK > 0 being a stabilization parameter.

Local problem

Write the local solution (Eλh ,Hλ

h ) on K as a function of λ (simplified for λh):(iωεr Eλ

h ,v)K − (Hλh ,curlv)K−< n× Hh,v >∂K = 0, ∀v ∈ V p

h (K ),

(iωµr Hλh ,v)K +(Eλ

h ,curlv)K−< λh,n×v >∂K = 0, ∀v ∈ Vph(K ).

HDG formulation

Enforcing a conservativity condition on Jn× HhK to obtain global consistency, we have the globalproblem: find (Eh,Hh,λh) ∈ V p

h ×Vph ×Mp

h such that:(iωεr Eh,v)Th

− (Hh,curlv)Th−< n× Hh,v >∂Th

= 0, ∀v ∈ V ph ,

(iωµr Hh,v)Th+(Eh,curlv)Th

−< λh,n×v >∂Th= 0, ∀v ∈ Vp

< Jn× HhK,η >Fh+ < λh,η >Γa

=< ginc,η >Γa, ∀η ∈ Mp

Notations:

(·, ·)Th= ∑

K∈Th

(·, ·)K , < ·, ·>∂Th= ∑

K∈Th

< ·, ·>∂K , < ·, ·>Fh= ∑

f∈Fh

< ·, ·>F ,

Local problem

Write the local solution (Eλh ,Hλ

h ) on K as a function of λ (simplified for λh):(iωεr Eλ

h ,v)K − (Hλh ,curlv)K−< n× Hh,v >∂K = 0, ∀v ∈ V p

h (K ),

(iωµr Hλh ,v)K +(Eλ

h ,curlv)K−< λh,n×v >∂K = 0, ∀v ∈ Vph(K ).

HDG formulation

Enforcing a conservativity condition on Jn× HhK to obtain global consistency, we have the globalproblem: find (Eh,Hh,λh) ∈ V p

h ×Vph ×Mp

− (Hh,curlv)Th−< n× Hh,v >∂Th

= 0, ∀v ∈ V ph ,

−< λh,n×v >∂Th= 0, ∀v ∈ Vp

< Jn× HhK,η >Fh+ < λh,η >Γa

=< ginc,η >Γa, ∀η ∈ Mp

Notations:

(·, ·)Th= ∑

K∈Th

(·, ·)K , < ·, ·>∂Th= ∑

K∈Th

< ·, ·>∂K , < ·, ·>Fh= ∑

f∈Fh

< ·, ·>F ,

For an interior face F = ∂K+∩∂K− we have:

< Jn× HhK,η >F = < Jn× (Hh + τ(Eh −λh)t)K,η >F

= < nK + ×H+,η >∂K + + < nK− ×H−,η >∂K−

−< τK +E+,η >∂K + −< τK−E−,η >∂K−

+ < τK +λh,η >∂K + + < τK−λh,η >∂K− ,

< Jn× HhK,η >Fh=< n×Hh,η >∂Th

−< τEh,η >∂Th+ < τλh,η >∂Th

HDG formulation

Applying integration by parts again to the first relation, we get the global HDG problem is to(Eh,Hh,λh) ∈ V p

h ×Vph ×Mp

− (curlHh,v)Th+ < τ(Eh −λh),v >∂Th

= 0, ∀v ∈ V ph ,

−< λh,n×v >∂Th= 0, ∀v ∈ vp

< n×Hh,η >∂Th−< τ(Eh −λh),η >∂Th

+ < λh,η >Γa−< ginc,η >Γa

= 0, ∀η ∈ Mph .

