On approximate cardinal preconditioning methods for solving PDEs with radial basis functions

Damian Brown*, Leevan LING, Edward Kansa, Jermy Levesley

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

74 Citations (Scopus)

Abstract

The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press], a preconditioning scheme that is based upon constructing the least-squares approximate cardinal basis function from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. This preconditioning technique is sufficiently general that it can be easily applied to many differential operators. In this paper, we review the ACBF preconditioning techniques previously used for interpolation problems and investigate a class of preconditioners based on the one proposed in [Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Adv Comput Math; in press] when a cardinality condition is enforced on different subsets. We numerically compare the ACBF preconditioners on several numerical examples of Poisson's, modified Helmholtz and Helmholtz equations, as well as a diffusion equation and discuss their performance.

Original languageEnglish
Pages (from-to)343-353
Number of pages11
JournalEngineering Analysis with Boundary Elements
Volume29
Issue number4
DOIs
Publication statusPublished - Apr 2005

Scopus Subject Areas

  • Analysis
  • Engineering(all)
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Cardinal basis function
  • Partial differential equation
  • Preconditioner
  • Radial basis function

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