Preconditioning for radial basis functions with domain decomposition methods

Leevan LING*, E. J. Kansa

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

104 Citations (Scopus)


In our previous work, an effective preconditioning scheme that is based upon constructing least-squares approximation cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. The preconditioner costs O(N2) flops to set up and O(N) storage. The preconditioning technique is sufficiently general that it can be applied to different types of different operators. This was applied to the 2D multiquadric method, with c∼1/√N on the Poisson test problem, the preconditioned GMRES converges in tens of iterations. In this paper, we combine the RBF methods and the ACBF preconditioning technique with the domain decomposition method (DDM). We studied different implementations of the ACBF-DDM scheme and provide numerical results for N > 10,000 nodes. We shall demonstrate that the efficiency of the ACBF-DDM scheme improves dramatically as successively finer partitions of the domain are considered.

Original languageEnglish
Pages (from-to)1413-1427
Number of pages15
JournalMathematical and Computer Modelling
Issue number13
Publication statusPublished - Dec 2004

Scopus Subject Areas

  • Modelling and Simulation
  • Computer Science Applications

User-Defined Keywords

  • Approximate cardinal basis function
  • Domain decomposition
  • Partial differential equation
  • Preconditioner
  • Radial basis function


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