For an interior face F = ∂K+∩∂K− we have:

< Jn× HhK,η >F = < Jn× (Hh + τ(Eh −λh)t)K,η >F

= < nK + ×H+,η >∂K + + < nK− ×H−,η >∂K−

−< τK +E+,η >∂K + −< τK−E−,η >∂K−

+ < τK +λh,η >∂K + + < τK−λh,η >∂K− ,

< Jn× HhK,η >Fh=< n×Hh,η >∂Th

−< τEh,η >∂Th+ < τλh,η >∂Th

HDG formulation

Applying integration by parts again to the first relation, we get the global HDG problem is to(Eh,Hh,λh) ∈ V p

h ×Vph ×Mp

− (curlHh,v)Th+ < τ(Eh −λh),v >∂Th

= 0, ∀v ∈ V ph ,

−< λh,n×v >∂Th= 0, ∀v ∈ vp

< n×Hh,η >∂Th−< τ(Eh −λh),η >∂Th

+ < λh,η >Γa−< ginc,η >Γa

= 0, ∀η ∈ Mph .

Link between DG and HDG

Eh = λh =1

τK+ + τK−(τK+ E+

h + τK−E−h )− 1

τK+ + τK−Jn×HhK on F ,

τK+ + τK−(τK−H+

h + τK+ H−h )+

τK+ τK−

τK+ + τK−JEhtK on F .

Upwind DG scheme ⇔ τK is uniformly equal to 1

Well-posedness of the local solver

Consider v = Eλh and v = Hλ

h , it is then obtained by adding the two relations together:

(iωεr Eλh ,Eλ

h )K − (curlHλh ,Eλ

h )K + τK < (Eλh −λ),Eλ

h >∂K

+(iωµr Hλh ,Hλ

h )K +(Eλh ,curlHλ

h )K − τK < λ ,Eλh >∂K = 0.

Taking λ = 0, it results in the following equality:

(iωεr E0h ,E0

h )K +(iωµr H0h,H

0h)K +2ℑ((E0

h ,curlH0h)K )+ τK < E0

h ,E0h >∂K = 0.

We cannot conclude on the general well-posedness because of possible resonant frequency.

The well-posedness of the local solver for the HDG-P1 and HDG-P2 method can be guaranteed,because all the degrees of freedom are on ∂K .

Link between DG and HDG

Eh = λh =1

τK+ + τK−(τK+ E+

h + τK−E−h )− 1

τK+ + τK−Jn×HhK on F ,

τK+ + τK−(τK−H+

h + τK+ H−h )+

τK+ τK−

τK+ + τK−JEhtK on F .

Upwind DG scheme ⇔ τK is uniformly equal to 1

Well-posedness of the local solver

Consider v = Eλh and v = Hλ

h , it is then obtained by adding the two relations together:

(iωεr Eλh ,Eλ

h )K − (curlHλh ,Eλ

h )K + τK < (Eλh −λ),Eλ

h >∂K

+(iωµr Hλh ,Hλ

h )K +(Eλh ,curlHλ

h )K − τK < λ ,Eλh >∂K = 0.

Taking λ = 0, it results in the following equality:

(iωεr E0h ,E0

h )K +(iωµr H0h,H

0h)K +2ℑ((E0

h ,curlH0h)K )+ τK < E0

h ,E0h >∂K = 0.

We cannot conclude on the general well-posedness because of possible resonant frequency.

The well-posedness of the local solver for the HDG-P1 and HDG-P2 method can be guaranteed,because all the degrees of freedom are on ∂K .

Characterization of the reduced systemExplicitly rewrite the reduced system:

ah(λh,η) = bh(η), ∀η ∈ Mph

with:ah(λh,η) =< Jn× Hλ

h K,η >Fh + < λ ,η >Γa

bh(η) =< ginc,η >Γa .

The sesquilinear form:

ah(λh,η) = (−iωµr Hλh ,Hη

h )Th +(iωεr Eλh ,Eη

h )Th + < τ(λ −Eλh ),(η −Eη

h ) >∂Th+ < λ ,η >Γa .

0 0.01 0.02 0.03−2.5

−1.5

−0.5

(a) HDG-P3, ω = 2π

0 0.005 0.01 0.015−0.05

−0.04

−0.03

−0.02

−0.01

(b) HDG-P3, ω = 2π , zoom in

0 0.01 0.02 0.03 0.04−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

(c) HDG-P3, ω = 8π

Performance results: propagation of a plane wave in vacuumNumerical convergence order of the HDG method

P1 P2 P3 P4E field 1.8 3.0 4.0 5.0H field 1.9 3.0 4.0 5.0

Figure: Convergence results on independently refined unstructured meshes.

