Trefftz solution for boundary value problem of three-dimensional Poisson equation

This paper describes the application of the Trefftz method to the solution of three-dimensional Poisson equation. An inhomogeneous term containing the unknown function is approximated with a polynomial function in the Cartesian coordinates to determine the particular solution for the Poisson equation. The solution of the problem is approximated with the superposition of the Trefftz functions of the Laplace equation and the derived particular solution. 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.