Preconditioning of the generalized Stokes problem arising from the approximation of the time-dependent Navier-Stokes equations

The paper compares standard iterative methods for solving the generalized Stokes problem arising from the time and space approximation of the time-dependent incompressible Navier-Stokes equations. Various preconditioning techniques are considered: (1) pressure Schur complement; (2) fully coupled system using an exact factorization as a basis for the preconditioner; (3) fully coupled system using norm equivalence considerations as a basis for the preconditioner; (4) in all the cases we also investigate the benefits of the augmented Lagrangian formulation. Our objective is to see whether one of these methods can compete with traditional pressure-correction and velocity-correction methods in terms of throughput (the throughput is the ratio of the number of degrees of freedom of the problem divided by the number of cores and the wall-clock time in second). Numerical tests on fine unstructured meshes (68 millions degrees of freedoms) demonstrate GMRES/CG convergence rates that are independent of the mesh size and improve with the Reynolds number for most methods. Three conclusions are drawn: (1) The throughputs of all the methods tested in the paper are similar up to mesh-independent multiplicative constants (see Fig. 6). (2) Although very good parallel scalability is observed for the augmented Lagrangian version of the generalized Stokes problem, the best throughputs are achieved without the augmented Lagrangian term. (3) The throughput of all the methods tested in the paper is on average 5 to 25 times slower than that of traditional pressure-correction and velocity-correction methods (on average 5 for the best one). Hence, although all these methods are very efficient for solving steady state problems, pressure-correction and velocity-correction methods are still faster for solving time-dependent problems.