Boundary Element Methods for Maxwell Transmission Problems in Lipschitz Domains

Abstract

We consider the Maxwell equations in a domain with Lipschitz boundary and the boundary integral operator A occuring in the Calderon projector. We prove an inf-sup condition for A using a Hodge decomposition. We apply this to two types of boundary value problems: the exterior scattering problem by a perfectly conducting body, and the dielectric problem with two different materials in the interior and exterior domain. In both cases we obtain an equivalent boundary equation which has a unique solution. We then consider Galerkin discretizations with Raviart-Thomas spaces. We show that these spaces have discrete Hodge decompositions which are in some sense close to the continuous Hodge decomposition. This property allows us to prove quasioptimal convergence of the resulting boundary element methods.