Performance results: propagation of a plane wave in vacuumMesh size Memory (MB) Tconstruction (s) Tsolution (s)

HDG DG upwind HDG DG upwind HDG DG upwindP1

0.14 2 5 0.00 0.00 0.01 0.030.071 5 19 0.01 0.00 0.02 0.100.035 20 85 0.03 0.01 0.04 0.640.018 86 389 0.09 0.03 0.52 3.87

P20.14 3 11 0.01 0.00 0.01 0.070.071 9 48 0.03 0.01 0.04 0.350.035 41 221 0.09 0.02 0.22 2.060.018 187 1024 0.37 0.08 1.27 13.33

P30.14 4 21 0.02 0.01 0.01 0.140.071 15 96 0.08 0.02 0.08 0.770.035 71 435 0.29 0.05 0.29 4.600.018 327 1955 1.16 0.19 2.54 31.1

P40.14 5 36 0.05 0.01 0.02 0.300.071 24 160 0.21 0.03 0.12 1.450.035 106 720 0.80 0.11 0.67 8.300.018 499 3258 3.17 0.40 4.54 51.4

Performance results: propagation of a plane wave in vacuumMesh size Memory (MB) Tconstruction (s) Tsolution (s)

HDG DG upwind HDG DG upwind HDG DG upwindP1

0.14 2 5 0.00 0.00 0.01 0.030.071 5 19 0.01 0.00 0.02 0.100.035 20 85 0.03 0.01 0.04 0.640.018 86 389 0.09 0.03 0.52 3.87

P20.14 3 11 0.01 0.00 0.01 0.070.071 9 48 0.03 0.01 0.04 0.350.035 41 221 0.09 0.02 0.22 2.060.018 187 1024 0.37 0.08 1.27 13.33

P30.14 4 21 0.02 0.01 0.01 0.140.071 15 96 0.08 0.02 0.08 0.770.035 71 435 0.29 0.05 0.29 4.600.018 327 1955 1.16 0.19 2.54 31.1

P40.14 5 36 0.05 0.01 0.02 0.300.071 24 160 0.21 0.03 0.12 1.450.035 106 720 0.80 0.11 0.67 8.300.018 499 3258 3.17 0.40 4.54 51.4

HDG on curvilinear domains – isoparametric elements

Performance results: scattering from a metallic cylinder

Affine map Quadratic map Cubic mapE H E H E H

P2 2.4 2.4 3.2 3.2 3.2 3.2P3 2.1 2.1 4.2 4.2 4.2 4.1P4 2.1 2.1 5.1 5.1 5.1 5.0

Figure: Affine map

Performance results: scattering from a metallic cylinder

(a) Quadratic map (b) Cubic map

3.25e-08 0.00419 0.00838

Error of Ez

(a) Affine map

0 5.03e-05 0.000101

Error of Ez

(b) Quadratic map

2.43e-08 0.0477 0.0953

Error of Hx

(c) Affine map

1e-10 2.64e-05 5.28e-05

Error of Hx

(d) Quadratic mapLiang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 26 / 40

Locally well-posed HDG

Formulation – adding a consistent stabilization term

Find (Eh,Hh,λh,Λh) ∈ V ph ×Vp

h ×Mph ×Mp

h such that:

