Convergence order estimates of meshless collocation methods using radial basis functions

We study meshless collocation methods using radial basis functions to approximate regular solutions of systems of equations with linear differential or integral operators. Our method can be interpreted as one of the emerging meshless methods, cf. T. Belytschko et al. (1996). Its range of application is not confined to elliptic problems. However, the application to the boundary value problem for an elliptic operator, connected with an integral equation, is given as an example. Although the method has been used for special cases for about ten years, cf E.J. Kansa (1990), there are no error bounds known. We put the main emphasis on detailed proofs of such error bounds, following the general outline described in C. Franke and R. Schaback (preprint).