A subspace of the DSSY nonconforming quadrilateral finite element space for the Stokes equations

In this paper, we propose a subspace of the DSSY nonconforming quadrilateral finite element space. The product of this space together with the piecewise constant space can be used for approximating the velocity and pressure variables, respectively, in solving Stokes problems. More precisely, this space consists of the P"1-nonconforming quadrilateral finite element space augmented by macro bubble functions based on the DSSY nonconforming quadrilateral space under a Hood-Taylor type assumption on meshes. It is shown that the pair satisfies the discrete inf-sup condition, using a boundedness estimate of an interpolation operator based on edge integrals. Numerical results are presented.