A Meshfree Method with Overlapping Domain Decomposition for Elliptic Boundary Value Problems

To appear in AIP Conf. Proc. of ICNAAM 2023


The method of fundamental solutions (MFS) is a meshfree numerical method that may be used for the approximatesolution of elliptic boundary value problems (BVPs). In its classical formulation, the MFS requires the solution of an ill-conditioned collocation linear system, which is usually regarded as one of the drawbacks of this method. When large-scale problems, posed in domains with a complex geometry, are considered, calculating an accurate solution of the MFS linear system becomes a real challenge which may aect significantly the precision of the BVP solution. A possible approach to improve the accuracy of the MFS consists in splitting the original BVP into a set of smaller scale sub-problems using a domain decomposition technique. In this work, we develop an iterative numerical scheme by coupling the MFS with the alternating Schwarz method. The accuracy of the proposed method is illustrated in the case of a BVP for the Laplace equation.