Preprint, 2012

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.

Wiener process

parabolic equation

stochastic

additive noise

error estimate

finite element

Cahn-Hilliard-Cook equation

heat equation

weak convergence

hyperbolic equation

wave equation

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

1-24

Computational Mathematics

Basic sciences