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The recently introduced divergence-conforming B-spline discretizations allowthe construction of smooth discrete velocity-pressure pairs for viscousincompressible flows that are at the same time inf-sup stable anddivergence-free. When applied to discretize Stokes equations, it generates asymmetric and indefinite linear system of saddle-point type. Krylov subspacemethods are usually the most efficient procedures to solve such systems. One ofsuch methods, for symmetric systems, is the Minimum Residual Method (MINRES).However, the efficiency and robustness of Krylov subspace methods is closelytied to appropriate preconditioning strategies. For the discrete Stokes system,in particular, block-diagonal strategies provide efficient preconditioners. Inthis paper, we compare the performance of block-diagonal preconditioners forseveral block choices. We verify how eigenvalue clustering promoted by thepreconditioning strategies affects MINRES convergence. We also compare thenumber of iterations and wall-clock timings. We conclude that an incompleteCholesky block-diagonal preconditioning strategy with relaxed inner conjugategradients iterations provides the best computational strategy when compared toother block-diagonal and global solution strategies.

Performance Evaluation of Block Diagonal Preconditioners for Divergence-Conforming B-Spline Discretization of Stokes Equations