American Institute of Mathematical Sciences

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

 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 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.

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. [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.
 [1] Zhong-Ci Shi, Xuejun Xu, Zhimin Zhang. The patch recovery for finite element approximation of elasticity problems under quadrilateral meshes. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 163-182. doi: 10.3934/dcdsb.2008.9.163 [2] Yinnian He, R. M.M. Mattheij. Reformed post-processing Galerkin method for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 369-387. doi: 10.3934/dcdsb.2007.8.369 [3] So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343 [4] A. Naga, Z. Zhang. The polynomial-preserving recovery for higher order finite element methods in 2D and 3D. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 769-798. doi: 10.3934/dcdsb.2005.5.769 [5] Yangyang Xu, Wotao Yin. A fast patch-dictionary method for whole image recovery. Inverse Problems and Imaging, 2016, 10 (2) : 563-583. doi: 10.3934/ipi.2016012 [6] Volodymyr Nekrashevych. The Julia set of a post-critically finite endomorphism of $\mathbb{PC}^2$. Journal of Modern Dynamics, 2012, 6 (3) : 327-375. doi: 10.3934/jmd.2012.6.327 [7] Shuhao Cao. A simple virtual element-based flux recovery on quadtree. Electronic Research Archive, 2021, 29 (6) : 3629-3647. doi: 10.3934/era.2021054 [8] Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637 [9] P. K. Jha, R. Lipton. Finite element approximation of nonlocal dynamic fracture models. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1675-1710. doi: 10.3934/dcdsb.2020178 [10] Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339 [11] Eduardo Casas, Mariano Mateos, Arnd Rösch. Finite element approximation of sparse parabolic control problems. Mathematical Control and Related Fields, 2017, 7 (3) : 393-417. doi: 10.3934/mcrf.2017014 [12] Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic and Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59 [13] Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks and Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1 [14] Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 [15] Gianmarco Manzini, Annamaria Mazzia. A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem. Journal of Computational Dynamics, 2022, 9 (2) : 207-238. doi: 10.3934/jcd.2021020 [16] Sören Bartels, Marijo Milicevic. Iterative finite element solution of a constrained total variation regularized model problem. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1207-1232. doi: 10.3934/dcdss.2017066 [17] Eduardo Casas, Boris Vexler, Enrique Zuazua. Sparse initial data identification for parabolic PDE and its finite element approximations. Mathematical Control and Related Fields, 2015, 5 (3) : 377-399. doi: 10.3934/mcrf.2015.5.377 [18] Fang Liu, Aihui Zhou. Localizations and parallelizations for two-scale finite element discretizations. Communications on Pure and Applied Analysis, 2007, 6 (3) : 757-773. doi: 10.3934/cpaa.2007.6.757 [19] Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641 [20] Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153

2020 Impact Factor: 1.327