Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise

In this paper, we analyze the weak error of a semi-discretization in time by the linear implicit Euler method for semilinear stochastic partial differential equations (SPDEs) with additive noise. The main result reveals how the weak order depends on the regularity of noise and that the order of weak convergence is twice that of strong convergence. In particular, the linear implicit Euler method for SPDEs driven by trace class noise achieves an almost optimal order 1−ϵ for arbitrarily small ϵ>0.