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

TL;DR

This paper develops an abstract framework to analyze the weak convergence of fully discrete finite element schemes for linear stochastic evolution equations with additive noise, demonstrating that weak convergence rates are twice the strong rates and exploring higher order schemes.

Contribution

It introduces a novel abstract representation formula for weak error analysis and applies it to wave and parabolic equations, highlighting the benefits of higher order time schemes.

Findings

Weak convergence rate is twice the strong convergence rate.

Higher order schemes like Crank-Nicolson are beneficial for less regular solutions.

The framework applies to wave, heat, and Cahn-Hilliard-Cook equations.

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 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.