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# A Modified Trefftz Method for Two-Dimensional Laplace Equation Considering the Domain's Characteristic Length

Chein-Shan Liu1
Department of Mechanical and Mechatronic Engineering, Taiwan Ocean University, Keelung, Taiwan. E-mail: csliu@mail.ntou.edu.tw

Computer Modeling in Engineering & Sciences 2007, 21(1), 53-66. https://doi.org/10.3970/cmes.2007.021.053

### Abstract

A newly modified Trefftz method is developed to solve the exterior and interior Dirichlet problems for two-dimensional Laplace equation, which takes the characteristic length of problem domain into account. After introducing a circular artificial boundary which is uniquely determined by the physical problem domain, we can derive a Dirichlet to Dirichlet mapping equation, which is an exact boundary condition. By truncating the Fourier series expansion one can match the physical boundary condition as accurate as one desired. Then, we use the collocation method and the Galerkin method to derive linear equations system to determine the Fourier coefficients. Here, the factor of characteristic length ensures that the modified Trefftz method is stable. We use a numerical example to explore why the conventional Trefftz method is failure and the modified one still survives. Numerical examples with smooth boundaries reveal that the present method can offer very accurate numerical results with absolute errors about in the orders from 10−10 to 10−16. The new method is powerful even for problems with complex boundary shapes, with discontinuous boundary conditions or with corners on boundary.

### Keywords

Laplace equation, Artificial boundary condition, Modified Trefftz method, Characteristic length, Collocation method, Galerkin method, DtD mapping

Liu, C. (2007). A Modified Trefftz Method for Two-Dimensional Laplace Equation Considering the Domain's Characteristic Length. CMES-Computer Modeling in Engineering & Sciences, 21(1), 53–66. 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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