[Show abstract][Hide abstract] ABSTRACT: The Poisson equation can be solved by first finding a particular solution and then solving the resulting Laplace equation. In this paper, a computational procedure based on the Trefftz method is developed to solve the Poisson equation for two-di-mensional domains. The radial basis function approach is used to find an approxi-mate particular solution for the Poisson equation. Then, two kinds of Trefftz methods, the T-Trefftz method and F-Trefftz method, are adopted to solve the resulting Laplace equation. In order to deal with the possible ill-posed behaviors existing in the Trefftz methods, the truncated singular value decomposition method and L-curve concept are both employed. The Poisson equation of the type, ∇ 2 u = f(x, u), in which x is the position and u is the dependent variable, is solved by the iterative procedure. Nu-merical examples are provided to show the validity of the proposed numerical meth-ods and some interesting phenomena are carefully discussed while solving the Helmholtz equation as a Poisson equation. It is concluded that the F-Trefftz method can deal with a multiply connected domain with genus p(p > 1) while the T-Trefftz method can only deal with a multiply connected domain with genus 1 if the domain partition technique is not adopted.
Journal of the Chinese Institute of Engineers. 01/2006; 29:989-1006.