  We propose two techniques aimed at improving the convergence rate of steady
state and eigenvalue solvers preconditioned by the inverse Stokes operator and
realized via time-stepping. First, we suggest a generalization of the Stokes
operator so that the resulting preconditioner operator depends on several
parameters and whose action preserves zero divergence and boundary conditions.
The parameters can be tuned for each problem to speed up the convergence of a
Krylov-subspace-based linear algebra solver. This operator can be inverted by
the Uzawa-like algorithm, and does not need a time-stepping. Second, we propose
to generate an initial guess of steady flow, leading eigenvalue and eigenvector
using orthogonal projection on a divergence-free basis satisfying all boundary
conditions. The approach, including the two proposed techniques, is illustrated
on the solution of the linear stability problem for laterally heated square and
cubic cavities.
