A Localized Version of the Method of Fundamental Solutions in a Multi-level Context

The Method of Fundamental Solutions is applied to the Laplace equation. Instead of using the traditional approach with external source points and boundary collocation points, the original domain decomposed into a lot of smaller, overlapping subdomains, and the Method of Fundamental Solutions is used to the individual local subdomains. After eliminating the local source points, local schemes are obtained. Instead of constructing a global scheme, the local subproblems are solved sequentially, in an iterative way. This mimics a multiplicative Schwarz method with overlapping subdomains, which assures the convergence of the method. Combining the iteration with a simple Seidel-type method, the resulting iteration is used as a smoothing procedure of a multi-level method. The points belonging to the coarse and fine levels are defined by a quadtree-generated cell system controlled by the boundary of the original domain. The multi-level character of the obtained method makes it possible to reduce the necessary number of iterations, that is, the overall computational cost can be significantly reduced. Moreover, the solution of large and ill-conditioned systems is completely avoided. The method is illustrated through several numerical test examples.