In recent years radial basis function collocation has become a useful alternative to finite difference and finite element methods for solving elliptic partial differential equations. RBF collocation methods have been shown numerically (see for example [17]) and theoretically (see [14, 13]) to be very accurate even for a small number of collocation points. In application finite difference methods often have a low approximation order and consequently can require a large grid and considerable computation to obtain a sufficiently accurate solution. RBF collocation has been applied to linear elliptic PDEs in R ² and R ³ [18], to time dependent problems [15, 16], and to non-linear problems [10].