Stabilized FEM-BEM Coupling for Maxwell Transmission Problems

We consider the scattering of monochromatic electromagnetic waves at a dielectric object with a non-smooth surface. This paper studies the discretization of this problem by means of coupling finite element methods (FEM) and boundary element methods (BEM). Straightforward symmetric coupling as in [R. Hiptmair, Coupling of finite elements and boundary elements in electromagnetic scattering, SIAM J. Num. Anal. 41 (2003), pp. 919-944] suffers from instabilities at wave numbers related to interior Dirichlet eigenvalues, the so-called spurious resonance phenomenon. A remedy is offered by adopting the idea underlying the widely used combined field integral equations (CFIE). These can be obtained from Robin-type trace operators, which ensure uniqueness of solutions of the associated interior boundary value problem for all frequencies. This implies uniqueness of solutions of the coupled problem. In the spirit of [R. Hiptmair and P. Meury, Stabilized FEM-BEM Coupling for Helmholtz Transmission Problems, SIAM J. Numer. Anal. 44 (2006), pp. 2106-2130], in order to get a coercive variational problem, we have to incorporate a regularizing operator into the modified traces. The discretization of the coupled variational problem is then based on curl-conforming finite elements inside the scatterer, divΓ-conforming boundary elements for the surface currents and curlΓ-conforming boundary elements for an auxiliary function on the boundary. Adapting a Helmholtz-type splitting to the discrete setting, permits us to show asymptotic optimality of the Galerkin-FEM-BEM solution.