Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Meshless Galerkin methods
using radial basis functions

Author: Holger Wendland
Journal: Math. Comp. 68 (1999), 1521-1531
MSC (1991): Primary 35A40, 35J50, 41A25, 41A30, 41A63, 65N15, 65N30
Published electronically: March 4, 1999
MathSciNet review: 1648419
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We combine the theory of radial basis functions with the field of Galerkin methods to solve partial differential equations. After a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions.

References [Enhancements On Off] (What's this?)

  • 1. R. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 56:9247
  • 2. T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, Meshless methods: an overview and recent developments, Computer Methods in Applied Mechanics and Engineering, special issue on Meshless Methods, vol 139 (1996), pp 3- 47.
  • 3. S. Brenner, L. Scott, The Mathematical Theory of Finite Element Methods, Springer Verlag, New York, 1994. MR 95f:65001
  • 4. J. G. Dix, R. O. Ogden, An interpolation scheme with radial basis in Sobolev spaces $H^s({\mathbb R}^n)$, Rocky Mountain Journal of Mathematics, Vol. 24, No. 4 (1994), pp 1319 - 1337. MR 95k:41003
  • 5. J. Duchon, Sur l'erreur d'interpolation des fonctions de plusieurs variables par les $D^m$-splines, R.A.I.R.O. Analyse numérique, Vol. 12, No. 4 (1978), pp 325 - 334. MR 80j:41052
  • 6. N. Dyn, Interpolation and approximation by radial and related functions, in: Approximation Theory VI, Vol. 1, C. K. Chui, L. L. Schumaker, J. D. Ward eds., Academic Press, Boston, 1983, pp 211-234. MR 92d:41002
  • 7. W. R. Madych, S. A. Nelson, Multivariate interpolation and conditionally positive definite fucntions, Approx. Theory and its Appl. 4.4 (1988), pp 77-89. MR 90e:41006
  • 8. M. J. D. Powell, The theory of radial basis function approximation in 1990, in: Advances in Numerical Analysis Vol. 2, W. Light, ed., Clarendon Press, Oxford, 1992, pp 105 - 210. MR 95c:41003
  • 9. E. Prugove\v{c}ki, Quantum Mechanics in Hilbert Space, Academic Press, New York, 1971. MR 58:14457
  • 10. R. Schaback, Creating surfaces from scattered data using radial basis functions, in: Mathematical Methods for Curves and Surfaces, M. Daehlen, T. Lyche, L. L. Schumaker, eds., Vanderbilt University Press, Nashville, 1995, pp 477-496. MR 96g:65025
  • 11. R. Schaback, Multivariate interpolation and approximation by translates of a basis function, in: Approximation Theory VIII, volume 1: Approximation and Interpolation, C. K. Chui, L. L. Schumaker, eds., Academic Press, Boston, 1995, pp 491 - 514. MR 98g:41036
  • 12. R. Schaback, Approximation by radial basis functions with finitely many centers, Constructive Approximation 12 (1996), pp 331-340. MR 97d:41013
  • 13. R. Schaback, Z. Wu, Operators on radial functions, J. of Computational and Applied Mathematics 73 (1996), pp 257-270. MR 97g:42002
  • 14. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Advances in Comp. Math. 4 (1995), pp 389 - 396. MR 96h:41025
  • 15. H. Wendland, Sobolev-type error estimates for interpolation by radial basis functions, in: Surface Fitting and Multiresolution Methods, A. LeMéhauté, C. Rabut, L. L. Schumaker, eds., Vanderbilt University Press, Nashville, 1997, pp 337-344.
  • 16. H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, Journal of Approx. Theory 93 (1998), pp 258-272. CMP 98:11
  • 17. Z. Wu, R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. of Numerical Analysis 13 (1993), pp 13-27. MR 93m:65012

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 35A40, 35J50, 41A25, 41A30, 41A63, 65N15, 65N30

Retrieve articles in all journals with MSC (1991): 35A40, 35J50, 41A25, 41A30, 41A63, 65N15, 65N30

Additional Information

Holger Wendland
Affiliation: Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Lotzestraße 16-18, D-37083 Göttingen, Germany

Keywords: Approximation orders, positive definite functions, PDE
Received by editor(s): April 1, 1997
Published electronically: March 4, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society