In this paper, some superconvergence results of high-degree finite element method are obtained for solving a second order elliptic equation with variable coefficients on the inner locally symmetric mesh with respect to a point x 0 for triangular meshes. By using of the weak estimates and local symmetric technique, we obtain improved discretization errors of O(h p+1 |ln h|2) and O(h p+2 |ln h|2) when p (≥ 3) is odd and p (≥ 4) is even, respectively. Meanwhile, the results show that the combination of the weak estimates and local symmetric technique is also effective for superconvergence analysis of the second order elliptic equation with variable coefficients.
@article{bwmeta1.element.doi-10_2478_s11533-014-0440-z,
author = {Xiaofei Guan and Mingxia Li and Wenming He and Zhengwu Jiang},
title = {Some superconvergence results of high-degree finite element method for a second order elliptic equation with variable coefficients},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {1733-1747},
zbl = {1302.65238},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0440-z}
}
Xiaofei Guan; Mingxia Li; Wenming He; Zhengwu Jiang. Some superconvergence results of high-degree finite element method for a second order elliptic equation with variable coefficients. Open Mathematics, Tome 12 (2014) pp. 1733-1747. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-014-0440-z/
[1] Brenner S.C., Scott L.R., The Mathematical Theory of Finite Element Methods, Texts Appl. Math., 15, Springer-Verlag, New York, 1994 | Zbl 0804.65101
[2] Ciarlet P. G., The Finite Element Method for Ellptic Problem, North-Holland, Amsterdam, 1978
[3] Guan X.F., Li M.X., Chen S.C., Some numerical quadrature schemes of a non-conforming quadrilateral finite element, Acta Math. Appl. Sin., 2012, 28, 117–126 http://dx.doi.org/10.1007/s10255-012-0127-9
[4] He W.M., Chen W.Q., Zhu Q.D., Local superconvergence of the derivative for tensor-product block FEM, Numer. Methods Partial Differential Equations, 2012, 28, 457–475 http://dx.doi.org/10.1002/num.20628 | Zbl 1244.65158
[5] He W.M., Cui J.Z., The local superconvergence of the linear finite element method for the Poisson problem, Numer. Methods Partial Differential Equations (in press), DOI: 10.1002/num.21842 | Zbl 1297.65133
[6] Krasovskiĭ J.P., Isolation of singularities of the Green’s function, Math. USSR Izv., 1967, 1, 935–966 http://dx.doi.org/10.1070/IM1967v001n05ABEH000594
[7] Lin Q., Lin J., Finite Element Methods: Accuracy and Improvement, Science Press, Beijing, 2006 (in Chinese)
[8] Lin Q., Yan N., Construction and Analysis of Finite Element Methods, Hebei University Press, 1996 (in Chinese)
[9] Lin Q., Zhang S., Yan N., An acceleration method for integral equations by using interpolation post-processing, Adv. Comp. Math., 1998, 9, 117–128 http://dx.doi.org/10.1023/A:1018925103993 | Zbl 0920.65087
[10] Lin Q., Zhou J., Superconvergence in high-order Galerkin finite element methods, Comput. Method Appl. Mech. Engrg., 2007, 196, 3779–3784 http://dx.doi.org/10.1016/j.cma.2006.10.027 | Zbl 1173.65370
[11] Lin Q., Zhu Q., The Preprocessing and Postprocessing for the Finite Element Methods, Scientific and Technical Publishers, Shanghai, 1994 (in Chinese)
[12] Mao S.P., Chen S.C., Shi D.Y., Convergence and superconvergence of a nonconforming finite element on anisotropic meshes, Int. J. Numer. Anal. Model., 2007, 4, 16–38 | Zbl 1122.65098
[13] Mao S.P., Chen S.C., Sun H., A quadrilateral, anisotropic, superconvergent, nonconforming double set parameter element, Appl. Numer. Math., 2006, 56, 937–961 http://dx.doi.org/10.1016/j.apnum.2005.07.005 | Zbl 1094.65120
[14] Mao S.P., Shi Z.-C., High accuracy analysis of two nonconforming plate elements, Numer. Math., 2009, 111, 407–443 http://dx.doi.org/10.1007/s00211-008-0190-6 | Zbl 1155.74045
[15] Meng L., Zhu Q., The ultraconvergence of derivative for bicubic finite element, Comput. Method Appl. Mech. Engrg., 2007, 196, 3771–3778 http://dx.doi.org/10.1016/j.cma.2006.10.026 | Zbl 1173.65358
[16] 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., 1996, 33, 505–521 http://dx.doi.org/10.1137/0733027 | Zbl 0855.65115
[17] Zhang Z., Ultraconvergence of the patch recovery technique, Math. Comp., 1996, 65, 1431–1437 http://dx.doi.org/10.1090/S0025-5718-96-00782-X | Zbl 0853.65116
[18] Zhang Z., Ultraconvergence of the patch recovery technique(II), Math. Comp., 2000, 69, 141–158 http://dx.doi.org/10.1090/S0025-5718-99-01205-3 | Zbl 0936.65132
[19] Zhang Z., Lin R., Ultraconvergence of ZZ patch recovery at mesh symmetry points, Numer. Math., 2003, 95, 781–801 http://dx.doi.org/10.1007/s00211-003-0457-x | Zbl 1045.65096
[20] Zhang Z., Zhu J., Analysis of the superconvergence patch recovery techniques and a posteriori error estimator in the finite element method(I), Comput. Methods Appl. Mech. Engrg., 1995, 123, 173–187 http://dx.doi.org/10.1016/0045-7825(95)00780-5
[21] Zhu Q., Superconvergence and Postprocessing Theory of Finite Element Theory, Science Press, 2008 (in Chinese)
[22] Zhu Q., Meng L., The derivative ultraconvergence for quadratic triangular finite elements, J. Comput. Math., 2004, 22, 857–864 | Zbl 1068.65124
[23] Zhu Q., Meng L., New construction and ultraconvergence of derivative recovery operator for odd-degree rectangular elements, Sci. China Math., 2004, 47, 940–949 http://dx.doi.org/10.1360/02ys0329 | Zbl 1079.65549
[24] Zhu Q., Zhao Q., SPR technique and finite element correction, Numer. Math., 2003, 96, 185–196 http://dx.doi.org/10.1007/s00211-003-0474-9 | Zbl 1053.65060