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

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Authors Mihaly Kovacs
Stig Larsson
Fredrik Lindgren
Published in arXiv:1203.2029v1 [math.NA]
Pages 1-24
Publication year 2012
Published at Department of Mathematical Sciences, Mathematics
Pages 1-24
Language en
Links arxiv.org/abs/1203.2029
https://gup.ub.gu.se/file/83797
Keywords finite element, parabolic equation, hyperbolic equation, stochastic, heat equation, Cahn-Hilliard-Cook equation, wave equation, additive noise, Wiener process, error estimate, weak convergence
Subject categories Numerical analysis

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 continu- ous 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 ap- ply 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.

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