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

TL;DR

This paper proves that a specific velocity-vorticity finite element scheme for the 2D Navier-Stokes equations remains stable over long times without timestep restrictions, with bounds depending polynomially on the Reynolds number.

Contribution

It introduces a decoupled velocity-vorticity discretization with unconditional long-time stability analysis avoiding Gronwall estimates.

Findings

Long-time stability in L2 and H1 norms for velocity and vorticity.

Stability bounds depend polynomially on Reynolds number.

Numerical experiments confirm effectiveness of the method.

Abstract

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 $L^2$ and $H^1$ 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.