$H^2$--Convergence of least-squares kernel collocation methods

TL;DR

This paper establishes $H^2$-convergence for least-squares kernel collocation methods solving second order elliptic PDEs, providing theoretical guarantees and analyzing the effects of different settings and kernels.

Contribution

It offers the first theoretical convergence analysis for least-squares kernel collocation methods, including conditions for $H^2$ convergence and optimal error rates.

Findings

Convergence in $H^2$ norm under certain conditions.

Optimal error behavior like $h^{m-2}$.

Impact of collocation settings and kernel choices on convergence.

Abstract

The strong-form asymmetric kernel-based collocation method, commonly referred to as the Kansa method, is easy to implement and hence is widely used for solving engineering problems and partial differential equations despite the lack of theoretical support. The simple least-squares (LS) formulation, on the other hand, makes the study of its solvability and convergence rather nontrivial. In this paper, we focus on general second order linear elliptic differential equations in $\Omega \subset R^d$ under Dirichlet boundary conditions. With kernels that reproduce $H^m(\Omega)$ and some smoothness assumptions on the solution, we provide denseness conditions for a constrained least-squares method and a class of weighted least-squares algorithms to be convergent. Theoretically, we identify some $H^2(\Omega)$ convergent LS formulations that have an optimal error behavior like $h^{m-2}$. We also demonstrate the effects of various collocation settings on the respective convergence rates, as well as how these formulations perform with high order kernels and when coupled with the stable evaluation technique for the Gaussian kernel.