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 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 Vβ⊂H with β∈[0,14) and that the explicit rate of convergence of our scheme depends on this parameter β.