On the parameter in augmented Lagrangian preconditioning for isogeometric discretizations of the Navier-Stokes equations

In this paper, we deal with the optimal choice of the parameter γ for augmented Lagrangian preconditioning of GMRES method for efficient solution of linear systems obtained from discretization of the incompressible Navier-Stokes equations. We consider discretization of the equations using the B-spline based isogeometric analysis approach. We are interested in the dependence of the convergence on the parameter γ for various problem parameters (Reynolds number, mesh refinement) and especially for various isogeometric discretizations (degree and interelement continuity of the B-spline discretization bases). The idea is to be able to determine the optimal value of γ for a problem that is relatively cheap to compute and, based on this value, predict suitable values for other problems, e.g., with finer mesh, different discretization, etc. The influence of inner solvers (direct or iterative based on multigrid method) is also discussed.