  We first establish the unique ergodicity of the Markov chain generated by the
stochastic theta method (STM) with $\theta \in [1/2, 1]$ for monotone SODEs,
without growth restriction on the coefficients, driven by nondegenerate
multiplicative noise. The main ingredient of the arguments lies in constructing
new Lyapunov functions involving the coefficients, the stepsize, and $\theta$,
and the irreducibility and the strong Feller property for the STM. We then
generalize the arguments to the temporal drift-implicit Euler (DIE) method and
its Galerkin-based full discretizations for a class of monotone SPDEs driven by
infinite-dimensional nondegenerate multiplicative trace-class noise. Applying
these results to the stochastic Allen--Cahn equation indicates that its DIE
scheme is uniquely ergodic for any interface thickness, which gives an
affirmative answer to a question proposed in (J. Cui, J. Hong, and L. Sun,
Stochastic Process. Appl. (2021): 55--93). Numerical experiments verify our
theoretical results.
