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

We discrete the ergodic semilinear stochastic partial differential equations in space dimension $d\le 3$ 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 $1-ϵ$ in space and $\frac{1}{2}-ϵ$ in time for the space-time white noise case and $2-ϵ$ in space and $1-ϵ$ in time for the trace class noise case in space dimension $d=1$, with arbitrarily small $ϵ>0$. Numerical results are finally reported to confirm these theoretical findings.