Finite element approximation of parabolic SPDEs with Whittle--Mat\'ern noise

TL;DR

This paper introduces a novel finite element method for approximating linear stochastic parabolic equations driven by Whittle--Matérn noise, achieving high convergence rates and verified through numerical experiments.

Contribution

It presents a new finite element discretization of the noise in stochastic PDEs, improving accuracy for equations with elliptic operator-based covariance structures.

Findings

Strong convergence rates up to order 2 in space and 1 in time.

Numerical experiments confirm theoretical convergence rates.

Applicable to equations with Whittle--Matérn type noise.

Abstract

We propose and analyse a new type of fully discrete finite element approximation of a class of linear stochastic parabolic evolution equations with additive noise. Our discretization differs from previous ones in that we use a finite element approximation of the noise, as opposed to an $L^2$ projection. This approximation is tailored for equations where the noise has covariance operator defined in terms of (negative powers of) elliptic operators, like Whittle--Mat\'ern random fields. Strong convergence rates up to order $2$ in space and $1$ in time are shown and verified by numerical experiments in dimension $1$ and $2$.