Skip to main content
SpringerLink
Log in

Local pointwise convergence of the 3D finite element

  • Published:
Applied Mathematics-A Journal of Chinese Universities Aims and scope Submit manuscript

Abstract

For an elliptic problem with variable coefficients in three dimensions, this article discusses local pointwise convergence of the three-dimensional (3D) finite element. First, the Green’s function and the derivative Green’s function are introduced. Secondly, some relationship of norms such as L2-norms, W1,∞-norms, and negative-norms in locally smooth subsets of the domain Ω is derived. Finally, local pointwise convergence properties of the finite element approximation are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J H Brandts, M Křížek. History and future of superconvergence in three-dimensional finite element methods, Proceedings of the Conference on Finite Element Methods: Three-dimensional Problems, GAKUTO International Series Mathematical Sciences and Applications, Gakkotosho, Tokyo, 2001, 15: 22–33.

    MathSciNet  MATH  Google Scholar 

  2. J H Brandts, M Křížek. Gradient superconvergence on uniform simplicial partitions of polytopes, IMA J Numer Anal, 2003, 23: 489–505.

    Article  MathSciNet  MATH  Google Scholar 

  3. J H Brandts, M Křížek. Superconvergence of tetrahedral quadratic finite elements, J Comput Math, 2005, 23: 27–36.

    MathSciNet  MATH  Google Scholar 

  4. C M Chen. Optimal points of stresses for the linear tetrahedral element, Natural Sci J Xiangtan Univ, 1980, 3: 16–24. (in Chinese)

    Google Scholar 

  5. C M Chen. Construction theory of superconvergence of finite elements, Hunan Science and Technology Press, Changsha, China, 2001. (in Chinese)

    Google Scholar 

  6. L Chen. Superconvergence of tetrahedral linear finite elements, Internat J Numer Anal Model, 2006, 3: 273–282.

    MathSciNet  MATH  Google Scholar 

  7. G Goodsell. Gradient superconvergence for piecewise linear tetrahedral finite elements, Technical Report RAL-90-031, Science and Engineering Research Council, Rutherford Appleton Laboratory, 1990.

  8. G Goodsell. Pointwise superconvergence of the gradient for the linear tetrahedral element, Numer Methods Partial Differential Equations, 1994, 10: 651–666.

    Article  MathSciNet  MATH  Google Scholar 

  9. A Hannukainen, S Korotov, M Křížek. Nodal O(h4)-superconvergence in 3D by averaging piece-wise linear, bilinear, and trilinear FE approximations, J Comp Math, 2010, 28: 1–10.

    Article  MATH  Google Scholar 

  10. W M He, X F Guan, J Z Cui. The local superconvergence of the trilinear element for the three-dimensional Poisson problem, J Math Anal Appl, 2012, 388: 863–872.

    Article  MathSciNet  MATH  Google Scholar 

  11. W M He, R C Lin, Z M Zhang. Ultraconvergence of finite element method by Richardson extrapolation for elliptic problems with constant coefficients, SIAM J Numer Anal, 2016, 54: 2302–2322.

    Article  MathSciNet  MATH  Google Scholar 

  12. W M He, Z M Zhang. 2k superconvergence of Q(k) finite elements by anisotropic mesh approximation in weighted Sobolev spaces, Math Comp, 2017, 86: 1693–1718.

    Article  MathSciNet  MATH  Google Scholar 

  13. V Kantchev, R D Lazarov. Superconvergence of the gradient of linear finite elements for 3D Poisson equation, Proceedings of the Conference on Optimal Algorithms, Bulgarian Academy of Sciences, Sofia, 1986, 172–182.

    MATH  Google Scholar 

  14. Q Lin, N N Yan. Construction and analysis of high efficient finite elements, Hebei University Press, Baoding, China, 1996. (in Chinese)

    Google Scholar 

  15. R C Lin, Z M Zhang. Natural superconvergent points in 3D finite elements, SIAM J Numer Anal, 2008, 46: 1281–1297.

    Article  MathSciNet  MATH  Google Scholar 

  16. J H Liu, B Jia, Q D Zhu. An estimate for the three-dimensional discrete Green’s function and applications, J Math Anal Appl, 2010, 370: 350–363.

    Article  MathSciNet  MATH  Google Scholar 

  17. J H Liu, Y S Jia. Estimates for the Green’s function of 3D elliptic equations, J Comp Anal Appl, 2017, 22: 1015–1022.

    MathSciNet  Google Scholar 

  18. J H Liu, Y S Jia. 3D Green’s function and its finite element error estimates, J Comp Anal Appl, 2017, 22: 1114–1123.

    MathSciNet  Google Scholar 

  19. J H Liu, H N Sun, Q D Zhu. Superconvergence of tricubic block finite elements, Sci China Ser A, 2009, 52: 959–972.

    Article  MathSciNet  MATH  Google Scholar 

  20. J H Liu, Q D Zhu. Maximum-norm superapproximation of the gradient for the trilinear block finite element, Numer Methods Partial Differential Equations, 2007, 23: 1501–1508.

    Article  MathSciNet  MATH  Google Scholar 

  21. J H Liu, Q D Zhu. Pointwise super closeness of tensor-product block finite elements, Numer Methods Partial Differential Equations, 2009, 25: 990–1008.

    Article  MathSciNet  Google Scholar 

  22. J H Liu, Q D Zhu. Pointwise supercloseness of pentahedral finite elements, Numer Methods Partial Differential Equations, 2010, 26: 1572–1580.

    Article  MathSciNet  MATH  Google Scholar 

  23. J H Liu, Q D Zhu. Maximum-norm superapproach of the gradient for quadratic finite elements in three dimensions, Acta Mathematica Scientia, 2006, 26: 458–466. (in Chinese)

    MATH  Google Scholar 

  24. A Pehlivanov. Superconvergence of the gradient for quadratic 3D simplex finite elements, Proceedings of the Conference on Numerical Methods and Application, Bulgarian Academy of Sciences, Sofia, 1989, 362–366.

    Google Scholar 

  25. A H Schatz, I H Sloan, L B Wahlbin. Superconvergence infinite element methods and meshes that are locally symmetric with respect to a point, SIAM J Numer Anal, 1996, 33: 505–521.

    Article  MathSciNet  MATH  Google Scholar 

  26. Z M Zhang, R C Lin. Locating natural superconvergent points of finite element methods in 3D, Internat J Numer Anal Model, 2005, 2: 19–30.

    MathSciNet  MATH  Google Scholar 

  27. Q D Zhu, Q Lin. Superconvergence theory of the finite element methods, Hunan Science and Technology Press, Changsha, China, 1989. (in Chinese)

    Google Scholar 

  28. M Zlámal. Superconvergence and reduced integration in the finite element method, Math Comp, 1978, 32: 663–685.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Science, Guangdong University of Petrochemical Technology, Maoming, 525000, China

    Jing-hong Liu

  2. School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, China

    Qi-ding Zhu

Authors
  1. Jing-hong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qi-ding Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Jing-hong Liu or Qi-ding Zhu.

Additional information

Conflict of interest

The authors declare no conflict of interest.

Supported by Special Projects in Key Fields of Colleges and Universities in Guangdong Province (2022ZDZX3016), and Projects of Talents Recruitment of GDUPT.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Jh., Zhu, Qd. Local pointwise convergence of the 3D finite element. Appl. Math. J. Chin. Univ. 38, 210–222 (2023). https://doi.org/10.1007/s11766-023-3911-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11766-023-3911-9

Keywords

MR Subject Classification

Search

Navigation