On the equivalence of MFS and Trefftz method for Laplace problems

In this paper, it is proved that the two approaches for Laplace problems, known in the literature as the method of fundamental solutions (MFS) and the Trefftz method, are mathematically equivalent in spite of their essentially minor and apparent differences in the formulation. It is interesting to find that the T-complete set in the Trefftz method for the interior and exterior problems are imbedded in the degenerate kernels of MFS. By designing circular-domain and circular-hole problems, the unknown coefficients of each method correlate by a mapping matrix after considering the degenerate kernels for the fundamental solutions in the MFS and the T-complete function in the Trefftz method. The mapping matrix is composed of a rotation matrix and a geometric matrix depends on the source location. The degenerate scale for the Laplace equation appears using the MFS when the geometric matrix is singular. The occurring mechanism of the degenerate scale in the MFS is studied by using circulants. The ill-posed problem in the MFS also stems from the ill-conditioned geometric matrix when the source is distributed far away from the real boundary. Several examples, interior and exterior problems with either simply- or doubly-connected domain, were solved by using the Trefftz method and the MFS. The comparison of efficiency between the two methods was addressed