We propose a new type of fully discrete finite element approximation of a class of semilinear stochastic parabolic equations with additive noise. Our discretization differs from the ones typically considered, in that we employ a nested finite element approximation of the noise. This is well suited for dealing with covariance operators defined in terms of (negative powers of) elliptic operators, like that of Whittle-Mat\'ern random fields. We derive strong and pathwise convergence rates for our proposed discretization, and our results are supported by numerical experiments.
Øyvind Stormark Auestad (Sun,) studied this question.