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

Mohamed El Bouajaji (1, 2) , Victorita Dolean (3, 4) , Martin J. Gander (5) , Stéphane Lanteri (6) , Ronan Perrussel (7)

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

We show in this paper how to properly discretize optimized Schwarz methods for the time-harmonic Maxwell equations 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 does not lead at convergence of the Schwarz method to the mono-domain DG discretization, which implies that for such discretizations, the DG error estimates do not hold when the Schwarz method has converged. We present an alternative discretization of the transmission conditions in the framework of a DG weak formulation, and prove that for this discretization the multidomain and mono-domain 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.

Domaines: Analyse numérique [math.NA]

Mohamed El Bouajaji, Victorita Dolean, Martin J. Gander, Stéphane Lanteri, Ronan Perrussel. Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell equations. 2014. ⟨hal-01062853⟩