Skip to main content
SpringerLink
Log in

A posteriori analysis of iterative algorithms for a nonlinear problem

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

Abstract

A posteriori error indicators have been studied in recent years owing to their remarkable capacity to enhance both speed and accuracy in computing. This work deals with a posteriori error estimation for the finite element discretization of a nonlinear problem. For a given nonlinear equation considering finite elements we solve the discrete problem using iterative methods involving some kind of linearization. For each of them, there are actually two sources of error, namely discretization and linearization. Balancing these two errors can be very important, since it avoids performing an excessive number of iterations. Our results lead to the construction of computable upper indicators for the full error. Several numerical tests are provided to evaluate the efficiency of our indicators.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev spaces. Acadamic Press, INC, Waltham (1978)

    Google Scholar 

  2. Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 4, 736–754 (1978)

    Google Scholar 

  3. Brezzi, F., Rappaz, J., Raviart, P.-A.: Finite dimensional approximation of nonlinear problems, part I: branches of nonsingular solutions. Numer. Math. 36, 1–25 (1980)

    Article  MATH  Google Scholar 

  4. Chaillou, A.-L., Suri, M.: Computable error estimators for the approximation of nonlinear problems by linearized models. Comput. Methods Appl. Mech. Eng. 196, 210–224 (2006)

    Article  MATH  Google Scholar 

  5. Chaillou, A.-L., Suri, M.: A posteriori estimation of the linearization error for strongly monotone nonlinear operators. Comput. Methods Appl. Mech. Eng. 205, 72–87 (2007)

    MATH  Google Scholar 

  6. El Alaoui, L., Ern, A., Vohralík, M.: Guaranteed and robust a posteriori error estimate and balancing discretization and linearization errore for monotone non linear problems. Comput. Methods Appl. Mech. Eng. 200, 2782–2795 (2011)

    Article  MATH  Google Scholar 

  7. Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations, theory and algorithms. Springer, Berlin-Heidelberg (1986)

  8. Hecht, F.: New development in freefem++. J. Numer. Math. 20, 251–266 (2012)

    Article  MATH  Google Scholar 

  9. Kavian, O.: Introduction à la théorie des points critiques et applications aux problèmes elliptiques. Springer, Berlin-Heidelberg (1993)

  10. Pousin, J., Rappaz, J.: Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numer. Math. 69(2), 213–231 (1994)

    Article  MATH  Google Scholar 

  11. Verfürth, R.: A posteriori error estimation techniques for finite element methods. Oxford University Press, Oxford (2013)

Download references

Author information

Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, C. N. R. S., Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252, Paris Cedex 05, France

    Christine Bernardi

  2. Faculté des Sciences, Université Saint-Joseph, B.P 11-514, Riad El Solh, Beirut, 1107 2050, Lebanon

    Jad Dakroub, Gihane Mansour & Toni Sayah

Authors
  1. Christine Bernardi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jad Dakroub
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gihane Mansour
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Toni Sayah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christine Bernardi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bernardi, C., Dakroub, J., Mansour, G. et al. A posteriori analysis of iterative algorithms for a nonlinear problem. J Sci Comput 65, 672–697 (2015). https://doi.org/10.1007/s10915-014-9980-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-014-9980-4

Keywords

Search

Navigation