A note on the importance of mass conservation in long-time stability of Navier–Stokes simulations using finite elements

We prove a long-time stability result for the finite element in space, linear extrapolated Crank–Nicolson in time discretization of the Navier–Stokes equations (NSE). From this result and a numerical experiment, we show the importance of discrete mass conservation in long-time simulations of the NSE. That is, we show that using elements that strongly enforce mass conservation can provide significantly more accurate solutions over long times, compared to those that enforce it weakly.