Unconditional long-time stability of a velocity–vorticity method for the 2D Navier–Stokes equations

We prove unconditional long-time stability for a particular velocity–vorticity discretization of the 2D Navier–Stokes equations. The scheme begins with a formulation that uses the Lamb vector to couple the usual velocity–pressure system to the vorticity dynamics equation, and then discretizes with the finite element method in space and implicit–explicit BDF2 in time, with the vorticity equation decoupling at each time step. We prove the method’s vorticity and velocity are both long-time stable in the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} and H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document} norms, without any timestep restriction. Moreover, our analysis avoids the use of Gronwall-type estimates, which leads us to stability bounds with only polynomial (instead of exponential) dependence on the Reynolds number. Numerical experiments are given that demonstrate the effectiveness of the method.