Stability analysis of method of fundamental solutions for mixed boundary value problems of Laplace's equation
Since the stability of the method of fundamental solutions (MFS) is a severe issue, the estimation on the bounds of condition number Cond is important to real application. In this paper, we propose the new approaches for deriving the asymptotes of Cond, and apply them for the Dirichlet problem of Laplace's equation, to provide the sharp bound of Cond for disk domains. Then the new bound of 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 Cond grow exponentially with respect to the number of fundamental solutions used. Note that there seems to exist no stability analysis for the MFS on non-disk (or non-elliptic) domains. Moreover, 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.
stability analysis method of fundamental solutions laplace's equation condition number motz's problem effective condition number mixed boundary problem