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In this paper, we consider the time-dependent Stokes problem in a three-dimensional domain with mixed boundary conditions. The discretization relies on spectral methods with respect to the space variables and Euler's implicit scheme with respect to the time variable, then by the second order BDF method. A detailed numerical analysis leads to a priori error estimates for each numerical scheme.
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 class of 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.
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