In this paper, we study
the linear Schrödinger equation over the d-dimensional torus,
with small values of the perturbing potential.
We consider numerical approximations of the associated solutions obtained
by a symplectic splitting method (to discretize the time variable) in combination with the
Fast Fourier Transform algorithm (to discretize the space variable).
In this fully discrete setting, we prove that the regularity of the initial
datum is preserved over long times, i.e. times that are exponentially long
with the time discretization parameter. We here refer to Gevrey regularity, and our estimates
turn out to be uniform in the space discretization parameter.
This paper extends [G. Dujardin and E. Faou, Numer. Math.
97 (2004) 493–535], where a similar result has been obtained in
the semi-discrete situation, i.e. when the mere time variable is discretized and space
is kept a continuous variable.