BIT Numerical Mathematics

, Volume 53, Issue 2, pp 497–525 | Cite as

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



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.


Finite element Parabolic equation Hyperbolic equation Stochastic Heat equation Cahn-Hilliard-Cook equation Wave equation Additive noise Wiener process Error estimate Weak convergence Rational approximation Time discretization 

Mathematics Subject Classification (2010)

65M60 60H15 60H35 65C30 


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Copyright information

© Springer Science+Business Media Dordrecht 2012

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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