Previous |  Up |  Next


finite element method; derivative recovery technique; superconvergence and ultraconvergence; elliptic boundary problems; numerical examples
A new finite element derivative recovery technique is proposed by using the polynomial interpolation method. We show that the recovered derivatives possess superconvergence on the recovery domain and ultraconvergence at the interior mesh points for finite element approximations to elliptic boundary problems. Compared with the well-known Z-Z patch recovery technique, the advantage of our method is that it gives an explicit recovery formula and possesses the ultraconvergence for the odd-order finite elements. Finally, some numerical examples are presented to illustrate the theoretical analysis.
[1] Chen, C. M., Huang, Y. Q.: High Accuracy Theory of Finite Element Methods. Hunan Science Press Hunan (1995), Chinese.
[2] Cairlet, P. G.: The Finite Element Methods for Elliptic Problems. North-Holland Publishing Amsterdam (1978).
[3] Hinton, E., Campbell, J. S.: Local and global smoothing of discontinuous finite element functions using a least squares method. Int. J. Numer. Methods Eng. 8 (1974), 461-480. DOI 10.1002/nme.1620080303 | MR 0411120 | Zbl 0286.73066
[4] Křížek, M., Neittaanmäki, P., (eds.), R. Stenberg: Finite Element Methods. Superconvergence, Postprocessing, and a Posteriori Estimates. Lecture Notes in Pure and Appl. Math., Vol. 196. Marcel Dekker New York (1998). MR 1602809
[5] Lin, Q., Zhu, Q. D.: The Preprocessing and Postprocessing for Finite Element Methods. Shanghai Sci. & Tech. Publishers Shanghai (1994), Chinese.
[6] Oden, J. T., Brauchli, H. J.: On the calculation of consistent stress distributions in finite element applications. Int. J. Numer. Methods Eng. 3 (1971), 317-325. DOI 10.1002/nme.1620030303
[7] Turner, M. J., Martin, H. C., Weikel, B. C.: Further developments and applications of stiffness method. Matrix Meth. Struct. Analysis 72 (1964), 203-266.
[8] Wahlbin, L. B.: Superconvergence in Galerkin Finite Element Methods. Lecture Notes in Mathematics, Vol. 1605. Springer Berlin (1995). MR 1439050
[9] Wilson, E. L.: Finite element analysis of two-dimensional structures. PhD. Thesis University of California Berkeley (1963).
[10] Wang, Z. X., Guo, D. R.: Special Functions. World Scientific Singapore (1989). MR 1034956 | Zbl 0724.33001
[11] Zienkiewicz, O. C., Zhu, J. Z.: The superconvergence patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int. J. Numer. Methods Eng. 33 (1992), 1331-1364. DOI 10.1002/nme.1620330702 | MR 1161557
[12] Zhang, Z.: Recovery techniques in finite element methods. Adaptive Computations: Theory and Algorithms T. Tang, J. C. Xu Science Press Beijing (2007).
[13] Zhang, Z.: Ultraconvergence of the patch recovery technique II. Math. Comput. 69 (2000), 141-158. DOI 10.1090/S0025-5718-99-01205-3 | MR 1680911 | Zbl 0936.65132
[14] Zhang, T., Lin, Y. P., Tait, R. J.: The derivative patch interpolation recovery technique for finite element approximations. J. Comput. Math. 22 (2004), 113-122. MR 2027918
[15] Zhang, T., Li, C. J., Nie, Y. Y.: Derivative superconvergence of linear finite elements by recovery techniques. Dyn. Contin. Discrete Impuls. Syst., Ser. A 11 (2004), 853-862. MR 2077127 | Zbl 1059.65096
[16] Zhang, T.: Finite Element Methods for Evolutionary Integro-Differential Equations. Northeastern University Press Shenyang (2002), Chinese.
[17] Zhu, Q. D., Meng, L. X.: New structure of the derivative recovery technique for odd-order rectangular finite elements and ultraconvergence. Science in China, Ser. A, Mathematics 34 (2004), 723-731 Chinese.
[18] Zhu, Q. D., Lin, Q.: Superconvergence Theory of Finite Element Methods. Hunan Science Press Hunan (1989), Chinese.
Partner of
EuDML logo