Published June 4, 2013 | Version v1
Publication

DG discretization of optimized Schwarz methods for Maxwell's equations

Others:
Robust control of infinite dimensional systems and applications (CORIDA) ; Institut Élie Cartan de Nancy (IECN) ; Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM) ; Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est ; Institut National de Recherche en Informatique et en Automatique (Inria)
Institut Élie Cartan de Lorraine (IECL) ; Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
Laboratoire Jean Alexandre Dieudonné (JAD) ; Université Nice Sophia Antipolis (1965 - 2019) (UNS) ; COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)
Section de mathématiques [Genève] ; Université de Genève = University of Geneva (UNIGE)
Numerical modeling and high performance computing for evolution problems in complex domains and heterogeneous media (NACHOS) ; Inria Sophia Antipolis - Méditerranée (CRISAM) ; Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD) ; Université Nice Sophia Antipolis (1965 - 2019) (UNS) ; COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS) ; COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)
Groupe de Recherche en Electromagnétisme (LAPLACE-GRE) ; LAboratoire PLasma et Conversion d'Energie (LAPLACE) ; Université Toulouse III - Paul Sabatier (UT3) ; Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP) ; Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse III - Paul Sabatier (UT3) ; Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP) ; Université Fédérale Toulouse Midi-Pyrénées

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Description

In the last decades, Discontinuous Galerkin (DG) methods have seen rapid growth and are widely used in various application domains (see [13] for an historical intro- duction). This is due to their main advantage of combining the best of finite element and finite volume methods. For the time-harmonic Maxwell equations, once the problem is discretized with a DG method, finding robust solvers is a difficult task since one has to deal with indefinite problems. From the pioneering work of Despre ́s [5] where the first provably convergent domain decomposition (DD) algorithm for the Helmholtz equation was proposed and then extended to Maxwell's equations in [6], other studies followed. Preliminary attempts to obtain better algorithms for this kind of equations were given in [3, 4, 12], where the first ideas of optimized Schwarz methods can be found. Then, the advantage of the optimization process was used for the second order Maxwell system in [1]. Later on, an entire hierarchy of optimized transmission conditions for the first order Maxwell's equations was proposed in [9, 11] . For the second order or curl-curl Maxwell's equations second order optimized transmission conditions can be found in [14, 15, 16, 17]. We study here optimized Schwarz DD methods for the time-harmonic Maxwell equations dis- cretized by a DG method. Due to the particularity of the latter, DG discretization ap- plied to more sophisticated Schwarz methods is not straightforward. In this work we show a strategy of discretization and prove the equivalence between multi-domain and single-domain solutions. The proposed discrete framework is then illustrated by some numerical results in the two-dimensional case.

Additional details

Created:
December 3, 2022
Modified:
December 1, 2023