Coupling of finite element and boundary element methods with regularization for a nonlinear interface problem with nonmonotone set-valued transmission conditions

TL;DR

This paper develops a novel numerical approach combining finite and boundary element methods with regularization to solve a complex nonlinear interface problem involving nonmonotone set-valued conditions, relevant to elastic media contact.

Contribution

It introduces a new coupling technique for FEM and BEM with regularization for hemivariational inequalities on unbounded domains.

Findings

Convergence of the approximation method is proven.

An asymptotic error estimate for the regularized problem is established.

The approach effectively handles nonmonotone set-valued transmission conditions.

Abstract

For the first time, a nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions is analyzed. The investigated problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models nonmonotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality, which lives on the unbounded domain, and so cannot be treated numerically in a direct way. By boundary integral methods the problem is transformed and a novel hemivariational inequality (HVI) is obtained that lives on the interior domain and on the coupling boundary, only. Thus for discretization the coupling of finite elements and boundary elements is the method of choice. In addition smoothing techniques of nondifferentiable optimization are adapted and the nonsmooth part in the HVI is regularized. Thus we reduce the original variational problem to a finite dimensional problem that can be solved by standard optimization tools. We establish not only convergence results for the total approximation procedure, but also an asymptotic error estimate for the regularized HVI.