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

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... 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.