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

Abstract

We discrete the ergodic semilinear stochastic partial differential equations in space dimension d≤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 12−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.