A taxonomy of consistently stabilized finite element methods for the Stokes problem

Stabilized mixed methods can circumvent the restrictive inf-sup condition without introducing penalty errors. For properly chosen stabilization parameters these methods are well-posed for all conforming velocity-pressure pairs. However, their variational forms have widely varying properties. First, stabilization offers a choice between weakly or strongly coercive bilinear forms that give rise to linear systems with identical solutions but very different matrix properties. Second, coercivity may be conditional upon a proper choice of a stabilizing parameter. Here we focus on how these two aspects of stabilized methods affect their accuracy and efficient iterative solution. We present results that indicate a preference of Krylov subspace solvers for strongly coercive formulations. Stability criteria obtained by finite element and algebraic analyses are compared with numerical experiments. While for two popular classes of stabilized methods, sufficient stability bounds correlate well with numerical stability, our experiments indicate the intriguing possibility that the pressure-stabilized Galerkin method is unconditionally stable.