A two-level pressure stabilization method for the generalized Stokes problem

Stabilization mixed methods that circumvent the restrictive inf-sup condition without introducing penalty errors have been developed for Stokes equations e.g. by Franca, Hughes and Stenberg in 1993, and Bonvin, Picasso and Stenberg in 2001. These methods consist of modifying the standard Galerkin formulation by adding mesh-dependent terms, which are weighted residuals of the original differential equations. The aim of the stabilization, however, is to select minimal terms that stabilize the approximation without losing the nice conservation properties. Although for properly chosen stabilization parameters these methods are well posed for all velocity-pressure pairs, numerical results reported by several researchers seem to indicate that these methods are sensitive to the choice of the stabilization parameters. A relatively recent stabilized finite-element formulation that seeems less sensitive to the choice of parameters and has better local conservation properties was developed and analysed by Codina and Blasco in 1997, Becker and Braack in 2001, and Nafa in 2004. This method consists of introducing the L2-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 are introduced into the continuity equation. In this paper, as done by Nafa in 2006, we analyse the pressure gradient stabilization method for the generalized Stokes problem and investigate its stability and convergence properties.