Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Linear finite element superconvergence on simplicial meshes


Authors: Jie Chen, Desheng Wang and Qiang Du
Journal: Math. Comp. 83 (2014), 2161-2185
MSC (2010): Primary 65N30, 65N50
DOI: https://doi.org/10.1090/S0025-5718-2014-02810-X
Published electronically: February 12, 2014
MathSciNet review: 3223328
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the linear finite element gradient superconvergence on special simplicial meshes which satisfy an edge pair condition. This special geometric condition implies that for most simplexes in the mesh, the lengths of each pair of opposite edges in each 3-face are assumed to differ only by $ O(h^{1+\alpha })$ for some constant $ \alpha >0$, with $ h$ being the mesh parameter. To analyze the interplant gradient superconvergence, we present a local error expansion formula in general $ n$ dimensional space which also motivates the condition on meshes. In the three dimensional space, we show that the gradient of the linear finite element solution $ u_h$ is superconvergent to the gradient of the linear interpolatant $ u_I$ with an order $ O(h^{1+\rho })$ for $ 0<\rho \leq \alpha $. Numerical examples are presented to verify the theoretical findings. While we illustrate that tetrahedral meshes satisfying the edge pair condition can often be produced in three dimension, we also show that this may not be the case in higher dimensional spaces.


References [Enhancements On Off] (What's this?)

