Unconditional convergence of high-order extrapolations of the Crank-Nicolson, finite element method for the Navier-Stokes equations

Error estimates for the Crank-Nicolson in time, finite element in space (CNFE) discretization of the Navier-Stokes equations require a discrete version of the Gronwall inequality, which leads to a time-step restriction. We prove herein that no restriction on the time-step is necessary for a linear, fully implicit variation of CN-FE obtained by extrapolation of the convecting velocity. Previous convergence analyses of CN-FE with linear extrapolation rely on a similar time-step restriction as the full CN-FE. We show: CN-FE with linear extrapolation is unconditionally convergent in the energy norm. We also show optimal convergence of CN-FE with extrapolation in a discrete L∞(H1)-norm and convergence of the corresponding discrete time derivative in a discrete L(L)-norm.