On the coupling of local discontinuous Galerkin and boundary element methods for non-linear exterior transmission problems

Abstract

In this paper, we apply the coupling of local discontinuous Galerkin and boundary element methods to solve a class of non-linear exterior transmission problems in the plane. As a model, we consider a non-linear elliptic equation in an annular polygonal domain coupled with the Poisson equation in the surrounding unbounded exterior region. In addition, we assume discontinuous transmission conditions on the interface boundary. Our approach constitutes an extension to non-linear problems of the a priori error analysis developed recently for linear exterior transmission problems. Most of the techniques employed here are similar to the linear case but some differences appear. In particular, because of the non-homogeneous transmission conditions, the so-called numerical fluxes need to be suitably defined. We prove stability of the resulting discrete scheme with respect to a mesh-dependent norm and derive a Strang-type estimate for the associated error. Then, we apply local and global approximation properties of the discrete spaces to obtain the a priori error estimates in the energy norm.