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

摘要

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_1b-P_1 element pair, which satisfies the discrete infsup condition. This method allows us to separate the computation of the velocity from the computation of the pressure with a larger time-step size Δt, so that the numerical velocity u_(∈h)~n and the pressure p_(∈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, the time-step size Δt and the mesh size h satisfy the following stability conditions: ∈c_1 ≤ 1, Δtκ _1 ≤ 1 and h~2 ≤ β_1Δt, 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.