POD-ROMs for Incompressible Flows Including Snapshots of the Temporal Derivative of the Full Order Solution

In this paper we study the influence of including snapshots that approach the velocitytime derivative in the numerical approximation of the incompressible Navier--Stokes equations bymeans of proper orthogonal decomposition (POD) methods. Our set of snapshots includes thevelocity approximation at the initial time from a full order mixed finite element method (FOM)together with approximations to the time derivative at different times. The approximation at theinitial velocity can be replaced by the mean value of the velocities at the different times so thatwhen implementing the method to the fluctuations, as done mostly in practice, only approximationsto the time derivatives are included in the set of snapshots. For the POD method we study thedifferences between projecting onto L2 and H1. In both cases pointwise in time error bounds canbe proved. Including grad-div stabilization in both the FOM and the POD methods, error boundswith constants independent of inverse powers of the viscosity can be obtained.