American Institute of Mathematical Sciences

Advanced
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

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 & 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.  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), 112-124. doi: 10.1137/0707006.  Google Scholar

[3]

C. M. Chen, Construction Theory of Superconvergence of Finite Elements, Hunan Science and Technology Press, Changsha, 2001 (in Chinese). Google Scholar

[4]

C. M. Chen and Y. Q. Huang, High Accuracy Theory of Finite Element Methods, Hunan Science and Technology Press, Changsha, 1995 (in Chinese). Google Scholar

[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.  Google Scholar

[6]

L. Chen, Superconvergence of tetrahedral linear finite elements, International Journal of Numerical Analysis and Modeling, 3 (2006), 273-282.  Google Scholar

[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.  Google Scholar

[8]

Q. Lin and N. N. Yan, Construction and Analysis of High Efficient Finite Elements, Hebei University Press, Baoding, 1996 (in Chinese). 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), 745-751.  Google Scholar

[10]

L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics vol. 1605, Springer-Verlag, Berlin, 1995.  Google Scholar

[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.  Google Scholar

[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.  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), 1192-1213. 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), 507-524. 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), 173-187. 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), 159-170. 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, Beijing, 2008 (in Chinese). Google Scholar

[18]

Q. D. Zhu and Q. Lin, The Superconvergence Theory of Finite Elements, Hunan Science and Technology Press, Changsha, 1989 (in Chinese).  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), 337-357. 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), 1331-1364. 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), 1365-1382. doi: 10.1002/nme.1620330703.  Google Scholar

show all references

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.  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), 112-124. doi: 10.1137/0707006.  Google Scholar

[3]

C. M. Chen, Construction Theory of Superconvergence of Finite Elements, Hunan Science and Technology Press, Changsha, 2001 (in Chinese). Google Scholar

[4]

C. M. Chen and Y. Q. Huang, High Accuracy Theory of Finite Element Methods, Hunan Science and Technology Press, Changsha, 1995 (in Chinese). Google Scholar

[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.  Google Scholar

[6]

L. Chen, Superconvergence of tetrahedral linear finite elements, International Journal of Numerical Analysis and Modeling, 3 (2006), 273-282.  Google Scholar

[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.  Google Scholar

[8]

Q. Lin and N. N. Yan, Construction and Analysis of High Efficient Finite Elements, Hebei University Press, Baoding, 1996 (in Chinese). 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), 745-751.  Google Scholar

[10]

L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics vol. 1605, Springer-Verlag, Berlin, 1995.  Google Scholar

[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.  Google Scholar

[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.  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), 1192-1213. 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), 507-524. 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), 173-187. 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), 159-170. 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, Beijing, 2008 (in Chinese). Google Scholar

[18]

Q. D. Zhu and Q. Lin, The Superconvergence Theory of Finite Elements, Hunan Science and Technology Press, Changsha, 1989 (in Chinese).  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), 337-357. 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), 1331-1364. 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), 1365-1382. doi: 10.1002/nme.1620330703.  Google Scholar

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