×

zbMATH — the first resource for mathematics

Finite element derivative interpolation recovery technique and superconvergence. (English) Zbl 1249.65258
Summary: 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.
MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] C.M. Chen, Y.Q. Huang: High Accuracy Theory of Finite Element Methods. Hunan Science Press, Hunan, 1995. (In Chinese.)
[2] P.G. Cairlet: The Finite Element Methods for Elliptic Problems. North-Holland Publishing, Amsterdam, 1978.
[3] E. Hinton, J. S. Campbell: Local and global smoothing of discontinuous finite element functions using a least squares method. Int. J. Numer. Methods Eng. 8 (1974), 461–480. · Zbl 0286.73066
[4] M. Křížek, P. Neittaanmäki, R. Stenberg (eds.): Finite Element Methods. Superconvergence, Postprocessing, and a Posteriori Estimates. Lecture Notes in Pure and Appl. Math., Vol. 196. Marcel Dekker, New York, 1998.
[5] Q. Lin, Q.D. Zhu: The Preprocessing and Postprocessing for Finite Element Methods. Shanghai Sci. & Tech. Publishers, Shanghai, 1994. (In Chinese.)
[6] J.T. Oden, H. J. Brauchli: On the calculation of consistent stress distributions in finite element applications. Int. J. Numer. Methods Eng. 3 (1971), 317–325. · Zbl 0251.73056
[7] M. J. Turner, H.C. Martin, B.C. Weikel: Further developments and applications of stiffness method. Matrix Meth. Struct. Analysis 72 (1964), 203–266.
[8] L.B. Wahlbin: Superconvergence in Galerkin Finite Element Methods. Lecture Notes in Mathematics, Vol. 1605. Springer, Berlin, 1995.
[9] E. L. Wilson: Finite element analysis of two-dimensional structures. PhD. Thesis. University of California, Berkeley, 1963.
[10] Z.X. Wang, D.R. Guo: Special Functions. World Scientific, Singapore, 1989.
[11] O.C. Zienkiewicz, J. Z. Zhu: The superconvergence patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int. J. Numer. Methods Eng. 33 (1992), 1331–1364. · Zbl 0769.73084
[12] Z. Zhang: Recovery techniques in finite element methods. Adaptive Computations: Theory and Algorithms (T. Tang, J.C. Xu, eds.). Science Press, Beijing, 2007.
[13] Z. Zhang: Ultraconvergence of the patch recovery technique II. Math. Comput. 69 (2000), 141–158. · Zbl 0936.65132
[14] T. Zhang, Y. P. Lin, R. J. Tait: The derivative patch interpolation recovery technique for finite element approximations. J. Comput. Math. 22 (2004), 113–122. · Zbl 1055.65132
[15] T. Zhang, C. J. Li, Y.Y. Nie: Derivative superconvergence of linear finite elements by recovery techniques. Dyn. Contin. Discrete Impuls. Syst., Ser. A 11 (2004), 853–862. · Zbl 1059.65096
[16] T. Zhang: Finite Element Methods for Evolutionary Integro-Differential Equations. Northeastern University Press, Shenyang, 2002. (In Chinese.)
[17] Q.D. Zhu, L.X. Meng: 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. (In Chinese.)
[18] Q.D. Zhu, Q. Lin: Superconvergence Theory of Finite Element Methods. Hunan Science Press, Hunan, 1989. (In Chinese.)
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.