Stable Mesh-Free Variational Radial Basis Function Approximation for Elliptic PDEs and Obstacle Problems

TL;DR

This paper explores radial basis function (RBF) methods for solving elliptic and obstacle boundary value problems, emphasizing accuracy, robustness, and efficiency through numerical experiments and stability analysis.

Contribution

It introduces a comprehensive study of RBF variational solvers, including techniques like TSVD to improve stability and detailed analysis of error trade-offs.

Findings

RBF variational solvers achieve high accuracy with lower computational cost.

Numerical experiments demonstrate fast error decay and stability improvements.

Trade-offs between approximation and truncation errors are characterized.

Abstract

We present a comprehensive study of radial basis function (RBF) approximations for elliptic and obstacle-type boundary value problems under a variational formulation. Our focus is on practical accuracy, robustness and efficiency. To address ill-conditioning in dense systems, we apply truncated singular value decomposition (TSVD) and investigate its effect on stability and accuracy trade-offs. Numerical experiments report benchmarks on accuracy and show fast error decay. We investigate the trade-off between approximation and truncation errors for practical settings for the number of basis functions, the oversampling ratio and the truncation threshold. In comparison with other methods, RBF variational solvers deliver high accuracy at similar or lower cost for boundary value problems.