Abstract In this paper, we study numerical methods for the stochastic Allen–Cahn equation driven by multiplicative trace-class noise. The temporal discretization uses a drift-implicit Euler scheme, and the spatial discretization employs a spectral Galerkin method. We show that the spatial weak convergence rate is nearly one order higher than the corresponding strong convergence rate for d=1, 2 d = 1, 2, and nearly 12 1 2 order higher than the corresponding strong convergence rate for d=3 d = 3 ; and that the temporal weak convergence rate is close to order one for d=1, 2 d = 1, 2 and close to 34 3 4 for d=3 d = 3. The weak error analysis is carried out by deriving a priori estimates for the solutions of the Kolmogorov equations associated with the spectral Galerkin semi-discretization. We also develop techniques to handle the trace of an operator involving a stochastic integral for the temporal weak error analysis. Finally, numerical experiments are presented to illustrate the theoretical results.
Zhang et al. (Wed,) studied this question.