Pointwise error estimates and local superconvergence of Jacobi expansions
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- by Shuhuang Xiang, Desong Kong, Guidong Liu and Li-Lian Wang HTML | PDF
- Math. Comp. 92 (2023), 1747-1778 Request permission
Abstract:
As one myth of polynomial interpolation and quadrature, Trefethen [Math. Today (Southend-on-Sea) 47 (2011), pp. 184–188] revealed that the Chebyshev interpolation of $|x-a|$ (with $|a|<1$) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about $95%$ range of $[-1,1]$ except for a small neighbourhood near the singular point $x=a.$ In this paper, we rigorously show that the Jacobi expansion for a more general class of $\Phi$-functions also enjoys such a local convergence behaviour. Our assertion draws on the pointwise error estimate using the reproducing kernel of Jacobi polynomials and the Hilb-type formula on the asymptotic of the Bessel transforms. We also study the local superconvergence and show the gain in order and the subregions it occurs. As a by-product of this new argument, the undesired $\log n$-factor in the pointwise error estimate for the Legendre expansion recently stated in Babus̆ka and Hakula [Comput. Methods Appl. Mech Engrg. 345 (2019), pp. 748–773] can be removed. Finally, all these estimates are extended to the functions with boundary singularities. We provide ample numerical evidences to demonstrate the optimality and sharpness of the estimates.References
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C., 1964.
- Ivo Babuška and Benqi Guo, Optimal estimates for lower and upper bounds of approximation errors in the $p$-version of the finite element method in two dimensions, Numer. Math. 85 (2000), no. 2, 219–255. MR 1754720, DOI 10.1007/PL00005387
- 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
- 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. II. Optimal rate of convergence of the $p$-version finite element solutions, Math. Models Methods Appl. Sci. 12 (2002), no. 5, 689–719. MR 1909423, DOI 10.1142/S0218202502001854
- Ivo Babuška and Harri Hakula, Pointwise error estimate of the Legendre expansion: the known and unknown features, Comput. Methods Appl. Mech. Engrg. 345 (2019), 748–773. MR 3892019, DOI 10.1016/j.cma.2018.11.017
- N. K. Bary, A treatise on trigonometric series. Vols. I, II, A Pergamon Press Book, The Macmillan Company, New York, 1964. Authorized translation by Margaret F. Mullins. MR 0171116
- S. Bernstein, On the best approximation of $|x-c|^{p}$, Doll. Akad. Nauk SSSR 18 (1938), 374–384.
- Waixiang Cao, Chi-Wang Shu, Yang Yang, and Zhimin Zhang, Superconvergence of discontinuous Galerkin methods for two-dimensional hyperbolic equations, SIAM J. Numer. Anal. 53 (2015), no. 4, 1651–1671. MR 3365565, DOI 10.1137/140996203
- Waixiang Cao, Chi-Wang Shu, Yang Yang, and Zhimin Zhang, Superconvergence of discontinuous Galerkin method for scalar nonlinear hyperbolic equations, SIAM J. Numer. Anal. 56 (2018), no. 2, 732–765. MR 3780748, DOI 10.1137/17M1128605
- Waixiang Cao, Zhimin Zhang, and Qingsong Zou, Superconvergence of discontinuous Galerkin methods for linear hyperbolic equations, SIAM J. Numer. Anal. 52 (2014), no. 5, 2555–2573. MR 3270187, DOI 10.1137/130946873
- Paul Castillo, Bernardo Cockburn, Dominik Schötzau, and Christoph Schwab, Optimal a priori error estimates for the $hp$-version of the local discontinuous Galerkin method for convection-diffusion problems, Math. Comp. 71 (2002), no. 238, 455–478. MR 1885610, DOI 10.1090/S0025-5718-01-01317-5
- G. Darboux, Mémoire sur l’approximation des fonctions de très-grands nombres et sur une classe étendue de développements en série, J. Math. Purer Appl. 4 (1978), 5–56, 377–416.
- Klaus-Jürgen Förster, Inequalities for ultraspherical polynomials and application to quadrature, Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), 1993, pp. 59–70. MR 1256011, DOI 10.1016/0377-0427(93)90135-X
- Jan S. Hesthaven, Sigal Gottlieb, and David Gottlieb, Spectral methods for time-dependent problems, Cambridge Monographs on Applied and Computational Mathematics, vol. 21, Cambridge University Press, Cambridge, 2007. MR 2333926, DOI 10.1017/CBO9780511618352
- E. Kruglov, Pointwise convergence of Jacobi polynomials, Master’s Thesis, Aalto University, 2018.
