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Convergence of non-stationary semi-discrete RBF schemes for the heat and wave equation

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Abstract

We give a detailed analysis of the convergence in Sobolev norm of the method of lines for the classical heat and wave equations on \(\mathbb {R }^n \) using non-stationary radial basis function interpolation on regular grids \(h \mathbb {Z }^n \) (scaled cardinal interpolation), for basis functions whose native space is a Sobolev space of order \(\nu / 2 \) with \(\nu > n + 2\).

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References

  1. Baxter, B., Brummelhuis, R.: Convergence estimates for stationary radial basis function interpolation and for semi-discrete collocation schemes. J. Fourier Anal. Appl. 28, 53 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bejancu, A.: Uniformly bounded Lebesgue constants for scaled cardinal interpolation with Matérn kernels. J. Approx. Th. 278 (2022)

  3. Bergh, J., Löfström, J.: Interpolation spaces, an introduction. Grundl. math. Wiss. 223, Springer-Verlag, Berlin Heidelberg. New York (1975)

  4. Buhmann, M.D.: Multivariate cardinal interpolation with radial basis functions. Constructive Approx. 6, 225–255 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buhmann, M.D.: Radial basis functions. Cambridge University Press, Cambridge Monographs on Applied and Computational Mathematics (2003)

  6. Bozzini, M., Rossini, M., Schaback, R.: Generalized Whittle-Matèrn and polyharmonic kernels. Adv. Comp. Math. 39, 129–141 (2013)

    Article  MATH  Google Scholar 

  7. Gohberg, I., Fel’dman, I.: Convolution equations and projection methods for their solutions, translated from the Russian by F. M. Goldware, Translations of Mathematical Monographs 41, American Mathematical Society, Providence, R. I. (1974)

  8. Gohberg, I., Goldberg, S., Kaashoek, M.A.: Basic classes of linear operators. Birkhäuser Verlag, Basel-Boston-Berlin (2003)

  9. Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of linear operators, Vol. II, Operator Theory Adv. & Appl. 63, Birkhäuser Verlag, Basel (1993)

  10. Hamm, K., Ledford, J.: Cardinal interpolation with general multiquadrics. Adv. Comp. Math. 42, 1149–1186 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hamm, K., Ledford, J.: Cardinal interpolation with general multiquadrics: convergence rates. Adv. Comp. Math. 44, 1205–1233 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hangelbroek, T., Madych, W., Narcowich, F., Ward, J.D.: Cardinal interpolation with Gaussian kernels. J. Fourier Anal. Appl. 18, 67–86 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hon, Y.C., Schaback, R., Zhong, M.: The meshless kernel-based method of lines for parabolic equations. Comp. Maths. Appl. 68, 2057–2067 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hubbert, S., Morton, T.M.: \(L_p \)-error estimates for radial basis function interpolation on the sphere. J. Approx. Th. 129, 58–77 (2004)

    Article  MATH  Google Scholar 

  15. Johnson, M.J.: Compactly supported, piecewise polyharmonic radial functions with prescribed regularity. Constructive Approx. 35, 201–223 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kansa, E.J.: Application of Hardy’s multiquadric interpolation to hydrodynamics. In: Proc. 1986 Simul. Conf., Vol. 4, pp. 111-117 (1986)

  17. Kansa, E.J.: Multiquadrics, a scattered data approximation scheme with applications to computational fluid dynamics II: solutions to parabolic, hyperbolic and elliptic partial differential equations. Comp. Maths. Appl. 19(8/9), 147–161 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Le Gia, Q.T.: Approximation of parabolic PDEs on spheres using spherical basis functions. Adv. Comp. Math. 22, 377–397 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. McLean, W.: Strongly elliptic systems and boundary integral equations. Cambridge University Press 2000

  20. Meyer, Y.: Wavelets and operators. Cambridge Studies in Advanced Mathematics 37, Cambridge University Press (1992)

  21. Narcowich, F.J., Schaback, R., Ward, J.D.: Approximation in Sobolev spaces by kernel expansions. J. Approx. Th. 114(1), 70–83 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rudin, W.: Fourier analysis on groups. Wiley Classics Library (1990)

  23. Schaback, R.: Improved error bounds for scattered data interpolation by radial basis functions. Maths. Comp. 68(nr. 225), 201–216 (1999)

  24. Schaback, R.: Convergence of unsymmetric kernel-based meshless collocation methods. SIAM J. Numer. Anal. 45(1), 333–351 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Schwartz, A.: The structure of the algebra of Hankel transforms and the algebra of Hankel-Stieltjes transforms. Canad. J. Math. 23, 236–246 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  26. Strohmer, Th.: Four short stories about Toeplitz matrix calculations. Lin. Alg. Appl. 343–344, 321–344 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Teschl, G.: Mathematical methods in quantum mechanics, Graduate Studies in Mathematics 157, 2nd edn. American Math. Soc. (2014)

  28. Ward, J.P., Unser, M.: Approximation properties of Sobolev splines and the construction of compactly supported equivalents. SIAM J. Math. Anal. 46(3), 1843–1858 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wendland, H.: Error estimates for interpolation by compactly supported radial basis functions of minimal degree. J. Approx. Th. 93, 258–272 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wendland, H.: Scattered data approximation. Cambridge University Press, Cambridge Monographs on Applied and Computational Mathematics (2004)

  31. Wendland, H.: A higher order approximation method for semilinear parabolic equations on spheres. Math. Comp. 82(281), 227–245 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wiener, N., Pitt, R.H.: On absolutely convergent Fourier-Stieltjes transforms. Duke Math. J. 4(2), 420–436 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wu, Z., Schaback, R.: Local error estimates for radial basis function interpolation of scattered data. IMA J. Numer. Anal. 13, 13–27 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

I would like to thank the anonymous referees for their comments and suggestions, in particular for pointing out that log-convexity of \(L^p \)-norms immediately provides \(L^p \)-estimates.

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  1. Laboratoire de Mathématiques de Reims, UMR 9008 du CNRS, Université de Reims Champagne-Ardenne, BP 1039, 51687 Cedex 2, Reims, France

    Raymond Brummelhuis

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Correspondence to Raymond Brummelhuis.

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Communicated by: Robert Schaback

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Brummelhuis, R. Convergence of non-stationary semi-discrete RBF schemes for the heat and wave equation. Adv Comput Math 49, 69 (2023). https://doi.org/10.1007/s10444-023-10058-8

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