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Analysis of recovery type a posteriori error estimators for mildly structured grids


Authors: Jinchao Xu and Zhimin Zhang
Journal: Math. Comp. 73 (2004), 1139-1152
MSC (2000): Primary 65N30; Secondary 65N50, 65N15, 65N12, 65D10, 74S05, 41A10, 41A25
DOI: https://doi.org/10.1090/S0025-5718-03-01600-4
Published electronically: August 19, 2003
MathSciNet review: 2047081
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Abstract | References | Similar Articles | Additional Information

Abstract: Some recovery type error estimators for linear finite elements are analyzed under $O(h^{1+\alpha})$ $(\alpha > 0)$ regular grids. Superconvergence of order $O(h^{1+\rho})$ $(0 < \rho \le \alpha)$ is established for recovered gradients by three different methods. As a consequence, a posteriori error estimators based on those recovery methods are asymptotically exact.


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  • 1. M. Ainsworth and J.T. Oden, A Posteriori Estimation in Finite Element Analysis, Wiley Interscience, New York, 2000. MR 2003b:65001
  • 2. I. Babuska and W.C. Rheinboldt, A Posteriori Error Estimates for the Finite Element Method, Internat. J. Numer. Methods Engrg., 12 (1978), pp.1597-1615.
  • 3. I. Babuska and T. Strouboulis, The Finite Element Method and its Reliability, Oxford University Press, London, 2001. MR 2002k:65001
  • 4. R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), pp.283-301. MR 86g:65207
  • 5. R.E. Bank and J. Xu, Asymptotically exact a posteriori error estimators, Part I: Grid with superconvergence, preprint, to appear in SIAM J. Numer. Anal.
  • 6. R.E. Bank and J. Xu, Asymptotically exact a posteriori error estimators, Part II: General Unstructured Grids, preprint, to appear in SIAM J. Numer. Anal.
  • 7. C. Carstensen and S. Bartels, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM, Math. Comp. 71 (2002), 945-969.
  • 8. C.M. Chen and Y.Q. Huang, High Accuracy Theory of Finite Element Methods. Hunan Science Press, Hunan, China, 1995 (in Chinese).
  • 9. W. Hoffmann, A.H. Schatz, L.B. Wahlbin, and G. Wittum, Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes I: A smooth problem and globally quasi-uniform meshes, Math. Comp., 70 (2001), pp.897-909. MR 2002a:65178
  • 10. M. Krízek, P. Neittaanmäki, and R. Stenberg (Eds.), Finite Element Methods: Superconvergence, Post-processing, and A Posteriori Estimates, Lecture Notes in Pure and Applied Mathematics Series, Vol.196, Marcel Dekker, New York, 1997. MR 98i:65003
  • 11. B. Li and Z. Zhang, Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements, Numerical Methods for Partial Differential Equations, 15 (1999), pp.151-167. MR 99m:65201
  • 12. Q. Lin and N. Yan, Construction and Analysis of High Efficient Finite Elements (in Chinese), Hebei University Press, P.R. China, 1996.
  • 13. A.H. Schatz and L.B. Wahlbin, Interior maximum norm estimates for finite element methods. Part II, Math.Comp., 64 (1995), pp.907-928. MR 95j:65143
  • 14. R. Verfürth, A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, Teubner Skripten zur Numerik, B.G. Teubner, Stuttgart, 1995.
  • 15. L.B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics, Vol.1605, Springer, Berlin, 1995. MR 98j:65083
  • 16. N. Yan and A. Zhou, Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes, Comput. Methods Appl. Mech. Engrg., 190 (2001), pp.4289-4299. MR 2002c:65189
  • 17. Z. Zhang, Ultraconvergence of the patch recovery technique II, Math. Comp., 69 (2000), pp.141-158. MR 2000j:65107
  • 18. Z. Zhang and H.D. Victory Jr., Mathematical Analysis of Zienkiewicz-Zhu's derivative patch recovery techniques, Numerical Methods for Partial Differential Equations, 12 (1996), pp.507-524. MR 98c:65191
  • 19. Q.D. Zhu and Q. Lin, Superconvergence Theory of the Finite Element Method, Hunan Science Press, China, 1989 (in Chinese).
  • 20. J.Z. Zhu and Z. Zhang, The relationship of some a posteriori error estimators, Comput. Methods Appl. Mech. Engrg., 176 (1999), pp.463-475. MR 2000f:65126
  • 21. O.C. Zienkiewicz and J.Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg., 24 (1987), pp.337-357. MR 87m:73055
  • 22. O.C. Zienkiewicz and J.Z. Zhu, The superconvergence patch recovery and a posteriori error estimates, Part 1: The recovery technique, Internat. J. Numer. Methods Engrg., 33 (1992), pp.1331-1364. MR 93c:73098
  • 23. O.C. Zienkiewicz and J.Z. Zhu, The superconvergence patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity, Internat. J. Numer. Methods Engrg., 33 (1992), pp.1365-1382. MR 93c:73099

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Additional Information

Jinchao Xu
Affiliation: Center for Computational Mathematics and Applications, Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Email: xu@math.psu.edu

Zhimin Zhang
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: zzhang@math.wayne.edu

DOI: https://doi.org/10.1090/S0025-5718-03-01600-4
Keywords: Gradient recovery, ZZ patch recovery, superconvergence, {\it a posteriori} error estimates
Received by editor(s): June 26, 2002
Received by editor(s) in revised form: December 15, 2002
Published electronically: August 19, 2003
Additional Notes: The work of the first author was supported in part by the National Science Foundation grant DMS-9706949 and the Center for Computational Mathematics and Applications, Penn State University
The work of the second author was supported in part by the National Science Foundation grants DMS-0074301, DMS-0079743, and INT-0196139
Article copyright: © Copyright 2003 American Mathematical Society

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