Stability analysis of method of fundamental solutions for mixed boundary value problems of Laplace's equation (Q975313)

Since the stability of the method of fundamental solutions (MFS) is a severe issue, the estimation on the bounds of the condition number \textit{Cond} is important to real application. In the present article, the authors propose new approaches for deriving the asymptotes of \textit{Cond}, and apply them for the Dirichlet problem of Laplace's equation, to provide the sharp bound of \textit{Cond} for disk domains. The new bound of \textit{Cond} is derived for bounded simply connected domains with mixed types of boundary conditions. Numerical results are reported for Motz's problem by adding singular functions. The values of \textit{Cond} grow exponentially with respect to the number of fundamental solutions used. There seems to exist no stability analysis for the MFS on non-disk (or non-elliptic) domains. Futhermore, the expansion coefficients obtained by the MFS are oscillatingly large, to cause the other kind of instability: subtraction cancelation errors in the final harmonic solutions.