A Cartesian-grid collocation method based on radial-basis-function networks for solving PDEs in irregular domains

  1. N. Mai-Duy*,
  2. T. Tran-Cong

Article first published online: 12 FEB 2007

DOI: 10.1002/num.20217

Issue

Numerical Methods for Partial Differential Equations

Numerical Methods for Partial Differential Equations

Volume 23, Issue 5, pages 1192–1210, September 2007

Additional Information

How to Cite

Mai-Duy, N. and Tran-Cong, T. (2007), A Cartesian-grid collocation method based on radial-basis-function networks for solving PDEs in irregular domains. Numer. Methods Partial Differential Eq., 23: 1192–1210. doi: 10.1002/num.20217

Author Information

  1. Faculty of Engineering and Surveying, The University of Southern Queensland, Toowoomba, Queensland 4350, Australia

*Faculty of Engineering and Surveying, The University of Southern Queensland, Toowoomba, Queensland 4350, Australia

Publication History

  1. Issue published online: 6 JUL 2007
  2. Article first published online: 12 FEB 2007
  3. Manuscript Accepted: 6 NOV 2006
  4. Manuscript Received: 24 APR 2006

Funded by

  • Australian Research Council

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Keywords:

  • integrated radial-basis-function network;
  • collocation method;
  • Cartesian grid;
  • irregular domain

Abstract

This paper reports a new Cartesian-grid collocation method based on radial-basis-function networks (RBFNs) for numerically solving elliptic partial differential equations in irregular domains. The domain of interest is embedded in a Cartesian grid, and the governing equation is discretized by using a collocation approach. The new features here are (a) one-dimensional integrated RBFNs are employed to represent the variable along each line of the grid, resulting in a significant improvement of computational efficiency, (b) the present method does not require complicated interpolation techniques for the treatment of Dirichlet boundary conditions in order to achieve a high level of accuracy, and (c) normal derivative boundary conditions are imposed by means of integration constants. The method is verified through the solution of second- and fourth-order PDEs; accurate results and fast convergence rates are obtained. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

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