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

Original Paper


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.


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

Mathematics Subject Classifications (2000)

65N30 60H35 60H15 35R60 


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Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  • Mihály Kovács
    • 1
  • 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

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