A stabilized DG-type method for solving efficiently Helmholtz problems

Mohamed Amara; Henri Calandra; Rabia Djellouli; Magdalena Grigoroscuta-Strugaru

inria-00537983, version 2

A stabilized DG-type method for solving efficiently Helmholtz problems

Mohamed Amara ()^a^, 1, 2, Henri Calandra () ³, Rabia Djellouli ()^b^, 4, Magdalena Grigoroscuta-Strugaru () ^1, 2, 5

N° RR-7461 (2010)

Abstract: We propose a stabilized discontinuous Galerkin-type method (SDGM) for solving efficiently Helmholtz problems. This mixed-hybrid formulation is a two-step procedure. Step 1 consists in solving well-posed problems at the element partition level of the computational domain, whereas Step 2 requires the solution of a global system whose unknowns are the Lagrange multipliers. The main features of SDGM include: (a) the resulting local problems are associated with small positive definite Hermitian matrices that can be solved in parallel, and (b) the matrix corresponding to the global linear system arising in Step 2 is Hermitian and positive semi-definite. Illustrative numerical results for two-dimensional waveguide problems highlight the potential of SDGM for solving efficiently Helmholtz problems in mid- and high-frequency regime.

a – Université de Pau et des Pays de l'Adour
b – California State University Northridge
1: Laboratoire de Mathématiques et de leurs Applications de Pau (LMA-PAU)
CNRS : UMR5142 – Université de Pau et des Pays de l'Adour [UPPA]
2: MAGIQUE-3D (INRIA Bordeaux - Sud-Ouest)
INRIA – CNRS – Université de Pau et des Pays de l'Adour [UPPA]
3: TOTAL-Scientific and Technical Center Jean Féger (CSTJF)
Total
4: Interdisciplinary Research Institute for the Sciences (IRIS)
California State University at Northridge
5: Basque Center for Applied Mathematics (BCAM)
Basque Center for Applied Mathematics

Collaboration : Associate Team MAGIC

inria-00537983, version 2
http://hal.inria.fr/inria-00537983
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From: Magdalena Grigoroscuta-Strugaru <>
Submitted on: Wednesday, 26 January 2011 11:33:31
Updated on: Wednesday, 26 January 2011 11:53:18

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%% inria-00537983, version 2
%% http://hal.inria.fr/inria-00537983
@techreport{amara:inria-00537983,
    hal_id = {inria-00537983},
    url = {http://hal.inria.fr/inria-00537983},
    title = {{A stabilized DG-type method for solving efficiently Helmholtz problems}},
    author = {Amara, Mohamed and Calandra, Henri and Djellouli, Rabia and Grigoroscuta-Strugaru, Magdalena},
    abstract = {{We propose a stabilized discontinuous Galerkin-type method (SDGM) for solving efficiently Helmholtz problems. This mixed-hybrid formulation is a two-step procedure. Step 1 consists in solving well-posed problems at the element partition level of the computational domain, whereas Step 2 requires the solution of a global system whose unknowns are the Lagrange multipliers. The main features of SDGM include: (a) the resulting local problems are associated with small positive definite Hermitian matrices that can be solved in parallel, and (b) the matrix corresponding to the global linear system arising in Step 2 is Hermitian and positive semi-definite. Illustrative numerical results for two-dimensional waveguide problems highlight the potential of SDGM for solving efficiently Helmholtz problems in mid- and high-frequency regime.}},
    keywords = {Helmholtz equation ; Discontinuous Galerkin ; Plane waves ; Lagrange multipliers ; Positive semi-definite Hermitian matrix ; Inf-sup condition ; Stability ; Waveguide problems},
    language = {English},
    affiliation = {Laboratoire de Math{\'e}matiques et de leurs Applications de Pau - LMA-PAU , MAGIQUE-3D - INRIA Bordeaux - Sud-Ouest , TOTAL-Scientific and Technical Center Jean F{\'e}ger - CSTJF , Interdisciplinary Research Institute for the Sciences - IRIS , Basque Center for Applied Mathematics - BCAM},
    pages = {30},
    type = {Research Report},
    institution = {INRIA},
    number = {RR-7461},
    collaboration = {Associate Team MAGIC },
    year = {2010},
    month = Nov,
    pdf = {http://hal.inria.fr/inria-00537983/PDF/RR-7461.pdf},
}

%F inria-00537983, version 2
%0 Report
%U http://hal.inria.fr/inria-00537983
%2 MATH:MATH_NA;MATH:MATH_AP
%T A stabilized DG-type method for solving efficiently Helmholtz problems
%A Amara, Mohamed
%A Calandra, Henri
%A Djellouli, Rabia
%A Grigoroscuta-Strugaru, Magdalena
%+ Laboratoire de Mathématiques et de leurs Applications de Pau - LMA-PAU , MAGIQUE-3D - INRIA Bordeaux - Sud-Ouest , TOTAL-Scientific and Technical Center Jean Féger - CSTJF , Interdisciplinary Research Institute for the Sciences - IRIS , Basque Center for Applied Mathematics - BCAM
%X We propose a stabilized discontinuous Galerkin-type method (SDGM) for solving efficiently Helmholtz problems. This mixed-hybrid formulation is a two-step procedure. Step 1 consists in solving well-posed problems at the element partition level of the computational domain, whereas Step 2 requires the solution of a global system whose unknowns are the Lagrange multipliers. The main features of SDGM include: (a) the resulting local problems are associated with small positive definite Hermitian matrices that can be solved in parallel, and (b) the matrix corresponding to the global linear system arising in Step 2 is Hermitian and positive semi-definite. Illustrative numerical results for two-dimensional waveguide problems highlight the potential of SDGM for solving efficiently Helmholtz problems in mid- and high-frequency regime.
%2 Nous proposons une méthode stabilisée de type Galerkin discontinu (SDGM) pour la résolution de problèmes de Helmholtz. Cette formulation mixte duale est une procédure en deux étapes. La première étape consiste à résoudre des problèmes bien posés au niveau des éléments du maillage. Dans la deuxième étape nous résolvons un système global dont les inconnues sont les multiplicateurs de Lagrange. Les propriétés principales de SDGM incluent: (a) les systèmes linéaires locaux sont Hermitiens, définis positifs et de petite taille; ils peuvent donc être résolus en parallèle et (b) la matrice du système linéaire global obtenu à l'étape 2 est Hermitienne et semi-définie positive. Nous présentons des résultats numériques pour des problèmes de guide d'ondes 2D qui montrent le potentiel de SDGM pour résoudre efficacement les problèmes de Helmholtz en régime moyenne et haute fréquence.
%2 G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.8: Partial Differential Equations, G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.2: Approximation
%G English
%2 Research Report
%P 30
%8 2010-11-00
%K Helmholtz equation ; Discontinuous Galerkin ; Plane waves ; Lagrange multipliers ; Positive semi-definite Hermitian matrix ; Inf-sup condition ; Stability ; Waveguide problems
%Z RR-7461
%Z Associate Team MAGIC

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