- Runchang Lin and Zhimin Zhang, Natural superconvergence points in three-dimensional finite elements, SIAM J. Numer. Anal. 46 (2008), no. 3, 1281–1297. MR 2390994, DOI 10.1137/070681168
- Wenjie Liu and Li-Lian Wang, Asymptotics of the generalized Gegenbauer functions of fractional degree, J. Approx. Theory 253 (2020), 105378, 25. MR 4064848, DOI 10.1016/j.jat.2020.105378
- Wenjie Liu, Li-Lian Wang, and Huiyuan Li, Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces, Math. Comp. 88 (2019), no. 320, 2857–2895. MR 3985478, DOI 10.1090/mcom/3456
- Wenjie Liu, Li-Lian Wang, and Boying Wu, Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation, Adv. Comput. Math. 47 (2021), no. 6, Paper No. 79, 32. MR 4329941, DOI 10.1007/s10444-021-09905-3
- Doron S. Lubinsky, A new approach to universality limits involving orthogonal polynomials, Ann. of Math. (2) 170 (2009), no. 2, 915–939. MR 2552113, DOI 10.4007/annals.2009.170.915
- Benjamin Muckenhoupt, Transplantation theorems and multiplier theorems for Jacobi series, Mem. Amer. Math. Soc. 64 (1986), no. 356, iv+86. MR 858466, DOI 10.1090/memo/0356
- Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
- E. B. Saff and V. Totik, Polynomial approximation of piecewise analytic functions, J. London Math. Soc. (2) 39 (1989), no. 3, 487–498. MR 1002461, DOI 10.1112/jlms/s2-39.3.487
- Jie Shen, Tao Tang, and Li-Lian Wang, Spectral methods, Springer Series in Computational Mathematics, vol. 41, Springer, Heidelberg, 2011. Algorithms, analysis and applications. MR 2867779, DOI 10.1007/978-3-540-71041-7
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Elias M. Stein and Rami Shakarchi, Real analysis, Princeton Lectures in Analysis, vol. 3, Princeton University Press, Princeton, NJ, 2005. Measure theory, integration, and Hilbert spaces. MR 2129625, DOI 10.1515/9781400835560
- Gabor Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, New York, 1939. MR 0000077, DOI 10.1090/coll/023
- Terence Tao, An introduction to measure theory, Graduate Studies in Mathematics, vol. 126, American Mathematical Society, Providence, RI, 2011. MR 2827917, DOI 10.1090/gsm/126
- Lloyd N. Trefethen, Six myths of polynomial interpolation and quadrature, Math. Today (Southend-on-Sea) 47 (2011), no. 4, 184–188. MR 2952622
- Lloyd N. Trefethen, Approximation theory and approximation practice, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. MR 3012510
- P. D. Tuan and David Elliott, Coefficients in series expansions for certain classes of functions, Math. Comp. 26 (1972), 213–232. MR 301440, DOI 10.1090/S0025-5718-1972-0301440-2
- Lars B. Wahlbin, A comparison of the local behavior of spline $L_2$-projections, Fourier series and Legendre series, Singularities and constructive methods for their treatment (Oberwolfach, 1983) Lecture Notes in Math., vol. 1121, Springer, Berlin, 1985, pp. 319–346. MR 806401, DOI 10.1007/BFb0076279
- Haiyong Wang, How much faster does the best polynomial approximation converge than Legendre projection?, Numer. Math. 147 (2021), no. 2, 481–503. MR 4215344, DOI 10.1007/s00211-021-01173-z
- Haiyong Wang, On the optimal estimates and comparison of Gegenbauer expansion coefficients, SIAM J. Numer. Anal. 54 (2016), no. 3, 1557–1581. MR 3504991, DOI 10.1137/15M102232X
- Shuhuang Xiang, Convergence rates on spectral orthogonal projection approximation for functions of algebraic and logarithmatic regularities, SIAM J. Numer. Anal. 59 (2021), no. 3, 1374–1398. MR 4259911, DOI 10.1137/20M134407X
- Shuhuang Xiang and Guidong Liu, Optimal decay rates on the asymptotics of orthogonal polynomial expansions for functions of limited regularities, Numer. Math. 145 (2020), no. 1, 117–148. MR 4091598, DOI 10.1007/s00211-020-01113-3
- Zhimin Zhang, Superconvergence points of polynomial spectral interpolation, SIAM J. Numer. Anal. 50 (2012), no. 6, 2966–2985. MR 3022250, DOI 10.1137/120861291
- Zhimin Zhang and Ahmed Naga, A new finite element gradient recovery method: superconvergence property, SIAM J. Sci. Comput. 26 (2005), no. 4, 1192–1213. MR 2143481, DOI 10.1137/S1064827503402837
- Xuan Zhao and Zhimin Zhang, Superconvergence points of fractional spectral interpolation, SIAM J. Sci. Comput. 38 (2016), no. 1, A598–A613. MR 3463700, DOI 10.1137/15M1011172
Additional Information
- Shuhuang Xiang
- Affiliation: School of Mathematics and Statistics, INP-LAMA, Central South University, Changsha 410083, People’s Republic of China
- ORCID: 0000-0002-6727-6170
- Email: xiangsh@csu.edu.cn
- Desong Kong
- Affiliation: School of Mathematics and Statistics, Central South University, Changsha 410083, People’s Republic of China
- MR Author ID: 1312014
- ORCID: 0000-0003-1995-8086
- Email: desongkong@csu.edu.cn
- Guidong Liu
- Affiliation: School of Statistics and Mathematics, Nanjing Audit University, Nanjing 211815, People’s Republic of China
- MR Author ID: 1237006
- Email: csu_guidongliu@163.com
- Li-Lian Wang
- Affiliation: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore
- MR Author ID: 681795
- Email: lilian@ntu.edu.sg
- Received by editor(s): June 16, 2021
- Received by editor(s) in revised form: July 30, 2022, and November 16, 2022
- Published electronically: March 21, 2023
- Additional Notes: The research of the first three authors was supported in part by the National Natural Foundation of China (No. 12271528 and No. 12001280). The research of the second author was supported in part by the Fundamental Research Funds for the Central Universities of Central South University (No. 2020zzts031). The research of the third author was supported in part by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 20KJB110012), and the Natural Science Foundation of Jiangsu Province (No. BK20211293). The research of the fourth author was supported in part by the Singapore MOE AcRF Tier 1 Grant: RG15/21.
The third author is the corresponding author - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 1747-1778
- MSC (2020): Primary 41A10, 41A25, 41A50, 65N35, 65M70
- DOI: https://doi.org/10.1090/mcom/3835
- MathSciNet review: 4570340