Desingularized meshless method for solving Laplace equation with over-specified boundary conditions using regularization techniques

Abstract

The desingularized meshless method (DMM) has been successfully used to solve boundary-value problems with specified boundary conditions (a direct problem) numerically. In this paper, the DMM is applied to deal with the problems with over-specified boundary conditions. The accompanied ill-posed problem in the inverse problem is remedied by using the Tikhonov regularization method and the truncated singular value decomposition method. The numerical evidences are given to verify the accuracy of the solutions after comparing with the results of analytical solutions through several numerical examples. The comparisons of results using Tikhonov method and truncated singular value decomposition method are also discussed in the examples.

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.…”

Boundary element methods for boundary condition inverse problems in elasticity using PCGM and CGM regularization

“…The desingularized meshless method (DMM) was used to solve boundary-value problems with specified boundary conditions. The accompanied ill-posed problem in the inverse problem was remedied by using the Tikhonov regularization method and the truncated singular value decomposition method [30]. Singh and Tanaka presented the PBCG and PCG algorithms in conjunction with the dual reciprocity boundary element method for solution of time-dependent inverse heat conduction problems [31].…”

The PCGM for Cauchy inverse problems in 3D potential

“…The desingularized meshless method was used to solve boundary-value problems with specified boundary conditions. The accompanied ill-posed problem in the inverse problem was remedied by using the Tikhonov regularization method and the truncated singular value decomposition method (TSVD) [11]. Mera et al [12] proposed an iterative boundary element for singular Cauchy problems in anisotropic heat conduction with an abrupt change in the boundary conditions and with a sharp re-entrant corner.…”