Advertisement

Numerical Algorithms

, Volume 53, Issue 2–3, pp 309–320 | Cite as

Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise

  • Mihály KovácsEmail author
  • Stig Larsson
  • Fredrik Lindgren
Original Paper

Abstract

We consider a semilinear parabolic PDE driven by additive noise. The equation is discretized in space by a standard piecewise linear finite element method. We show that the orthogonal expansion of the finite-dimensional Wiener process, that appears in the discretized problem, can be truncated severely without losing the asymptotic order of the method, provided that the kernel of the covariance operator of the Wiener process is smooth enough. For example, if the covariance operator is given by the Gauss kernel, then the number of terms to be kept is the quasi-logarithm of the number of terms in the original expansion. Then one can reduce the size of the corresponding linear algebra problem enormously and hence reduce the computational complexity, which is a key issue when stochastic problems are simulated.

Keywords

Finite element Semilinear parabolic equation Wiener process Error estimate Stochastic partial differential equation Truncation 

Mathematics Subject Classifications (2000)

65N30 60H35 60H15 35R60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Da Prato, G.: Kolmogorov Equations for Stochastic PDEs (x+182 pp.). Birkhäuser, Basel (2004)zbMATHGoogle Scholar
  2. 2.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions (xviii+454 pp.). Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  3. 3.
    Hausenblas, E.: Numerical analysis of semilinear stochastic evolution equations in Banach spaces. J. Comput. Appl. Math. 147, 485–516 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Hausenblas, E.: Approximation for semilinear stochastic evolution equations. Potential Anal. 18, 141–186 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Johnson, C., Larsson, S., Thomée, V., Wahlbin, L.B.: Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data. Math. Comput. 49, 331–357 (1987)zbMATHCrossRefGoogle Scholar
  6. 6.
    Prévôt, C., Röckner, M.: A consise course on stochastic partial differential equations. In: Lecture Notes in Mathematics, vol. 1905, vi+144 pp. Springer, Berlin (2007)Google Scholar
  7. 7.
    Schwab, Ch., Todor, R.A.: Karhunen-Loève approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217, 100–122 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems (2nd ed., xii+370 pp). Springer, Berlin (2006)zbMATHGoogle Scholar
  9. 9.
    Todor, R.A., Robust eigenvalue computation for smoothing operators. SIAM J. Numer. Anal. 44, 865–878 (2006) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Weidmann, J.: Linear Operators in Hilbert Spaces (xiii+402 pp). Springer, New York (1980)zbMATHGoogle Scholar
  11. 11.
    Yan, Y.: Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise. BIT 44, 829–847 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Yan, Y.: Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 43, 1363–1384 (2005) (electronic).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  • Mihály Kovács
    • 1
    Email author
  • Stig Larsson
    • 2
  • Fredrik Lindgren
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of OtagoDunedinNew Zealand
  2. 2.Department of Mathematical SciencesChalmers University of Technology and University of GothenburgGöteborgSweden

Personalised recommendations