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 (Schur complement, fully coupled system, with and without augmented Lagrangian). One investigates whether these methods can compete with traditional pressure-correction and velocity-correction methods in terms of throughput (number of degrees of freedom per time step per core per 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) Whether solving the pressure Schur complement or the fully coupled system does not make any significant difference in terms of throughput. (2) Although very good parallel scalability is observed for the augmented Lagrangian method, the best throughput is achieved without using the augmented Lagrangian formulation. (3) The throughput of all the methods tested in the paper are on average 25 times slower than that of traditional pressure-correction and velocity-correction methods. Hence, although all these methods are very efficient for solving steady state problems, none of them is unfortunately competitive for solving time-dependent problems.