On node distributions for interpolation and spectral methods

Hoang, N.

doi:10.1090/mcom/3018

On node distributions for interpolation and spectral methods
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by N. S. Hoang PDF

Math. Comp. 85 (2016), 667-692 Request permission

Abstract:

A scaled Chebyshev node distribution is studied in this paper. It is proved that the node distribution is optimal for interpolation in $C_M^{s+1}[-1,1]$, the set of $(s+1)$-time differentiable functions whose $(s+1)$-th derivatives are bounded in absolute value by a constant $M>0$. Node distributions for computing spectral differentiation matrices and integration matrices are proposed and studied. Numerical experiments have shown that the proposed node distributions can yield results of higher accuracy than those obtained by the most commonly used Chebyshev-Gauss-Lobatto node distribution.

References

L. Brutman, Lebesgue functions for polynomial interpolation—a survey, Ann. Numer. Math. 4 (1997), no. 1-4, 111–127. The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin. MR 1422674
Bengt Fornberg, A practical guide to pseudospectral methods, Cambridge Monographs on Applied and Computational Mathematics, vol. 1, Cambridge University Press, Cambridge, 1996. MR 1386891, DOI 10.1017/CBO9780511626357
R. Günttner, On asymptotics for the uniform norms of the Lagrange interpolation polynomials corresponding to extended Chebyshev nodes, SIAM J. Numer. Anal. 25 (1988), no. 2, 461–469. MR 933735, DOI 10.1137/0725030
Myron S. Henry, Approximation by polynomials: interpolation and optimal nodes, Amer. Math. Monthly 91 (1984), no. 8, 497–499. MR 761404, DOI 10.2307/2322572
J. S. Hesthaven, From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J. Numer. Anal. 35 (1998), no. 2, 655–676. MR 1618874, DOI 10.1137/S003614299630587X
Rainer Kress, Numerical analysis, Graduate Texts in Mathematics, vol. 181, Springer-Verlag, New York, 1998. MR 1621952, DOI 10.1007/978-1-4612-0599-9
Heinz-Joachim Rack, An example of optimal nodes for interpolation, Internat. J. Math. Ed. Sci. Tech. 15 (1984), no. 3, 355–357. MR 750010, DOI 10.1080/0020739840150312
Simon J. Smith, Lebesgue constants in polynomial interpolation, Ann. Math. Inform. 33 (2006), 109–123. MR 2385471
Lloyd N. Trefethen, Spectral methods in MATLAB, Software, Environments, and Tools, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. MR 1776072, DOI 10.1137/1.9780898719598
P. Vértesi, Optimal Lebesgue constant for Lagrange interpolation, SIAM J. Numer. Anal. 27 (1990), no. 5, 1322–1331. MR 1061132, DOI 10.1137/0727075
Zhimin Zhang, Superconvergence of a Chebyshev spectral collocation method, J. Sci. Comput. 34 (2008), no. 3, 237–246. MR 2377618, DOI 10.1007/s10915-007-9163-7