We discrete the ergodic semilinear stochastic partial differential equations in space dimension $d \leq 3$ with additive noise, spatially by a spectral Galerkin method and temporally by an exponential Euler scheme. It is shown that both the spatial semi-discretization and the spatio-temporal full discretization are ergodic. Further, convergence orders of the numerical invariant measures, depending on the regularity of noise, are recovered based on an easy time-independent weak error analysis without relying on Malliavin calculus. To be precise, the convergence order is $1-��$ in space and $\frac{1}{2}-��$ in time for the space-time white noise case and $2-��$ in space and $1-��$ in time for the trace class noise case in space dimension $d = 1$, with arbitrarily small $��>0$. Numerical results are finally reported to confirm these theoretical findings.

27 pages, to appear in: Applied Numerical Mathematics