On the selective local stabilization of the mixed Q1–P0 element

In this paper we study the stability and performance of the quadrilateral finite element ??₁–??₀ (bili‐ near/constant) for the Stokes equations. We set up a framework to show the stability of the element for a wide range of meshes with macroelement patches. We apply the new theory to show the stability of ??₁–??₀ elements on some previously studied meshes and on some newly suggested meshes. Nevertheless such earlier and newly suggested meshes are not effective in practice, compared to the traditional unstable meshes for the ??₁–??₀ element. The new theory leads naturally to a general idea in treating instability of square ??₁–??₀ elements by the local stabilization on macroelement patches of larger, but fixed sizes. The good performance of the traditional ??₁–??₀ square elements with filtering can be kept in some cases after the local stabilization. Some numerical tests are provided to support the theory and to show the performance of stabilized ??₁–??₀ elements. Copyright © 2007 John Wiley & Sons, Ltd.