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 Delta t, so that the numerical velocity u(epsilon h)(n) and the pressure p(epsilon 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 epsilon, the time-step size Delta t and the mesh size h satisfy the following stability conditions: epsilon c(1) <= 1, Delta t kappa(1) <= 1 and h(2) <= beta(1)Delta t respectively, for some positive constants c(1), kappa(1) and beta(1). Finally, some numerical tests to confirm the theoretical results of the penalty finite element method are provided. (C) 2010 Elsevier B.V. All rights reserved.