RBF-FD formulas and convergence properties

Authors:
Victor Bayona;Miguel Moscoso;Manuel Carretero;Manuel Kindelan
Affiliations:
Gregorio Millán Institute, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain;Gregorio Millán Institute, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain;Gregorio Millán Institute, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain;Gregorio Millán Institute, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain
Venue:
Journal of Computational Physics
Year:
2010

Citing 7
Cited 3

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Abstract

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.