The method of fundamental solutions (MFS), also known as the method of auxiliary sources (MAS), 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 demonstrated that it is possible for the MAS solution to converge to the true solution even when the $N$ auxiliary sources diverge and oscillate. The present paper extends the past studies by demonstrating this possibility within the context of Laplace's equation with Neumann boundary conditions. One can thus obtain the correct solution 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 that were studied in the past.

翻译：基本解方法（MFS），亦称辅助源方法（MAS），是一种求解边值问题的著名计算方法。一旦求得 $N$ 个辅助“MAS源”的强度，即可获得最终解（“MAS解”）。以往研究表明，即使这 $N$ 个辅助源发散并振荡，MAS解仍有可能收敛到真实解。本文通过将这一可能性置于具有诺伊曼边界条件的拉普拉斯方程背景下进行探讨，从而拓展了先前的研究。由此，我们可以从那些在 $N$ 较大时必须被视为非物理的源中，得到正确的解。我们仔细解释了产生非物理结果的深层原因，将其与可能同时出现的其他困难区分开来，并指出了与以往研究的时间依赖问题之间的显著差异。