The method of fundamental solutions for Poisson's equation

We show how to extend the method of fundamental solutions (MFS) to solve Poisson's equation in R(2) and R(2) without boundary or domain discretization. To do this an approximate particular solution is found by approximating the right hand side by thin plate splines. The particular solution is then subtracted from the complete solution and Laplace's equation is solved by the usual MFS. Numerical results are obtained for a number of standard boundary value problems with 3-4 figure accuracy attainable by solving fewer than 20 linear equations.