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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)
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 2 / 40
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
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−20
−15
−10
−5
0
5
10
15
θ (Degrees)
RC
S (
dB)
φ=0
φ=90Y X
Z
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 3 / 40
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
−25
−20
−15
−10
−5
0
5
10
15
θ (Degrees)
RC
S (
dB)
φ=0
φ=90Y X
Z
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 3 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 4 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 4 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 5 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 5 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 6 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 7 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 7 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 8 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 8 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 9 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 9 / 40
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 10 / 40
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)
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 10 / 40
Example - exposure of head tissues
Assessment:
complexity in modeling - discontinuous Galerkin FEM
large number of DOFs - hybridization to reduce the DOFs
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 11 / 40
Example - exposure of head tissues
Assessment:
complexity in modeling - discontinuous Galerkin FEM
large number of DOFs - hybridization to reduce the DOFs
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 11 / 40
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
)T,
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.
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 12 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 13 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 13 / 40
Discontinuous Galerkin (DG) formulation
Discontinuous finite element spaces:V p
h =
v ∈ L2(Ω) | v |K ∈ V ph (K ), ∀K ∈Th
,
Vph =
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 =
∫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.
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 14 / 40
Discontinuous Galerkin (DG) formulation
Discontinuous finite element spaces:V p
h =
v ∈ L2(Ω) | v |K ∈ V ph (K ), ∀K ∈Th
,
Vph =
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 =
∫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.
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 14 / 40
Discontinuous Galerkin (DG) formulation
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−
2,
vF=v+ + v−
2,
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 15 / 40
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)
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 16 / 40
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)
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 16 / 40
Hybridizable discontinuous Galerkin (HDG) methods
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:
Mph =
η ∈ L2(Fh) | η |F ∈ Pp(F), ∀F ∈Fh and η |Γm
= 0
.
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.
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 17 / 40
Hybridizable discontinuous Galerkin (HDG) methods
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:
Mph =
η ∈ L2(Fh) | η |F ∈ Pp(F), ∀F ∈Fh and η |Γm
= 0
.
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.
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 17 / 40
Hybridizable discontinuous Galerkin (HDG) methods
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
h,
< Jn× HhK,η >Fh+ < λh,η >Γa
=< ginc,η >Γa, ∀η ∈ Mp
h ,
Notations:
(·, ·)Th= ∑
K∈Th
(·, ·)K , < ·, ·>∂Th= ∑
K∈Th
< ·, ·>∂K , < ·, ·>Fh= ∑
f∈Fh
< ·, ·>F ,
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 18 / 40
Hybridizable discontinuous Galerkin (HDG) methods
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
h,
< Jn× HhK,η >Fh+ < λh,η >Γa
=< ginc,η >Γa, ∀η ∈ Mp
h ,
Notations:
(·, ·)Th= ∑
K∈Th
(·, ·)K , < ·, ·>∂Th= ∑
K∈Th
< ·, ·>∂K , < ·, ·>Fh= ∑
f∈Fh
< ·, ·>F ,
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 18 / 40
Hybridizable discontinuous Galerkin (HDG) methods
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− ,
thus:
< 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
h such that:(iωεr Eh,v)Th
− (curlHh,v)Th+ < τ(Eh −λh),v >∂Th
= 0, ∀v ∈ V ph ,
(iωµr Hh,v)Th+(Eh,curlv)Th
−< λh,n×v >∂Th= 0, ∀v ∈ vp
h,
< n×Hh,η >∂Th−< τ(Eh −λh),η >∂Th
+ < λh,η >Γa−< ginc,η >Γa
= 0, ∀η ∈ Mph .
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 19 / 40
Hybridizable discontinuous Galerkin (HDG) methods
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− ,
thus:
< 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
h such that:(iωεr Eh,v)Th
− (curlHh,v)Th+ < τ(Eh −λh),v >∂Th
= 0, ∀v ∈ V ph ,
(iωµr Hh,v)Th+(Eh,curlv)Th
−< λh,n×v >∂Th= 0, ∀v ∈ vp
h,
< n×Hh,η >∂Th−< τ(Eh −λh),η >∂Th
+ < λh,η >Γa−< ginc,η >Γa
= 0, ∀η ∈ Mph .
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 19 / 40
Hybridizable discontinuous Galerkin (HDG) methods
Link between DG and HDG
Eh = λh =1
τK+ + τK−(τK+ E+
h + τK−E−h )− 1
τK+ + τK−Jn×HhK on F ,
Hh =1
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 .
