On exponential convergence of Gegenbauer interpolation and spectral differentiation

Xie, Ziqing; Wang, Li-Lian; Zhao, Xiaodan

doi:10.1090/S0025-5718-2012-02645-7

On exponential convergence of Gegenbauer interpolation and spectral differentiation
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by Ziqing Xie, Li-Lian Wang and Xiaodan Zhao;

Math. Comp. 82 (2013), 1017-1036

DOI: https://doi.org/10.1090/S0025-5718-2012-02645-7

Published electronically: August 21, 2012

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Abstract:

This paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the Gegenbauer-Gauss and Gegenbauer-Gauss-Lobatto points, when the underlying function is analytic on and within an ellipse. Sharp error estimates in the maximum norm are derived.

References

George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Ivo Babuška and Benqi Guo, Direct and inverse approximation theorems for the $p$-version of the finite element method in the framework of weighted Besov spaces. I. Approximability of functions in the weighted Besov spaces, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1512–1538. MR 1885705, DOI 10.1137/S0036142901356551
Christine Bernardi, Monique Dauge, and Yvon Maday, Spectral methods for axisymmetric domains, Series in Applied Mathematics (Paris), vol. 3, Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris; North-Holland, Amsterdam, 1999. Numerical algorithms and tests due to Mejdi Azaïez. MR 1693480
Christine Bernardi and Yvon Maday, Spectral methods, Handbook of numerical analysis, Vol. V, Handb. Numer. Anal., V, North-Holland, Amsterdam, 1997, pp. 209–485. MR 1470226, DOI 10.1016/S1570-8659(97)80003-8
John P. Boyd, The rate of convergence of Fourier coefficients for entire functions of infinite order with application to the Weideman-Cloot sinh-mapping for pseudospectral computations on an infinite interval, J. Comput. Phys. 110 (1994), no. 2, 360–372. MR 1267887, DOI 10.1006/jcph.1994.1032
John P. Boyd, Chebyshev and Fourier spectral methods, 2nd ed., Dover Publications, Inc., Mineola, NY, 2001. MR 1874071
John P. Boyd, Trouble with Gegenbauer reconstruction for defeating Gibbs’ phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations, J. Comput. Phys. 204 (2005), no. 1, 253–264. MR 2121910, DOI 10.1016/j.jcp.2004.10.008
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods, Scientific Computation, Springer-Verlag, Berlin, 2006. Fundamentals in single domains. MR 2223552
Yanping Chen and Tao Tang, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comp. 79 (2010), no. 269, 147–167. MR 2552221, DOI 10.1090/S0025-5718-09-02269-8
Philip J. Davis, Interpolation and approximation, Dover Publications, Inc., New York, 1975. Republication, with minor corrections, of the 1963 original, with a new preface and bibliography. MR 380189
Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 760629
E. H. Doha, The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function, J. Comput. Appl. Math. 89 (1998), no. 1, 53–72. MR 1625979, DOI 10.1016/S0377-0427(97)00228-8
E. H. Doha and A. H. Bhrawy, Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials, Numer. Algorithms 42 (2006), no. 2, 137–164. MR 2262512, DOI 10.1007/s11075-006-9034-6
Eid H. Doha and Ali H. Bhrawy, A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations, Numer. Methods Partial Differential Equations 25 (2009), no. 3, 712–739. MR 2510756, DOI 10.1002/num.20369
E. H. Doha, A. H. Bhrawy, and W. M. Abd-Elhameed, Jacobi spectral Galerkin method for elliptic Neumann problems, Numer. Algorithms 50 (2009), no. 1, 67–91. MR 2487228, DOI 10.1007/s11075-008-9216-5
Moshe Dubiner, Spectral methods on triangles and other domains, J. Sci. Comput. 6 (1991), no. 4, 345–390. MR 1154903, DOI 10.1007/BF01060030
David Elliott, Uniform asymptotic expansions of the Jacobi polynomials and an associated function, Math. Comp. 25 (1971), 309–315. MR 294737, DOI 10.1090/S0025-5718-1971-0294737-5
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
Daniele Funaro, Polynomial approximation of differential equations, Lecture Notes in Physics. New Series m: Monographs, vol. 8, Springer-Verlag, Berlin, 1992. MR 1176949
A. Gelb and Z. Jackiewicz, Determining analyticity for parameter optimization of the Gegenbauer reconstruction method, SIAM J. Sci. Comput. 27 (2005), no. 3, 1014–1031. MR 2199918, DOI 10.1137/040603814