(iωεr Eh,v)Th − (curlHh,v)Th + < τ(Eh −λh),v >∂Th= 0, ∀v ∈ V p

(iωµr Hh,v)Th +(Eh,curlv)Th−< λh,n×v >∂Th

+ < n×Hh,n×v >∂Th−< Λh,nF ×v >∂Th

= 0, ∀v ∈ Vph,

< n×Hh,η >∂Th+ < τ(λh −Eh),η >∂Th

+ < λh,η >Γa −< ginc,η >Γa = 0 ∀η ∈ Mph ,

< Λh −nF ×Hh,ζ >∂Th= 0 ∀ζ ∈ Mp

Numerical results

HDG-P1 HDG-P3Mesh size ‖E −Eh‖2 ‖H−Hh‖2 ‖E −Eh‖2 ‖H−Hh‖2

error order error order error order error order0.3310 1.16e-1 – 1.50e-1 – 8.56e-4 – 1.22e-3 –0.1514 2.84e-2 1.8 3.41e-2 1.9 3.98e-5 3.9 5.34e-5 4.00.0731 7.04e-3 1.9 8.27e-3 1.9 2.22e-6 4.0 2.95e-6 4.00.0285 1.25e-3 1.8 1.47e-3 1.8 6.65e-8 3.7 8.75e-8 3.7

Generalized locally well-posed HDG formulation

In order to obtain the locally well-posed HDG formulation, one could also define:Eh = λh +(Λh(t×n)−n×Hh),

Hh = Hh + τ(Eh −λh)t,

and enforce the physical transmission conditions:JEhK = 0,

Jn× HhK = 0,

on the interior faces.

Define: Eh = λh +α

E1 (Eh −λh)+α

E2 (Λh(tF ×n)−n×Hh),

Hh = Λh(−tF )+αH1 (Hh −Λh(−tF ))+α

H2 (Eh −λh)t.

Choices of the parameters:

HDG: αE1 = αE

2 = 0,αH1 = 1, and αH

2 = τ

Locally well-posed HDG: αE1 = 0,αE

2 = 1,αH1 = 1, and αH

2 = τ

Generalized locally well-posed HDG formulation

In order to obtain the locally well-posed HDG formulation, one could also define:Eh = λh +(Λh(t×n)−n×Hh),

Hh = Hh + τ(Eh −λh)t,

and enforce the physical transmission conditions:JEhK = 0,

Jn× HhK = 0,

on the interior faces.

Define: Eh = λh +α

E1 (Eh −λh)+α

E2 (Λh(tF ×n)−n×Hh),

Hh = Λh(−tF )+αH1 (Hh −Λh(−tF ))+α

H2 (Eh −λh)t.

Choices of the parameters:

HDG: αE1 = αE

2 = 0,αH1 = 1, and αH

2 = τ

Locally well-posed HDG: αE1 = 0,αE

2 = 1,αH1 = 1, and αH

2 = τ

Generalized locally well-posed HDG formulation - cont.

Transmission Conditions – Global Problem

The transmission conditions on a face F = ∂K+∩∂K−:

JEhK = JαE1 EhK− Jα

E1 Kλh

+(αE ,+2 +α

E ,−2 )Λh −nF × (α

E ,+2 H+

h +αE ,−2 H−

h ) = 0,

Jn× HhK = JαH1 (n×Hh)K− Jα

H1 KΛh

+(αH,+2 +α

H,−2 )λh − (α

H,+2 E+

h +αH,−2 E−

h ) = 0.

If we take the coefficients αE ,H1,2 as constants in the whole domain, then we obtain:

JEhK = αE1 JEhK+α

E2 (2Λh −nF × (H+

h +H−h )) = 0,

Jn× HhK = αH1 Jn×HhK+α

H2 (2λh − (E+

h +E−h )) = 0.

A Special Case

Eh = Eh + τ(Λh(tF ×n)−n×H),

Hh = Hh + τ(Eh −λh)t.

Transmission Conditions – Global Problem

The transmission conditions on a face F = ∂K+∩∂K−:

JEhK = JαE1 EhK− Jα

E1 Kλh

+(αE ,+2 +α

E ,−2 )Λh −nF × (α

E ,+2 H+

h +αE ,−2 H−

h ) = 0,

Jn× HhK = JαH1 (n×Hh)K− Jα

H1 KΛh

+(αH,+2 +α

H,−2 )λh − (α

H,+2 E+

h +αH,−2 E−

h ) = 0.

