A fully discrete Galerkin scheme for a two-fold saddle point formulation of an exterior nonlinear problem

We analyze a fully discrete Galerkin method for the coupling of mixed finite elements and boundary elements as applied to an exterior nonlinear transmission problem arising in potential theory. We first show that the corresponding continuous formulation becomes a well posed two-fold saddle point problem. Our discrete approach uses Raviart-Thomas elements of lowest order and is based on simple quadrature formulas for the interior and boundary terms. We prove that, if the parameter of discretization is sufficiently small, the fully discrete Galerkin scheme is uniquely solvable and leads to optimal error estimates.