On the ellipticity of coupled finite element and one-equation boundary element methods for boundary value problems

In this paper we extend some recent results on the stability of the Johnson–Nédelec coupling of finite and boundary element methods in the case of boundary value problems. In Of and Steinbach (Z Angew Math Mech 93:476–484, 2013), Sayas (SIAM J Numer Anal 47:3451–3463, 2009) and Steinbach (SIAM J Numer Anal 49:1521–1531, 2011), the case of a free-space transmission problem was considered, and sufficient and necessary conditions are stated which ensure the ellipticity of the bilinear form for the coupled problem. The proof was based on considering the energies which are related to both the interior and exterior problem. In the case of boundary value problems for either interior or exterior problems, additional estimates are required to bound the energy for the solutions of related subproblems. Moreover, several techniques for the stabilization of the coupled formulations are analysed. Applications involve boundary value problems with either hard or soft inclusions, exterior boundary value problems, and macro-element techniques