Error Analysis of Proper Orthogonal Decomposition Stabilized Methods for Incompressible Flows

Abstract

Proper orthogonal decomposition (POD) stabilized methods for the Navier--Stokesequations are considered and analyzed. We consider two cases: the case in which the snapshots arebased on a non inf-sup stable method and the case in which the snapshots are based on an inf-supstable method. For both cases we construct approximations to the velocity and the pressure. Forthe first case, we analyze a method in which the snapshots are based on a stabilized scheme withequal 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 LPSstabilization for the gradient of the velocity and the pressure as the direct method, together withgrad-div stabilization. In the second case, the snapshots are based on an inf-sup stable Galerkinmethod with grad-div stabilization and for the POD model we also apply grad-div stabilization.In this case, since the snapshots are discretely divergence-free, the pressure can be removed fromthe formulation of the POD approximation to the velocity. To approximate the pressure, needed inmany engineering applications, we use a supremizer pressure recovery method. Error bounds withconstants independent of inverse powers of the viscosity parameter are proved for both methods.Numerical experiments show the accuracy and performance of the schemes.