On weak and strong convergence of numerical approximations of stochastic partial differential equations

Doktorsavhandling, 2012

This thesis is concerned with numerical approximation of linear stochastic partial differential equations driven by additive noise. In the first part, we develop a framework for the analysis of weak convergence and within this framework we analyze the stochastic heat equation, the stochastic wave equation, and the linearized stochastic Cahn-Hilliard, or the linearized Cahn-Hilliard-Cook equation. The general rule of thumb, that the rate of weak convergence is twice the rate of strong convergence, is confirmed. In the second part, we investigate various ways to approximate the driving noise and analyze its effect on the rate of strong convergence. First, we consider the use of frames to represent the noise. We show that if the frame is chosen in a way that is well suited for the covariance operator, then the number of elements of the frame needed to represent the noise without effecting the overall convergence rate of the numerical method may be quite low. Second, we investigate the use of finite element approximations of the eigenpairs of the covariance operator. It turns out that if the kernel of the operator is smooth, then the number of basis functions needed may be substantially reduced. Our analysis is done in a framework based on operator semigroups. It is performed in a way that reduces our results to results about approximation of the respective (deterministic) semigroup.