Skip to main content
SpringerLink
Log in

On error functionals

  • Published:
SeMA Journal Aims and scope Submit manuscript

Abstract

After exploring the particular situation of a non-variational elliptic equation, we introduce the formal concept of an error functional as a generalization of the intuitive idea of a non-negative functional whose only possible critical value is zero. The main result we prove is that such an error is a true measure of how far we are from the zero set of the error. It turns out that an error functional always complies with the Palais–Smale condition. We illustrate the abstract results with a specific example.

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

Similar content being viewed by others

References

  1. Bochev, P., Gunzburger, M.D.: Least-squares finite element methods. Applied Mathematical Sciences, 166. Springer, New York (2009)

  2. Bochev, P., Gunzburger, M.: Least-squares finite element methods. In: International Congress of Mathematicians, vol. III, pp. 1137–1162. European Mathematical Society, Zurich (2006)

  3. Costa, D.: An invitation to Variational Methods in Differential Equations. Birkhäuser, Boston (2007)

    Book  MATH  Google Scholar 

  4. Dacorogna, B.: Direct methods in the Calculus of Variations, 2nd edn. Springer, Berlin (2008)

  5. Evans, L.C.: Partial differential equations, 2nd edn. Graduate Studies in Mathematics, vol. 19, AMS, Providence (2010)

  6. Hecht, F.: New development in freefem++. J. Numer. Math. 20(3–4), 251–265 (2012)

    MATH  MathSciNet  Google Scholar 

  7. Glowinski, R.: Numerical methods for nonlinear variational problems. Springer series in computational physics (1983)

  8. Pedregal, P.: Efectos sorprendentes relacionados con la homogeneización. Bol. Soc. Esp. Mat. Appl. 46, 9–28 (2009)

    MATH  MathSciNet  Google Scholar 

  9. Willem, M.: Lectures on critical point theory, Trabalhos de Matemática. University of Brasilia, Brasilia (1983)

Download references

Author information

Authors and Affiliations

  1. E.T.S. Ingenieros Industriales, Universidad de Castilla-La Mancha, Campus de Ciudad Real, Ciudad Real, Spain

    Pablo Pedregal

Authors
  1. Pablo Pedregal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pablo Pedregal.

Additional information

Research supported in part by MTM2010-19739 of the MCyT (Spain).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pedregal, P. On error functionals. SeMA 65, 13–22 (2014). https://doi.org/10.1007/s40324-014-0016-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40324-014-0016-7

Keywords

Mathematics Subject Classification

Search

Navigation