  The aim of this work is to consider multiscale algorithms for solving 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 RBFs overlapping the boundary in our stability analysis, which has not
been considered in the literature, to the best of our knowledge.
