In this paper, we address the study of the time-dependent Stokes system with boundary conditions involving the pressure. We obtain existence and uniqueness for a classof Lipschitz-continuous domains. Next, a spectral discretizations of the problem is proposed combined with the backward Euler scheme. The discrete spaces are defined in a way to give exactly divergence-free discrete approximations for the velocity. Then, we prove the associated discrete inf-sup condition and derive a priori error estimates. Finally, some numerical experiments are presented
