  The recently introduced divergence-conforming B-spline discretizations allow
the construction of smooth discrete velocity-pressure pairs for viscous
incompressible flows that are at the same time inf-sup stable and
divergence-free. When applied to discretize Stokes equations, it generates a
symmetric and indefinite linear system of saddle-point type. Krylov subspace
methods are usually the most efficient procedures to solve such systems. One of
such methods, for symmetric systems, is the Minimum Residual Method (MINRES).
However, the efficiency and robustness of Krylov subspace methods is closely
tied to appropriate preconditioning strategies. For the discrete Stokes system,
in particular, block-diagonal strategies provide efficient preconditioners. In
this paper, we compare the performance of block-diagonal preconditioners for
several block choices. We verify how eigenvalue clustering promoted by the
preconditioning strategies affects MINRES convergence. We also compare the
number of iterations and wall-clock timings. We conclude that an incomplete
Cholesky block-diagonal preconditioning strategy with relaxed inner conjugate
gradients iterations provides the best computational strategy when compared to
other block-diagonal and global solution strategies.
