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

Mihaly Kovacs ; Stig Larsson (Institutionen för matematiska vetenskaper, matematik) ; Fredrik Lindgren (Institutionen för matematiska vetenskaper, matematik)
arXiv:1203.2029v1 [math.NA] p. 1-24. (2012)
[Preprint]

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.

Nyckelord: finite element, parabolic equation, hyperbolic equation, stochastic, heat equation, Cahn-Hilliard-Cook equation, wave equation, additive noise, Wiener process, error estimate, weak convergence



Denna post skapades 2012-03-19. Senast ändrad 2014-09-02.
CPL Pubid: 156010

 

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Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)

Ämnesområden

Numerisk analys

Chalmers infrastruktur