Zhu, Qiding A review of two different approaches for superconvergence analysis. (English) Zbl 0938.65128 Appl. Math., Praha 43, No. 6, 401-411 (1998). The paper surveys several methods leading to superconvergence phenomena when solving linear second-order elliptic boundary value problems by the finite element method. These methods include Green’s function method, tensor convolution methods, averaging convolution methods, energy-orthogonalization methods, interpolate postprocessing methods and asymptotic expansion methods. The author compares theoretical results obtained in China and “Western” countries. The Chinese approach enables us to prove superconvergence up to the boundary of a domain in question for a smooth solution. Triangulations used can be, e.g., quasi-uniform, which means that any two adjacent triangles form an approximate parallelogram. In this case triangulations are not locally symmetric with respect to a point. Superconvergence phenomena can also be obtained for piecewise quasi-uniform meshes, which is important from a practical point of view. Reviewer: M.Křížek (Praha) Cited in 1 Document MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs Keywords:finite element method; superconvergence error estimates PDF BibTeX XML Cite \textit{Q. Zhu}, Appl. Math., Praha 43, No. 6, 401--411 (1998; Zbl 0938.65128) Full Text: DOI EuDML 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. · Zbl 0353.65064 [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. · Zbl 0511.65080 [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. · Zbl 0645.65075 [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. [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. [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. · Zbl 0315.65062 [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. · Zbl 0624.65107 [12] Li, B.: Superconvergence for higher-order triangular finite elements. Chinese J. Numer. Math. Appl. 12 (1990), 75-79. [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. · Zbl 0563.65070 [14] Lin, Q., Xu, J. C.: Linear finite elements with high accuracy. J. Comput. Math. 3. (1985), 115-133. · Zbl 0577.65094 [15] Lin, Q., Yan, N. N.: Construction and Analysis for Efficient Finite Element Method. Hebei University Press, 1996. [16] Lin, Q., Zhu, Q. D.: Asymptotic expansion for the derivative of finite elements. J. Comput. Math. 2 (1984), 361-363. · Zbl 0563.65069 [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. [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. · Zbl 0855.65115 [20] Schatz, A. H., Wahlbin, L. B.: Interior maximum norm estimates for finite element methods, Part II. Math. Comp (1995). · Zbl 0826.65091 [21] Thomée, V.: High order local approximation to derivatives in the finite element method. Math. Comp. 31 (1977), 652-660. · Zbl 0367.65055 [22] Wahlbin, L. B.: Superconvergence in Galerkin Finite Element Methods. LN in Math. 1605, Springer, Berlin, 1995. · 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. · 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. · Zbl 0611.65074 [26] Zhu, Q. D.: Uniform superconvergence estimates of derivatives for the finite element method. Numer. Math. J. Xiangtan Univ. 5. · 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. · Zbl 0618.65092 [28] Zhu, Q. D., Lin, Q.: The Superconvergence Theory of Finite Element Methods. Hunan Scientific and Technical Publishers, Changsha, 1989. [29] Zhu, Q. D.: The superconvergence for the 3rd order triangular finite elements. (1997) [30] Zlámal, M.: Some superconvergence results in the finite element method, LN in Math. 606. (1977, 353-362), Springer, Berlin. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.