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

TL;DR

This paper investigates how finite element discretisation affects the stationary distribution of SPDEs, showing that under certain conditions, the discretised stationary distribution converges to that of the continuous SPDE.

Contribution

It provides a rigorous analysis demonstrating convergence of the stationary distribution of finite element discretisations to that of the original SPDE.

Findings

Stationary distribution of discretised SPDE converges to the continuous case.

Convergence occurs in total variation norm under specific assumptions.

Finite element discretisation preserves key distributional properties.

Abstract

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.