Optimal-Order Convergence of a Two-Step BDF Method for the Navier–Stokes Equations with H^1 Initial Data

In this paper, we study the convergence of a fully discrete linearly extrapolated two-step backward difference time-stepping scheme, with finite element method in space, for the two-dimensional Navier–Stokes equations with H^1 initial data (without any additional compatibility conditions), i.e., u_0∈ [H_0^1(Ω )]^2 , and ∇· u_0 = 0 . By using properly designed variable time stepsizes locally refined towards t=0 , we prove second-order convergence of the method in both time and space without any CFL conditions. Numerical examples are provided to illustrate the convergence of the method.