Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T04:16:02.409Z Has data issue: false hasContentIssue false
  • English
  • Français

The a Posteriori Error Estimates for Chebyshev-Galerkin Spectral Methods in One Dimension

Published online by Cambridge University Press:  23 March 2015

Jianwei Zhou
Jianwei Zhou*
Affiliation:
Department of Mathematics, Linyi University, Shandong 276005, China
*
*Corresponding author. Email: jwzhou@yahoo.com (J. W. Zhou)
Get access

Abstract

In this paper, the Chebyshev-Galerkin spectral approximations are employed to investigate Poisson equations and the fourth order equations in one dimension. Meanwhile, p-version finite element methods with Chebyshev polynomials are utilized to solve Poisson equations. The efficient and reliable a posteriori error estimators are given for different models. Furthermore, the a priori error estimators are derived independently. Some numerical experiments are performed to verify the theoretical analysis for the a posteriori error indicators and a priori error estimations.

Keywords

65N3065N3565M70Chebyshev-Galerkin spectral approximationChebyshev polynomiala posteriori error indicatorp-version finite element method
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 2 , April 2015 , pp. 145 - 157
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adams, R. A., Sobolev Spaces, Academic Press, 1978.Google Scholar
[2]Ainsworth, M. and Oden, J. T., A posteriori error estimators in finite element analysis, Comput. Methods Appl. Mech. Eng., 142 (1997), pp. 88–88.Google Scholar
[3]Babuška, I. and Suri, M., The optimal covergence rate of the p-version of the finite element method, SIAM J. Numer. Anal., 24 (1987), pp. 776–776.Google Scholar
[4]Bernardi, C., Indicateurs d’erreur en h-N version des element spaectraux, M2AN, 30(1996), pp. 38–38.Google Scholar
[5]Bernardi, C. and Maday, Y., Spectral methods, in Handbook of Numerical Analysis, Ciarlet, P. G. and Lions, J.-L., eds., Elsevier, Amsterdam, 1997, pp. 486–486.Google Scholar
[6]Bium, H. and Rannacher, R., On mixed finite element methods in plate beding analysis, Comput. Mech., 6 (1990), pp. 236–236.Google Scholar
[7]Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer, New York, 1994.Google Scholar
[8]Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods in Fluid Dynamics, Springer-Verlag, 1987.Google Scholar
[9]Charbonneau, A., Dossou, K., and Pierre, R., A residual-based a posteriori error estimator for the Ciarlet-Raviart formulation of the first Biharmonic problem, Numerical Methods for Partial Differential Equations, 13 (1997), pp. 93–111.Google Scholar
[10]Cheng, X. L., Han, W. M. and Huang, H. C.,, Some mixed finite element methods for biharmonic equation, J. Comput. Appl. Math., 126 (2000), pp. 91–109.Google Scholar
[11]Ciarlet, P. G., The Finite Element Methods for Elliptic Problems, North-Holland Publishing Company, Amsterdam, 1978.Google Scholar
[12]Evans, LAWRENCE C., Partial Differential Equations, Graduate Studies in Mathematics, 19 (1997).Google Scholar
[13]Gui, C. and Babuška, I., The h, p and h-p versions of the finite element method in 1 dimension, Part 1: the error analysis of the p-version; Part 2: The error analysis of the h- and h-p version; Part 3: The adaptive h-p version, Numer. Math., 49 (1986), pp. 577–683.Google Scholar
[14]Guo, B. Q., Recent progress in a-posteriori error estimation for the p-version of finite element method, Recent Advances in Adaptive, eds. Shi, Z-C., Chen, Z., Tang, T. and Yu, D., AMS, Comtemporary Mathematics, 383 (2005), pp. 47–61.Google Scholar
[15]Gottlieb, D. and Orszag, S. A., Numerical Analysis of Spectral Methods: Theory and Applications, SIAM-CBMS, Philadelphia, 1977.Google Scholar
[16]Maday, Y., Analysis of spectral projectors in one-dimensional domains, Math. Comput., 55 (1990), pp. 537–562.CrossRefGoogle Scholar
[17]Melenk, M. and Wohlmuth, B., On residual-based a posteriori error estimation in hp-FEM, Adv. Comput. Math., 15 (2001), pp. 311–331.Google Scholar
[18]Monk, P., A mixed finite element method for the biharmonic equation, SIAM J. Numer. Anal., 24 (1987), pp. 737–749.Google Scholar
[19]Oden, T., Demokowicz, L., Rachowicz, W. and Westermann, T. A., Towards a unversal hp-adaptive finite element method, II. a posteriori error estimation, Comput. Meth. Appl. Mech. Eng., 77 (1984), pp. 113–180.Google Scholar
[20]Shen, J., Efficient Spectral-Galerkin method II. direct solvers of the second and fourth order equations by using Chebyshev polynomials, SIAM J. Sci. Comput., 16 (1995), pp. 74–87.CrossRefGoogle Scholar
[21]Shen, J. and Wang, L. L., Spectral approximation of the Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 43 (2005), pp. 623–644.Google Scholar
[22]Scott, L. R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary condition, Math. Comput., 54 (1990), pp. 483–493.Google Scholar
[23]Xiang, X. M., Numerical Analysis of Spectral Methods, Science Press, Beijing, 2000.Google Scholar
[24]Zhou, J. W. and Yang, D. P., An improved a posteriori error estimate for the Galerkin spectral method in one dimension, Comput. Math. Appl., 61 (2011), pp. 334–340.Google Scholar
[25]Zhou, J. W. and Yang, D. P., Spectral mixed Galerkin method for state constrained optimal control problem governed by the first bi-harmonic equation, Int. J. Comput. Math., 88 (2011), pp. 2988–3011.Google Scholar