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

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.