If we take the coefficients αE ,H1,2 as constants in the whole domain, then we obtain:

JEhK = αE1 JEhK+α

E2 (2Λh −nF × (H+

h +H−h )) = 0,

Jn× HhK = αH1 Jn×HhK+α

H2 (2λh − (E+

h +E−h )) = 0.

A Special Case

Eh = Eh + τ(Λh(tF ×n)−n×H),

Hh = Hh + τ(Eh −λh)t.

Local Problem of the Special Case(iωεr Eh,v)K − (curlHh,v)K −〈τ(Eh −λh),v〉∂K = 0, ∀v ∈ V p(K ),

(iωµr Hh,v)K +(curlEh,v)K

+ 〈τn× (Hh −Λh(−tF )),n×v〉∂K = 0, ∀v ∈ Vp(K ).

Local Problem of the Generalized Case(iωεr Eh,v)K − (curlHh,v)K + 〈αH

2 (Eh −λh),v〉∂K

+ 〈(1−αH1 )n× (Hh −Λh(−tF )),v〉∂K = 0, ∀v ∈ V p(K ),

(iωµr Hh,v)K +(Eh,curlv)K + 〈αE2 n× (Hh −Λh(−tF )),n×v〉∂K

−〈(1−αE1 )λh +α

E1 Eh,n×v〉∂K = 0, ∀v ∈ Vp(K ).

Local Problem of the Special Case(iωεr Eh,v)K − (curlHh,v)K −〈τ(Eh −λh),v〉∂K = 0, ∀v ∈ V p(K ),

(iωµr Hh,v)K +(curlEh,v)K

+ 〈τn× (Hh −Λh(−tF )),n×v〉∂K = 0, ∀v ∈ Vp(K ).

Local Problem of the Generalized Case(iωεr Eh,v)K − (curlHh,v)K + 〈αH

2 (Eh −λh),v〉∂K

+ 〈(1−αH1 )n× (Hh −Λh(−tF )),v〉∂K = 0, ∀v ∈ V p(K ),

(iωµr Hh,v)K +(Eh,curlv)K + 〈αE2 n× (Hh −Λh(−tF )),n×v〉∂K

−〈(1−αE1 )λh +α

E1 Eh,n×v〉∂K = 0, ∀v ∈ Vp(K ).

Local well-posedness

Considering λh = 0 and Λh = 0, and taking the test functions v and v to be the local solutions

E(0,0)h and H(0,0)

h , we obtain:

(iωεr E(0,0)h ,E(0,0)

h )K +(iωµr H(0,0)h ,H(0,0)

h )K +2ℑ((E(0,0)h ,curlH(0,0)

h )K )

2(1−αH1 )ℑ

(〈n×H(0,0)

h ,E(0,0)h 〉∂K

)+(1−α

H1 −α

E1 )〈E(0,0)

h ,n×H(0,0)h 〉∂K

+αH2 〈E

(0,0)h ,E(0,0)

h 〉∂K +αE2 〈n×H(0,0)

h ,n×H(0,0)h 〉∂K = 0

Comparing the Real Parts

αH2 ‖E(0,0)

h ‖2L2(∂K ) +α

E2 ‖n×H(0,0)

h ‖2L2(∂K )

+(1−αH1 −α

E1 )ℜ

(〈E(0,0)

h ,n×H(0,0h 〉∂K

Note that we have:

αH2 αE

∣∣∣ℜ(〈E(0,0)

h ,n×H(0,0)h 〉∂K

)∣∣∣ 6 αH2 ‖E(0,0)

h ‖2L2(∂K ) +α

E2 ‖n×H(0,0)

h ‖2L2(∂K ).

