  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.
