Subgrid and interelement interactions affect discretisations of stochastically forced diffusion

Abstract

Constructing discrete models of stochastic partial differential equations is very delicate. I apply dynamical systems theory to support spatial discretisations of the stochastically forced diffusion equation. To apply stochastic centre manifold theory, divide the physical domain into finite sized elements by introducing insulating internal boundaries which are later removed to fully couple the dynamical interactions between neighbouring elements. The approach automatically parametrises the stochastically forced microscale, subgrid structures within each element. The crucial aspect of this work is that we explore how a multitude of noise processes interact within and between neighbouring elements. Noise processes with coarse structure across a finite element are the most significant noises for the discrete model. Their influence also diffuses away to weakly correlate the noise in a spatial discretisation.