Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T17:34:34.379Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Weighted Averaging Methods for Gradient Recovery

Published online by Cambridge University Press:  03 June 2015

Yunqing Huang ,
Kai Jiang  and
Nianyu Yi
Yunqing Huang*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China
Kai Jiang*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China
Nianyu Yi*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China
*
Corresponding author. URL: http://math.xtu.edu.cn/myphp/math/personal/huangyq/index.htm, Email: huangyq@xtu.edu.cn
Email: jkxtu@yahoo.com.cn
Email: yinianyu365109@126.com
Get access

Abstract

We propose some new weighted averaging methods for gradient recovery, and present analytical and numerical investigation on the performance of these weighted averaging methods. It is shown analytically that the harmonic averaging yields a superconvergent gradient for any mesh in one-dimension and the rectangular mesh in two-dimension. Numerical results indicate that these new weighted averaging methods are better recovered gradient approaches than the simple averaging and geometry averaging methods under triangular mesh.

Keywords

65N1265N1565N3065D1074S0541A1041A25Finite element methodweighted averaginggradient recovery
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 2 , April 2012 , pp. 131 - 155
Copyright
Copyright © Global-Science Press 2012

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] Ainsworth, M. and Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis, New York, Wiley Interscience, 2000.CrossRefGoogle Scholar
[2] Babuska, I. and Strouboulis, T., The Finite Element Method and Its Reliability, Numerical Mathematics and Scientific Computation, 2001.CrossRefGoogle Scholar
[3] Bank, R. E. and Xu, J., Asymptotically exact a posteriori error estimators, part I: grids with superconvergence, SIAM J. Numer. Anal., 41 (2003), pp. 2294–2312.CrossRefGoogle Scholar
[4] Bramble, J. H. and Schatz, A. H., Higher order local accuracy by averaging in the finite element method, Math. Comput., 31 (1977), pp. 74–111.CrossRefGoogle Scholar
[5] Carstensen, C. and Bartels, S., Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids: part I: low order conforming, nonconforming, and mixed FEM, Math. Comput., 71 (2002), pp. 945–969.CrossRefGoogle Scholar
[6] Carstensen, C. and Bartels, S., Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids: part II: higher order FEM, Math. Comput., 71 (2002), pp. 971–994.CrossRefGoogle Scholar
[7] Chen, C. M. and Huang, Y., High Accuracy Theory of Finite Element Methods, Hunan Science Press, Hunan, China (in Chinese), 1995.Google Scholar
[8] Heimsund, B., Tai, X. and Wang, J., Superconvergence for the gradient of the finite element approximations by L2-projections, SIAM J. Numer. Anal., 40 (2002), pp. 1538–1560.CrossRefGoogle Scholar
[9] Hinton, E., Least Square Analysis Using Finite Elements, M.Sc. Thesis, Civ. Engng. Dept., University College of Swansea, 1968.Google Scholar
[10] Hinton, E. and Campbell, J. S., Local and global smoothing of discontinuous finite element functions using a least-square method, Int. J. Numer. Methods. Eng., 8 (1974), pp. 461–480.CrossRefGoogle Scholar
[11] Hlavacek, I., Krizek, M. and Praha, V. P., How to recovery the gradient of linear elements on nonuniform triangulations, App. Mat., 41 (1996), pp. 241–267.CrossRefGoogle Scholar
[12] Huang, Y. and Yi, N., The superconvergent cluster recovery method, J. Sci. Comput., 44 (2010), pp. 301–322.CrossRefGoogle Scholar
[13] Irons, B.M., Least square surface fitting by finite elements, and an application to stess smoothing, Aero. Stree. Memo., ASM 1524, Rolls-Royce, 1967.Google Scholar
[14] Lin, Q. and Yan, N., Construction and Analysis of High Eficient Finite Elements (in Chinese), Hebei University Press, China, 1996.Google Scholar
[15] Maisano, G., Micheletti, S., Perotto, S. and Bottasso, C. L., On some new recovery-based a posteriori error estimators, Comput. Methods. Appl. Mech. Eng., 195 (2006), pp. 4794–4815.CrossRefGoogle Scholar
[16] Naga, A. and Zhang, Z., A Posteriori error estimates based on polynomial preserving recovery, SIAM J. Numer. Anal., 42 (2004), pp. 1780–1800.CrossRefGoogle Scholar
[17] Oden, J. T. and Brauchli, H. J., On the calculation of consistent stess distributions in finite element applications, Int. J. Numer. Meth. Eng., 3 (1971), pp. 317–325.CrossRefGoogle Scholar
[18] Tabbara, M., Blacker, T. and Belytschko, T., Finite element derivative recovery by moving least square interpolants, Comput. Methods. Appl. Mech. Eng., 117 (1994), pp. 211–223.CrossRefGoogle Scholar
[19] Wahlbin, L. B., Superconvergence in Galkerkin Finite Element Methods, Springer Verlag, Berlin, 1995.CrossRefGoogle Scholar
[20] Wang, J., A superconvergence analysis for finite element solutions by the least-squares surface fitting on irregular meshes for smooth problems, J. Math. Study., 33 (2000), pp. 229–243.Google Scholar
[21] Xu, J. and Zhang, Z. M., Analysis of recovery type a posteriori error estimators for mildly structured grids, Math. Comput., (2003), pp. 781–801.Google Scholar
[22] Yan, N.N., Superconvergence Analysis and a Posteriori Error Estimation in Finite Element Methods, Science Press, 2008.Google Scholar
[23] Zhang, Z. and Naga, A., A new finite element gradient recovery method: superconvergence property, SIAM J. Sci. Comput., 26 (2005), pp. 1192–1213.CrossRefGoogle Scholar
[24] Zhang, Z. and Zhu, J. Z., Superconvergent patch recovery technique and a posteriori error estimator in the finite element method (I), Comput. Methods. Appl. Mech. Eng., 123 (1995), pp. 173–187.CrossRefGoogle Scholar
[25] Zhu, J. Z. and Zienkiewicz, O. C., Supercovergence recovery technique and a posteriori error estimates, Int. J. Numer. Meth. Eng., 30 (1990), pp. 1321–1339.CrossRefGoogle Scholar
[26] Zienkiewicz, O. C. and Zhu, J. Z., The superconvergence patch recovery (SPR) and adaptive finite element refinement, Comput. Methods. Appl. Mech. Eng., 101 (1992), pp. 207–224.CrossRefGoogle Scholar
[27] Zienkiewicz, O. C. and Zhu, J. Z., The supercovergent patch recovery and a posteriori error estimates, Int. J. Numer. Methods. Eng., 33 (1992), pp. 1331–1382.CrossRefGoogle Scholar