Skip to main content
SpringerLink
Log in

A Posteriori Error Estimates for Parabolic Variational Inequalities

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We study a posteriori error estimates in the energy norm for some parabolic obstacle problems discretized with a Euler implicit time scheme combined with a finite element spatial approximation. We discuss the reliability and efficiency of the error indicators, as well as their localization properties. Apart from the obstacle resolution, the error indicators vanish in the so-called full contact set. The case when the obstacle is piecewise affine is studied before the general case. Numerical examples are given.

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. Bergam, A., Bernardi, C., Mghazli, Z.: A posteriori analysis of the finite element discretization of some parabolic equations. Math. Comput. 74(251), 1117–1138 (2005) (electronic)

    MATH  MathSciNet  Google Scholar 

  2. Binev, P., Dahmen, W., DeVore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 97(2), 219–268 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, Z., Nochetto, R.H.: Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84(4), 527–548 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, vol. II, pp. 17–351. North-Holland, Amsterdam (1991)

    Google Scholar 

  5. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    Book  MATH  Google Scholar 

  6. Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal. 28(1), 43–77 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  7. Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems. II. Optimal error estimates in L ∞ L 2 and L ∞ L ∞. SIAM J. Numer. Anal. 32(3), 706–740 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hecht, F., Pironneau, O.: FREEFEM. www.ann.jussieu.fr

  9. Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13(3), 865–888 (2003) (electronic)

    Article  MATH  Google Scholar 

  10. Hoppe, R.H.W., Kornhuber, R.: Adaptive multilevel methods for obstacle problems. SIAM J. Numer. Anal. 31(2), 301–323 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ito, K., Kunisch, K.: Semi-smooth Newton methods for variational inequalities of the first kind. M2AN Math. Model. Numer. Anal. 37(1), 41–62 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Johnson, C.: Adaptive finite element methods for the obstacle problem. Math. Models Methods Appl. Sci. 2(4), 483–487 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kornhuber, R.: A posteriori error estimates for elliptic variational inequalities. Comput. Math. Appl. 31(8), 49–60 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nochetto, R.H., Siebert, K.G., Veeser, A.: Pointwise a posteriori error control for elliptic obstacle problems. Numer. Math. 95(1), 163–195 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Nochetto, R.H., Siebert, K.G., Veeser, A.: Fully localized a posteriori error estimators and barrier sets for contact problems. SIAM J. Numer. Anal. 42(5), 2118–2135 (2005) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  16. Veeser, A.: Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39(1), 146–167 (2001) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  17. Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, New York (1996)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. UFR Mathématiques, Université Paris Diderot, Case 7012, 75251, Paris Cedex 05, France

    Yves Achdou

  2. Laboratoire Jacques-Louis Lions, Université Paris 6, 75252, Paris Cedex 05, France

    Yves Achdou & Frédéric Hecht

  3. Projet MATHFI–INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153, Le Chesnay Cedex, France

    David Pommier

Authors
  1. Yves Achdou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frédéric Hecht
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Pommier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yves Achdou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Achdou, Y., Hecht, F. & Pommier, D. A Posteriori Error Estimates for Parabolic Variational Inequalities. J Sci Comput 37, 336–366 (2008). https://doi.org/10.1007/s10915-008-9215-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-008-9215-7

Keywords

Search

Navigation