  Reduced order methods (ROMs) for the incompressible Navier--Stokes equations,
based on proper orthogonal decomposition (POD), are studied that include
snapshots which approach the temporal derivative of the velocity from a full
order mixed finite element method (FOM). In addition, the set of snapshots
contains the mean velocity of the FOM. Both the FOM and the POD-ROM are
equipped with a grad-div stabilization. A velocity error analysis for this
method can be found already in the literature. The present paper studies two
different procedures to compute approximations to the pressure and proves error
bounds for the pressure that are independent of inverse powers of the
viscosity. Numerical studies support the analytic results and compare both
methods.
