A review of two different approaches for superconvergence analysis

Institute of Mathematics, Czech Academy of Sciences - Tập 43 Số 6 - Trang 401-411 - 1998
Quanxin Zhu1
1Institute of Computer Science, Hunan Normal University, Changsha, China

Tóm tắt

Từ khóa


Tài liệu tham khảo

Bramble, J. H., Schatz A. H.: High order local accuracy by averaging in the finite element method. Math. Comp. 31 (1977), 94–111.

Chen, C. M.: Optimal points of the stresses approximated by triangular linear element in FEM. Natur. Sci. J. Xiangtan Univ. 1 (1978), 77–90.

Chen, C. M.: Superconvergence of finite element solution and its derivatives. Numer. Math. J. Chinese Univ. 3:2 (1981), 118–125.

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.

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.

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.

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.

Douglas, J. Jr., Dupont, T., Wheeler, M. F.: An L ∞ 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.

He, W. M.: A derivative extrapolation for second order triangular element. Master thesis.

Jia, Z. P.: The high accuracy arithmetic for k-th order rectangular finite element. Master thesis.

Křížek, M., Neittaanmäki, P.: On superconvegence techniques. Acta Appl. Math. 9 (1987), 175–198.

Li, B.: Superconvergence for higher-order triangular finite elements. Chinese J. Numer. Math. Appl. 12 (1990), 75–79.

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.

Lin, Q., Xu, J. C.: Linear finite elements with high accuracy. J. Comput. Math. 3.

Lin, Q., Yan, N. N.: Construction and Analysis for Efficient Finite Element Method. Hebei University Press, 1996. (In Chinese.)

Lin, Q., Zhu, Q. D.: Asymptotic expansion for the derivative of finite elements. J. Comput. Math. 2 (1984), 361–363.

Lin, Q., Zhu, Q. D.: The Preprocessing and Postprocessing for the Finite Element Method. Shanghai Scientific & Technical Publishers, 1994.

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.

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.

Schatz, A. H., Wahlbin, L. B.: Interior maximum norm estimates for finite element methods, Part II. Math. Comp (1995).

Thomée, V.: High order local approximation to derivatives in the finite element method. Math. Comp. 31 (1977), 652–660.

Wahlbin, L. B.: Superconvergence in Galerkin Finite Element Methods. LN in Math. 1605, Springer, Berlin, 1995.

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.

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.

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.

Zhu, Q. D.: Uniform superconvergence estimates of derivatives for the finite element method. Numer. Math. J. Xiangtan Univ. 5.

Zhu, Q. D.: Uniform superconvergence estimates for the finite element method. Natur. Sci. J. Xiangtan Univ. (1983), 311–318.

Zhu, Q. D., Lin, Q.: The Superconvergence Theory of Finite Element Methods. Hunan Scientific and Technical Publishers, Changsha, 1989. (In Chinese).

Zhu, Q. D.: The superconvergence for the 3rd order triangular finite elements. To appear.

Zlámal, M.: Some superconvergence results in the finite element method, LN in Math. 606. Springer, Berlin.