Convergence, divergence, and inherent oscillations in MAS solutions of 2D Laplace-Neumann problems

The method of auxiliary sources (MAS), also known as the method of fundamental solutions (MFS), is a well-known computational method for the solution of boundary-value problems. The final solution (“MAS solution”) is obtained once we have found the amplitudes of N auxiliary “MAS sources.” Past studies have shown that it is possible for the MAS solution to converge to the true solution even when the N auxiliary sources diverge and oscillate. Here, we extend the past studies by demonstrating this possibility within the context of Laplace's equation with Neumann boundary conditions. The correct solution can thus be obtained from sources that, when N is large, must be considered unphysical. We carefully explain the underlying reasons for the unphysical results, distinguish from other difficulties that might concurrently arise, and point to significant differences with time-dependent problems studied in the past.