On the influence of the nonlinear term in the numerical approximation of Incompressible Flows by means of proper orthogonal decomposition methods

TL;DR

This paper investigates how different discretizations of the nonlinear term affect the accuracy of POD methods for incompressible Navier-Stokes equations, showing that the convergence rate remains stable despite additional error terms.

Contribution

It demonstrates the impact of using different discretizations for the nonlinear term in FOM and POD, and analyzes stabilization effects on error bounds.

Findings

Additional error term appears with different discretizations.

Convergence rate of POD is barely affected by the extra error.

Grad-div stabilization improves error bounds independent of viscosity.

Abstract

We consider proper orthogonal decomposition (POD) methods to approximate the incompressible Navier-Stokes equations. We study the case in which one discretization for the nonlinear term is used in the snapshots (that are computed with a full order method (FOM)) and a different discretization of the nonlinear term is applied in the POD method. We prove that an additional error term appears in this case, compared with the case in which the same discretization of the nonlinear term is applied for both the FOM and the POD methods. However, the added term has the same size as the error coming from the FOM so that the rate of convergence of the POD method is barely affected. We analyze the case in which we add grad-div stabilization to both the FOM and the POD methods because it allows to get error bounds with constants independent of inverse powers of the viscosity. We also study the case in which no stabilization is added. Some numerical experiments support the theoretical analysis.