  Proper orthogonal decomposition (POD) stabilized methods for the
Navier-Stokes equations are considered and analyzed. We consider two cases, the
case in which the snapshots are based on a non inf-sup stable method and the
case in which the snapshots are based on an inf-sup stable method. For both
cases we construct approximations to the velocity and the pressure. For the
first case, we analyze a method in which the snapshots are based on a
stabilized scheme with equal order polynomials for the velocity and the
pressure with Local Projection Stabilization (LPS) for the gradient of the
velocity and the pressure. For the POD method we add the same kind of LPS
stabilization for the gradient of the velocity and the pressure than the direct
method, together with grad-div stabilization. In the second case, the snapshots
are based on an inf-sup stable Galerkin method with grad-div stabilization and
for the POD model we apply also grad-div stabilization. In this case, since the
snapshots are discretely divergence-free, the pressure can be removed from the
formulation of the POD approximation to the velocity. To approximate the
pressure, needed in many engineering applications, we use a supremizer pressure
recovery method. Error bounds with constants independent on inverse powers of
the viscosity parameter are proved for both methods. Numerical experiments show
the accuracy and performance of the schemes.
