On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation

Authors:
Oleg Davydov;Dang Thi Oanh
Affiliations:
Department of Mathematics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, Scotland, United Kingdom;Department of Computer Science, Faculty of Information Technology - Thai Nguyen University, Quyet Thang Commune, Thai Nguyen City, Viet Nam
Venue:
Computers & Mathematics with Applications
Year:
2011

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Abstract

We investigate the influence of the shape parameter in the meshless Gaussian radial basis function finite difference (RBF-FD) method with irregular centres on the quality of the approximation of the Dirichlet problem for the Poisson equation with smooth solution. Numerical experiments show that the optimal shape parameter strongly depends on the problem, but insignificantly on the density of the centres. Therefore, we suggest a multilevel algorithm that effectively finds a near-optimal shape parameter, which helps to significantly reduce the error. Comparison to the finite element method and to the generalised finite differences obtained in the flat limits of the Gaussian RBF is provided.