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 20 / 40
Hybridizable discontinuous Galerkin (HDG) methods
Link between DG and HDG
Eh = λh =1
τK+ + τK−(τK+ E+
h + τK−E−h )− 1
τK+ + τK−Jn×HhK on F ,
Hh =1
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 .
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 20 / 40
Hybridizable discontinuous Galerkin (HDG) methods
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
−2
−1.5
−1
−0.5
0
0.5
(a) HDG-P3, ω = 2π
0 0.005 0.01 0.015−0.05
−0.04
−0.03
−0.02
−0.01
0
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
0
0.1
(c) HDG-P3, ω = 8π
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 21 / 40
Hybridizable discontinuous Galerkin (HDG) methods
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.
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 22 / 40
Hybridizable discontinuous Galerkin (HDG) methods
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 23 / 40
Hybridizable discontinuous Galerkin (HDG) methods
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 23 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 24 / 40
HDG on curvilinear domains – isoparametric elements
Performance results: scattering from a metallic cylinder
(a) Quadratic map (b) Cubic map
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 25 / 40
HDG on curvilinear domains – isoparametric elements
3.25e-08 0.00419 0.00838
Error of Ez
X
Y
Z
(a) Affine map
0 5.03e-05 0.000101
Error of Ez
X
Y
Z
(b) Quadratic map
2.43e-08 0.0477 0.0953
Error of Hx
X
Y
Z
(c) Affine map
1e-10 2.64e-05 5.28e-05
Error of Hx
X
Y
Z
(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
h ,
(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
h .
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 27 / 40
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.
Idea
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 28 / 40
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.
Idea
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 28 / 40
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.
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 29 / 40
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.
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 29 / 40
Generalized locally well-posed HDG formulation - cont.
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 ).
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 30 / 40
Generalized locally well-posed HDG formulation - cont.
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 ).
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 30 / 40
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
)= 0.
Note that we have:
2√
αH2 αE
2
∣∣∣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 ).
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 31 / 40
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
)= 0.
Note that we have:
2√
αH2 αE
2
∣∣∣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 ).
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 31 / 40
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
)= 0.
Note that we have:
2√
αH2 αE
2
∣∣∣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 ).
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 31 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 32 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 32 / 40
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
h,
(iωµr Hh,v)Th +(Eh,curlv)Th−< Eh,n×v >∂Th= 0, ∀v ∈ Vp
h,
< 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,
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 33 / 40
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
h,
(iωµr Hh,v)Th +(Eh,curlv)Th−< Eh,n×v >∂Th= 0, ∀v ∈ Vp
h,
< 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,
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 33 / 40
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
h,
n21× (E2h + τn21× (Λ2
h −Ht,2h ))−ZΛ2
h =
n21× (E1h + τn12× (Λ1
h −Ht,1h ))−ZΛ1
h,
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
h.
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 34 / 40
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
h,
n21× (E2h + τn21× (Λ2
h −Ht,2h ))−ZΛ2
h =
n21× (E1h + τn12× (Λ1
h −Ht,1h ))−ZΛ1
h,
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
h.
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 34 / 40
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
h ,
n21×E2,n+1h + τ(Ht,2,n+1
h −Λ2,n+1h )−ZΛ2,n+1
h =
n21×E1,nh − τ(Ht,1,n
h −Λ1,nh )−ZΛ1,n
h .
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,
+Boundary conditions on ∂Ω2∩∂Ω,
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 35 / 40
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
h ,
n21×E2,n+1h + τ(Ht,2,n+1
h −Λ2,n+1h )−ZΛ2,n+1
h =
n21×E1,nh − τ(Ht,1,n
h −Λ1,nh )−ZΛ1,n
h .
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,
+Boundary conditions on ∂Ω2∩∂Ω,
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 35 / 40
Schwarz algorithm - cont.
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
Λ1h
Λ2h
U1h
U2h
W1h
W2h
=
f 1h
f 2h
0
0
0
0
,
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.
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 36 / 40
Schwarz algorithm - cont.
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
Λ1h
Λ2h
U1h
U2h
W1h
W2h
=
f 1h
f 2h
0
0
0
0
,
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.
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 36 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 37 / 40
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
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 38 / 40
Electric Fields in a Head
ht
Y X
Z
XY
Z
Y X
Z
Y X
Z
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 39 / 40
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!
Liang Li (SMS, UESTC) HDG for Maxwell’s equations 2012, July 3rd 40 / 40
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