Combining the augmented Lagrangian preconditioner withe the SIMPLE Schur complement approximation

The augmented Lagrangian (AL) preconditioner and its variant have been successfully applied to solve saddle point systems arising from the incompressible Navier-Stokes equations discretized by the finite element method. Attractive features are the purely algebraic construction and robustness with respect to the Reynolds number and mesh renement. In this report, we reconsider the application of the AL preconditioner in the context of the stabilized finite volume methods and present the extension to the
Reynolds-Averaged Navier-Stokes (RANS) equations, which are used to model turbulent flows in industrial applications. Furthermore, we propose a new variant of the AL preconditioner, obtained by substituting the approximation of the Schur complement from the SIMPLE preconditioner into the inverse of the Schur complement for the AL preconditioner. This new variant is applied to both Navier-Stokes and RANS equations to compute laminar and turbulent boundary-layer flows on grids with large aspect ratios. Spectral analysis shows that the new variant yields a more clustered spectrum of eigenvalues away from zero, which explains why it outperforms the existing variants in terms of the number of the Krylov subspace iterations.