A comparison of block preconditioners for isogeometric analysis discretizations of the incompressible Navier–Stokes equations

We deal with numerical solution of the incompressible Navier–Stokes equations discretized using the isogeometric analysis (IgA) approach. Similarly to finite elements, the discretization leads to sparse nonsymmetric saddle-point linear systems. The IgA discretization basis has several specific properties different from standard FEM basis, most importantly a higher interelement continuity leading to denser matrices. We are interested in iterative solution of the resulting linear systems using a Krylov subspace method (GMRES) preconditioned with several state-of-the-art block preconditioners. We compare the efficiency of the ideal versions of these preconditioners for three model problems (for both steady and unsteady flow in two and three dimensions) and investigate their properties with focus on the IgA specifics, that is, various degree and continuity of the discretization basis. Our experiments show that the block preconditioners can be successfully applied to the systems arising from high continuity IgA, moreover, that the high continuity can bring some benefits in this context. For example, some of the preconditioners, whose convergence is h-dependent in the steady case, seem to be less sensitive to the mesh refinement for higher continuity discretizations. In the unsteady case, we generally get faster convergence for higher continuity than for C⁰ continuous discretizations of the same degree for most of the preconditioners.