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

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 $2$.