Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise

A unified approach is given for the analysis of the weak error of spatially semidiscrete finite element methods for linear stochastic partial differential equations driven by additive noise. An error representation formula is found in an abstract setting based on the semigroup formulation of stochastic evolution equations. This is then applied to the stochastic heat, linearized Cahn-Hilliard, and wave equations. In all cases it is found that the rate of weak convergence is twice the rate of strong convergence, sometimes up to a logarithmic factor, under the same or, essentially the same, regularity requirements.