Document Type

Journal Article

Department/Unit

Department of Mathematics

Language

English

Abstract

We proposed ways to implement meshless collocation methods extrinsically for solving elliptic PDEs on smooth, closed, connected, and complete Riemannian manifolds with arbitrary codimensions. Our methods are based on strong-form collocations with oversampling and least-squares minimizations, which can be implemented either analytically or approximately. By restricting global kernels to the manifold, our methods resemble their easy-to-implement domain-type analogies, i.e., Kansa methods. Our main theoretical contribution is the robust convergence analysis under some standard smoothness assumptions for high-order convergence. Numerical demonstrations are provided to verify the proven convergence rates, and we simulate reaction-diffusion equations for generating Turing patterns on manifolds.

Publication Date

2020

Source Publication Title

SIAM Journal on Numerical Analysis

Volume

58

Issue

2

Start Page

988

End Page

1007

Publisher

Society for Industrial and Applied Mathematics

DOI

10.1137/17M1158641

Link to Publisher's Edition

https://doi.org/10.1137/17M1158641

ISSN (print)

00361429

ISSN (electronic)

10957170

Included in

Mathematics Commons

Share

COinS