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

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 equa- tion, which is an exact boundary condition. By truncating the Fourier series expansion one can match the physical boundary condition as accu- rate as one desired. Then, we use the colloca- tion 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 ex- plore why the conventional Trefftz method is fail- ure and the modified onestill survives. Numerical examples with smooth boundaries reveal that the present method can offer very accurate numeri- cal results with absoluteerrors about in the orders from 10 −10 to 10 −16 . The new method is pow- erful even for problems with complex boundary shapes, with discontinuous boundary conditions or with corners on boundary. Keyword: Laplace equation, Artificial bound- ary condition, Modified Trefftz method, Char- acteristic length, Collocation method, Galerkin method, DtD mapping