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Improving the Ill-conditioning of the Method of Fundamental Solutions for 2D Laplace Equation

Improving the Ill-conditioning of the Method of Fundamental Solutions for 2D Laplace Equation,Chein-Shan Liu

Improving the Ill-conditioning of the Method of Fundamental Solutions for 2D Laplace Equation   (Citations: 9)
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The method of fundamental solu- tions (MFS) is a truly meshless numerical method widely used in the elliptic type boundary value problems, of which the approximate solution is expressed as a linear combination of fundamental solutions and the unknown coefficients are deter- mined from the boundary conditions by solving a linear equations system. However, the accuracy of MFS is severely limited by its ill-conditioning of the resulting linear equations system. This paper is motivated by the works of Chen, Wu, Lee and Chen (2007) and Liu (2007a). The first paper proved an equivalent relation of the Tre- fftz method and MFS for circular domain, while the second proposed a modified Trefftz method (MTM). We first prove an equivalent relation of MTM and MFS for arbitrary plane domain. Due to the well-posedness of MTM, we can alleviate the ill-conditioning of MFS through a new linear equations system of the modified MFS (MMFS). In doing so we can raise the accuracy of MMFS over four orders more than the original MFS. Nu- merical examples indicate that the MMFS can attain highly accurate numerical solutions with accuracy over the order of 10−10. The present method is fully not similar to the preconditioning technique as used to solve the ill-conditioned lin- ear equations system.
Published in 2009.
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