Error Analysis of Supremizer Pressure Recovery for POD based Reduced Order Models of the time-dependent Navier-Stokes Equations

TL;DR

This paper analyzes and compares two pressure recovery techniques in POD-based reduced order models for Navier-Stokes equations, providing theoretical stability results and numerical performance evaluations.

Contribution

It offers a rigorous stability and convergence analysis of the supremizer stabilized pressure recovery method, which was previously mainly studied numerically.

Findings

Supremizer approach is stable and convergent under certain conditions.

Pressure Poisson method faces challenges in stability.

Supremizer method outperforms Pressure Poisson in numerical tests.

Abstract

For incompressible flow models, the pressure term serves as a Lagrange multiplier to ensure that the incompressibility constraint is satisfied. In engineering applications, the pressure term is necessary for calculating important quantities based on stresses like the lift and drag. For reduced order models generated via a Proper orthogonal decomposition, it is common for the pressure to drop out of the equations and produce a velocity-only reduced order model. To recover the pressure, many techniques have been numerically studied in the literature; however, these techniques have undergone little rigorous analysis. In this work, we examine two of the most popular approaches: pressure recovery through the Pressure Poisson equation and recovery via the momentum equation through the use of a supremizer stabilized velocity basis. We examine the challenges that each approach faces and prove stability and convergence results for the supremizer stabilized approach. We also investigate numerically the stability and convergence of the supremizer based approach, in addition to its performance against the Pressure Poisson method.