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

Xu, Jinchao; Zhang, Zhimin

doi:10.1090/S0025-5718-03-01600-4

Analysis of recovery type a posteriori error estimators for mildly structured grids
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by Jinchao Xu and Zhimin Zhang;

Math. Comp. 73 (2004), 1139-1152

DOI: https://doi.org/10.1090/S0025-5718-03-01600-4

Published electronically: August 19, 2003

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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.

References

Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
I. Babuška and W.C. Rheinboldt, A Posteriori Error Estimates for the Finite Element Method, Internat. J. Numer. Methods Engrg., 12 (1978), pp.1597-1615.
Ivo Babuška and Theofanis Strouboulis, The finite element method and its reliability, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2001. MR 1857191
R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985), no. 170, 283–301. MR 777265, DOI 10.1090/S0025-5718-1985-0777265-X
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.
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.
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.
C.M. Chen and Y.Q. Huang, High Accuracy Theory of Finite Element Methods. Hunan Science Press, Hunan, China, 1995 (in Chinese).
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), no. 235, 897–909. MR 1826572, DOI 10.1090/S0025-5718-01-01286-8
M. Křížek, P. Neittaanmäki, and R. Stenberg (eds.), Finite element methods, Lecture Notes in Pure and Applied Mathematics, vol. 196, Marcel Dekker, Inc., New York, 1998. Superconvergence, post-processing, and a posteriori estimates; Papers from the conference held at the University of Jyväskylä, Jyväskylä, 1997. MR 1602809
Bo Li and Zhimin Zhang, Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements, Numer. Methods Partial Differential Equations 15 (1999), no. 2, 151–167. MR 1674357, DOI 10.1002/(SICI)1098-2426(199903)15:2<151::AID-NUM2>3.0.CO;2-O
Q. Lin and N. Yan, Construction and Analysis of High Efficient Finite Elements (in Chinese), Hebei University Press, P.R. China, 1996.
A. H. Schatz and L. B. Wahlbin, Interior maximum-norm estimates for finite element methods. II, Math. Comp. 64 (1995), no. 211, 907–928. MR 1297478, DOI 10.1090/S0025-5718-1995-1297478-7
R. Verfürth, A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques, Teubner Skripten zur Numerik, B.G. Teubner, Stuttgart, 1995.
Lars B. Wahlbin, Superconvergence in Galerkin finite element methods, Lecture Notes in Mathematics, vol. 1605, Springer-Verlag, Berlin, 1995. MR 1439050, DOI 10.1007/BFb0096835
Ningning Yan and Aihui Zhou, Gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes, Comput. Methods Appl. Mech. Engrg. 190 (2001), no. 32-33, 4289–4299. MR 1832657, DOI 10.1016/S0045-7825(00)00319-4
Zhimin Zhang, Ultraconvergence of the patch recovery technique. II, Math. Comp. 69 (2000), no. 229, 141–158. MR 1680911, DOI 10.1090/S0025-5718-99-01205-3
Zhimin Zhang and Harold Dean Victory Jr., Mathematical analysis of Zienkiewicz-Zhu’s derivative patch recovery technique, Numer. Methods Partial Differential Equations 12 (1996), no. 4, 507–524. MR 1396469, DOI 10.1002/(SICI)1098-2426(199607)12:4<507::AID-NUM6>3.0.CO;2-Q
Q.D. Zhu and Q. Lin, Superconvergence Theory of the Finite Element Method, Hunan Science Press, China, 1989 (in Chinese).
J. Z. Zhu and Zhimin Zhang, The relationship of some a posteriori estimators, Comput. Methods Appl. Mech. Engrg. 176 (1999), no. 1-4, 463–475. New advances in computational methods (Cachan, 1997). MR 1665357, DOI 10.1016/S0045-7825(98)00349-1
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), no. 2, 337–357. MR 875306, DOI 10.1002/nme.1620240206
O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Internat. J. Numer. Methods Engrg. 33 (1992), no. 7, 1331–1364. MR 1161557, DOI 10.1002/nme.1620330702
O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. II. Error estimates and adaptivity, Internat. J. Numer. Methods Engrg. 33 (1992), no. 7, 1365–1382. MR 1161558, DOI 10.1002/nme.1620330703