A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier–Stokes equations

Abstract:

Recently, Douglas et al. [4] introduced a new, low-order, nonconforming rectangular element for scalar elliptic equations. Here, we apply this element in the approximation of each component of the velocity in the stationary Stokes and Navier–Stokes equations, along with a piecewise-constant element for the pressure. We obtain a stable element in both cases for which optimal error estimates for the approximation of both the velocity and pressure in L ² can be established, as well as one in a broken H ¹-norm for the velocity.