Some Regularized Versions of the Method of Fundamental Solutions
A powerful method of the solution of homogeneous equations is considered. Using the traditional approach of the Method of Fundamental Solutions, the fundamental solution has to be shifted to external source points. This is inconvenient from computational point of view, moreover, the resulting linear system can easily become severely ill-conditioned. To overcome this difficulty, a special regularization technique is applied. In this approach, the original second-order elliptic problem (a modified Helmholtz problem in the paper) is approximated by a fourth-order multi-elliptic boundary interpolation problem. To perform this boundary interpolation, either the Method of Fundamental Solutions, or a direct multi-elliptic interpolation can be used. In the paper, a priori error estimations are deduced. A numerical example is also presented.
KeywordsMethod of fundamental solutions Multi-elliptic interpolation Regularization Radial basis functions Quadtree Multigrid
The research was partly supported by the European Union (co-financed by the European Social Fund) under the project TÁMOP 4.2.1/B-09/1/KMR-2010-0003.
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