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

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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.

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References

  1. Da Prato, G.: Kolmogorov Equations for Stochastic PDEs (x+182 pp.). Birkhäuser, Basel (2004)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Hausenblas, E.: Numerical analysis of semilinear stochastic evolution equations in Banach spaces. J. Comput. Appl. Math. 147, 485–516 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hausenblas, E.: Approximation for semilinear stochastic evolution equations. Potential Anal. 18, 141–186 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. Schwab, Ch., Todor, R.A.: Karhunen-Loève approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217, 100–122 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  9. Todor, R.A., Robust eigenvalue computation for smoothing operators. SIAM J. Numer. Anal. 44, 865–878 (2006) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  10. Weidmann, J.: Linear Operators in Hilbert Spaces (xiii+402 pp). Springer, New York (1980)

    MATH  Google Scholar 

  11. Yan, Y.: Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise. BIT 44, 829–847 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Yan, Y.: Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 43, 1363–1384 (2005) (electronic).

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Author notes

  1. Mihály Kovács

    Present address: Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96, Göteborg, Sweden

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
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  2. Stig Larsson
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  3. Fredrik Lindgren
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Corresponding author

Correspondence to Mihály Kovács.

Additional information

M. Kovács was partially supported by the Swedish Research Council (VR). S. Larsson was partially supported by the Swedish Foundation for Strategic Research through GMMC, the Gothenburg Mathematical Modeling Centre and the Swedish Research Council (VR).

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Kovács, M., Larsson, S. & Lindgren, F. Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise. Numer Algor 53, 309–320 (2010). https://doi.org/10.1007/s11075-009-9281-4

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  • DOI: https://doi.org/10.1007/s11075-009-9281-4

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