Andrew Chernih

Andrew Chernih, Quoc Thong Le Gia

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.

Andrew Chernih, Quoc Thong Le Gia

In this paper, we investigate the application of radial basis functions (RBFs) for the approximation with collocation of the Stokes problem. The approximate solution is constructed in a multi-level fashion, each level using compactly supported radial basis functions with decreasing scaling factors. We use symmetric collocation and give sufficient conditions for convergence and stability analysis is also presented. Numerical experiments support the theoretical results.