# FE-/BE-Coupling for Strongly Nonlinear Transmission Problems with Contact

Heiko Gimperlein^{*}, Matthias Maischak and Ernst P. Stephan

We analyze an adaptive finite element/boundary element formulation for nonlinear transmission and contact problems. In the model problem, the p-Laplacian or a double-well potential in a bounded domain $\Omega$ is coupled to the homogeneous Laplace equation in $\mathbb{R}^n\setminus \overline{\Omega}$. The exterior problem is reduced to an integral equation on $\partial \Omega$, and an equivalent coupled boundary/domain variational inequality is solved. We discuss the well-posedness in suitable $L^2$--$L^p$ Sobolev spaces, the appearance of microstructure in the nonconvex case and a priori/a posteriori error estimates for Galerkin approximations. Extensions to strongly anisotropic coupling problems will be considered.

Mathematics Subject Classification: 65N30 65N38 65K15 74M10

Keywords: transmission problem; variational inequality; finite element method; boundary element method

Minisymposion: Nonsmooth and Unilateral Problems - Modelling, Analysis and Optimization Methods