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Approximating the invariant measure and the expectation of the functionalsfor parabolic stochastic partial differential equations (SPDEs) withnon-globally Lipschitz coefficients is an active research area and is far frombeing well understood. In this article, we study such problem in terms of afull discretization based on the spectral Galerkin method and the temporalimplicit Euler scheme. By deriving the a priori estimates and regularityestimates of the numerical solution via a variational approach and Malliavincalculus, we establish the sharp weak convergence rate of the fulldiscretization. When the SPDE admits a unique $V$-uniformly ergodic invariantmeasure, we prove that the invariant measure can be approximated by the fulldiscretization. The key ingredients lie on the time-independent weakconvergence analysis and time-independent regularity estimates of thecorresponding Kolmogorov equation. Finally, numerical experiments confirm thetheoretical findings.

Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE