# Numerical analysis of a projection-based stabilized POD-ROM for incompressible flows

## TL;DR

This paper introduces a new stabilized projection-based POD reduced order model for incompressible flows that improves computational efficiency and accuracy by relaxing traditional stability conditions and incorporating pressure modes.

## Contribution

The paper presents a novel Local Projection Stabilization ROM that avoids the discrete inf-sup condition and allows use of non-divergence-free snapshots, enhancing CFD simulations.

## Key findings

- The new LPS-ROM provides accurate velocity and pressure approximations.
- Numerical results demonstrate improved stability and efficiency.
- The method effectively models flow past a circular obstacle.

## Abstract

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.