Skip to main content
SpringerLink
Log in

Stability and error of the variable two-step BDF for semilinear parabolic problems

  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

The temporal discretisation of a moderate semilinear parabolic problem in an abstract setting by the two-step backward differentiation formula with variable step sizes is analysed. Stability as well as optimal smooth data error estimates are derived if the ratios of adjacent step sizes are bounded from above by 1.91.

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. J. Becker,A second order backward difference method with variable steps for a parabolic problem, BIT,38 (1998) 4, pp. 644–662.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Crouzeix and V. Thomée,On the discretization in time of semilinear parabolic equations with nonsmooth initial data, Math. Comp.,49 (1987) 180, pp. 359–377.

    Article  MATH  MathSciNet  Google Scholar 

  3. E. Emmrich,Stability and convergence of the two-step BDF for the incompressible Navier-Stokes problem, Int. J. Nonlinear Sci. Numer. Simul.,5 (2004) 3, pp. 199–209.

    Google Scholar 

  4. E. Emmrich,Error of the two-step BDF for the incompressible Navier-Stokes problem, M2AN, Math. Model. Numer. Anal.38 (2004) 5, 757–764.

    Article  MATH  MathSciNet  Google Scholar 

  5. D. Estep and S. Larsson,The discontinuous Galerkin method for semilinear parabolic problems, M2AN, Math Model. Numer. Anal.,27 (1993) 1, 35–54.

    MATH  MathSciNet  Google Scholar 

  6. H. Fujita and T. Suzuki,Evolution Problems, in Handbook of Numerical Analysis, Vol. II: Finite Element Methods (Part 1), P. G. Ciarlet and J.-L. Lions, eds. Elsevier, Amsterdam, 1991, pp. 789–928.

    Google Scholar 

  7. C. González, A. Ostermann, C. Palencia, and M. Thalhammer,Backward Euler discretization of fully nonlinear parabolic problems. Math. Comput.,71 (2002) 237, pp. 125–145.

    MATH  Google Scholar 

  8. R. D. Grigorieff,Stability of multistep-methods on variable grids, Numer. Math.,42 (1983), pp. 359–377.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. D. Grigorieff,Time discretization of semigroups by the variable two-step BDF method, in Numerical treatment of differential equations, Sel. Papers NUMDIFF-5, Halle, 1989, B. G. Teubner, Leipzig, 1991, pp. 204–216.

    Google Scholar 

  10. R. D. Grigorieff,On the variable grid two-step BDF method for parabolic equations, Preprint 426, Fachber. Mathematik, Techn. Univ. Berlin, 1995.

  11. M.-N. Le Roux,Variable step size multistep methods for parabolic problems, SIAM J. Numer. Anal.,19 (1982) 4, pp. 725–741.

    Article  MATH  MathSciNet  Google Scholar 

  12. J.-L. Lions,Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthier-Villars, Paris, 1969.

    MATH  Google Scholar 

  13. C. Lubich and A. Ostermann,Linearly implicit time discretization of non-linear parabolic equations, IMA J. Numer. Anal.,15 (1995) 4, pp. 555–583.

    Article  MATH  MathSciNet  Google Scholar 

  14. C. Lubich and A. Ostermann,Runge-Kutta time discretization of reaction-diffusion and Navier-Stokes equations: nonsmooth-data error estimates and applications to long-time behaviour, Appl. Numer. Math.,22 (1996), pp. 279–292.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Temam,Behaviour at time t=0 of the solutions of semi-linear evolution equations, J. Diff. Eqs.,43 (1982), pp. 73–92.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Ostermann and M. Thalhammer,Non-smooth data error estimates for linearly implicit Runge-Kutta methods, IMA J. Numer. Anal.,20 (2000) 20, pp. 167–184.

    Article  MATH  MathSciNet  Google Scholar 

  17. C. Palencia and B. García-Archilla,Stability of linear multistep methods for sectorial operators in Banach spaces, Appl. Numer. Math.,12 (1993) 6, pp. 503–520.

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Slodička,Smoothing effect and discretization in time to semilinear parabolic equations with nonsmooth data, Comment. Math. Univ. Carol.,32 (1991) 4, 703–713.

    MATH  Google Scholar 

  19. M. Slodička,Semigroup formulation of Rothe's method: application to parabolic problems, Comment. Math. Univ. Carol.,33 (1992) 2, 245–260.

    MATH  Google Scholar 

  20. V. Thomée,Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin, 1997.

    MATH  Google Scholar 

  21. E. Zeidler,Nonlinear Functional Analysis and its Applications I, Springer, New York, 1992.

    Google Scholar 

  22. M. Zlámal,Finite element methods for nonlinear parabolic equations, RAIRO Analyse Numérique,11 (1977) 1, pp. 93–107.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Etienne Emmrich

Authors
  1. Etienne Emmrich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Etienne Emmrich.

Additional information

Etienne Emmrich received his Diploma degree in mathematics from the Magdeburg University in Germany and his Dr. rer. nat. from the Technical University Berlin. In his research, he is in particular interested in the numerical analysis of differential equations.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Emmrich, E. Stability and error of the variable two-step BDF for semilinear parabolic problems. JAMC 19, 33–55 (2005). https://doi.org/10.1007/BF02935787

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02935787

AMS Mathematics Subject Classification

Key words and phrases

Search

Navigation