FE-BE coupling for a transmission problem involving microstructure

TL;DR

This paper presents a finite element/boundary element coupling method for solving a non-convex contact problem with microstructure, focusing on convergence analysis and error estimation of the numerical solution.

Contribution

It introduces a novel coupling approach for a complex non-convex problem, including a discretized saddle point formulation and convergence results.

Findings

Convergence of Galerkin approximations to macroscopic quantities

Development of an a posteriori error estimate

Numerical solution of a boundary/domain variational inequality

Abstract

We analyze a finite element/boundary element procedure to solve a non-convex contact problem for the double-well potential. After relaxing the associated functional, the degenerate minimization problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which may then be solved numerically. The convergence of the Galerkin approximations to certain macroscopic quantities and a corresponding a posteriori estimate for the approximation error are discussed.