A Reduced basis stabilization for the unsteady Stokes and Navier-Stokes equations

TL;DR

This paper develops a stabilized Reduced Basis method for unsteady Stokes and Navier-Stokes equations, ensuring stability and reducing computational costs in parametric reduced order modeling.

Contribution

It extends residual-based stabilization techniques to Reduced Basis methods for unsteady fluid flow problems, addressing stability issues in parametric settings.

Findings

The stabilized RB method maintains inf-sup stability.

Numerical experiments show reduced online computational costs.

Comparison with supremizer enrichment demonstrates effectiveness.

Abstract

In the Reduced Basis approximation of Stokes and Navier-Stokes problems, the Galerkin projection on the reduced spaces does not necessarily preserved the inf-sup stability even if the snapshots were generated through a stable full order method. Therefore, in this work we aim at building a stabilized Reduced Basis (RB) method for the approximation of unsteady Stokes and Navier-Stokes problems in parametric reduced order settings. This work extends the results presented for parametrized steady Stokes and Navier-Stokes problems in a work of ours \cite{Ali2018}. We apply classical residual-based stabilization techniques for finite element methods in full order, and then the RB method is introduced as Galerkin projection onto RB space. We compare this approach with supremizer enrichment options through several numerical experiments. We are interested to (numerically) guarantee the parametrized reduced inf-sup condition and to reduce the online computational costs.