  In this paper, several projection method based preconditioners for various
incompressible flow models are studied. In particular, we are interested in the
theoretical analysis of a pressure-correction projection method based
preconditioner \cite{griffith2009accurate}. For both the steady and unsteady
Stokes problems, we will show that the preconditioned systems are well
conditioned. Moreover, when the flow model degenerates to the mixed form of an
elliptic operator, the preconditioned system is an identity no matter what type
of boundary conditions are imposed; when the flow model degenerates to the
steady Stokes problem, the multiplicities of the non-unitary eigenvalues of the
preconditioned system are derived. These results demonstrate the effects of
boundary treatments and are related to the stability of the staggered grid
discretization. To further investigate the effectiveness of these projection
method based preconditioners, numerical experiments are given to compare their
performances. Generalizations of these preconditioners to other saddle point
problems will also be discussed.
