Analysis of augmented lagrangian-based preconditioners for the steady incompressible navier-stokes equations

We analyze a class of modified augmented Lagrangian-based preconditioners for both stable and stabilized finite element discretizations of the steady incompressible Navier-Stokes equations. We study the eigenvalues of the preconditioned matrices obtained from Picard linearization, and we devise a simple and effective method for the choice of the augmentation parameter ? based on Fourier analysis. Numerical experiments on a wide range of model problems demonstrate the robustness of these preconditioners, yielding fast convergence independent of mesh size and only mildly dependent on viscosity on both uniform and stretched grids. Good results are also obtained on linear systems arising from Newton linearization. We also show that performing inexact preconditioner solves with an algebraic multigrid algorithm results in excellent scalability. Comparisons of the modified augmented Lagrangian preconditioners with other state-of-the-art techniques show the competitiveness of our approach. \ua9 2011 Society for Industrial and Applied Mathematics