E(0,0)h and H(0,0)

h , we obtain:

(iωεr E(0,0)h ,E(0,0)

h )K +(iωµr H(0,0)h ,H(0,0)

h )K +2ℑ((E(0,0)h ,curlH(0,0)

h )K )

2(1−αH1 )ℑ

(〈n×H(0,0)

h ,E(0,0)h 〉∂K

)+(1−α

H1 −α

E1 )〈E(0,0)

h ,n×H(0,0)h 〉∂K

+αH2 〈E

(0,0)h ,E(0,0)

h 〉∂K +αE2 〈n×H(0,0)

h ,n×H(0,0)h 〉∂K = 0

αH2 ‖E(0,0)

h ‖2L2(∂K ) +α

E2 ‖n×H(0,0)

h ‖2L2(∂K )

+(1−αH1 −α

E1 )ℜ

(〈E(0,0)

h ,n×H(0,0h 〉∂K

Note that we have:

αH2 αE

∣∣∣ℜ(〈E(0,0)

h ,n×H(0,0)h 〉∂K

)∣∣∣ 6 αH2 ‖E(0,0)

h ‖2L2(∂K ) +α

E2 ‖n×H(0,0)

h ‖2L2(∂K ).

E(0,0)h and H(0,0)

h , we obtain:

(iωεr E(0,0)h ,E(0,0)

h )K +(iωµr H(0,0)h ,H(0,0)

h )K +2ℑ((E(0,0)h ,curlH(0,0)

h )K )

2(1−αH1 )ℑ

(〈n×H(0,0)

h ,E(0,0)h 〉∂K

)+(1−α

H1 −α

E1 )〈E(0,0)

h ,n×H(0,0)h 〉∂K

+αH2 〈E

(0,0)h ,E(0,0)

h 〉∂K +αE2 〈n×H(0,0)

h ,n×H(0,0)h 〉∂K = 0

αH2 ‖E(0,0)

h ‖2L2(∂K ) +α

E2 ‖n×H(0,0)

h ‖2L2(∂K )

+(1−αH1 −α

E1 )ℜ

(〈E(0,0)

h ,n×H(0,0h 〉∂K

Note that we have:

αH2 αE

∣∣∣ℜ(〈E(0,0)

h ,n×H(0,0)h 〉∂K

)∣∣∣ 6 αH2 ‖E(0,0)

h ‖2L2(∂K ) +α

E2 ‖n×H(0,0)

h ‖2L2(∂K ).

Conclusions on Locally Well-posed HDG

Conditions for the local problem being well-posed:

αH2 and αE

2 are strictly positive

the coefficients αH1 and αE

1 satisfy:

1−2√

αH2 αE

2 < αE1 +α

H1 < 1+2

√αH

2 αE2 ,

Some Existing HDG formulations

the first HDG formulation: αE1 = αE

2 = 0,αH1 = 1, and αH

2 = τ

second HDG formulation (locally well-posed):αE1 = 0,αE

2 = 1,αH1 = 1, and αH

2 = τ

The HDG formulation with symmetry: τ > 12

Conclusions on Locally Well-posed HDG

Conditions for the local problem being well-posed:

αH2 and αE

2 are strictly positive

the coefficients αH1 and αE

1 satisfy:

1−2√

αH2 αE

2 < αE1 +α

H1 < 1+2

√αH

2 αE2 ,

Some Existing HDG formulations

the first HDG formulation: αE1 = αE

2 = 0,αH1 = 1, and αH

2 = τ

second HDG formulation (locally well-posed):αE1 = 0,αE

2 = 1,αH1 = 1, and αH

2 = τ

The HDG formulation with symmetry: τ > 12

Domain Decomposition Methods

HDGM for 3D problem

Introducing a hybrid term Λh:Λh := Ht

h, ∀F ∈Fh,

we find (Eh,Hh,Λh) ∈ Vph ×Vp

h ×Mph such that:

(iωεr Eh,v)Th − (Hh,curlv)Th + < Λh,n×v >∂Th= 0, ∀v ∈ Vp

(iωµr Hh,v)Th +(Eh,curlv)Th−< Eh,n×v >∂Th= 0, ∀v ∈ Vp

< Jn× EhK,η >Fh −< Λh,η >Γa =< ginc,η >Γa , ∀η ∈ Mph,

with Eh = Eh + τK n× (Λh −Hth).

