On the symmetric boundary element method and the symmetric coupling of boundary elements and finite elements

Abstract

The finite element method and the boundary element method are among the most frequently applied tools in the numerical treatment of partial differential equations. However. their properties appear to be complementary: while the boundary element method is appropriate for the most important linear partial differential equations with constant coefficients in bounded or unbounded domains, the finite element method seems to be more appropriate for inhomogeneous or even nonlinear problems, but is somehow restricted to bounded domains. The symmetric coupling of the two methods inherits the advantages of both methods. This paper treats the symmetric coupling of finite elements and boundary elements for a model transmission problem in two and three dimensions where we have two domains: a bounded domain with nonlinear, even plastic material behaviour, is surrounded by an unbounded, exterior, domain with isotropic homogeneous linear elastic material. Practically, the coupling is performed such that the boundary element method contributes a macro-element, like a large finite element, within a standard finite element analysis program. Emphasis is on two-dimensional problems where the approach using the Poincare-Steklov operator seems to be impossible at first glance.