  We present a new hybrid direct/iterative approach to the solution of a
special class of saddle point matrices arising from the discretization of the
steady incompressible Navier-Stokes equations on an Arakawa C-grid. The
two-level method introduced here has the following properties: (i) it is very
robust, even close to the point where the solution becomes unstable; (ii) a
single parameter controls fill and convergence, making the method
straightforward to use; (iii) the convergence rate is independent of the number
of unknowns; (iv) it can be implemented on distributed memory machines in a
natural way; (v) the matrix on the second level has the same structure and
numerical properties as the original problem, so the method can be applied
recursively; (vi) the iteration takes place in the divergence- free space, so
the method qualifies as a 'constraint preconditioner'; (vii) the approach can
also be applied to Poisson problems.
  This work is also relevant for problems in which similar saddle point
matrices occur, for instance when simulating electrical networks, where one has
to satisfy Kirchhoff's conservation law for currents.
