Stabilized reduced basis methods for parametrized steady Stokes and Navier-Stokes equations

TL;DR

This paper introduces residual-based stabilization techniques for reduced basis methods applied to parametrized steady Stokes and Navier-Stokes equations, enabling stable pressure approximation without supremizer enrichment.

Contribution

It proposes novel offline-only and offline-online stabilization methods that maintain inf-sup stability at reduced order, reducing the need for supremizer functions.

Findings

Stabilization methods achieve stable pressure approximation.

Reduced basis spaces are smaller with the new methods.

Comparable or improved accuracy compared to supremizer enrichment.

Abstract

It is well known in the Reduced Basis approximation of saddle point problems that the Galerkin projection on the reduced space does not guarantee the inf-sup approximation stability even if a stable high fidelity method was used to generate snapshots. For problems in computational fluid dynamics, the lack of inf-sup stability is reflected by the inability to accurately approximate the pressure field. In this context, inf-sup stability is usually recovered through the enrichment of the velocity space with suitable supremizer functions. The main goal of this work is to propose an alternative approach, which relies on the residual based stabilization techniques customarily employed in the Finite Element literature, such as Brezzi-Pitkaranta, Franca-Hughes, streamline upwind Petrov-Galerkin, Galerkin Least Square. In the spirit of \textit{offline-online} reduced basis computational splitting, two such options are proposed, namely \textit{offline-only stabilization} and \textit{offline-online stabilization}. These approaches are then compared to (and combined with) the state of the art supremizer enrichment approach. Numerical results are discussed, highlighting that the proposed methodology allows to obtain smaller reduced basis spaces (i.e., neglecting supremizer enrichment) for which a modified inf-sup stability is still preserved at the reduced order level.