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 P1b–P1P1b–P1 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 ΔtΔt, so that the numerical velocity uϵhn and the pressure pϵhn 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 ϵϵ, the time-step size ΔtΔt and the mesh size hh satisfy the following stability conditions: ϵc1≤1ϵc1≤1, Δtκ1≤1Δtκ1≤1 and h2≤β1Δth2≤β1Δt, respectively, for some positive constants c1c1, κ1κ1 and β1β1. Finally, some numerical tests to confirm the theoretical results of the penalty finite element method are provided.