Skip to main content
SpringerLink
Log in

Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation.

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. Andersson, A., Larsson, S.: Weak error for the approximation in space of the non-linear stochastic heat equation. Preprint (2012)

  2. Baker, G.A., Bramble, J.H.: Semidiscrete and single step fully discrete approximations for second order hyperbolic equations. RAIRO. Anal. Numér. 13, 76–100 (1979)

    MathSciNet  Google Scholar 

  3. Brenner, P., Thomée, V.: On rational approximations of groups of operators. SIAM J. Numer. Anal. 17, 119–125 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brzeźniak, Z.: Some remarks on Itô and Stratonovich integration in 2-smooth Banach spaces. In: Probabilistic Methods in Fluids, pp. 48–69. World Scientific, River Edge (2003)

    Google Scholar 

  5. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992), xviii+454 pp.

    Book  MATH  Google Scholar 

  6. Da Prato, G., Zabczyk, J.: Second Order Partial Differential Equations in Hilbert Spaces. Cambridge University Press, Cambridge (2002), xvi+379 pp.

    Book  MATH  Google Scholar 

  7. Da Prato, G., Jentzen, A., Roekner, M.: A mild Ito formula for SPDEs. arXiv:1009.3526v4 [math.PR]

  8. de Bouard, A., Debussche, A.: Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation. Appl. Math. Optim. 54, 369–399 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Debussche, A.: Weak approximation of stochastic partial differential equations: the non linear case. Math. Comput. 80, 89–117 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Debussche, A., Printems, J.: Weak order for the discretization of the stochastic heat equation. Math. Comput. 78, 845–863 (2009)

    MathSciNet  MATH  Google Scholar 

  11. Elliott, C.M., Larsson, S.: Error estimates with smooth and nonsmooth data for the finite element method for the Cahn-Hilliard equation. Math. Comput. 58, 603–630 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hausenblas, E.: Weak approximation for semilinear stochastic evolution equations. In: Stochastic Analysis and Related Topics VIII. Progr. Probab., vol. 53, pp. 111–128 (2003)

    Chapter  Google Scholar 

  13. Hausenblas, E.: Weak approximation of the stochastic wave equation. J. Comput. Appl. Math. 235, 33–58 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Geissert, M., Kovács, M., Larsson, S.: Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise. BIT Numer. Math. 49, 343–356 (2009)

    Article  MATH  Google Scholar 

  15. Kovács, M., Larsson, S., Saedpanah, F.: Finite element approximation of the linear stochastic wave equation with additive noise. SIAM J. Numer. Anal. 48, 408–427 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kovács, M., Larsson, S., Lindgren, F.: Weak convergence of finite element approximations of stochastic evolution equations with additive noise. BIT Numer. Math. 52, 85–108 (2012)

    Article  MATH  Google Scholar 

  17. Larsson, S., Mesforush, A.: Finite element approximation of the linear stochastic Cahn-Hilliard equation. IMA J. Numer. Anal. 31, 1315–1333 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lax, P.D.: Functional Analysis. Wiley-Interscience, New York (2002), xx+580 pp.

    MATH  Google Scholar 

  19. Le Roux, M.N.: Semidiscretization in time for parabolic problems. Math. Comput. 33, 919–931 (1979)

    MathSciNet  MATH  Google Scholar 

  20. Lindner, F., Schilling, R.L.: Weak order for the discretization of the stochastic heat equation driven by impulsive noise. Potential Analysis. doi:10.1007/s11118-012-9276-y

  21. Nørsett, S.P., Wanner, G.: The real-pole sandwich for rational approximations and oscillation equations. BIT Numer. Math. 19, 79–94 (1979)

    Article  Google Scholar 

  22. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983), viii+279 pp.

    Book  MATH  Google Scholar 

  23. Quer-Sardanyons, L., Sanz-Solé, M.: Space semi-discretisations for a stochastic wave equation. Potential Anal. 24, 303–332 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, 2nd edn. Springer, Berlin (2006), xii+370 pp.

    MATH  Google Scholar 

  25. Walsh, J.B.: On numerical solutions of the stochastic wave equation. Ill. J. Math. 50, 991–1018 (2006)

    MATH  Google Scholar 

  26. Weidmann, J.: Linear Operators in Hilbert Spaces. Springer, Berlin (1980), xiii+402 pp.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand

    Mihály Kovács

  2. Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96, Göteborg, Sweden

    Stig Larsson & Fredrik Lindgren

Authors
  1. Mihály Kovács
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stig Larsson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fredrik Lindgren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mihály Kovács.

Additional information

Communicated by Desmond Higham.

S. Larsson and F. Lindgren are supported by the Swedish Research Council (VR).

S. Larsson is supported by the Swedish Foundation for Strategic Research (SSF) through GMMC, the Gothenburg Mathematical Modelling Centre.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kovács, M., Larsson, S. & Lindgren, F. Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes. Bit Numer Math 53, 497–525 (2013). https://doi.org/10.1007/s10543-012-0405-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-012-0405-1

Keywords

Mathematics Subject Classification (2010)

Search

Navigation