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arXiv · 0803.1905

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

Takemi Shigeta·D. L. Young
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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.

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Takemi Shigeta, D. L. Young. 2008-03-13. Method of Fundamental Solutions with Optimal Regularization Techniques for the Cauchy Problem of the Laplace Equation with Singular Points. https://doi.org/10.1016/j.jcp.2008.11.018

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