A Connection Between Scott–Vogelius and Grad-Div Stabilized Taylor–Hood FE Approximations of the Navier–Stokes Equations

Abstract

This article studies two methods for obtaining excellent mass conservation in finite element computations of the Navier–Stokes equations using continuous velocity fields. With a particular mesh construction, the Scott–Vogelius element pair has recently been shown to be inf-sup stable and have optimal approximation properties, while also providing pointwise mass conservation. We present herein the first numerical tests of this element pair for the time dependent Navier–Stokes equations. We also prove that the limit of the grad-div stabilized Taylor–Hood solutions to the Navier–Stokes problem converges to the Scott–Vogelius solution as the stabilization parameter tends to infinity. That is, we provide theoretical justification that choosing the grad-div parameter large does not destroy the solution. Numerical tests are provided which verify the theory and show how both Scott–Vogelius and grad-div stabilized Taylor–Hood (with large stabilization parameter) elements can provide accurate results with excellent mass conservation for Navier–Stokes approximations.