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Multiscale methods with compactly supported radial basis functions for the Stokes problem on bounded domains

Advances in Computational Mathematics volume 42, pages 1187–1208 (2016)Cite this article

Abstract

In this paper, we investigate the application of radial basis functions (RBFs) for the approximation with collocation of the Stokes problem. The approximate solution is constructed in a multi-level fashion, each level using compactly supported radial basis functions with decreasing scaling factors. We use symmetric collocation and give sufficient conditions for convergence and consider stability analysis. Numerical experiments support the theoretical results.

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References

  1. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3 edn. Texts in Applied Mathematics Springer (2008)

  2. Burman, E.: Pressure projection stabilizations for Galerkin approximations of stokes’ and darcy’s problem. Numer Methods Partial Differ. Equ 24, 127–143 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chernih, A., Le Gia, Q.T.: Multiscale methods with compactly supported radial basis functions for Galerkin approximation of elliptic PDEs. IMA J. Numer. Anal 34(2), 569–591 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Devore, R.A., Sharpley, R.C.: Besov spaces on domains in ℝd. Trans. Amer. Math. Soc 335, 843–864 (1993)

    MathSciNet  MATH  Google Scholar 

  5. Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB Interdisciplinary Mathematical Sciences, vol. 6. World Scientific Publishing Co., Singapore (2007)

    Book  Google Scholar 

  6. Floater, M.S., Iske, A.: Multistep scattered data interpolation using compactly supported radial basis functions. J. Comput. Appl. Math 73, 65–78 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fuselier, E.: Improved stability estimates and a characterization of the native space for matrix-valued rbfs. Adv. Comput. Math 29, 269–290 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fuselier, E.: Sobolev-type approximation rates for divergence-free and curl-free rbf interpolants. Math. Comput 77, 1407–1423 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Giesl, P., Wendland, H.: Meshless collocation: Error estimates with application to dynamical systems. SIAM J. Numer. Anal 45, 1723–1741 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Le Gia, Q.T., Sloan, I.H., Wendland, H.: Multiscale analysis in Sobolev spaces on the sphere. SIAM J. Numer. Anal 48, 2065–2090 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Le Gia, Q.T., Sloan, I.H., Wendland, H.: Multiscale RBF collocation for solving PDEs on spheres. Numer. Math 121, 99–125 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lowitzsch, S.: A density theorem for matrix-valued radial basis functions. Numer. Algorithms 39, 253–256 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lowitzsch, S.: Error estimates for matrix-valued radial basis function interpolation. J. Approx. Theory 137, 234–249 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Narcowich, F.J., Ward, J.D.: Generalized Hermite interpolation via matrix-valued conditionally positive definite functions. Math. Comput 63, 661–687 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Narcowich, F.J., Ward, J.D., Wendland, H.: Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting. Math. Comput 74, 743–763 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Narcowich, F.J., Ward, J.D., Wendland, H.: Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions. Constr. Approx 24, 175–186 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Schaback, R.: On the efficiency of interpolation by radial basis functions. In: Le Méhauté, C.R. A., Schumaker, L.L. (eds.) Surface Fitting and Multiresolution Methods, 309-318. Vanderbilt University Press (1996)

  18. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  19. Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis. AMS Chelsea Publishing, North-Holland (1977)

    MATH  Google Scholar 

  20. Wendland, H.: Scattered Data Approximation Cambridge Monographs on Applied and Computational Mathematics, vol. 17. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  21. Wendland, H.: Divergence-free kernel methods for approximating the Stokes problem. SIAM J. Numer. Anal 47, 3158–3179 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wendland, H.: Multiscale analysis in Sobolev spaces on bounded domains. Numer. Math 116, 493–517 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    A. Chernih & Q. T. Le Gia

Authors
  1. A. Chernih
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  2. Q. T. Le Gia
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Corresponding author

Correspondence to A. Chernih.

Additional information

Communicated by: Robert Schaback

This work was supported by the Australian Research Council.

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Chernih, A., Le Gia, Q.T. Multiscale methods with compactly supported radial basis functions for the Stokes problem on bounded domains. Adv Comput Math 42, 1187–1208 (2016). https://doi.org/10.1007/s10444-016-9458-z

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  • DOI: https://doi.org/10.1007/s10444-016-9458-z

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