RBF-FD Formulas and Convergence Properties Articles uri icon

publication date

  • noviembre 2010

start page

  • 8281

end page

  • 8295

issue

  • 22

volume

  • 229

international standard serial number (ISSN)

  • 0021-9991

electronic international standard serial number (EISSN)

  • 1090-2716

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.