Stabilized Finite Element Methods

Abstract

Introduction Physically the Stokes equations model “slow” flows of incompressible fluids or alternatively isotropic incompressible elastic materials. In Computational Fluid Dynamics, however, the Stokes equations have become an important model problem for designing and analyzing finite element algorithms. The reason being, that some of the problems encountered when solving the full Navier-Stokes equations are already present in the more simple Stokes equations. In particular, it gives the right setting for studying the stability problem connected with the choice of finite element spaces for the velocity and the pressure. It is well known that these spaces cannot be chosen independently when the discretization is based on the “Galerkin” variational form. This method belongs to the class of saddle-point problems for which an abstract theory has been developed by Brezzi [1974] and Babuska [1973]. The theory shows that the method is optimally convergent if the finite element spaces for velocity and pressure satisfy the “Babuska-Brezzi” or “inf-sup” condition. In computations the violation of this condition often leads to unphysical pressure oscillations and a “locking” of the velocity field, cf. Hughes [1987]. During the last decade this problem has been studied thoroughly and various velocity-pressure combinations have been shown to satisfy the Babuska-Brezzi condition. Unfortunately, however, it has turned out that many seemingly natural combinations do not satisfy it. (See Girault and Raviart [1986], Brezzi and Fortin [1991], and references therein.) In this chapter we will review a recent technique of “stabilizing” mixed methods. In this approach the standard Galerkin form is modified by the addition of mesh-dependent terms which are weighted residuals of the differential equations.