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

The finite element method and the boundary element method areamong the most frequently applied tools in the numerical treatmentof partial differential equations. However, their propertiesappear to be complementary: while the boundary element methodis appropriate for the most important linear partial differentialequations with constant coefficients in bounded or unboundeddomains, the finite element method seems to be more appropriatefor inhomogeneous or even nonlinear problems. but is somehowrestricted to bounded domains. The symmetric coupling of thetwo methods inherits the advantages of both methods. This paper treats the symmetric coupling of finite elementsand boundary elements for a model transmission problem in twoand three dimensions where we have two domains: a bounded domainwith nonlinear, even plastic material behaviour, is surroundedby an unbounded, exterior, domain with isotropic homogeneouslinear elastic material. Practically. the coupling is performedsuch that the boundary element method contributes a macro-element,like a large finite element, within a standard finite elementanalysis program. Emphasis is on two-dimensional problems wherethe approach using the Poincaré-Steklov operator seemsto be impossible at first glance. E-mail: cc{at}numerik.uni-kiel.de