The Effect of Finite Element Discretisation on the Stationary Distribution of SPDEs

This article studies the effect of discretisation error on the stationary distribution of stochastic partial differential equations (SPDEs). We restrict the analysis to the effect of space discretisation, performed by finite element schemes. The main result is that under appropriate assumptions the stationary distribution of the finite element discretisation converges in total variation norm to the stationary distribution of the full SPDE.