Surface finite element approximation of parabolic SPDEs with Whittle--Matérn noise

We propose and analyse a new type of fully discrete surface finite element approximation of a class of linear parabolic stochastic evolution equations with additive noise. Our discretization uses a surface finite element approximation of the noise, and is tailored for equations with noise having covariance operator defined by (negative powers of) elliptic operators, like Whittle--Mat\'ern random fields. We derive strong and pathwise convergence rates of our approximation, and verify these by numerical experiments.