A penalty finite element method based on the Euler implicit/explicit scheme for the time-dependent Navier–Stokes equations

A fully discrete penalty finite element method is presented for the two-dimensional time-dependent Navier–Stokes equations, where the time discretization is based on the Euler implicit/explicit scheme with some implicit linear terms and an explicit nonlinear term, and the finite element spatial discretization is based on the P 1 b – P 1 element pair, which satisfies the discrete inf–sup condition. This method allows us to separate the computation of the velocity from the computation of the pressure with a larger time-step size Δ t , so that the numerical velocity u ϵ h n and the pressure p ϵ h n are easily computed. An optimal error estimate of the numerical velocity and the pressure is provided for the fully discrete penalty finite element method when the penalty parameter ϵ , the time-step size Δ t and the mesh size h satisfy the following stability conditions: ϵ c 1 ≤ 1 , Δ t κ 1 ≤ 1 and h 2 ≤ β 1 Δ t , respectively, for some positive constants c 1 , κ 1 and β 1 . Finally, some numerical tests to confirm the theoretical results of the penalty finite element method are provided.