Trefftz solution of boundary-value problem with Poisson equation (in case of non-homogeneous term including derivatives of unknown function)

摘要

This paper describes the application of Trefftz method to the boundary value problem governed with the two-dimensional Poisson equation. Trefftz method is formulated with regular Trefftz functions (T-complete functions) satisfying the governing equation. When Trefftz method is applied to the Poisson equation, it is generally difficult to derive the Trefftz functions for the problem. For overcoming this difficulty, this paper presents the following scheme. A non-homogeneous term containing an unknown function and it derivatives are approximated by the polynomial in the Cartesian coordinates and then, the solution for the Poisson equation is approximated with the superposition of the Trefftz function of the Laplace equation and the particular solutions related to the approximate polynomal. Unknown parameters included in the approximate solution are determined so that the solution satisfies the boundary conditions. The present scheme is applied to some examples in order to study the numerical properties.