TY - EJOU
AU - Liu, Chein-Shan
TI - A Modified Trefftz Method for Two-Dimensional Laplace Equation Considering the Domain's Characteristic Length
T2 - Computer Modeling in Engineering \& Sciences
PY - 2007
VL - 21
IS - 1
SN - 1526-1506
AB - 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.
KW - Laplace equation
KW - Artificial boundary condition
KW - Modified Trefftz method
KW - Characteristic length
KW - Collocation method
KW - Galerkin method
KW - DtD mapping
DO - 10.3970/cmes.2007.021.053