Multiscale methods with compactly supported radial basis functions for Galerkin approximation of elliptic PDEs

The aim of this work is to consider multiscale algorithms for solving partial differential equations (PDEs) with Galerkin methods on bounded domains. We provide results on convergence and condition numbers. We show how to handle PDEs with Dirichlet boundary conditions. We also investigate convergence in terms of the mesh norms and the angles between subspaces to better understand the differences between the algorithms and the observed results. We also consider the issue of the supports of the radial basis funtions overlapping the boundary in our stability analysis, which has not been considered in the literature to the best of our knowledge.