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

Chein-Shan Liu1
Department of Mechanical & Mechatronic Engineering, Department of Harbor & River Engineering, Taiwan Ocean University, Keelung, Taiwan. E-mail:

Computer Modeling in Engineering & Sciences 2008, 28(2), 77-94.


The method of fundamental solutions (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 determined 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 Trefftz 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. Numerical 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 linear equations system.


Laplace equation, Modified Trefftz method, Collocation method, Method of fundamental solutions, Modified MFS (MMFS)

Cite This Article

Liu, C. (2008). Improving the Ill-conditioning of the Method of Fundamental Solutions for 2D Laplace Equation. CMES-Computer Modeling in Engineering & Sciences, 28(2), 77–94.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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