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# Article

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Keywords:
finite element method; superconvergence error estimates
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.
References:
[1] Bramble, J. H., Schatz A. H.: High order local accuracy by averaging in the finite element method. Math. Comp. 31 (1977), 94–111. DOI 10.1090/S0025-5718-1977-0431744-9 | MR 0431744
[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.
[3] Chen, C. M.: Superconvergence of finite element solution and its derivatives. Numer. Math. J. Chinese Univ. 3:2 (1981), 118–125. MR 0635547
[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
[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.
[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
[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
[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
[9] He, W. M.: A derivative extrapolation for second order triangular element. (1997), Master thesis.
[10] Jia, Z. P.: The high accuracy arithmetic for $k$-th order rectangular finite element. (1990), Master thesis.
[11] Křížek, M., Neittaanmäki, P.: On superconvegence techniques. Acta Appl. Math. 9 (1987), 175–198. DOI 10.1007/BF00047538
[12] Li, B.: Superconvergence for higher-order triangular finite elements. Chinese J. Numer. Math. Appl. 12 (1990), 75–79. MR 1118707
[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.
[14] Lin, Q., Xu, J. C.: Linear finite elements with high accuracy. J. Comput. Math. 3. (1985), 115–133. MR 0854355
[15] Lin, Q., Yan, N. N.: Construction and Analysis for Efficient Finite Element Method. Hebei University Press, 1996. (Chinese)
[16] Lin, Q., Zhu, Q. D.: Asymptotic expansion for the derivative of finite elements. J. Comput. Math. 2 (1984), 361–363. MR 0869509
[17] Lin, Q., Zhu, Q. D.: The Preprocessing and Postprocessing for the Finite Element Method. Shanghai Scientific & Technical Publishers, 1994.
[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. DOI 10.1016/0041-5553(69)90159-1
[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. DOI 10.1137/0733027 | MR 1388486
[20] Schatz, A. H., Wahlbin, L. B.: Interior maximum norm estimates for finite element methods, Part II. Math. Comp (1995). MR 0431753
[21] Thomée, V.: High order local approximation to derivatives in the finite element method. Math. Comp. 31 (1977), 652–660. DOI 10.2307/2005998 | MR 0438664
[22] Wahlbin, L. B.: Superconvergence in Galerkin Finite Element Methods. LN in Math. 1605, Springer, Berlin, 1995. MR 1439050 | Zbl 0826.65092
[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. MR 1602738 | Zbl 0902.65046
[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.
[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. MR 0754041 | Zbl 0611.65074
[26] Zhu, Q. D.: Uniform superconvergence estimates of derivatives for the finite element method. Numer. Math. J. Xiangtan Univ. 5. MR 0745576 | Zbl 0549.65073
[27] Zhu, Q. D.: Uniform superconvergence estimates for the finite element method. Natur. Sci. J. Xiangtan Univ. (19851983), 10–26 311–318. MR 0890708
[28] Zhu, Q. D., Lin, Q.: The Superconvergence Theory of Finite Element Methods. Hunan Scientific and Technical Publishers, Changsha, 1989. (Chinese)
[29] Zhu, Q. D.: The superconvergence for the 3rd order triangular finite elements. (1997) (to appear).
[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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