Strong Approximation of Monotone SPDEs Driven by Multiplicative Noise: Exponential Ergodicity and Uniform Estimates

We analyze the long-time behavior of numerical schemes, studied by \cite{LQ21} in a finite time horizon, for a class of monotone SPDEs driven by multiplicative noise. We derive several time-independent a priori estimates for both the exact and numerical solutions and establish time-independent strong error estimates between them. These uniform estimates, in combination with ergodic theory of Markov processes, are utilized to establish the exponential ergodicity of these numerical schemes with an invariant measure. Applying these results to the stochastic Allen--Cahn equation indicates that these numerical schemes always have at least one invariant measure, respectively, and converge strongly to the exact solution with sharp time-independent rates. We also show that the invariant measures of these schemes are also exponentially ergodic and thus give an affirmative answer to a question proposed in \cite{CHS21}, provided that the interface thickness is not too small.