Skip to main content
SpringerLink
Log in

Error estimates for the numerical solution of a time-periodic linear parabolic problem

  • Part II Numerical Mathematics
  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

In this paper we consider the numerical solution of a time-periodic linear parabolic problem. We derive optimal order error estimates inL 2(Ω) for approximate solutions obtained by discretizing in space by a Galerkin finite-element method and in time by single-step finite difference methods, using known estimates for the associated initial value problem. We generalize this approach and obtain error estimates for more general discretization methods in the norm of a Banach spaceB ⊂L 2(Ω), e.g.,B=H 10 (Ω) orL ∞(Ω). Finally, we consider some computational aspects and give a numerical 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

Similar content being viewed by others

References

  1. O. Axelsson and V. A. Barker,Finite Element Solution of Boundary Value Problems, Academic Press, London, 1984.

    Google Scholar 

  2. G. A. Baker, J. H. Bramble and V. Thomée,Single step Galerkin approximations for parabolic problems, Math. Comp. 31 (1977), 818–847.

    Google Scholar 

  3. C. Bernardi,Numerical approximation of a periodic linear parabolic problem, SIAM J. Numer. Anal. 19 (1982), 1196–1207.

    Google Scholar 

  4. P. Brenner, M. Crouzeix and V. Thomée,Single step methods for inhomogeneous linear differential equations in Banach space, RAIRO Anal. Numér. 16 (1982), 5–26.

    Google Scholar 

  5. A. Carasso,On least squares methods for parabolic equations and the computation of time-periodic solutions, SIAM J. Numer. Anal. 11 (1974), 1181–1192.

    Google Scholar 

  6. J. Douglas, Jr. and T. Dupont,Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 7 (1970), 575–626.

    Google Scholar 

  7. W. Hackbusch,Fast numerical solution of time-periodic parabolic problems by a multigrid method, SIAM J. Sci. Stat. Comput. 2 (1981), 198–206.

    Google Scholar 

  8. W. Hackbusch,Multi-Grid Methods and Applications, Springer-Verlag, Berlin Heidelberg, 1985.

    Google Scholar 

  9. A. Hansbo,Numerical solution of a time-periodic linear parabolic problem, Preprint 1990-03, Dept. of Math., Chalmers Univ. of Technology and the Univ. of Göteborg.

  10. J. L. Lions and E. Magenes,Problèmes aux Limites non Homogènes et Applications, vol. I and II, Dunod, Paris, 1968.

    Google Scholar 

  11. M. Luskin and R. Rannacher,On the smoothing property of the Galerkin method for parabolic equations, SIAM J. Numer. Anal. 19 (1982), 93–113.

    Google Scholar 

  12. A. H. Schatz, V. Thomée, and L. B. Wahlbin,Maximum norm stability and error estimates in parabolic finite element equations, Comm. Pure Appl. Math. 33 (1980), 265–304.

    Google Scholar 

  13. V. Thomée,Galerkin Finite Element Method for Parabolic Problems, Lecture Notes in Mathematics, No 1054, Springer-Verlag, Berlin Heidelberg, 1984.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Chalmers University of Technology and the University of Göteborg, S-412 96, Göteborg, Sweden

    Anita Hansbo

Authors
  1. Anita Hansbo
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hansbo, A. Error estimates for the numerical solution of a time-periodic linear parabolic problem. BIT 31, 664–685 (1991). https://doi.org/10.1007/BF01933180

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01933180

AMS 1980 subject classification

Keywords

Search

Navigation