A Numerical Study of some Radial Basis Function based Solution Methods for Elliptic PDEs

During the last decade, three main variations have been proposed for solving elliptic PDEs by means of collocation with radial basis functions (RBFs). In this study, we have implemented them for infinitely smooth RBFs, and then compared them across the full range of values for the shape parameter of the RBFs. This was made possible by a recently discovered numerical procedure that bypasses the ill-conditioning, which has previously limited the range that could be used for this parameter. We find that the best values for it often fall outside the range that was previously available. We have also looked at piecewise smooth versus infinitely smooth RBFs, and found that for PDE applications with smooth solutions, the infinitely smooth RBFs are preferable, mainly because they lead to higher accuracy. In a comparison of RBF-based methods against two standard techniques (a second-order finite difference method and a pseudospectral method), the former gave a much superior accuracy.