On the grad-div stabilization for the steady Oseen and Navier-Stokes equations

This paper deals with the choice of stabilization parameter for the grad-div stabilization applied to the generalized Oseen equations. In particular, inf-sup stable conforming pairs of finite element are used to derive the stabilization parameter on the basis of minimizing the H1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1({\varOmega })$$\end{document} error of the velocity. For the proposed choice of the parameter, the H1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1({\varOmega })$$\end{document} error of the velocity is derived that shows a direct dependence on the viscosity coefficient. Differences and common features with the Stokes equations are discussed. Numerical studies are presented which confirm the theoretical results. Moreover, for the Navier-Stokes equations, numerical simulations are performed on a two-dimensional flow past a circular cylinder. It turns out that, for the MINI element, the best results are achieved without grad-div stabilization.