Numerical Analysis of a Projection-Based Stabilized POD-ROM for Incompressible Flows

Abstract

In this paper, we propose a new stabilized projection-based proper orthogonal de-composition reduced order model (POD-ROM) for the numerical simulation of incompressible flows.The new method draws inspiration from successful numerical stabilization techniques used in thecontext 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 thereduced order pressure needed for instance in the computation of relevant quantities, such as dragand lift forces on bodies in the flow. The new LPS-ROM circumvents the standard discrete inf-supcondition for the POD velocity-pressure spaces, whose fulfillment can be rather expensive in realisticapplications in computational fluid dynamics (CFD). Also, the velocity modes do not have to beeither strongly or weakly divergence-free, which allows the use of 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 errorestimates. Numerical studies are performed to discuss the accuracy and performance of the newLPS-ROM on a two-dimensional laminar unsteady flow past a circular obstacle.