Two-level Penalized Finite Element Methods for the Stationary Navier-stoke Equations

In this article, we first consider some penalized finite element methods for the stationary Navier-stokes equations, based on the finite element space pair (Xh, Mh) which satisfies the discrete inf-sup condition for the P2−P0 element or does not satisfy the discrete inf-sup condition for the P1−P0 element. Then, we consider two-level penalty finite element methods which involve solving one small Navier-Stokes problem on a coarse mesh with mesh size H, a large Stokes problem on a fine mesh with mesh size h = O(2i/2−1H2) for the Pi − P0 element with i = 1, 2, where 0 < 2 < 1 is a penalty parameter. The methods we study provide an approximate solution (u2 , p h 2 ) with the convergence rate of same order as the penalty finite element solution (u2h, p2h), which involves solving one large Navier-Stokes problem on a fine mesh with mesh size h. Hence, our method can save a large amount of computational time.