# Numerical Analysis of the Leray Reduced Order Model

## TL;DR

This paper provides a detailed numerical analysis of the Leray Reduced Order Model, including error estimates for its filtering process and overall accuracy in simulating 2D Navier-Stokes equations, addressing oscillation issues in convection-dominated flows.

## Contribution

It introduces rigorous error estimates for the Leray ROM's filtering and overall performance, supported by numerical experiments, advancing the understanding of regularized ROMs.

## Key findings

- Error estimates for the ROM differential filter are established.
- Error bounds for the Leray ROM are derived and validated.
- Numerical experiments confirm the theoretical error estimates.

## Abstract

Standard ROMs generally yield spurious numerical oscillations in the simulation of convection-dominated flows. Regularized ROMs use explicit ROM spatial filtering to decrease these spurious numerical oscillations. The Leray ROM is a recently introduced regularized ROM that utilizes explicit ROM spatial filtering of the convective term in the Navier-Stokes equations. This paper presents the numerical analysis of the finite element discretization of the Leray ROM. Error estimates for the ROM differential filter, which is the explicit ROM spatial filter used in the Leray ROM, are proved. These ROM filtering error estimates are then used to prove error estimates for the Leray ROM. Finally, both the ROM filtering error estimates and the Leray ROM error estimates are numerically investigated in the simulation of the two-dimensional Navier-Stokes equations with an analytic solution.