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Title: A review of two different approaches for superconvergence analysis (English)
Author: Zhu, Qiding
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 43
Issue: 6
Year: 1998
Pages: 401-411
Summary lang: English
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Category: math
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Summary: In 1995, Wahbin presented a method for superconvergence analysis called “Interior symmetric method,” and declared that it is universal. In this paper, we carefully examine two superconvergence techniques used by mathematicians both in China and in America. We conclude that they are essentially different. (English)
Keyword: finite element method
Keyword: superconvergence error estimates
MSC: 65N12
MSC: 65N30
idZBL: Zbl 0938.65128
idMR: MR1652096
DOI: 10.1023/A:1023220520477
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Date available: 2009-09-22T17:59:06Z
Last updated: 2020-07-02
Stable URL: http://hdl.handle.net/10338.dmlcz/134397
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Reference: [1] Bramble, J. H., Schatz A. H.: High order local accuracy by averaging in the finite element method.Math. Comp. 31 (1977), 94–111. MR 0431744, 10.1090/S0025-5718-1977-0431744-9
Reference: [2] Chen, C. M.: Optimal points of the stresses approximated by triangular linear element in FEM.Natur. Sci. J. Xiangtan Univ. 1 (1978), 77–90.
Reference: [3] Chen, C. M.: Superconvergence of finite element solution and its derivatives.Numer. Math. J. Chinese Univ. 3:2 (1981), 118–125. MR 0635547
Reference: [4] Chen, C. M., Liu, J. G.: Superconvergence of gradient of triangular linear element in general domain.Natur. Sci. J. Xiangtan Univ. 1 (1987), 114–127. MR 0899930
Reference: [5] Chen, C. M., Zhu Q. D.: A new estimate for the finite element method and optimal point theorem for stresses.Natur. Sci. J. Xiangtan Univ. 1 (1978), 10–20.
Reference: [6] Ding, X. X., Jiang, L.S., Lin, Q., Luo, P. Z.: The finite element method for 4th order non-linear differential equation.Acta Mathematica Sinica 20:2 (1977), 109–118. MR 0657978
Reference: [7] Douglas, J. Jr., Dupond, T.: Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary value problems.Topics in Numerical Analysis, Academic Press, 1973, pp. 89–92. MR 0366044
Reference: [8] Douglas, J. Jr., Dupont, T., Wheeler, M. F.: An $L^{\infty }$ estimate and a superconvergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials.RAIRO Anal. Numér. 8 (1974), 61–66. MR 0359358
Reference: [9] He, W. M.: A derivative extrapolation for second order triangular element.(1997), Master thesis.
Reference: [10] Jia, Z. P.: The high accuracy arithmetic for $k$-th order rectangular finite element.(1990), Master thesis.
Reference: [11] Křížek, M., Neittaanmäki, P.: On superconvegence techniques.Acta Appl. Math. 9 (1987), 175–198. 10.1007/BF00047538
Reference: [12] Li, B.: Superconvergence for higher-order triangular finite elements.Chinese J. Numer. Math. Appl. 12 (1990), 75–79. MR 1118707
Reference: [13] Lin, Q., Lu, T., Shen, S. M.: Maximum norm estimates extrapolation and optimal points of stresses for the finite element methods on the strongly regular triangulation.J. Comput. Math. 1 (1983), 376–383.
Reference: [14] Lin, Q., Xu, J. C.: Linear finite elements with high accuracy. J. Comput. Math. 3.(1985), 115–133. MR 0854355
Reference: [15] Lin, Q., Yan, N. N.: Construction and Analysis for Efficient Finite Element Method.Hebei University Press, 1996. (Chinese)
Reference: [16] Lin, Q., Zhu, Q. D.: Asymptotic expansion for the derivative of finite elements.J. Comput. Math. 2 (1984), 361–363. MR 0869509
Reference: [17] Lin, Q., Zhu, Q. D.: The Preprocessing and Postprocessing for the Finite Element Method.Shanghai Scientific & Technical Publishers, 1994.
Reference: [18] Oganesyan, L. A., Rukhovetz, L. A.: A study of the rate of convergence of variational difference schemes for second order elliptic equations in a two-dimensional region with a smooth boundary.U.S.S.R. Comput. Math. and Math. Phys. 9 (1969), 158–183. 10.1016/0041-5553(69)90159-1
Reference: [19] Schatz, A. H., Sloan, I. H., Wahlbin, L. B.: Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point.SIAM J. Numer. Anal. 33 (1996), 505–521. MR 1388486, 10.1137/0733027
Reference: [20] Schatz, A. H., Wahlbin, L. B.: Interior maximum norm estimates for finite element methods, Part II..Math. Comp (1995). MR 0431753
Reference: [21] Thomée, V.: High order local approximation to derivatives in the finite element method.Math. Comp. 31 (1977), 652–660. MR 0438664, 10.2307/2005998
Reference: [22] Wahlbin, L. B.: Superconvergence in Galerkin Finite Element Methods.LN in Math. 1605, Springer, Berlin, 1995. Zbl 0826.65092, MR 1439050
Reference: [23] Wahlbin, L. B.: General principles of superconvergence in Galerkin finite element methods.In Finite element methods: superconvergence, post-processing and a posteriori estimates, M. Křížek, P. Neittaanmäki, R. Stenberg (eds.), Marcel Dekker, New York, 1998, pp. 269–285. Zbl 0902.65046, MR 1602738
Reference: [24] Zhu, Q. D.: The derivative optimal point of the stresses for second order finite element method.Natur. Sci. J. Xiangtan Univ. 3 (1981), 36–45.
Reference: [25] Zhu, Q. D.: Natural inner superconvergence for the finite element method.In Proc. of the China-France Symposium on Finite Element Methods (Beijing 1982), Science Press, Gorden and Breach, Beijing, 1983, pp. 935–960. Zbl 0611.65074, MR 0754041
Reference: [26] Zhu, Q. D.: Uniform superconvergence estimates of derivatives for the finite element method.Numer. Math. J. Xiangtan Univ. 5. Zbl 0549.65073, MR 0745576
Reference: [27] Zhu, Q. D.: Uniform superconvergence estimates for the finite element method.Natur. Sci. J. Xiangtan Univ. (19851983), 10–26 311–318. MR 0890708
Reference: [28] Zhu, Q. D., Lin, Q.: The Superconvergence Theory of Finite Element Methods.Hunan Scientific and Technical Publishers, Changsha, 1989. (Chinese)
Reference: [29] Zhu, Q. D.: The superconvergence for the 3rd order triangular finite elements.(1997) (to appear).
Reference: [30] Zlámal, M.: Some superconvergence results in the finite element method, LN in Math. 606.(1977, 353–362), Springer, Berlin. MR 0488863
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