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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: [email protected] (J. W. Zhou)
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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 

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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