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

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.

On numerical experiments for Cauchy problems of elliptic operators

“…In [9]: u à ¼ xy, In [16]: u à ¼ expð0:5xÞsinð0:5yÞ and u à ¼ x þ y, In [17]: u à ¼ x 3 À3xy 2 þ expð2yÞsinð2xÞÀexpðxÞcosðyÞ, In [19]: u à ¼ 10yÀ9, and others in [1][2][3][4][13][14][15], the tested Cauchy problems with globally harmonic solutions may also have numerically redundant Cauchy data.…”

The determination temperature-dependent thermal conductivity as inverse steady heat conduction problem

“…As of right now, the MFS was applied in inverse heat conduction problems involving the identification of heat sources [49][50][51][52], boundary heat flux [53][54][55] or Cauchy problem [56][57][58][59]. To the best knowledge of the authors, this paper is a first application of this method to the inverse heat conduction concerned with the identification of thermal conductivity coefficient.…”

Inverse source identification by Green's function

“…The truly meshless computational advantage of the MFS has attracted the attentions of many authors. Recent works on using the MFS for solving inverse problems can be found in literatures such as inverse heat conduction [34,42,43,56]; cavity reconstruction [5]; eigenfrequencies and eigenmodes [3]; elasticity and elastrostatics [40,45]; free boundary determination [27]; and Cauchy problems for Laplace [44,53,59,60], Helmholtz [33,41,46], and biharmonic [47] operators. It is well known that the placement of source points in the MFS plays an important role in the accuracy of the method.…”