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Mathematics of Computation

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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
DOI: https://doi.org/10.1090/S0025-5718-99-01102-3
Published electronically: March 4, 1999
MathSciNet review: 1648419
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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.


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  • 1. Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • 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. Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258
  • 4. Julio G. Dix and Robert D. Ogden, An interpolation scheme with radial basis in Sobolev spaces 𝐻^{𝑠}(𝑅ⁿ), Rocky Mountain J. Math. 24 (1994), no. 4, 1319–1337. MR 1322230, https://doi.org/10.1216/rmjm/1181072340
  • 5. Jean Duchon, Sur l’erreur d’interpolation des fonctions de plusieurs variables par les 𝐷^{𝑚}-splines, RAIRO Anal. Numér. 12 (1978), no. 4, 325–334, vi (French, with English summary). MR 519016
  • 6. N. Dyn, Interpolation and approximation by radial and related functions, Approximation theory VI, Vol. I (College Station, TX, 1989) Academic Press, Boston, MA, 1989, pp. 211–234. MR 1090994
  • 7. W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive definite functions, Approx. Theory Appl. 4 (1988), no. 4, 77–89. MR 986343
  • 8. Will Light (ed.), Advances in numerical analysis. Vol. II, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. Wavelets, subdivision algorithms, and radial basis functions. MR 1172118
  • 9. Eduard Prugovečki, Quantum mechanics in Hilbert space, Academic Press, New York-London, 1971. Pure and Applied Mathematics, Vol. 41. MR 0495809
  • 10. R. Schaback, Creating surfaces from scattered data using radial basis functions, Mathematical methods for curves and surfaces (Ulvik, 1994) Vanderbilt Univ. Press, Nashville, TN, 1995, pp. 477–496. MR 1356989
  • 11. Robert Schaback, Multivariate interpolation and approximation by translates of a basis function, Approximation theory VIII, Vol. 1 (College Station, TX, 1995) Ser. Approx. Decompos., vol. 6, World Sci. Publ., River Edge, NJ, 1995, pp. 491–514. MR 1471761
  • 12. R. Schaback, Approximation by radial basis functions with finitely many centers, Constr. Approx. 12 (1996), no. 3, 331–340. MR 1405002, https://doi.org/10.1007/s003659900017
  • 13. Robert Schaback and Z. Wu, Operators on radial functions, J. Comput. Appl. Math. 73 (1996), no. 1-2, 257–270. MR 1424880, https://doi.org/10.1016/0377-0427(96)00047-7
  • 14. Holger Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995), no. 4, 389–396. MR 1366510, https://doi.org/10.1007/BF02123482
  • 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. Zong Min Wu and Robert Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993), no. 1, 13–27. MR 1199027, https://doi.org/10.1093/imanum/13.1.13

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Additional Information

Holger Wendland
Affiliation: Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Lotzestraße 16-18, D-37083 Göttingen, Germany
Email: wendland@math.uni-goettingen.de

DOI: https://doi.org/10.1090/S0025-5718-99-01102-3
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