RBF Multiscale Collocation for Second Order Elliptic Boundary Value Problems

Patricio Farrell, Holger Wendland

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

Abstract

In this paper, we discuss multiscale radial basis function collocation methods for solving elliptic partial differential equations on bounded domains. The approximate solution is constructed in a multilevel fashion, each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh. On each level, standard symmetric collocation is employed. A convergence theory is given, which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions. We are able to show that the convergence is linear in the number of levels. We also discuss the condition numbers of the arising systems and the effect of simple, diagonal preconditioners, now proving rigorously previous numerical observations. © 2013 Society for Industrial and Applied Mathematics.
Original languageEnglish (US)
Pages (from-to)2403-2425
Number of pages23
JournalSIAM Journal on Numerical Analysis
Volume51
Issue number4
DOIs
StatePublished - Jan 2013
Externally publishedYes

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