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We discrete the ergodic semilinear stochastic partial differential equationsin space dimension $d \leq 3$ with additive noise, spatially by a spectralGalerkin method and temporally by an exponential Euler scheme. It is shown thatboth the spatial semi-discretization and the spatio-temporal fulldiscretization are ergodic. Further, convergence orders of the numericalinvariant measures, depending on the regularity of noise, are recovered basedon an easy time-independent weak error analysis without relying on Malliavincalculus. To be precise, the convergence order is $1-\epsilon$ in space and$\frac{1}{2}-\epsilon$ in time for the space-time white noise case and$2-\epsilon$ in space and $1-\epsilon$ in time for the trace class noise casein space dimension $d = 1$, with arbitrarily small $\epsilon>0$. Numericalresults are finally reported to confirm these theoretical findings.

A full-discrete exponential Euler approximation of invariant measure for parabolic stochastic partial differential equations