Document Type

Journal Article

Department/Unit

Department of Mathematics

Language

English

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 effects of various collocation settings on the respective convergence rates.

Keywords

meshfree method, radial basis function, Kansa method, overdetermined collocation

Publication Date

2-2018

Source Publication Title

SIAM Journal on Numerical Analysis

Volume

56

Issue

1

Start Page

614

End Page

633

Publisher

Society for Industrial and Applied Mathematics

Peer Reviewed

1

Copyright

© 2018, Society for Industrial and Applied Mathematics

Funder

This work was partially supported by a Hong Kong Research Grant Council GRF grant, a Hong Kong Baptist University FRG grant.

DOI

10.1137/16M1072863

Link to Publisher's Edition

http://dx.doi.org/10.1137/16M1072863

ISSN (print)

00361429

ISSN (electronic)

10957170

Included in

Mathematics Commons

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