POD model order reduction with space-adapted snapshots for incompressible flows

TL;DR

This paper develops stable POD-Galerkin reduced-order models for unsteady incompressible Navier-Stokes equations using space-adapted snapshots, introducing two approaches to ensure stability and handling inhomogeneous boundary conditions.

Contribution

It introduces two novel methods for stable POD-Galerkin model reduction with space-adapted snapshots, including pressure stabilization and supremizer enrichment.

Findings

Both methods produce stable reduced-order models.

Numerical tests show comparable accuracy and stability.

Approaches effectively handle inhomogeneous boundary conditions.

Abstract

We consider model order reduction based on proper orthogonal decomposition (POD) for unsteady incompressible Navier-Stokes problems, assuming that the snapshots are given by spatially adapted finite element solutions. We propose two approaches of deriving stable POD-Galerkin reduced-order models for this context. In the first approach, the pressure term and the continuity equation are eliminated by imposing a weak incompressibility constraint with respect to a pressure reference space. In the second approach, we derive an inf-sup stable velocity-pressure reduced-order model by enriching the velocity reduced space with supremizers computed on a velocity reference space. For problems with inhomogeneous Dirichlet conditions, we show how suitable lifting functions can be obtained from standard adaptive finite element computations. We provide a numerical comparison of the considered methods for a regularized lid-driven cavity problem.