Stable MFS Solution to Singular Direct and Inverse Problems Associated with the Laplace Equation Subjected to Noisy Data

Abstract

In this paper, a meshless method for the stable solution of direct and inverse problemsassociatedwiththe two-dimensionalLaplace equationinthe pres- ence of boundarysingularitiesand noisyboundarydata is proposed. Thegoverning equation and boundary conditions are discretized by the method of fundamental solutions (MFS), whilst the existence of the boundary singularity is taken into ac- count by subtracting from the original MFS solution the corresponding singular solutions, as given by the asymptotic expansion of the solution near the singular point. However, even in the case when the boundary singularity is accounted for, the numerical solutions obtained by the direct inversion of the associated MFS lin- ear algebraic system are still inaccurate and unstable. Therefore, the regularization of the aforementioned problems is required and this is realized by employing either the Tikhonov regularization method (TRM), or the singular value decomposition (SVD), with the corresponding optimal regularization parameter given by the L- curve method. Numerical experiments show that the proposed method is stable with respect to the noise added into the boundary data, highly accurate and compu- tationally very efficient.