  • [1] Jan Brandts and Michal Křížek, Gradient superconvergence on uniform simplicial partitions of polytopes, IMA J. Numer. Anal. 23 (2003), no. 3, 489-505. MR 1987941 (2004i:65105), https://doi.org/10.1093/imanum/23.3.489
  • [2] Jan Brandts and Michal Křížek, History and future of superconvergence in three-dimensional finite element methods, Finite element methods (Jyväskylä, 2000) GAKUTO Internat. Ser. Math. Sci. Appl., vol. 15, Gakkōtosho, Tokyo, 2001, pp. 22-33. MR 1896264
  • [3] C. Chen and Y. Q. Huang, High accuracy theory of finite element methods, Hunan Science Press, Hunan, China, 1995 (in Chinese).
  • [4] Jie Chen and Desheng Wang, Three-dimensional finite element superconvergent gradient recovery on Par6 patterns, Numer. Math. Theory Methods Appl. 3 (2010), no. 2, 178-194. MR 2682792 (2011f:65250), https://doi.org/10.4208/nmtma.2010.32s.4
  • [5] Long Chen, Superconvergence of tetrahedral linear finite elements, Int. J. Numer. Anal. Model. 3 (2006), no. 3, 273-282. MR 2237882 (2007a:65191)
  • [6] George Goodsell, Pointwise superconvergence of the gradient for the linear tetrahedral element, Numer. Methods Partial Differential Equations 10 (1994), no. 5, 651-666. MR 1290950 (95e:65101), https://doi.org/10.1002/num.1690100511
  • [7] V. K. Kantchev and R. D. Lazarov, Superconvergence of the gradient of linear finite elements for 3-d Poisson equation, Proceeding of the Conference on Optimal Algorithms, 1986, pp. 172-182.
  • [8] Michal Křížek, Superconvergence phenomena on three-dimensional meshes, Int. J. Numer. Anal. Model. 2 (2005), no. 1, 43-56. MR 2112657 (2005j:65143)
  • [9] A. H. Schatz, I. H. Sloan, and L. B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal. 33 (1996), no. 2, 505-521. MR 1388486 (98f:65112), https://doi.org/10.1137/0733027
  • [10] Q. Zhu and Q Lin, Superconvergence theory of the finite element method, 1989, Hunan Science Press, China.
  • [11] Randolph E. Bank and Jinchao Xu, Asymptotically exact a posteriori error estimators. I. Grids with superconvergence, SIAM J. Numer. Anal. 41 (2003), no. 6, 2294-2312 (electronic). MR 2034616 (2004k:65194), https://doi.org/10.1137/S003614290139874X
  • [12] O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg. 24 (1987), no. 2, 337-357. MR 875306 (87m:73055), https://doi.org/10.1002/nme.1620240206
  • [13] O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Internat. J. Numer. Methods Engrg. 33 (1992), no. 7, 1331-1364. MR 1161557 (93c:73098), https://doi.org/10.1002/nme.1620330702
  • [14] O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. II. Error estimates and adaptivity, Internat. J. Numer. Methods Engrg. 33 (1992), no. 7, 1365-1382. MR 1161558 (93c:73099), https://doi.org/10.1002/nme.1620330703
  • [15] Zhimin Zhang and Ahmed Naga, A new finite element gradient recovery method: superconvergence property, SIAM J. Sci. Comput. 26 (2005), no. 4, 1192-1213 (electronic). MR 2143481 (2006d:65137), https://doi.org/10.1137/S1064827503402837
  • [16] D. SOMMERVILLE, Space-filling Tetrahedra in Euclidean Space, Proc. Edinburgh Math. Soc., 41 (1923), pp.49-57.
  • [17] Qiang Du, Vance Faber, and Max Gunzburger, Centroidal Voronoi tessellations: applications and algorithms, SIAM Rev. 41 (1999), no. 4, 637-676. MR 1722997 (2001b:52025), https://doi.org/10.1137/S0036144599352836
  • [18] Qiang Du and Desheng Wang, The optimal centroidal Voronoi tessellations and the Gersho's conjecture in the three-dimensional space, Comput. Math. Appl. 49 (2005), no. 9-10, 1355-1373. MR 2149486 (2006d:65016), https://doi.org/10.1016/j.camwa.2004.12.008
  • [19] Qiang Du and Desheng Wang, Recent progress in robust and quality Delaunay mesh generation, J. Comput. Appl. Math. 195 (2006), no. 1-2, 8-23. MR 2244622, https://doi.org/10.1016/j.cam.2005.07.014
  • [20] Jinchao Xu and Zhimin Zhang, Analysis of recovery type a posteriori error estimators for mildly structured grids, Math. Comp. 73 (2004), no. 247, 1139-1152 (electronic). MR 2047081 (2005f:65141), https://doi.org/10.1090/S0025-5718-03-01600-4
  • [21] D. J. Naylor, Filling space with tetrahedra, Internat. J. Numer. Methods Engrg. 44 (1999), no. 10, 1383-1395. MR 1678387 (99k:65119), https://doi.org/10.1002/(SICI)1097-0207(19990410)44:10$ \langle $1383::AID-NME616$ \rangle $3.3.CO;2-9
  • [22] Allen Gersho, Asymptotically optimal block quantization, IEEE Trans. Inform. Theory 25 (1979), no. 4, 373-380. MR 536229 (80h:94023), https://doi.org/10.1109/TIT.1979.1056067
  • [23] Donald J. Newman, The hexagon theorem, IEEE Trans. Inform. Theory 28 (1982), no. 2, 137-139. MR 651808 (83d:94011), https://doi.org/10.1109/TIT.1982.1056492
  • [24] Yunqing Huang, Hengfeng Qin, Desheng Wang, and Qiang Du, Convergent adaptive finite element method based on centroidal Voronoi tessellations and superconvergence, Commun. Comput. Phys. 10 (2011), no. 2, 339-370. MR 2799645 (2012i:65267), https://doi.org/10.4208/cicp.030210.051110a
  • [25] Robert M. Gray and David L. Neuhoff, Quantization, IEEE Trans. Inform. Theory 44 (1998), no. 6, 2325-2383. Information theory: 1948-1998. MR 1658787 (99i:94029), https://doi.org/10.1109/18.720541
  • [26] A. Naga and Z. Zhang, The polynomial-preserving recovery for higher order finite element methods in 2D and 3D, Discrete Contin. Dyn. Syst. Ser. B 5 (2005), no. 3, 769-798. MR 2151732 (2006h:65192), https://doi.org/10.3934/dcdsb.2005.5.769
  • [27] Leonard M. Blumenthal, Theory and applications of distance geometry, Oxford, at the Clarendon Press, 1953. MR 0054981 (14,1009a)
  • [28] G. M. Crippen and T. F. Havel, Distance geometry and molecular conformation, Chemometrics Series, vol. 15, Research Studies Press Ltd., Chichester, 1988. MR 975025 (90a:92082)
  • [29] Timothy F. Havel, Some examples of the use of distances as coordinates for Euclidean geometry, J. Symbolic Comput. 11 (1991), no. 5-6, 579-593. Invariant-theoretic algorithms in geometry (Minneapolis, MN, 1987). MR 1122640 (92j:51033), https://doi.org/10.1016/S0747-7171(08)80120-4

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N30, 65N50

Retrieve articles in all journals with MSC (2010): 65N30, 65N50


Additional Information

Jie Chen
Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
Email: majchen@ust.hk

Desheng Wang
Affiliation: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371
Email: desheng@ntu.edu.sg

Qiang Du
Affiliation: Pennsylvania State University, University Park, Pennsylvania 16802
Email: qdu@math.psu.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02810-X
Keywords: Superconvergence, finite element methods, simplicial meshes, edge patches, edge pair condition
Received by editor(s): September 29, 2012
Received by editor(s) in revised form: December 24, 2012
Published electronically: February 12, 2014
Additional Notes: This work was supported by Singapore AcRF RG59/08 (M52110092) and Singapore NRF 2007 IDM-IDM002-010, and US NSF DMS-1318586.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society