  We introduce a robust and efficient preconditioner for a hybrid Newton-GMRES
method for solving the nonlinear systems arising from incompressible
Navier-Stokes equations. When the Reynolds number is relatively high, these
systems often involve millions of degrees of freedom (DOFs), and the nonlinear
systems are difficult to converge, partially due to the strong asymmetry of the
system and the saddle-point structure. In this work, we propose to alleviate
these issues by leveraging a multilevel ILU preconditioner called HILUCSI,
which is particularly effective for saddle-point problems and can enable robust
and rapid convergence of the inner iterations in Newton-GMRES. We further use
Picard iterations with the Oseen systems to hot-start Newton-GMRES to achieve
global convergence, also preconditioned using HILUCSI. To further improve
efficiency and robustness, we use the Oseen operators as physics-based
sparsifiers when building preconditioners for Newton iterations and introduce
adaptive refactorization and iterative refinement in HILUCSI. We refer to the
resulting preconditioned hybrid Newton-GMRES as HILUNG. We demonstrate the
effectiveness of HILUNG by solving the standard 2D driven-cavity problem with
Re 5000 and a 3D flow-over-cylinder problem with low viscosity. We compare
HILUNG with some state-of-the-art customized preconditioners for INS, including
two variants of augmented Lagrangian preconditioners and two physics-based
preconditioners, as well as some general-purpose approximate-factorization
techniques. Our comparison shows that HILUNG is much more robust for solving
high-Re problems and it is also more efficient in both memory and runtime for
moderate-Re problems.
