H^{2}–Convergence of least-squares kernel collocation methods

# H2–Convergence of least-squares kernel collocation methods

Ka Chun Cheung Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong. \qquad Leevan Ling11footnotemark: 1 Correspondence to L. Ling (E-mail: lling@hkbu.edu.hk) \qquad Robert Schaback Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Germany.
###### 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 under Dirichlet boundary conditions. With kernels that reproduce 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 convergent LS formulations that have an optimal error behavior like . 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.

Key words. Meshfree, radial basis function, Kansa method, overdetermined collocation.

AMS subject classifications. 65D15, 65N35, 41A63.

## 1 Introduction

Mathematical models or differential equations are meaningful only if they can somehow mirror the overly complicated real world. Similarly, numerical methods are useful only if they can produce approximations guaranteed to converge to the outcome that the mathematical model predicts. It could take tens of years for some good numerical strategies to mature and become a well-established class of numerical methods with a complete and rigid theoretical framework. Take the finite element method as an example. It waited for a quarter of a century to get its rigorous mathematical foundation. This paper aims to continue our theoretical contributions to the unsymmetric radial basis function collocation method, which is also known as the Kansa method in the community and we shall use this name throughout this paper for brevity.

To quickly overview the development of the Kansa method and its connection to the radial basis function (RBF) scattered data interpolation problem, let us look at some of its cornerstones [Fasshauer-Meshapprmethwith:07, Fasshauer+McCourt-KernApprMethusin:15, Wendland-ScatDataAppr:05]. An RBF is a smooth scalar function , which usually is induced from a kernel function in today’s applications, such that the interpolant of an interpolation problem is given as a linear combination

 u=nZ∑j=1λjϕ(∥⋅−zj∥2)=nZ∑j=1λjΦ(⋅,zj), \hb@xt@.01(1.1)

of shifted RBFs in which the set contains trial centers that specify the shifts of the kernel function in the expansion. Dealing with scaling has been another huge topic in Kansa methods [Golbabai+MohebianfarETAL-varishapparastra:15, Kansa+Carlson-Impraccumultinte:92, Tsai+KolibalETAL-goldsectsearalgo:10] for a decade, but we will ignore this point for the sake of simplicity.

Impressed by the meshfree nature, simplicity to program, dimension independence, and arbitarily high convergence rates interpolations, E.J. Kansa [Kansa-Multscatdataappr:90, Kansa-Multscatdataappr:90a] proposed to modify the RBF interpolation method to solve partial differential equations (PDEs) in the early 90s. Using the same RBF expansion (LABEL:RBFexpansion), Kansa imposed strong-form collocation conditions instead of interpolation conditions for identifying the unknown coefficients. Consider a PDE given by in and on . The Kansa method collocates the PDE at the trial centers to yield exactly conditions:

 Lu(zi)=∑λjLϕ(∥zi−zj∥),for zi∈Z∩Ω,Bu(zi)=∑λjBϕ(∥zi−zj∥),for zi∈Z∩Γ,\omit\span\omit\span\@@LTX@noalign\omit
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