 
Summary: STABLE AND UNSTABLE CROSSGRID PkQl MIXED FINITE ELEMENTS
FOR THE STOKES PROBLEM
MAR´IA G. ARMENTANO AND JORDI BLASCO
Abstract. 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 crossgrid PkQl. A stability analysis of these ap
proximations 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.
1. Introduction
In order to approximate the solution of the Stokes problem by finite element methods, there
are basically two approaches. The first one consists in approximating the two independent
variables, velocity and pressure, using different spaces for each one. This leads to mixed finite
element methods, examples of which can be found in [4], [6], [8], [10], [16], [27] and in the
