Weak Convergence of Drift-Implicit Euler and Spectral Galerkin Approximation to Stochastic Allen–Cahn Equation Driven by Multiplicative Trace-Class Noise

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 1/2 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 3/4 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.