Skip to main content
Springer Nature Link
Log in

A posteriori error estimations of residual type for Signorini's problem

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

This paper presents an a posteriori error analysis for the linear finite element approximation of the Signorini problem in two space dimensions. A posteriori estimations of residual type are defined and upper and lower bounds of the discretization error are obtained. We perform several numerical experiments in order to compare the convergence of the terms in the error estimator with the discretization error.

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

Similar content being viewed by others

References

  1. Ainsworth, M., Oden, J.T., Lee, C.: Local a posteriori error estimators for variational inequalities. Numer. Meth. PDE 9, 23–33 (1993)

    MATH  MathSciNet  Google Scholar 

  2. Ben Belgacem, F.: Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element method. SIAM J. Numer. Anal. 37, 1198–1216 (2000)

    MATH  MathSciNet  Google Scholar 

  3. Chen, Z., Nochetto, R.H.: Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84, 527–548 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chikouche, W., Mercier, D., Nicaise, S.: Regularity of the solution of some unilateral boundary value problems in polygonal and polyhedral domains. Commun. PDE 29, 43–70 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ciarlet, P.G.: The finite element method for elliptic problems (North-Holland, Amsterdam, 1978)

  6. Clément, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numer. 2, 77–84 (1975)

    MATH  Google Scholar 

  7. Fierro, F., Veeser, A.: A posteriori error estimators for regularized total variation of characteristic functions. SIAM J. Numer. Anal. 41, 2032–2055 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Grisvard, P.: Elliptic problems in nonsmooth domains, Vol. 24 of Monographs and Studies in Mathematics. (Pitman, Boston-London-Melbourne, 1985)

  9. Hoppe, R.H.W., Kornhuber, R.: Adaptive multilevel methods for obstacle problems. SIAM J. Numer. Anal. 31, 301–323 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Johnson, C. : Adaptive finite element methods for the obstacle problem. Math. Models Methods Appl. Sci. 2, 483–487 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Mercier, D., Nicaise, S.: Regularity of the solution of some unilateral boundary value problems in polygonal domains. Math. Nachrichten, 2004

  12. Nochetto, R.H., Siebert, K., Veeser, A.: Pointwise a posteriori error control for elliptic obstacle problems. Numer Math. 95, 163–195 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Nochetto, R.H., Siebert, K., Veeser, A.: Fully localized a posteriori error estimators and barrier sets for contact problems. SIAM J. Numer. Anal. (to appear)

  14. Scott, L.R., Zhang, S.: Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp. 54, 483–493 (1990)

    MATH  MathSciNet  Google Scholar 

  15. Veeser, A.: Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39, 146–167 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley and Teubner, Chichester and Stuttgart, 1996

Download references

Author information

Authors and Affiliations

  1. Université de Franche-Comté, Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, 16 route de Gray, 25030, Besançon, France

    Patrick Hild

  2. Université de Valenciennes et du Hainaut Cambrésis, MACS, ISTV, 59313, Valenciennes Cedex 9, France

    Serge Nicaise

Authors
  1. Patrick Hild
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Serge Nicaise
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Patrick Hild.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hild, P., Nicaise, S. A posteriori error estimations of residual type for Signorini's problem. Numer. Math. 101, 523–549 (2005). https://doi.org/10.1007/s00211-005-0630-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-005-0630-5

Mathematics Subject Classification (2000)