$H^2$-Convergence of Least-Squares Kernel Collocation Methods

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 $\Omega \subset \mathbb{R}^d$ under Dirichlet boundary conditions. With kernels that reproduce $H^m(\Omega)$ and some smoothness assumptions on the solution, we provide conditions for a constrained LS method and a class of weighted LS algorithms to be convergent. Theoretically, for ${\max(2,\,\lceil (d+1)/2 \rceil)\leq \nu \leq m}$, we identify some $H^\nu(\Omega)$ convergent LS formulations that have an optimal error behavior like $h^{{m-\nu}}$. For $d\leq3$, the proposed methods are optimal in $H^2(\Omega)$. We demonstrate the effect...