A method of solving an exterior three-dimensional boundary value problem for the Laplace equation

We develop and experimentally study the algorithms for solving three-dimensional boundary value problems for the Laplace equation in unbounded domains. The algoriths combinef the finite element method and the integral representation of the solution in homogeneous media. The proposed approach is based on the Schwarz alternating method and the consecutive solution of the interior and exterior boundary value problems in subdomains with intersection such that some iterable interface conditions are imposed on the adjacent boundaries. The convergence of the method is proved. The convergence rate of the iterative process is studied analytically in the case that the subdomains are spherical layers with known exact representations of all consecutive approximations. In this model situation, the impact is analyzed of the parameters of the algorithm on the efficiency of the method. The above approach is implemented for solving a problem with a complicated configuration of the boundary. Also, the algorithbm uses high precision finite element methods for solving the interior boundary problems. The convergence rate of the iterations and the achieved accuracy of the computations are illustrated by a series of numerical experiments.