Combinations of method of fundamental solutions for Laplace's equation with singularities

Since the fundamental solutions (FS) are smooth, the method of FS (MFS) is poor in accuracy for singularity problems, except for some techniques, such as local refinements or adding singular particular solutions (SPS). To better deal with singularity problems, in this paper, we combine the FS and the SPS in the Trefftz methods (TM), explore efficient coupling techniques, and derive error bounds. Moreover, we also combine the MFS with the popular finite element method (FEM), as error bounds are provided. Numerical experiments are reported for Motz's problem by the TM with both FS and SPS. It is due to the algorithms and the error analysis in this paper that the MFS can be combined with other numerical methods, to greatly extend its application for engineering problems.