Skip to main content
Springer Nature Link
Log in

A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Bartels, S., Carstensen, C.: Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids I and II. Math. Comp. 71(945–969), 971–994 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beirao da Veiga, L., Niiranen, J., Stenberg, R.: A posteriori error estimates for the Morley plate bending element. Numer. Math. 106, 165–179 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beirao da Veiga, L., Niiranen, J., Stenberg, R.: A posteriori error analysis for the Morley plate element with general boundary conditions. Int. J. Numer. Methods Eng. 83, 1–26 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Brenner, S.C.: A two-level additive Schwarz preconditioner for nonconforming plate elements. Numer. Math. 72(4), 419–447 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Berlin, New York (2008)

    Book  MATH  Google Scholar 

  6. Brenner, S.C., Gudi, T., Sung, L.Y.: An a posteriori error estimator for a quadratic \(C^0\) interior penalty method for the biharmonic problem. IMA J. Numer. Anal. 30, 777–798 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carstensen, C.: A unifying theory of a posteriori finite element error control. Numer. Math. 100, 617–637 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carstensen, C., Gedicke, J., Rim, D.: Explicit error estimates for Courant, Crouzeix-Raviart and Raviart-Thomas finite element methods. J. Comput. Math. 30(4), 337–353 (2012)

    Article  MathSciNet  Google Scholar 

  9. Carstensen, C., Hu, J.: A unifying theory of a posteriori error control for nonconforming finite element methods. Numer. Math. 107, 473–502 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carstensen, C., Hu, J., Orlando, A.: Framework for the a posteriori error analysis of nonconforming finite elements. SIAM J. Numer. Anal. 45, 62–82 (2007)

    Article  MathSciNet  Google Scholar 

  11. Charbonneau, A., Dossou, K., Pierre, R.: A residual-based a posteriori error estimator for the Ciarlet-Raviart formulation of the first biharmonic problem. Numer. Methods Partial Differ. Equ. 13, 93–111 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ciarlet, P.G.: The finite element method for elliptic problems, North-Holland (1978); reprinted as SIAM Classics in, Applied Mathematics (2002)

  13. Clément, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9, 77–84 (1975)

    MATH  Google Scholar 

  14. Georgoulis, E.H., Houston, P., Virtanen, J.: An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems. IMA J. Numer. Anal. 31, 281–298 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grisvard, P.: Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] 22 (1992)

  16. Gudi, T.: Residual-based a posteriori error estimator for the mixed finite element approximation of the biharmonic equation. Numer. Methods Partial Differ. Equ. 27, 315–328 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hansbo, P., Larson, M.G.: A posteriori error estimates for continuous/discontinuous Galerkin approximations of the Kirchhoff-Love plate. http://www.math.chalmers.se/Math/Research/Preprints/2008/10.pdf (2008)

  18. Hu, J., Huang, Y.Q., Zhang, S.Y.: The lowest order differentiable finite element on rectangular grids. SIAM J. Numer. Anal. 49, 1350–1368 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hu, J., Shi, Z.C.: A new a posteriori error estimate for the Morley element. Numer. Math. 112, 25–40 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hu, J., Shi, Z.C., Xu, J.C.: Convergence and optimality of the adaptive Morley element method, Numer. Math., 121, pp. 731–752 (2012). (See J. Hu, Z. C. Shi, J. C. Xu, Convergence and optimality of the adaptive Morley element method, Research Report 19(2009), School of Mathematical Sciences and Institute of Mathematics, Peking University, Also available online from May 2009. http://www.math.pku.edu.cn:8000/var/preprint/7280.pdf

  21. Huang, J.G., Huang, X.H., Xu, Y.F.: Convergence of an adaptive mixed finite element method for Kirchhoff plate bending problems. SIAM J. Numer. Anal. 49, 574–607 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lascaux, P., Lesaint, P.: Some nonconforming finite elements for the plate bending problem. RAIRO Anal. Numer. 1, 9–53 (1985)

    Google Scholar 

  23. Morley, L.S.D.: The triangular equilibrium element in the solutions of plate bending problem. Aero. Quart. 19, 149–169 (1968)

    Google Scholar 

  24. Shi, Z.C.: On the convergence of the incomplete biqudratic nonconforming plate element. Math. Numer. Sinica 8, 53–62 (1986)

    MathSciNet  MATH  Google Scholar 

  25. Shi, Z.C., Wang, M.: The finite element method. Science Press, Beijing (2010)

    Google Scholar 

  26. Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner, New York (1996)

    MATH  Google Scholar 

  27. Wang, M., Zhang, S.: Local a priori and a posteriori error estimates of finite elements for biharmonic equation, research report 13. Peking University, School of Mathematical Sciences and Institute of Mathematics, Beijing (2006)

  28. Wang, M., Xu, J.C.: The Morley element for fourth order elliptic equations in any dimensions. Numer. Math. 103, 155–169 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, M., Xu, J.C.: Some tetrahedron nonconforming elements for fourth order elliptic equations. Math. Comp. 76, 1–18 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, M., Shi, Z.C., Xu, J.C.: Some n-rectangle nonconforming elements for fourth order elliptic equations. J. Comp. Math. 25, 408–420 (2007)

    MathSciNet  MATH  Google Scholar 

  31. Wu, M.Q.: The incomplete biquadratic nonconforming plate element. J. Suzhou Univ. 1, 20–29 (1983)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 , Berlin, Germany

    Carsten Carstensen & Dietmar Gallistl

  2. Department of Computational Science and Engineering, Yonsei University, 120-749 , Seoul, Korea

    Carsten Carstensen

  3. LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, Peoples Republic of China

    Jun Hu

Authors
  1. Carsten Carstensen
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Dietmar Gallistl
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Jun Hu
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding authors

Correspondence to Carsten Carstensen or Jun Hu.

Additional information

Dedicated to Professor Z.-C. Shi in honour of his 80th birthday.

This project was supported by the Chinesisch-Deutsches Zentrum project GZ578. The first two authors were partially supported by the DFG Research Center MATHEON. The research of the third author was supported by the NSFC Project 10971005, partially supported by the NSFC Project 11271035 and by the NSFC Key Project 11031006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carstensen, C., Gallistl, D. & Hu, J. A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles. Numer. Math. 124, 309–335 (2013). https://doi.org/10.1007/s00211-012-0513-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-012-0513-5

Mathematics Subject Classification (2000)