DG discretization of optimized Schwarz methods for Maxwell's equations

Mohamed El Bouajaji (1, 2) , Victorita Dolean (3) , Martin J. Gander (4) , Stephane Lanteri (5) , Ronan Perrussel (6)

1 CORIDA - Robust control of infinite dimensional systems and applications
2 IECL - Institut Élie Cartan de Lorraine
3 LJAD - Laboratoire Jean Alexandre Dieudonné
4 Section de mathématiques [Genève]
5 NACHOS - Numerical modeling and high performance computing for evolution problems in complex domains and heterogeneous media
6 LAPLACE-GRE - Groupe de Recherche en Electromagnétisme

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.

Domaines: Analyse numérique [math.NA]

Mohamed El Bouajaji, Victorita Dolean, Martin J. Gander, Stephane Lanteri, Ronan Perrussel. DG discretization of optimized Schwarz methods for Maxwell's equations. 2013. ⟨hal-00830274⟩