Stable and unstable cross-grid PkQl mixed finite elements for the Stokes problem

In this paper we develop and analyze a family of mixed finite element methods for the numerical solution of the Stokes problem in two space dimensions. In these schemes, the pressure is interpolated on a mesh of quadrilateral elements, while the velocity is approximated on a triangular mesh obtained by dividing each quadrilateral into four triangles by its diagonals. Continuous interpolations of degrees k for the velocity and l for the pressure are considered, so that the new finite elements are called cross-grid PkQl. A stability analysis of these approximations is provided, based on the macroelement technique of Stemberg. The lowest order P1Q1 and P2Q1 cases are analyzed in detail; in the first case, a global spurious pressure mode is shown to exist, so that this element is unstable. In the second case, however, stability is rigorously proved. Numerical results obtained in these two cases (with both rectangular and general quadrilateral elements) are also presented, which confirm the existence of the spurious pressure mode for the P1Q1 element and the stability of the P2Q1 element.