Multiscale methods with compactly supported radial basis functions for the Stokes problem on bounded domains

TL;DR

This paper explores a multiscale collocation approach using compactly supported radial basis functions to efficiently approximate solutions to the Stokes problem on bounded domains, with theoretical guarantees and numerical validation.

Contribution

It introduces a multilevel RBF collocation method with convergence and stability analysis for the Stokes problem, advancing numerical techniques for fluid dynamics.

Findings

Convergence conditions are established for the multilevel RBF method.

Numerical experiments confirm the theoretical stability and accuracy.

The method effectively approximates solutions to the Stokes problem.

Abstract

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.