COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol. 4 (2004), No. 1, pp. 3–22 c  2004 Editorial Board of the Journal ”Computational Methods in Applied Mathematics” Joint Co. Ltd. OVERLAPPED BEM–FEM AND SOME SCHWARZ ITERATIONS 1 RICARDO CELORRIO Dep. Matem´ atica Aplicada, Universidad de Zaragoza, EUITIZ, 50018 Zaragoza, Spain V ´ ICTOR DOM ´ INGUEZ Dep. Matem´ atica e Inform´ atica, Universidad P´ ublica de Navarra Campus de Arrosad´ ıa s/n, 31.006 Pamplona, Spain FRANCISCO-JAVIER SAYAS Dep. Matem´ atica Aplicada, Universidad de Zaragoza C.P.S., 50018 Zaragoza, Spain Abstract — In this work we consider the numerical solution of the Laplace’s equation in a domain with holes by means of the overlapping of finite and boundary elements. The essence of the method is the consideration of the finite element solution of the Laplace’s equation in the domain without holes and the exterior single–layer solution on the unbounded domain around these holes. This solution can be viewed as a limit of a discretized interior–exterior Schwarz–type iteration. A convergence analysis of both the iteration and the discrete solution is carried out, taking full generality in the BEM scheme. Some numerical experiments are also given. 2000 Mathematics Subject Classification: 65N30, 65N38, 65F10. Keywords: finite elements, boundary elements, Schwarz method. 1. Introduction The present work is concerned with the numerical solution of a model problem, namely Laplace’s equation with Dirichlet boundary conditions, on a domain with holes (henceforth referred to as obstacles), in two or three dimensions. The basic idea is to take out the interior obstacles and make a triangulation of the simpler remaining domain using finite elements on it. At the same time we will consider problems exterior to the set of boundaries of the obstacles and use boundary elements based on an indirect single layer potential formulation for this kind of problem. Then we iterate between both problems, interchanging traces, i.e., the finite element solutions generate traces on the boundaries of the obstacles and the boundary element solution is plugged into the single layer potential to obtain values on the outer boundary. The iteration is then carried out till the process becomes stationary. 1 Research partially supported by MCYT Project BFM2001–2521 and MAT2002-04153