Solving Partial Differential Equations with Multiscale Radial Basis Functions

The goal of this paper is to review, discuss and extend the current theory on multiscale radial basis functions for solving elliptic partial differential equations. Multiscale radial basis functions provide approximation spaces using different scales and shifts of a compactly supported, positive definite function in an orderly fashion. In this paper, both collocation and Galerkin approximation are described and analysed. To this end, first symmetric and non-symmetric recovery is discussed. Then, error estimates for both schemes are derived, though special emphasis is given to Galerkin approximation, since the current situation here is not as clear as in the case of collocation. We will distinguish between stationary and non-stationary multiscale approximation spaces and multilevel approximation schemes. For Galerkin approximation, we will establish error bounds in the stationary setting based upon Cea’s lemma showing that the approximation spaces are indeed rich enough. Unfortunately, convergence of a simple residual correction algorithm, which is often applied in this context to compute the approximation, can only be shown for a non-stationary multiscale approximation space.