David Gottlieb and Steven A. Orszag, Numerical analysis of spectral methods: theory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1977. MR 520152
David Gottlieb and Chi-Wang Shu, On the Gibbs phenomenon. IV. Recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function, Math. Comp. 64 (1995), no. 211, 1081–1095. MR 1284667, DOI 10.1090/S0025-5718-1995-1284667-0
David Gottlieb and Chi-Wang Shu, On the Gibbs phenomenon. V. Recovering exponential accuracy from collocation point values of a piecewise analytic function, Numer. Math. 71 (1995), no. 4, 511–526. MR 1355052, DOI 10.1007/s002110050155
David Gottlieb and Chi-Wang Shu, On the Gibbs phenomenon. III. Recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise analytic function, SIAM J. Numer. Anal. 33 (1996), no. 1, 280–290. MR 1377254, DOI 10.1137/0733015
David Gottlieb and Chi-Wang Shu, On the Gibbs phenomenon and its resolution, SIAM Rev. 39 (1997), no. 4, 644–668. MR 1491051, DOI 10.1137/S0036144596301390
David Gottlieb, Chi-Wang Shu, Alex Solomonoff, and Hervé Vandeven, On the Gibbs phenomenon. I. Recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function, J. Comput. Appl. Math. 43 (1992), no. 1-2, 81–98. Orthogonal polynomials and numerical methods. MR 1193295, DOI 10.1016/0377-0427(92)90260-5
B.Q. Guo, Mathematical Foundation and Applications of the $p$ and $h$-$p$ Finite Element Methods, Series in Analysis, vol. 4, World Scientific, 2011.
Ben-yu Guo, Gegenbauer approximation and its applications to differential equations with rough asymptotic behaviors at infinity, Appl. Numer. Math. 38 (2001), no. 4, 403–425. MR 1852266, DOI 10.1016/S0168-9274(01)00039-3
Guo Ben-yu and Wang Li-lian, Jacobi interpolation approximations and their applications to singular differential equations, Adv. Comput. Math. 14 (2001), no. 3, 227–276. MR 1845244, DOI 10.1023/A:1016681018268
Ben-yu Guo and Li-lian Wang, Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory 128 (2004), no. 1, 1–41. MR 2063010, DOI 10.1016/j.jat.2004.03.008
Bertil Gustafsson, The work of David Gottlieb: a success story, Commun. Comput. Phys. 9 (2011), no. 3, 481–496. MR 2726814, DOI 10.4208/cicp.010110.010310s
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
George Em Karniadakis and Spencer J. Sherwin, Spectral/$hp$ element methods for computational fluid dynamics, 2nd ed., Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2005. MR 2165335, DOI 10.1093/acprof:oso/9780198528692.001.0001
Ilia Krasikov, An upper bound on Jacobi polynomials, J. Approx. Theory 149 (2007), no. 2, 116–130. MR 2374601, DOI 10.1016/j.jat.2007.04.008
J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1937591
Paul Nevai, Tamás Erdélyi, and Alphonse P. Magnus, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), no. 2, 602–614. MR 1266580, DOI 10.1137/S0036141092236863
S. C. Reddy and J. A. C. Weideman, The accuracy of the Chebyshev differencing method for analytic functions, SIAM J. Numer. Anal. 42 (2005), no. 5, 2176–2187. MR 2139243, DOI 10.1137/040603280
Theodore J. Rivlin, Chebyshev polynomials, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. From approximation theory to algebra and number theory. MR 1060735
J. Shen, T. Tang, and L.L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Series in Computational Mathematics, vol. 41, Springer-Verlag, Berlin, Heidelberg, 2011.
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, RI, 1975. MR 372517
Eitan Tadmor, The exponential accuracy of Fourier and Chebyshev differencing methods, SIAM J. Numer. Anal. 23 (1986), no. 1, 1–10. MR 821902, DOI 10.1137/0723001
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
Lloyd N. Trefethen, Is Gauss quadrature better than Clenshaw-Curtis?, SIAM Rev. 50 (2008), no. 1, 67–87. MR 2403058, DOI 10.1137/060659831
Haiyong Wang and Shuhuang Xiang, On the convergence rates of Legendre approximation, Math. Comp. 81 (2012), no. 278, 861–877. MR 2869040, DOI 10.1090/S0025-5718-2011-02549-4
Zhong-Qing Wang and Ben-Yu Guo, Jacobi rational approximation and spectral method for differential equations of degenerate type, Math. Comp. 77 (2008), no. 262, 883–907. MR 2373184, DOI 10.1090/S0025-5718-07-02074-1
G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
Zhimin Zhang, Superconvergence of spectral collocation and $p$-version methods in one dimensional problems, Math. Comp. 74 (2005), no. 252, 1621–1636. MR 2164089, DOI 10.1090/S0025-5718-05-01756-4
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