Skip to main content
SpringerLink
Log in

Convergence order estimates of meshless collocation methods using radial basis functions

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We study meshless collocation methods using radial basis functions to approximate regular solutions of systems of equations with linear differential or integral operators. Our method can be interpreted as one of the emerging meshless methods, cf. T. Belytschko et al. (1996). Its range of application is not confined to elliptic problems. However, the application to the boundary value problem for an elliptic operator, connected with an integral equation, is given as an example. Although the method has been used for special cases for about ten years, cf. E.J. Kansa (1990), there are no error bounds known. We put the main emphasis on detailed proofs of such error bounds, following the general outline described in C. Franke and R. Schaback (preprint).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. T. Belytschko, Y. Krongauz, D.J. Organ, M. Fleming and P. Krysl, Meshless methods: An overview and recent developments, Comput. Methods Appl. Mech. Engrg. 139 (1996) 3–47. Preprint: http: //tam6.mech.nwu.edu/mfleming/EFG/meshless.ps.

    Article  MATH  Google Scholar 

  2. S. Bochner, Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse, Math. Ann. 108 (1933) 378–410.

    Article  MATH  MathSciNet  Google Scholar 

  3. S.C. Brenner, Multigrid methods for the computation of singular solutions and stress intensity factors II: Crack singularities, BIT 37(3) (1997) 623–643.

    MATH  MathSciNet  Google Scholar 

  4. G.E. Fasshauer, Solving partial differential equations by collocation with radial basis functions, in: Surface Fitting and Multiresolution Methods, eds. A. Le Méhauté, C. Rabut and L.L. Schumaker (Vanderbilt University Press, Nashville, TN, 1997) pp. 131–138. Preprint: http://www.math. nwu.edu/"fass/collocate.ps.

    Google Scholar 

  5. G.E. Fasshauer and J.W. Jerome, Multistep approximation algorithms: Improved convergence rates through postconditioning with smoothing kernels (1995). Preprint: http://www.math.nwu. edu/"fass/smooth.dvi.

  6. C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basis functions, Appl. Math. Comput., to appear. Preprint: http://www.num.math. uni-goettingen.de/schaback/research/papers/SPDEbCuRBF.ps.

  7. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224 (Springer, Berlin, Heidelberg, 2nd ed., 1983).

    Google Scholar 

  8. M.A. Golberg, Recent developments in the numerical evaluation of particular solutions in the boundary element method, Appl. Math. Comput. 75 (1996) 91–101.

    Article  MATH  MathSciNet  Google Scholar 

  9. C. Großmann and H.-G. Roos, Numerik Partieller Differentialgleichungen (Teubner, Stuttgart, 2nd ed., 1994).

    Google Scholar 

  10. A. Iske, Reconstruction of functions from generalized Hermite–Birkhoff data, in: Approximation Theory VIII, Vol. 1, Approximation and Interpolation, eds. C.K. Chui and L.L. Schumaker (World Scientific, Singapore, 1995) pp. 257–264.

    Google Scholar 

  11. E.J. Kansa, Multiquadrics – a scattered data approximation scheme with applications to computational fluid-dynamics – I: Surface approximations and partial derivative estimates, Comput. Math. Appl. 19(8/9) (1990) 127–145.

    Article  MATH  MathSciNet  Google Scholar 

  12. F.J. Narcowich, R. Schaback and J.D. Ward, Multilevel interpolation and approximation (1997). Preprint: http://www.num.math.uni-goettingen.de/schaback/research/ papers/MIaA.ps.

  13. R. Schaback, Multivariate interpolation and approximation by translates of a basis function, in: Approximation Theory VIII, Vol. 1, Approximation and Interpolation, eds. C.K. Chui and L.L. Schumaker (World Scientific, Singapore, 1995) pp. 491–514.

    Google Scholar 

  14. R. Schaback, On the efficiency of interpolation by radial basis functions, in: Surface Fitting and Multiresolution Methods, eds. A. LeMéhauté, C. Rabut and L.L. Schumaker (Vanderbilt University Press, Nashville, TN, 1997) pp. 309–328. Preprint: ftp://ftp.gwdg.de/pub/numerik/ schaback/efficiency1.dvi.Z.

    Google Scholar 

  15. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995) 389–396.

    Article  MATH  MathSciNet  Google Scholar 

  16. H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory, to appear. Preprint: http://www.num.math. uni-goettingen.de/wendland/error.ps.gz.

  17. H. Wendland, Sobolev-type error estimates for interpolation by radial basis functions, in: Surface Fitting and Multiresolution Methods, eds. A. Le Méhauté, C. Rabut and L.L. Schumaker (Vanderbilt University Press, Nashville, TN, 1997) pp. 337–344. Preprint: http://www.num.math. uni-goettingen.de/wendland/sobtyp.ps.gz.

    Google Scholar 

  18. J.T. Wloka, Partial Differential Equations (Cambridge University Press, Cambridge, 1987).

    Google Scholar 

  19. Z. Wu, Hermite–Birkhoff interpolation of scattered data by radial basis functions, Approx. Theory Appl. 8(2) (1992) 1–10.

    MathSciNet  Google Scholar 

  20. Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993) 13–27. Preprint: ftp://ftp.gwdg.de/pub/numerik/ schaback/wu1.dvi.Z.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Numerische und Angewandte Mathematik der Universität Göttingen, Lotzestraße 16-18, D-37083, Göttingen, Germany

    Carsten Franke & Robert Schaback

Authors
  1. Carsten Franke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Schaback
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Franke, C., Schaback, R. Convergence order estimates of meshless collocation methods using radial basis functions. Advances in Computational Mathematics 8, 381–399 (1998). https://doi.org/10.1023/A:1018916902176

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1018916902176

Search

Navigation