  In this paper we consider the Navier-Stokes equations supplemented with
either the Dirichlet or vorticity-based Navier boundary conditions. We prove
that weak solutions obtained as limits of solutions to the Navier-Stokes-Voigt
model satisfy the local energy inequality. Moreover, in the periodic setting we
prove that if the parameters are chosen in an appropriate way, then we can
construct suitable weak solutions trough a Fourier-Galerkin finite-dimensional
approximation in the space variables.
