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A-Posteriori Error Estimates for Uniform p-Version Finite Element Methods in Square

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Partial differential equations, boundary value problems

Published online by Cambridge University Press:  09 January 2017

Jianwei Zhou ,
Danping Yang  and
Yujie Liu
Jianwei Zhou*
Affiliation:
Department of Mathematics, Linyi University, Shandong 276005, China
Danping Yang*
Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200241, China
Yujie Liu*
Affiliation:
School of Data and Computational Science, Sun Yat-sen University, Guangzhou, Guangdong 510275, China
*
*Corresponding author. Email:[email protected] (J. Zhou), [email protected] (D. Yang), [email protected] (Y. Liu)
*Corresponding author. Email:[email protected] (J. Zhou), [email protected] (D. Yang), [email protected] (Y. Liu)
*Corresponding author. Email:[email protected] (J. Zhou), [email protected] (D. Yang), [email protected] (Y. Liu)
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Abstract

In this work, the a-posteriori error indicator with an explicit formula for p-version finite element methods in square is investigated, and its reliable and efficient properties are deduced. Especially, this a-posteriori error indicator is determined by the right hand itemof themodel. We reformulate this a-posteriori error indicator with finite coefficients, which can be easily calculated during applications.

Keywords

a-posteriori error indicatorLegendre polynomialp-version finite element method

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M15: Error bounds
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 2 , April 2017 , pp. 272 - 281
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Adams, R. A. and Fournier, J. J., Sobolev Spaces, Academic Press, 2003.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. 1–88.Google Scholar
[3] Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer Science & Business Media, 2008.CrossRefGoogle Scholar
[4] Babuška, L. and Suri, M., The optimal convergence rate of the p-version of the finite element method, SIAM J. Numer. Anal., 24 (1987), pp. 750–776.Google Scholar
[5] Bernardi, C. and Maday, Y., Polynomial approximation of some singular functions, Appl. Anal., 42 (1991), pp. 1–32.Google Scholar
[6] Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A., Spectral Methods in Fluid Dynamics, Springer-Verlag, 1987.Google Scholar
[7] Chen, Y. P. and Tang, T., Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comput., 79 (2010), pp. 147–167.Google Scholar
[8] Chen, Y. P., Xia, N. S. and Yi, N. Y., A Legendre Galerkin spectral method for optimal control problems, J. Syst. Sci. Complex., 24 (2011), pp. 663–671.Google Scholar
[9] Ciarlet, P. G., The Finite Element Methods for Elliptic Problems, SIAM, 2002.CrossRefGoogle Scholar
[10] Düster, A., Bröker, H. and Rank, E., The p-version of the finite element method for three-dimensional curved thin walled structures, Int. J. Numer. Meth. Eng., 52 (2001), pp. 673–703.Google Scholar
[11] Gui, W. and Babuška, I., The h, p and hp versions of the finite element method in 1 dimension: Part 1. The error analysis of the p-version (No. BN-1036), Maryland Univ. College Park Lab for Numerical Analysis, 1985.Google Scholar
[12] Guo, B. Q., Recent progress on a-posteriori error analysis for the p and hp finite element method, Contem. Math., 383 (2005), pp. 47–62.Google Scholar
[13] Kelly, D. W., Gago, D. S., Zienkiewicz, O. C. and Babuška, I., A posteriori error analysis and adaptive processes in the finite element method: Part I–error analysis, Int. J. Numer. Meth. Eng., 19 (1983), pp. 1593–1619.Google Scholar
[14] Melenk, J. M., hp-interpolation of nonsmooth functions and an application to hp-a posteriori error estimation, SIAM J. Numer. Anal., 43 (2005), pp. 127–155.CrossRefGoogle Scholar
[15] Melenk, J. M. and Wohlmuth, B. I., On residual-based a posteriori error estimation in hp- FEM, Adv. Comput. Math., 15 (2001), pp. 311–331.Google Scholar
[16] Oden, J. T., Demokowicz, L., Rachowicz, W. and Westermann, T. A., Towards a universal hp-adaptive finite element method II: a posteriori error estimation, Comput. Meth. Appl. Mech. Eng., 77 (1989), pp. 113–180.CrossRefGoogle Scholar
[17] Schmidt, A. and Siebert, K. G., A posteriori estimators for the hp version of the finite element method in 1d, Appl. Numer. Math., 35 (2000), pp. 43–66.CrossRefGoogle Scholar
[18] Shen, J., Efficient spectral-Galerkin method I: direct solvers for second and fourth order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), pp. 1489–1505.Google Scholar
[19] 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
[20] Vejchodský, T. and Šolín, P., Discrete maximum principle for Poisson equation with mixed boundary conditions solved by hp-FEM, Adv. Appl. Math. Mech., 1 (2009), pp. 201–214.Google Scholar
[21] Verfürth, R., A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, John Wiley & Sons Inc., 1996.Google Scholar
[22] Wei, Y. X. and Chen, Y. P., Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions, Adv. Appl. Math. Mech., 4 (2012), pp. 1–20.Google Scholar
[23] Yang, J. M. and Chen, Y. P., A posteriori error analysis for a fully discrete discontinuous Galerkin approximation to a kind of reactive transport problems, J. Syst. Sci. Complex., 25 (2012), pp. 398–409.CrossRefGoogle Scholar
[24] Zhou, J. W. and Yang, D. P., Improved a posteriori error estimate for Galerkin spectral method in one dimension, Comput. Math. Appl., 61 (2011), pp. 334–340.CrossRefGoogle 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