Optimal Order Multigrid Methods for Solving Exterior Boundary Value Problems

Abstract

The coupling of boundary elements and finite elements combines the advantage of boundary elements for treating domains extended to infinity and that of finite elements in treating the nonhomogeneity of equations and the complexity of domains. In the case of the Laplacian, by taking a circle or a sphere as the artificial coupling boundary, it is shown that the corresponding boundary integral equation can be solved without any cost and the coupled system is reduced to a simple finite element system. Two multigrid methods are proposed to solve this finite element linear system. Both methods are of optimal order and can be used to solve such finite element equations as efficiently as to solve those arising from interior boundary value problems. Numerical experiments are included to show the efficiency and advantages of the methods. An apparent significance of the methods is that the boundary elements appear neither in the discretization nor in the coding.