Journal
ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
Volume 34, Issue 9, Pages 802-809Publisher
ELSEVIER SCI LTD
DOI: 10.1016/j.enganabound.2010.03.002
Keywords
Multiquadric collocation method; Meshless method; Error estimate; Arbitrary precision computation; Increasingly flat radial basis function
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For the interpolation of continuous functions and the solution of partial differential equation (PDE) by radial basis function (RBF) collocation, it has been observed that solution becomes increasingly more accurate as the shape of the RBF is flattened by the adjustment of a shape parameter. In the case of interpolation of continuous functions, it has been proven that in the limit of increasingly flat RBF, the interpolant reduces to Lagrangian polynomials. Does this limiting behavior implies that RBFs can perform no better than Lagrangian polynomials in the interpolation of a function, as well as in the solution of PDE? In this paper, arbitrary precision computation is used to test these and other conjectures. It is found that RBF in fact performs better than polynomials, as the optimal shape parameter exists not in the limit, but at a finite value. (C) 2010 Elsevier Ltd. All rights reserved.
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