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

TL;DR

This paper develops multiscale Galerkin algorithms using compactly supported radial basis functions for elliptic PDEs, analyzing convergence, stability, and boundary support issues to improve numerical solutions.

Contribution

It introduces new stability analysis considering boundary support overlaps and compares multiscale algorithms in terms of convergence and condition numbers.

Findings

Convergence depends on mesh norms and subspace angles.

Handling boundary support overlaps improves stability.

Algorithms show favorable condition numbers for elliptic PDEs.

Abstract

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.