Stable and unstable cross-grid Pk Ql mixed finite elements for the Stokes problem - Armentano, M.G. (2010)
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 rectangular elements, while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals. Continuous interpolations of degrees k for the velocity and l for the pressure are considered, so the new finite elements are called cross-grid Pk Ql. A stability analysis of these approximations is provided, based on the macroelement technique of Stenberg. The lowest order P1 Q1 and P2 Q1 cases are analyzed in detail; in the first case, a global spurious pressure mode is shown to exist, so this element is unstable. In the second case, however, stability is rigorously proved. Numerical results obtained in these two cases are also presented, which confirm the existence of the spurious pressure mode for the P1 Q1 element and the stability of the P2 Q1 element. © 2010 Elsevier B.V. All rights reserved.