Error analysis of proper orthogonal decomposition stabilized methods for incompressible flows

TL;DR

This paper analyzes the error behavior of POD stabilized methods for incompressible Navier-Stokes flows, comparing stable and non-stable snapshot bases, and introduces pressure recovery techniques with proven error bounds.

Contribution

It provides a comprehensive error analysis of POD stabilized methods for both stable and non-stable snapshot bases, including pressure approximation strategies.

Findings

Error bounds with viscosity-independent constants are established.

Numerical experiments confirm the accuracy and effectiveness of the proposed schemes.

Pressure recovery via supremizer methods is successfully integrated into the POD framework.

Abstract

Proper orthogonal decomposition (POD) stabilized methods for the Navier-Stokes equations are considered and analyzed. We consider two cases, the case in which the snapshots are based on a non inf-sup stable method and the case in which the snapshots are based on an inf-sup stable method. For both cases we construct approximations to the velocity and the pressure. For the first case, we analyze a method in which the snapshots are based on a stabilized scheme with equal order polynomials for the velocity and the pressure with Local Projection Stabilization (LPS) for the gradient of the velocity and the pressure. For the POD method we add the same kind of LPS stabilization for the gradient of the velocity and the pressure than the direct method, together with grad-div stabilization. In the second case, the snapshots are based on an inf-sup stable Galerkin method with grad-div stabilization and for the POD model we apply also grad-div stabilization. In this case, since the snapshots are discretely divergence-free, the pressure can be removed from the formulation of the POD approximation to the velocity. To approximate the pressure, needed in many engineering applications, we use a supremizer pressure recovery method. Error bounds with constants independent on inverse powers of the viscosity parameter are proved for both methods. Numerical experiments show the accuracy and performance of the schemes.