Advertisement

Nitsche’s Method for the Obstacle Problem of Clamped Kirchhoff Plates

  • Tom Gustafsson
  • Rolf StenbergEmail author
  • Juha Videman
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

The theory behind Nitsche’s method for approximating the obstacle problem of clamped Kirchhoff plates is reviewed. A priori estimates and residual-based a posteriori error estimators are presented for the related conforming stabilised finite element method and the latter are used for adaptive refinement in a numerical experiment.

Notes

Acknowledgements

The authors are grateful for the financial support from the Portuguese Science Foundation (FCOMP-01-0124-FEDER-029408), Tekes (Decision number 3305/31/2015), the Finnish Academy of Science and Letters, and the Finnish Cultural Foundation.

References

  1. 1.
    R. Becker, P. Hansbo, R. Stenberg, A finite element method for domain decomposition with non-matching grids. ESAIM: Math. Model. Numer. Anal. 37, 209–225 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    S. Brenner, L.-Y. Sung, Y. Zhang, Finite element methods for the displacement obstacle problem of clamped plates. Math. Comput. 81, 1247–1262 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    S. Brenner, L.-Y. Sung, H. Zhang, Y. Zhang, A Morley finite element method for the displacement obstacle problem of clamped Kirchhoff plates. J. Comput. Appl. Math. 254, 31–42 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Brenner, J. Gedicke, L.-Y. Sung, Y. Zhang, An a posteriori analysis of C 0 interior penalty methods for the obstacle problem of clamped Kirchhoff plates. SIAM J. Numer. Anal. 55, 87–108 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    E. Burman, A penalty-free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions. SIAM J. Numer. Anal. 50, 1959–1981 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    E. Burman, P. Hansbo, M.G. Larson, The penalty-free Nitsche’s method and non-conforming finite elements for the Signorini problem. SIAM J. Numer. Anal 55, 2523–2539 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    L.A. Caffarelli, A. Friedman, The obstacle problem for the biharmonic operator. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6, 151–183 (1979)Google Scholar
  8. 8.
    F. Chouly, P. Hild, A Nitsche-based method for unilateral contact problems: numerical analysis. SIAM J. Numer. Anal. 51, 1295–1307 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    F. Chouly, P. Hild, Y. Renard, Symmetric and non-symmetric variants of Nitsche’s method for contact problems in elasticity: theory and numerical experiments. Math. Comput. 84, 1089–1112 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    F. Chouly, M. Fabre, P. Hild, J. Pousin, Y. Renard, Residual-based a posteriori error estimation for contact problems approximated by Nitsche’s method. IMA J. Numer. Anal. 38, 921–954 (2018). https://doi.org/10.1093/imanum/drx024MathSciNetCrossRefGoogle Scholar
  11. 11.
    I. Ekeland, R. Temam, Convex Analysis and Variational Problems (SIAM, Philadelphia, 1999)CrossRefGoogle Scholar
  12. 12.
    K. Feng, Z.-C. Shi, Mathematical Theory of Elastic Structures (Springer/Science Press, Berlin/Beijing, 1996)Google Scholar
  13. 13.
    L.P. Franca, R. Stenberg, Error analysis of Galerkin least squares methods for the elasticity equations. SIAM J. Numer. Anal. 28, 1680–1697 (1991)MathSciNetCrossRefGoogle Scholar
  14. 14.
    J. Frehse, Zum Differenzierbarkeitsproblem bei Variationsungleichungen höherer Ordnung. Abh. Math. Sem. Univ. Hamburg 36, 140–149 (1971)MathSciNetCrossRefGoogle Scholar
  15. 15.
    T. Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comput. 79, 2169–2189 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    T. Gudi, K. Porwal, A C 0 interior penalty method for a fourth-order variational inequality of the second kind. Numer. Methods Partial Differ. Equ. 32, 36–59 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    T. Gustafsson, R. Stenberg, J. Videman, Mixed and stabilized finite element methods for the obstacle problem. SIAM J. Numer. Anal. 55, 2718–2744 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    T. Gustafsson, R. Stenberg, J. Videman, A posteriori estimates for conforming Kirchhoff plate elements. arXiv preprint: 1707.08396Google Scholar
  19. 19.
    T. Gustafsson, R. Stenberg, J. Videman, A stabilized finite element method for the plate obstacle problem. arXiv preprint: 1711.04166Google Scholar
  20. 20.
    M. Juntunen, R. Stenberg, Nitsche’s method for general boundary conditions. Math. Comput. 78, 1353–1374 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    J.L. Lions, G. Stampacchia, Variational inequalities. Commun. Pure Appl. Math. 20, 493–519 (1967)CrossRefGoogle Scholar
  22. 22.
    H. Melzer, R. Rannacher, Spannungskonzentrationen in Eckpunkten der vertikal belasteten Kirchhoffschen Platte. Bauingenieur 55, 181–189 (1980)Google Scholar
  23. 23.
    J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Univ. Hamburg 36, 9–15 (1970/1971)MathSciNetCrossRefGoogle Scholar
  24. 24.
    R. Scholz, Mixed finite element approximation of a fourth order variational inequality by the penalty method. Numer. Funct. Anal. Optim. 9, 233–247 (1987)MathSciNetCrossRefGoogle Scholar
  25. 25.
    R. Stenberg, On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math. 63, 139–148 (1995)MathSciNetCrossRefGoogle Scholar
  26. 26.
    R. Stenberg, J. Videman, On the error analysis of stabilized finite element methods for the Stokes problem. SIAM J. Numer. Anal. 53, 2626–2633 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Systems AnalysisAalto UniversityAaltoFinland
  2. 2.CAMGSD/Departamento de MatemáticaInstituto Superior Técnico, Universidade de LisboaLisbonPortugal

Personalised recommendations