Pressure stabilized finite element approximation for the generalized stokes problem

Stabilization methods are often used in connection with fluid flow problems to circumvent the difficulties associated with the stability of mixed finite element methods. The aim of the stabilization, of course, is to select minimal terms that stabilize the approximation without loosing the nice conservation properties. In this paper we analyze the finite element approximation of the generalized Stokes problem. First we review the main ingredients of the general least squares formulation(GLS)(see, [7], [4], and [5]). Then, we discuss the pressure stabilization formulation, which consists in introducing the L-2-projection of the pressure gradient as a new unknown of the problem. Hence, a third equation to enforce the projection property is added to the original discrete equations, and a weighted difference of the pressure gradient and its projection is introduced into the continuity equation. Using appropriate norms, the resulting formulation is shown to be stable. As a result, optimal rates of convergence are obtained for both velocity and pressure approximation (c) 2004 WILEY-VCH Verlag GmbH & Co. KGaA. Weinheim.