On each interior face of the computational domain the HDG scheme satisfies followingconditions:

Jn× EhK = 0, and JΛhK = 0

Transmission Conditions

Jn× EhK+Z1JΛhK = 0 and Jn× EhK+Z2JΛhK = 0,

Domain Decomposition Methods

HDGM for 3D problem

Introducing a hybrid term Λh:Λh := Ht

h, ∀F ∈Fh,

we find (Eh,Hh,Λh) ∈ Vph ×Vp

h ×Mph such that:

(iωεr Eh,v)Th − (Hh,curlv)Th + < Λh,n×v >∂Th= 0, ∀v ∈ Vp

(iωµr Hh,v)Th +(Eh,curlv)Th−< Eh,n×v >∂Th= 0, ∀v ∈ Vp

< Jn× EhK,η >Fh −< Λh,η >Γa =< ginc,η >Γa , ∀η ∈ Mph,

with Eh = Eh + τK n× (Λh −Hth).

On each interior face of the computational domain the HDG scheme satisfies followingconditions:

Jn× EhK = 0, and JΛhK = 0

Transmission Conditions

Jn× EhK+Z1JΛhK = 0 and Jn× EhK+Z2JΛhK = 0,

Schwarz algorithm

In the case of the Schwarz algorithm, based on the transmission operator in terms of theimpedance condition, and using the definitions Eh and Ht

h and the fact that n×Eth = n×Eh, we

want to enforce n12× (E1

h + τn12× (Λ1h −Ht,1

h ))−ZΛ1h =

n12× (E2h + τn21× (Λ2

h −Ht,2h ))−ZΛ2

n21× (E2h + τn21× (Λ2

n21× (E1h + τn12× (Λ1

Simplified versionn12×E1

h + τ(Ht,1h −Λ1

h)−ZΛ1h = n12×E2

h − τ(Ht,2h −Λ2

h)−ZΛ2h,

n21×E2h + τ(Ht,2

h −Λ2h)−ZΛ2

h = n21×E1h − τ(Ht,1

h −Λ1h)−ZΛ1

Schwarz algorithm

In the case of the Schwarz algorithm, based on the transmission operator in terms of theimpedance condition, and using the definitions Eh and Ht

h and the fact that n×Eth = n×Eh, we

want to enforce n12× (E1

h + τn12× (Λ1h −Ht,1

h ))−ZΛ1h =

n12× (E2h + τn21× (Λ2

n21× (E2h + τn21× (Λ2

n21× (E1h + τn12× (Λ1

Simplified versionn12×E1

h + τ(Ht,1h −Λ1

h)−ZΛ1h = n12×E2

h − τ(Ht,2h −Λ2

h)−ZΛ2h,

n21×E2h + τ(Ht,2

h −Λ2h)−ZΛ2

h = n21×E1h − τ(Ht,1

h −Λ1h)−ZΛ1

Schwarz algorithm - cont.

In the Schwarz iterative algorithm, the transmission conditions will translate asn12×E1,n+1

h + τ(Ht,1,n+1h −Λ1,n+1

h )−ZΛ1,n+1h =

n12×E2,nh − τ(Ht,2,n

h −Λ2,nh )−ZΛ2,n

n21×E2,n+1h + τ(Ht,2,n+1

h −Λ2,n+1h )−ZΛ2,n+1

Interface systemL (U1,n+1,Λ1,n+1) = f 1, in Ω1,

Bn12(U1,n+1) = W1,n, on Γ12,

+Boundary conditions on ∂Ω1∩∂Ω,

L (U2,n+1,Λ2,n+1) = f 2, in Ω2,

Bn21(U2,n+1) = W2,n, on Γ21,

In the Schwarz iterative algorithm, the transmission conditions will translate asn12×E1,n+1

h + τ(Ht,1,n+1h −Λ1,n+1

h )−ZΛ1,n+1h =

n21×E2,n+1h + τ(Ht,2,n+1

h −Λ2,n+1h )−ZΛ2,n+1

Interface systemL (U1,n+1,Λ1,n+1) = f 1, in Ω1,

Bn12(U1,n+1) = W1,n, on Γ12,

L (U2,n+1,Λ2,n+1) = f 2, in Ω2,

Bn21(U2,n+1) = W2,n, on Γ21,

Schwarz algorithm in concise formW1,n+1 = Bn12(U2(W2,n, f 2),Λ2,n+1),W2,n+1 = Bn21(U1(W1,n, f 1),Λ1,n+1),

