Skip to main content Accessibility help

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

  • Jianwei Zhou (a1), Danping Yang (a2) and Yujie Liu (a3)


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.


Corresponding author

*Corresponding author. Email: (J. Zhou), (D. Yang), (Y. Liu)


Hide All
[1] Adams, R. A. and Fournier, J. J., Sobolev Spaces, Academic Press, 2003.
[2] Ainsworth, M. and Oden, J. T., A posteriori error estimators in finite element analysis, Comput. Methods Appl. Mech. Eng., 142 (1997), pp. 188.
[3] Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer Science & Business Media, 2008.
[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. 750776.
[5] Bernardi, C. and Maday, Y., Polynomial approximation of some singular functions, Appl. Anal., 42 (1991), pp. 132.
[6] Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A., Spectral Methods in Fluid Dynamics, Springer-Verlag, 1987.
[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. 147167.
[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. 663671.
[9] Ciarlet, P. G., The Finite Element Methods for Elliptic Problems, SIAM, 2002.
[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. 673703.
[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.
[12] Guo, B. Q., Recent progress on a-posteriori error analysis for the p and hp finite element method, Contem. Math., 383 (2005), pp. 4762.
[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. 15931619.
[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. 127155.
[15] Melenk, J. M. and Wohlmuth, B. I., On residual-based a posteriori error estimation in hp- FEM, Adv. Comput. Math., 15 (2001), pp. 311331.
[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. 113180.
[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. 4366.
[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. 14891505.
[19] Shen, J. and Wang, L. L., Spectral approximation of the Helmholtz equation with high wave numbers, SIAM J. Numer. Anal., 43 (2005), pp. 623644.
[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. 201214.
[21] Verfürth, R., A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, John Wiley & Sons Inc., 1996.
[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. 120.
[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. 398409.
[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. 334340.
[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. 29883011.


MSC classification

Related content

Powered by UNSILO

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

  • Jianwei Zhou (a1), Danping Yang (a2) and Yujie Liu (a3)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.