Convergence analysis of a Galerkin boundary element method for electromagnetic resonance problems

Abstract

In this paper a convergence analysis of a Galerkin boundary element method for resonance problems arising from the time harmonic Maxwell’s equations is presented. The cavity resonance problem with perfect conducting boundary conditions and the scattering resonance problem for impenetrable and penetrable scatterers are treated. The considered boundary integral formulations of the resonance problems are eigenvalue problems for holomorphic Fredholm operator-valued functions, where the occurring operators satisfy a so-called generalized Gårding’s inequality. The convergence of a conforming Galerkin approximation of this kind of eigenvalue problems is in general only guaranteed if the approximation spaces fulfill special requirements. We use recent abstract results for the convergence of the Galerkin approximation of this kind of eigenvalue problems in order to show that two classical boundary element spaces for Maxwell’s equations, the Raviart–Thomas and the Brezzi–Douglas–Marini boundary element spaces, satisfy these requirements. Numerical examples are presented, which confirm the theoretical results.