Numerische Mathematik

, Volume 124, Issue 2, pp 309–335 | Cite as

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

Article

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.

Mathematics Subject Classification (2000)

65N10 65N15 35J25 

References

  1. 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)MathSciNetMATHCrossRefGoogle Scholar
  2. 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)MathSciNetMATHCrossRefGoogle Scholar
  3. 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)MathSciNetMATHGoogle Scholar
  4. 4.
    Brenner, S.C.: A two-level additive Schwarz preconditioner for nonconforming plate elements. Numer. Math. 72(4), 419–447 (1996)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Berlin, New York (2008)MATHCrossRefGoogle Scholar
  6. 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)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Carstensen, C.: A unifying theory of a posteriori finite element error control. Numer. Math. 100, 617–637 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Carstensen, C., Hu, J.: A unifying theory of a posteriori error control for nonconforming finite element methods. Numer. Math. 107, 473–502 (2007)MathSciNetMATHCrossRefGoogle Scholar
  10. 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)MathSciNetCrossRefGoogle Scholar
  11. 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)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Ciarlet, P.G.: The finite element method for elliptic problems, North-Holland (1978); reprinted as SIAM Classics in, Applied Mathematics (2002)Google Scholar
  13. 13.
    Clément, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9, 77–84 (1975)MATHGoogle Scholar
  14. 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)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Grisvard, P.: Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] 22 (1992)Google Scholar
  16. 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)MathSciNetMATHCrossRefGoogle Scholar
  17. 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. 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)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Hu, J., Shi, Z.C.: A new a posteriori error estimate for the Morley element. Numer. Math. 112, 25–40 (2009)MathSciNetMATHCrossRefGoogle Scholar
  20. 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. 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)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Lascaux, P., Lesaint, P.: Some nonconforming finite elements for the plate bending problem. RAIRO Anal. Numer. 1, 9–53 (1985)Google Scholar
  23. 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. 24.
    Shi, Z.C.: On the convergence of the incomplete biqudratic nonconforming plate element. Math. Numer. Sinica 8, 53–62 (1986)MathSciNetMATHGoogle Scholar
  25. 25.
    Shi, Z.C., Wang, M.: The finite element method. Science Press, Beijing (2010)Google Scholar
  26. 26.
    Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner, New York (1996)MATHGoogle Scholar
  27. 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)Google Scholar
  28. 28.
    Wang, M., Xu, J.C.: The Morley element for fourth order elliptic equations in any dimensions. Numer. Math. 103, 155–169 (2006)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Wang, M., Xu, J.C.: Some tetrahedron nonconforming elements for fourth order elliptic equations. Math. Comp. 76, 1–18 (2007)MathSciNetMATHCrossRefGoogle Scholar
  30. 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)MathSciNetMATHGoogle Scholar
  31. 31.
    Wu, M.Q.: The incomplete biquadratic nonconforming plate element. J. Suzhou Univ. 1, 20–29 (1983)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Department of Computational Science and EngineeringYonsei UniversitySeoulKorea
  3. 3.LMAM and School of Mathematical SciencesPeking UniversityBeijingPeoples Republic of China

Personalised recommendations