Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes
Journal article, 2013

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.

Author

Mihaly Kovacs

University of Otago

Stig Larsson

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Fredrik Lindgren

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

BIT Numerical Mathematics

0006-3835 (ISSN) 1572-9125 (eISSN)

Vol. 53 2 497-525

Subject Categories

Computational Mathematics

Roots

Basic sciences

DOI

10.1007/s10543-012-0405-1

More information

Created

10/7/2017