This paper presents a new method for the solution of multiscale stochastic differential equations at
the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale
method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper
we introduce a new numerical methodology that is based on a spectral method. In particular, we use
an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation,
which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence
is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate
the performance of the method. A comparison with the HMM and an application to singularly
perturbed stochastic PDEs are also presented.