RBF Multiscale Collocation for Second Order Elliptic Boundary Value Problems

In this talk, we discuss multiscale radial basis function collocation methods for solving elliptic partial differential equations on bounded domains. The approximate solution is constructed in a multi-level fashion, each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh. On each level, standard symmetric collocation is employed. A convergence theory is given, which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions. If time permits, we also discuss the condition numbers of the arising systems as well as the effect of simple, diagonal preconditioners. In particular, we present results that prove previous numerical observations made by Fasshauer [1]. More than a decade ago, he observed: