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.
Highlights
The numerical solution of electromagnetic resonance problems is an important task in different fields of engineering and technology
In this paper we show that these conditions are satisfied for the Galerkin approximation of the proposed boundary integral formulations of the considered electromagnetic resonance problems when classical boundary elements of Raviart–Thomas or Brezzi–Douglas–Marini type are used
In this subsection we summarize the properties of the trace spaces which we need for the analysis of the boundary integral formulations of the resonance problems
Summary
The numerical solution of electromagnetic resonance problems is an important task in different fields of engineering and technology. In this paper we consider for a given bounded, connected Lipschitz domain Ωi ⊂ R3 the cavity resonance problem and the scatter-. This article is part of the topical collection “Waves 2019 – invited papers” edited by Manfred Kaltenbacher and Markus Melenk.
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