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

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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

Jinghong Liu ^1, and Yinsuo Jia ^2,

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

Abstract
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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.

Keywords: superconvergent patch recovery, post-processing, supercloseness., Pentahedral finite element.

Mathematics Subject Classification: Primary: 65N30; Secondary: 65N1.

Citation: Jinghong Liu, Yinsuo Jia. Gradient superconvergence post-processing of the tensor-product quadratic pentahedral finite element. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 495-504. doi: 10.3934/dcdsb.2015.20.495

References:

[1]	I. Babuška and T. Strouboulis, The finite element method and its reliability, in Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, 2001.
[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), 112-124. doi: 10.1137/0707006.
[3]	C. M. Chen, Construction Theory of Superconvergence of Finite Elements, Hunan Science and Technology Press, Changsha, 2001 (in Chinese).
[4]	C. M. Chen and Y. Q. Huang, High Accuracy Theory of Finite Element Methods, Hunan Science and Technology Press, Changsha, 1995 (in Chinese).
[5]	J. Chen and D. S. Wang, Three-dimensional finite element superconvergent gradient recovery on Par6 patterns, Numerical Mathematics: Theory, Methods and Applications, 3 (2010), 178-194. doi: 10.4208/nmtma.2010.32s.4.
[6]	L. Chen, Superconvergence of tetrahedral linear finite elements, International Journal of Numerical Analysis and Modeling, 3 (2006), 273-282.
[7]	G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element, Numerical Methods for Partial Differential Equations, 10 (1994), 651-666. doi: 10.1002/num.1690100511.
[8]	Q. Lin and N. N. Yan, Construction and Analysis of High Efficient Finite Elements, Hebei University Press, Baoding, 1996 (in Chinese).
[9]	J. H. Liu, Superconvergence of tensor-product quadratic pentahedral elements for variable coefficient elliptic equations, Journal of Computational Analysis and Applications, 14 (2012), 745-751.
[10]	L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics vol. 1605, Springer-Verlag, Berlin, 1995.
[11]	Z. M. Zhang, Ultraconvergence of the patch recovery technique, Mathematics of Computation, 65 (1996), 1431-1437. doi: 10.1090/S0025-5718-96-00782-X.
[12]	Z. M. Zhang, Ultraconvergence of the patch recovery technique II, Mathematics of Computation, 69 (2000), 141-158. doi: 10.1090/S0025-5718-99-01205-3.
[13]	Z. M. Zhang and A. Naga, A new finite element gradient recovery method: Superconvergence property, SIAM Journal on Scientific Computing, 26 (2005), 1192-1213. doi: 10.1137/S1064827503402837.
[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), 507-524. doi: 10.1002/(SICI)1098-2426(199607)12:4<507::AID-NUM6>3.0.CO;2-Q.
[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), 173-187. doi: 10.1016/0045-7825(95)00780-5.
[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), 159-170. doi: 10.1016/S0045-7825(98)00010-3.
[17]	Q. D. Zhu, High Accuracy Post-Processing Theory of the Finite Element Method, Science Press, Beijing, 2008 (in Chinese).
[18]	Q. D. Zhu and Q. Lin, The Superconvergence Theory of Finite Elements, Hunan Science and Technology Press, Changsha, 1989 (in Chinese).
[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), 337-357. doi: 10.1002/nme.1620240206.
[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), 1331-1364. doi: 10.1002/nme.1620330702.
[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), 1365-1382. doi: 10.1002/nme.1620330703.

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References:

[1]	I. Babuška and T. Strouboulis, The finite element method and its reliability, in Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, 2001.
[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), 112-124. doi: 10.1137/0707006.
[3]	C. M. Chen, Construction Theory of Superconvergence of Finite Elements, Hunan Science and Technology Press, Changsha, 2001 (in Chinese).
[4]	C. M. Chen and Y. Q. Huang, High Accuracy Theory of Finite Element Methods, Hunan Science and Technology Press, Changsha, 1995 (in Chinese).
[5]	J. Chen and D. S. Wang, Three-dimensional finite element superconvergent gradient recovery on Par6 patterns, Numerical Mathematics: Theory, Methods and Applications, 3 (2010), 178-194. doi: 10.4208/nmtma.2010.32s.4.
[6]	L. Chen, Superconvergence of tetrahedral linear finite elements, International Journal of Numerical Analysis and Modeling, 3 (2006), 273-282.
[7]	G. Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element, Numerical Methods for Partial Differential Equations, 10 (1994), 651-666. doi: 10.1002/num.1690100511.
[8]	Q. Lin and N. N. Yan, Construction and Analysis of High Efficient Finite Elements, Hebei University Press, Baoding, 1996 (in Chinese).
[9]	J. H. Liu, Superconvergence of tensor-product quadratic pentahedral elements for variable coefficient elliptic equations, Journal of Computational Analysis and Applications, 14 (2012), 745-751.
[10]	L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics vol. 1605, Springer-Verlag, Berlin, 1995.
[11]	Z. M. Zhang, Ultraconvergence of the patch recovery technique, Mathematics of Computation, 65 (1996), 1431-1437. doi: 10.1090/S0025-5718-96-00782-X.
[12]	Z. M. Zhang, Ultraconvergence of the patch recovery technique II, Mathematics of Computation, 69 (2000), 141-158. doi: 10.1090/S0025-5718-99-01205-3.
[13]	Z. M. Zhang and A. Naga, A new finite element gradient recovery method: Superconvergence property, SIAM Journal on Scientific Computing, 26 (2005), 1192-1213. doi: 10.1137/S1064827503402837.
[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), 507-524. doi: 10.1002/(SICI)1098-2426(199607)12:4<507::AID-NUM6>3.0.CO;2-Q.
[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), 173-187. doi: 10.1016/0045-7825(95)00780-5.
[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), 159-170. doi: 10.1016/S0045-7825(98)00010-3.
[17]	Q. D. Zhu, High Accuracy Post-Processing Theory of the Finite Element Method, Science Press, Beijing, 2008 (in Chinese).
[18]	Q. D. Zhu and Q. Lin, The Superconvergence Theory of Finite Elements, Hunan Science and Technology Press, Changsha, 1989 (in Chinese).
[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), 337-357. doi: 10.1002/nme.1620240206.
[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), 1331-1364. doi: 10.1002/nme.1620330702.
[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), 1365-1382. doi: 10.1002/nme.1620330703.

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