Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise

In this paper, we consider the ergodic semilinear stochastic partial differential equation driven by additive noise and the long-time behavior of its full discretization realized by a spectral Galerkin method in spatial direction and an Euler scheme in the temporal direction, which admits a unique invariant probability measure. Under the condition that the nonlinearity is once differentiable, the optimal convergence orders of the numerical invariant measures are obtained based on the time-independent weak error, but not relying on the associated Kolmogorov equation. More precisely, the obtained convergence orders are O(λN−γ) in space and O(τγ) in time, where γ∈(0,1] from the assumption ∥Aγ−12Q12∥L2 is used to characterize the spatial correlation of the noise process. Finally, numerical examples confirm the theoretical findings.