Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points

✍ Scribed by Takemi Shigeta; D.L. Young

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 659 KB
• Volume: 228
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1016/j.jcp.2008.11.018

✦ Synopsis

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 the 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 the 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.