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

Abstract

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.