Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces

The present paper is devoted to the numerical approximation of an abstract stochastic nonlinear evolution equation in a separable Hilbert space {$\mathrm{H}$}. Examples of equations which fall into our framework include the GOY and Sabra shell models and { a class of nonlinear heat equations.} The space-time numerical scheme is defined in terms of a Galerkin approximation in space and a { semi-implicit Euler--Maruyama scheme in time}. {We prove the convergence in probability of our scheme by means of an estimate of the error on a localized set of arbitrary large probability.} Our error estimate is shown to hold in a more regular space $\mathrm{V}_{\beta}\subset \mathrm{H}$ with $\beta \in [0,\frac14)$ and { that the explicit rate of convergence of our scheme depends on this parameter $\beta$. }