FEM and BEM Coupling for a Nonlinear Transmission Problem with Signorini Contact

This paper is concerned with the interface problem consisting of a nonlinear uniformly monotone partial differential equation in some bounded Lipschitz domain $\Omega$ in $\Reals^n$ ($n=2$ or $n \geq 3$) and the Laplace equation with some radiation condition in the unbounded exterior domain $\Omega_c := \Reals^n\setminus\overline{\Omega}$. The two problems are coupled by transmission conditions and Signorini contact conditions on the interface $\Gamma=\partial \Omega$. The exterior part of the interface problem is rewritten in terms of boundary integral operators. This leads to a variational inequality with a nonlinear monotone operator. By this approach, existence and uniqueness of a solution in appropriate Sobolev spaces are obtained. Its approximation is performed by coupling the finite element method (FEM) (in $\Omega$) and the boundary element method (BEM) (the latter living on the interface $\Gamma$), yielding a discrete monotone variational inequality. We present an abstract Cea-type error estimate and derive asymptotic error estimates.