Stability of some mixed finite element methods for Stokesian flows

In so-called primitive variable formulations of problems of flow of viscous, incompressible, Stokesian fluids, two fields appear as unknowns: the velocity field u and the pressure field p, the latter representing a Lagrange multiplier associated with the incompressibility constraint, div u = 0. Finite element methods based on such formulations were first introduced over a decade ago [18]. Since the mid-1970s, interest in these methods was rekindled by the appearance of several new techniques which provided for very efficient calculation of the element pressures. These included mixed methods which employ pressure approximations which are discontinuous at interelement boundaries as well as the closely related mixed-type methods which employ an exterior penalty approximation of the incompressibility condition and reduced integration of the penalty terms. All of these methods have the attractive feature that the discontinuous element pressures can be eliminated element by element, reducing the problem to one only involving velocities. Upon determining velocities, element pressures can then be evaluated through a simple post-processing operation. Methods of this type were developed and discussed by several authors, and we mention in particular the papers of Malkus [14, 151, Hughes [12], Malkus and Hughes [17], Reddy [24], Bercovier [2], the book of Engleman and Sani [6], and the references therein. In 1980, however, mathematical analyses indicated that some of the more popular discontinuouspressure/mixed methods might be numerically unstable [W-22]. It was discovered that while certain of these methods perform well in problems with smooth solutions for which regular uniform meshes are employed, serious oscillations in the pressure approximation can occur when the data or the mesh pattern are mildly irregular, and these oscillations increase in amplitude as the mesh is refined. Oden, Kikuchi and Song [22] attributed the deficiency of these unstable methods to their failure to satisfy a key stability criterion which they referred to as the ‘LBB condition’, making reference to the work of Ladyszhenskaya [13] on existence theorems of viscous flow problems and of BabuSka [l] and Brezzi [3] on the approximation of elliptic problems with constraints. The discrete LBB condition of Oden, Kikuchi and Song is basically the requirement that the discrete approximation BZ of the transpose B* of the constraint operator B = div be bounded