A Coercive Combined Field Integral Equation for Electromagnetic Scattering

Many boundary integral equation methods used in the simulation of direct electromagnetic scattering of a time-harmonic wave at a perfectly conducting obstacle break down when applied at frequencies close to a resonant frequency of the obstacle. A remedy is offered by special indirect boundary element methods based on the so-called combined field integral equation. However, hitherto no theoretical results about the convergence of discretized combined field integral equations have been available. In this paper we propose a new combined field integral equation, convert it into variational form, establish its coercivity in the natural trace spaces for electromagnetic fields, and conclude existence and uniqueness of solutions for any frequency. Moreover, a conforming Galerkin discretization of the variational equations by means of ${\rm div}_\Gamma$-conforming boundary elements can be shown to be asymptotically quasi-optimal. This permits us to derive quantitative convergence rates on sufficiently fine, uniformly shape-regular sequences of surface triangulations.