Method of Fundamental Solutions with Optimal Regularization Techniques for the Cauchy Problem of the Laplace Equation with Singular Points

TL;DR

This paper introduces an enhanced method combining the method of fundamental solutions with optimal regularization to accurately and efficiently solve the Cauchy problem of the Laplace equation, even with singular points and noisy data.

Contribution

It proposes a novel combination of MFS and Tikhonov regularization with optimal parameter selection for high-accuracy solutions of the Laplace Cauchy problem involving singularities.

Findings

Optimal regularization parameter coincides with the L-curve method.

The method effectively handles singular points and irregular domains.

Numerical results demonstrate high accuracy with noisy Cauchy data.

Abstract

The purpose of this study is to propose a high-accuracy and fast numerical method for the Cauchy problem of the Laplace equation. Our problem is directly discretized by the method of fundamental solutions (MFS). The Tikhonov regularization method stabilizes a numerical solution of the problem for given Cauchy data with high noises. The accuracy of the numerical solution depends on a regularization parameter of the Tikhonov regularization technique and some parameters of MFS. The L-curve determines a suitable regularization parameter for obtaining an accurate solution. Numerical experiments show that such a suitable regularization parameter coincides with the optimal one. Moreover, a better choice of the parameters of MFS is numerically observed. It is noteworthy that a problem whose solution has singular points can successfully be solved. It is concluded that the numerical method proposed in this paper is effective for a problem with an irregular domain, singular points, and the Cauchy data with high noises.