Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell's equations

Mohamed El Bouajaji, Victorita Dolean Maini, Martin J. Gander, Stephane Lanteri, Ronan Perrussel

Research output: Contribution to journalArticle

Abstract

We show in this paper how to properly discretize optimized Schwarz methods for the time-harmonic Maxwell's equations in two and three spatial dimensions using a discontinuous Galerkin (DG) method. Due to the multiple traces between elements in the DG formulation, it is not clear a priori how the more sophisticated transmission conditions in optimized Schwarz methods should be discretized, and the most natural approach, at convergence of the Schwarz method, does not lead to the monodomain DG solution, which implies that for such discretizations, the DG error estimates do not hold when the Schwarz method has converged. We present here a consistent discretization of the transmission conditions in the framework of a DG weak formulation, for which we prove that the multidomain and monodomain solutions for the Maxwell's equations are the same. We illustrate our results with several numerical experiments of propagation problems in homogeneous and heterogeneous media.
LanguageEnglish
Pages572-592
Number of pages21
JournalETNA - Electronic Transactions on Numerical Analysis
Volume44
Publication statusPublished - 3 Nov 2015

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Schwarz Methods
Discontinuous Galerkin
Maxwell's equations
Harmonic
Discretization
Transmission Conditions
Heterogeneous Media
Weak Formulation
Inhomogeneous Media
Discontinuous Galerkin Method
Error Estimates
Trace
Numerical Experiment
Propagation
Imply
Formulation

Keywords

  • computational electromagnetism
  • time harmonic Maxerll's equations
  • discontinuous Galerkin method
  • optimized Schwarz methods
  • transmission conditions

Cite this

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title = "Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell's equations",
abstract = "We show in this paper how to properly discretize optimized Schwarz methods for the time-harmonic Maxwell's equations in two and three spatial dimensions using a discontinuous Galerkin (DG) method. Due to the multiple traces between elements in the DG formulation, it is not clear a priori how the more sophisticated transmission conditions in optimized Schwarz methods should be discretized, and the most natural approach, at convergence of the Schwarz method, does not lead to the monodomain DG solution, which implies that for such discretizations, the DG error estimates do not hold when the Schwarz method has converged. We present here a consistent discretization of the transmission conditions in the framework of a DG weak formulation, for which we prove that the multidomain and monodomain solutions for the Maxwell's equations are the same. We illustrate our results with several numerical experiments of propagation problems in homogeneous and heterogeneous media.",
keywords = "computational electromagnetism, time harmonic Maxerll's equations, discontinuous Galerkin method, optimized Schwarz methods, transmission conditions",
author = "{El Bouajaji}, Mohamed and {Dolean Maini}, Victorita and Gander, {Martin J.} and Stephane Lanteri and Ronan Perrussel",
year = "2015",
month = "11",
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pages = "572--592",
journal = "ETNA - Electronic Transactions on Numerical Analysis",
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Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell's equations. / El Bouajaji, Mohamed; Dolean Maini, Victorita; Gander, Martin J.; Lanteri, Stephane; Perrussel, Ronan.

In: ETNA - Electronic Transactions on Numerical Analysis, Vol. 44, 03.11.2015, p. 572-592.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell's equations

AU - El Bouajaji, Mohamed

AU - Dolean Maini, Victorita

AU - Gander, Martin J.

AU - Lanteri, Stephane

AU - Perrussel, Ronan

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N2 - We show in this paper how to properly discretize optimized Schwarz methods for the time-harmonic Maxwell's equations in two and three spatial dimensions using a discontinuous Galerkin (DG) method. Due to the multiple traces between elements in the DG formulation, it is not clear a priori how the more sophisticated transmission conditions in optimized Schwarz methods should be discretized, and the most natural approach, at convergence of the Schwarz method, does not lead to the monodomain DG solution, which implies that for such discretizations, the DG error estimates do not hold when the Schwarz method has converged. We present here a consistent discretization of the transmission conditions in the framework of a DG weak formulation, for which we prove that the multidomain and monodomain solutions for the Maxwell's equations are the same. We illustrate our results with several numerical experiments of propagation problems in homogeneous and heterogeneous media.

AB - We show in this paper how to properly discretize optimized Schwarz methods for the time-harmonic Maxwell's equations in two and three spatial dimensions using a discontinuous Galerkin (DG) method. Due to the multiple traces between elements in the DG formulation, it is not clear a priori how the more sophisticated transmission conditions in optimized Schwarz methods should be discretized, and the most natural approach, at convergence of the Schwarz method, does not lead to the monodomain DG solution, which implies that for such discretizations, the DG error estimates do not hold when the Schwarz method has converged. We present here a consistent discretization of the transmission conditions in the framework of a DG weak formulation, for which we prove that the multidomain and monodomain solutions for the Maxwell's equations are the same. We illustrate our results with several numerical experiments of propagation problems in homogeneous and heterogeneous media.

KW - computational electromagnetism

KW - time harmonic Maxerll's equations

KW - discontinuous Galerkin method

KW - optimized Schwarz methods

KW - transmission conditions

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