  Factorization of the incompressible Stokes operator linking pressure and
velocity is revisited. The main purpose is to use the inverse of the Stokes
operator with a large time step as a preconditioner for Newton and Arnoldi
iterations applied to computation of steady three-dimensional flows and to
study of their stability. It is shown that the Stokes operator can be inversed
within an acceptable computational effort. This inverse includes fast direct
inverses of several Helmholtz operators and iterative inverse of the pressure
matrix. It is shown, additionally, that fast direct solvers can be attractive
for the inverse of the Helmholtz and Laplace operators on fine grids and at
large Reynolds numbers, as well as for other problems where convergence of
iterative methods slows down. Implementation of the Stokes operator inverse to
time-stepping-based formulation of the Newton and Arnoldi iterations is
discussed.
