RBF-FD formulas and convergence properties

The local RBF is becoming increasingly popular as an alternative to the global version that suffers from ill-conditioning. In this paper, we study analytically the convergence behavior of the local RBF method as a function of the number of nodes employed in the scheme, the nodal distance, and the sh...

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Bibliographic Details
Published in:Journal of computational physics 2010-11, Vol.229 (22), p.8281-8295
Main Authors: Bayona, Victor, Moscoso, Miguel, Carretero, Manuel, Kindelan, Manuel
Format: Article
Language:English
Subjects:
Computation
Computational techniques
Convergence
Derivatives
Error analysis
Exact sciences and technology
Mathematical analysis
Mathematical methods in physics
Mathematical models
Mesh-free
Optimization
Physics
Radial basis functions
Two dimensional
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Summary:The local RBF is becoming increasingly popular as an alternative to the global version that suffers from ill-conditioning. In this paper, we study analytically the convergence behavior of the local RBF method as a function of the number of nodes employed in the scheme, the nodal distance, and the shape parameter. We derive exact formulas for the first and second derivatives in one dimension, and for the Laplacian in two dimensions. Using these formulas we compute Taylor expansions for the error. From this analysis, we find that there is an optimal value of the shape parameter for which the error is minimum. This optimal parameter is independent of the nodal distance. Our theoretical results are corroborated by numerical experiments.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2010.07.008