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

This thesis is concerned with numerical approximation of linear stochastic partialdifferential equations driven by additive noise. In the first part, we develop aframework for the analysis of weak convergence and within this framework weanalyze the stochastic heat equation, the stochastic wave equation, and the linearizedstochastic Cahn-Hilliard, or the linearized Cahn-Hilliard-Cook equation.The general rule of thumb, that the rate of weak convergence is twice the rate ofstrong convergence, is confirmed.In the second part, we investigate various ways to approximate the drivingnoise and analyze its effect on the rate of strong convergence. First, we considerthe use of frames to represent the noise. We show that if the frame is chosen in away that is well suited for the covariance operator, then the number of elementsof the frame needed to represent the noise without effecting the overall convergencerate of the numerical method may be quite low. Second, we investigate theuse 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 basisfunctions needed may be substantially reduced.Our analysis is done in a framework based on operator semigroups. It is performedin a way that reduces our results to results about approximation of therespective (deterministic) semigroup