Maximizing the projection/minimizing the mass gap to choose optimal source points in the MFS for 2D and 3D Laplace equations

Abstract

Two novel methods are developed to generate the optimal method of fundamental solutions (MFS), of which the offset to locate source points from the domain's boundary is optimized. First the maximal projection method (MPM), together with a new idea of a simple substitution function being inserted into the derived merit function, can choose a better offset; then a better numerical solution is achieved. On the other hand, we derive a gap functional in terms of the conservation of mass. When the mass gap is minimized, the optimal source points can be chosen to improve the accuracy, which is labeled a mass gap method (MGM); it is applicable to solve the mixed type and the Cauchy type boundary value problems, but not the Dirichlet problem. When the optimal source points in the MFS are obtained, both the MPM and MGM can efficiently solve the 2D and 3D Laplace equations with good accuracy.