The discretization of the global problem can be written in the matrix form:

K1 0 0 0 R1 00 K2 0 0 0 R2

C1 0 A1 0 0 00 C2 0 A2 0 00 −B12 0 −B12 I 0

−B21 0 −B21 0 0 I

where Ki is the coefficient matrix of the disrectized reduced system on each subdomain, Ai is thelocal matrix for the solution of Ui with block diagonal structure, Ci express the coupling betweenthe hybrid term Λi and the fields Ui , Ri represents the coupling between hybrid variable and theinterface unknowns, B12 and B21 are the discretized interface operators.

Schwarz algorithm in concise formW1,n+1 = Bn12(U2(W2,n, f 2),Λ2,n+1),W2,n+1 = Bn21(U1(W1,n, f 1),Λ1,n+1),

The discretization of the global problem can be written in the matrix form:

K1 0 0 0 R1 00 K2 0 0 0 R2

C1 0 A1 0 0 00 C2 0 A2 0 00 −B12 0 −B12 I 0

−B21 0 −B21 0 0 I

where Ki is the coefficient matrix of the disrectized reduced system on each subdomain, Ai is thelocal matrix for the solution of Ui with block diagonal structure, Ci express the coupling betweenthe hybrid term Λi and the fields Ui , Ri represents the coupling between hybrid variable and theinterface unknowns, B12 and B21 are the discretized interface operators.

Numerical results of Schwarz algorithm

Wave propagation in vacuum

Table: Propagation of a plane wave in vacuum: comparisons between HDG and upwind flux-based DGmethods based on memory and CPU times.

Mesh Ns Strategy #it CPU (min/max) REALHDG UF HDG UF

M1 16 DD-bicgl 7 38.9 s/39.2 s 103.2 s/104.9 s 40.3 s 107.1 s- DD-gmres 7 20.5 s/20.8 s 53.9 s/54.2 s 21.4 s 55.3 s- DD-itref 7 13.2 s/13.5 s 34.9 s/35.3 s 13.9 s 36.2 s

32 DD-bicgl 9 24.4 s/24.8 s 54.3 s/55.2 s 25.6 s 55.6 s- DD-gmres 9 13.0 s/13.3 s 28.5 s/28.7 s 13.7 s 29.5 s- DD-itref 9 7.3 s/7.6 s 18.2 s/18.4 s 7.9 s 19.0 s

M2 32 DD-bicgl 9 89.8 s/91.2 s 261.5 s/267.7 s 94.0 s 272.8 s- DD-gmres 9 47.2 s/47.9 s 137.0 s/137.6 s 49.5 s 140.3 s- DD-itref 9 32.4 s/33.8 s 87.2 s/88.4 s 35.2 s 90.4 s

Numerical results of Schwarz algorithm

Wave propagation in head tissue

Table: Meshes used for propagation of a plane wave in a heterogeneous medium.

Mesh # Vertices # Tetrahedra # FacesM1 60,590 361,848 725,136

Table: Computation times and memory requirement for storing the L and U factors.

Mesh NS CPU (min/max) RAM (min/max)HDG UF HDG UF

M1 64 21.84 s/37.94 s 398.43 s/774.25 s 179 MB/265 MB 961 MB/1590 MB

Table: Computation times (solution phase).

Mesh NS Strategy #it CPU (min/max)HDG UF HDG UF

M1 64 DD-itref 42 39 132.99 s/146.14 s 480.38 s/577.34

Electric Fields in a Head

Last page

Conclusions

an HDG method which has optimal convergence order

curvilinear domains – isoparametric elements

Work on locally well-posed HDGM

Schwarz methods

Undergoing and future work

optimal Schwarz method

HDG/BEM

Multigrid solver for HDG

· · ·

Thank you for your attention!

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