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

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

2008-03-13

arxiv.org/abs/0803.1905v1

Mathematics

Numerical Analysis

20 pages, 15 figures, 1 table

Scientific paper

10.1016/j.jcp.2008.11.018

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.

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