Stabilization of low-order cross-grid PkQl mixed finite elements

Abstract

In this paper we analyze a low-order family of mixed finite element methods for the numerical solution of the Stokes problem and a second order elliptic problem, in two space dimensions. In these schemes, the pressure is interpolated on a mesh of rectangular elements, while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals. For the lowest order P1Q0, a global spurious pressure mode is shown to exist and so this element, as P1Q1 case analyzed in Armentano and Blasco (2010), is unstable. However, following the ideas given in Bochev et al. (2006), a simple stabilization procedure can be applied, when we approximate the solution of the Stokes problem, such that the new P1Q0 and P1Q1 methods are unconditionally stable, and achieve optimal accuracy with respect to solution regularity with simple and straightforward implementations. Moreover, we analyze the application of our P1Q1 element to the mixed formulation of the elliptic problem. In this case, by introducing the modified mixed weak form proposed in Brezzi et al. (1993), optimal order of accuracy can be obtained with our stabilized P1Q1 elements. Numerical results are also presented, which confirm the existence of the spurious pressure mode for the P1Q0 element and the excellent stability and accuracy of the new stabilized methods.