Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T06:55:01.215Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access

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 

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. 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