On the quadrilateral Q2-P1 element for the stokes problem

The Q2 - P1 approximation is one of the most popular Stokes elements. Two possible choices are given for the definition of the pressure space: one can either use a global pressure approximation (that is on each quadrilateral the finite element space is spanned by 1 and by the global co-ordinates x and y) or a local approach (consisting in generating the local space by means of the constants and the local curvilinear co-ordinates on each quadrilateral ξ and η). The former choice is known to provide optimal error estimates on general meshes. This has been shown, as it is standard, by proving a discrete inf-sup condition. In the present paper we check that the latter approach satisfies the inf-sup condition as well. However, recent results on quadrilateral finite elements bring to light a lack in the approximation properties for the space coming out from the local pressure approach. Numerical results actually show that the second choice (local or mapped pressure approximation) is suboptimally convergent. Copyright © 2002 John Wiley & Sons, Ltd.