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Point collocation methods using the fast moving least‐square reproducing kernel approximation
Author(s) -
Kim Do Wan,
Kim Yongsik
Publication year - 2003
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.618
Subject(s) - mathematics , interpolation (computer graphics) , collocation (remote sensing) , kernel (algebra) , collocation method , meshfree methods , convergence (economics) , piecewise , function (biology) , mathematical analysis , computer science , finite element method , differential equation , animation , ordinary differential equation , physics , computer graphics (images) , machine learning , evolutionary biology , biology , economic growth , economics , thermodynamics , combinatorics
A pseudo‐spectral point collocation meshfree method is proposed. We apply a scheme of approximating derivatives based on the moving least‐square reproducing kernel approximations. Using approximated derivatives, we propose a new point collocation method. Unlike other meshfree methods that require direct calculation of derivatives for shape functions, with the proposed scheme, approximated derivatives are obtained in the process of calculating the shape function itself without further cost. Moreover, the scheme does not require the regularity of the window function, which ensures the regularity of shape functions. In this paper, we show the reproducing property and the convergence of interpolation for approximated derivatives of shape functions. As numerical examples of the proposed scheme, Poisson and Stokes problems are considered in various situations including the case of randomly generated node sets. In short, the proposed scheme is efficient and accurate even for complicated geometry such as the flow past a cylinder. Copyright © 2003 John Wiley & Sons, Ltd.