Boundary element methods: foundation and error analysis G. C. Hsiao 1 and W. L. Wendland 2 1 Department of Mathematical Sciences, University of Delaware Newark, Delaware 19716-2533, USA 2 Institut f¨ ur Angewandte Analysis und numerische Simulation, Universit¨at Stuttgart D-70569 Stuttgart, Germany ABSTRACT This chapter gives an expository introduction to the Galerkin-BEM for the elliptic boundary value problems from the mathematical point of view. Emphasis will be placed upon the variational formulations of the boundary integral equations and the general error estimates for the approximate solution in appropriate Sobolev spaces. A classification of boundary integral equations will be given based on the Sobolev index. The simple relations between the variational formulations of the boundary integral equations and the corresponding partial differential equations under consideration will be indicated. Basic concepts such as stability, consistency, convergence as well as the condition numbers, ill–posedness will be discussed by using elementary examples. key words: Boundary integral equations; fundamental solutions; variational formulations; G˚ arding’s inequality; Galerkin’s method; boundary elements; stability; ill–posedness and asymptotic error estimates 1. INTRODUCTION In essence, the boundary element method (BEM) may be considered as application of finite element method (FEM), designed originally for the numerical solutions of partial differential equations in the domains, to the boundary integral equations on closed boundary manifolds. The terminology of BEM originated from the practice of discretizing the boundary manifold of the solution domain for the boundary integral equation into boundary elements, resembling the term of finite elements in FEM. As in FEM, in the literature, the use of the terminology boundary element is in two different contexts; the boundary manifolds are decomposed into boundary elements which are geometric objects, while the boundary elements for approximating solutions of boundary integral equations are actually the finite element functions defined on the boundaries. Looking through the literature, it is difficult to trace back one fundamental research paper and individuals who were responsible for the historical development of the BEM. However, from the computational point of view, the work by Hess and Smith deserves mentioning as one of the corner stones work. In their 1966 paper (Hess and Smith (1966)), boundary elements (or surface elements rather) have been used to approximate various types of bodies and to calculate potential flow about arbitrary bodies. On the other Encyclopedia of Computational Mechanics. Edited by Erwin Stein, Ren´ e de Borst and Thomas J.R. Hughes. c  2004 John Wiley & Sons, Ltd.