ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes

Fezoui, Loulaa1, Lanteri, Stéphanea1, Lohrengel, Stéphaniea2 and Piperno, Sergea1

a1 CERMICS, INRIA, BP93, 06902 Sophia-Antipolis Cedex, France. Serge.Piperno@cermics.enpc.fr

a2 Dieudonné Lab., UNSA, UMR CNRS 6621, Parc Valrose, 06108 Nice Cedex 2, France.

Abstract

A Discontinuous Galerkin method is used for to the numerical solution of the time-domain Maxwell equations on unstructured meshes. The method relies on the choice of local basis functions, a centered mean approximation for the surface integrals and a second-order leap-frog scheme for advancing in time. The method is proved to be stable for cases with either metallic or absorbing boundary conditions, for a large class of basis functions. A discrete analog of the electromagnetic energy is conserved for metallic cavities. Convergence is proved for $\mathbb{P}_k$ Discontinuous elements on tetrahedral meshes, as well as a discrete divergence preservation property. Promising numerical examples with low-order elements show the potential of the method.

(Received August 17 2004)

(Revised July 2 2005)

(Online publication November 15 2005)

Key Words:

  • Electromagnetics;
  • finite volume methods;
  • discontinuous Galerkin methods;
  • centered fluxes;
  • leap-frog time scheme;
  • L 2 stability;
  • unstructured meshes;
  • absorbing boundary condition;
  • convergence;
  • divergence preservation.

Mathematics Subject Classification:

  • 65M12;
  • 65M60;
  • 78-08;
  • 78A40
Metrics