  In this paper, we propose a new stabilized projection-based POD-ROM for the
numerical simulation of incompressible flows. The new method draws inspiration
from successful numerical stabilization techniques used in the context of
Finite Element (FE) methods, such as Local Projection Stabilization (LPS). In
particular, the new LPS-ROM is a velocity-pressure ROM that uses pressure modes
as well to compute the reduced order pressure, needed for instance in the
computation of relevant quantities, such as drag and lift forces on bodies in
the flow. The new LPS-ROM circumvents the standard discrete inf-sup condition
for the POD velocity-pressure spaces, whose fulfillment can be rather expensive
in realistic applications in Computational Fluid Dynamics (CFD). Also, the
velocity modes does not have to be neither strongly nor weakly divergence-free,
which allows to use snapshots generated for instance with penalty or
projection-based stabilized methods. The numerical analysis of the fully
Navier-Stokes discretization for the new LPS-ROM is presented, by mainly
deriving the corresponding error estimates. Numerical studies are performed to
discuss the accuracy and performance of the new LPS-ROM on a two-dimensional
laminar unsteady flow past a circular obstacle.
