Performance comparison between the BEM and the MFS solving Laplace’s problems

This paper compares the numerical performances of the boundary element method (BEM) and the method of fundamental solutions (MFS) for solving problems governed by the Laplace equation in two dimensions. They are similar methods since both use the idea of a Green fundamental solution, but they also have some essential distinctions. BEM is currently a powerful, well-known numerical technique applied to important engineering fields. BEM solves differential equations using an integral formulation that results in boundary integrations. The MFS is a meshless technique that works directly with discrete points. Nowadays, the MFS is experiencing a rediscovery due to the intensification of research on meshless techniques. It has also been used as an auxiliary tool for other methods, including BEM, modelling intermediary steps in broader, more complex discrete cases, for example, calculating fluxes and stresses from knowledge of primal variables. Aiming to understand these relevant techniques better, this article compares some characteristics of the two methods, analysing each technique’s accuracy and robustness and giving special attention to MFS particularities for solving Laplace's problems. This approach is suitable since the Laplacian is one the most sensitive differential operators within the scalar field theory, generating ill-conditioning in the resulting discrete matrix system.