# American Institute of Mathematical Sciences

March  2015, 20(2): 495-504. doi: 10.3934/dcdsb.2015.20.495

## Gradient superconvergence post-processing of the tensor-product quadratic pentahedral finite element

 1 Department of Fundamental Courses, Ningbo Institute of Technology, Zhejiang University, Ningbo, 315100, China 2 School of Mathematics and Computer Science, Shangrao Normal University, Shangrao, 334001, China

Received  August 2013 Revised  October 2014 Published  January 2015

In this article, using the well-known Superconvergent Patch Recovery (SPR) method, we present a gradient superconvergence post-processing scheme for the tensor-product quadratic pentahedral finite element approximation to the solution of a general second-order elliptic boundary value problem in three dimensions over fully uniform meshes. The supercloseness property of the gradients between the finite element solution $u_h$ and the tensor-product quadratic interpolation $\Pi u$ is first given. Then we show that the gradient recovered from the finite element solution by using the SPR method is superconvergent to $\nabla u$ at interior vertices.
Citation: Jinghong Liu, Yinsuo Jia. Gradient superconvergence post-processing of the tensor-product quadratic pentahedral finite element. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 495-504. doi: 10.3934/dcdsb.2015.20.495
##### References:
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##### References:
 [1] I. Babuška and T. Strouboulis, The finite element method and its reliability,, in Numerical Mathematics and Scientific Computation, (2001).   Google Scholar [2] J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation,, SIAM Journal on Numerical Analysis, 7 (1970), 112.  doi: 10.1137/0707006.  Google Scholar [3] C. M. Chen, Construction Theory of Superconvergence of Finite Elements,, Hunan Science and Technology Press, (2001).   Google Scholar [4] C. M. Chen and Y. Q. Huang, High Accuracy Theory of Finite Element Methods,, Hunan Science and Technology Press, (1995).   Google Scholar [5] J. Chen and D. S. Wang, Three-dimensional finite element superconvergent gradient recovery on Par6 patterns,, Numerical Mathematics: Theory, 3 (2010), 178.  doi: 10.4208/nmtma.2010.32s.4.  Google Scholar [6] L. Chen, Superconvergence of tetrahedral linear finite elements,, International Journal of Numerical Analysis and Modeling, 3 (2006), 273.   Google Scholar [7] G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element,, Numerical Methods for Partial Differential Equations, 10 (1994), 651.  doi: 10.1002/num.1690100511.  Google Scholar [8] Q. Lin and N. N. Yan, Construction and Analysis of High Efficient Finite Elements,, Hebei University Press, (1996).   Google Scholar [9] J. H. Liu, Superconvergence of tensor-product quadratic pentahedral elements for variable coefficient elliptic equations,, Journal of Computational Analysis and Applications, 14 (2012), 745.   Google Scholar [10] L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods,, Lecture Notes in Mathematics vol. 1605, (1605).   Google Scholar [11] Z. M. Zhang, Ultraconvergence of the patch recovery technique,, Mathematics of Computation, 65 (1996), 1431.  doi: 10.1090/S0025-5718-96-00782-X.  Google Scholar [12] Z. M. Zhang, Ultraconvergence of the patch recovery technique II,, Mathematics of Computation, 69 (2000), 141.  doi: 10.1090/S0025-5718-99-01205-3.  Google Scholar [13] Z. M. Zhang and A. Naga, A new finite element gradient recovery method: Superconvergence property,, SIAM Journal on Scientific Computing, 26 (2005), 1192.  doi: 10.1137/S1064827503402837.  Google Scholar [14] Z. M. Zhang and H. D. Victory Jr., Mathematical analysis of Zienkiewicz-Zhu's derivative patch recovery technique,, Numerical Methods for Partial Differential Equations, 12 (1996), 507.  doi: 10.1002/(SICI)1098-2426(199607)12:4<507::AID-NUM6>3.0.CO;2-Q.  Google Scholar [15] Z. M. Zhang and J. Z. Zhu, Analysis of the superconvergence patch recovery techniques and a posteriori error estimator in the finite element method (I),, Computer Methods in Applied Mechanics and Engineering, 123 (1995), 173.  doi: 10.1016/0045-7825(95)00780-5.  Google Scholar [16] Z. M. Zhang and J. Z. Zhu, Analysis of the superconvergence patch recovery techniques and a posteriori error estimator in the finite element method (II),, Computer Methods in Applied Mechanics and Engineering, 163 (1998), 159.  doi: 10.1016/S0045-7825(98)00010-3.  Google Scholar [17] Q. D. Zhu, High Accuracy Post-Processing Theory of the Finite Element Method,, Science Press, (2008).   Google Scholar [18] Q. D. Zhu and Q. Lin, The Superconvergence Theory of Finite Elements,, Hunan Science and Technology Press, (1989).   Google Scholar [19] O. C. Zienkiewicz and J. Z. Zhu, A simple estimator and adaptive procedure for practical engineering analysis,, International Journal for Numerical Methods in Engineering, 24 (1987), 337.  doi: 10.1002/nme.1620240206.  Google Scholar [20] O. C. Zienkiewicz and J. Z. Zhu, The superconvergence patch recovery and a posteriori error estimates. Part 1: The recovery techniques,, International Journal for Numerical Methods in Engineering, 33 (1992), 1331.  doi: 10.1002/nme.1620330702.  Google Scholar [21] O. C. Zienkiewicz and J. Z. Zhu, The superconvergence patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity,, International Journal for Numerical Methods in Engineering, 33 (1992), 1365.  doi: 10.1002/nme.1620330703.  Google Scholar
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