Stability of some low-order approximations for the Stokes problem

Two-level low-order finite element approximations are considered for the inhomogeneous Stokes equations. The elements introduced are attractive because of their simplicity and computational efficiency. In this paper, the stability of a Q₁(h)-Q₁(2h) approximation is analysed for general geometries. Using the macroelement technique, we prove the stability condition for both two- and three-dimensional problems. As a result, optimal rates of convergence are found for the velocity and pressure approximations. Numerical results for three test problems are presented. We observe that for the computed examples, the accuracy of the two-level bilinear approximation is compared favourably with some standard finite elements.