There has been considerable discussion of the relative merits of multigrid and Newton-based methods for solving the steady state incompressible Navier Stokes equations. The full approximation storage method (FAS) promises cheap iterations and high rates of convergence, but fast convergence is difficult to achieve without the correct combination of interlevel transfers and smoothing operations. While superlinear rates of convergence are attractive, Newton-based iterations are more expensive, both in terms of operations and storage. Further, performance can be sensitive to the choice of an initial approximation, even with globalization strategies. Recent advances in the development of inexact Newton methods have made a Newton-based approach more competetive, but a good preconditioner is still necessary to achieve satisfactory performance.
This debate is largely misdirected. Since both strategies require selection of components whose effectiveness is highly problem-dependent, it is unlikely that one strategy will emerge as the method of choice for a broad class of applications. In fact, the two approaches have complementary strengths and weaknesses that can be exploited to compose efficient and robust solvers. The expense of a Newton-Krylov method can be mitigated by using it as the coarse grid solver for FAS, which will often provide a good initial approximation. An improved coarse grid solver can also improve the robustness of a nonlinear multigrid scheme. Linear multigrid methods can be used as preconditioners for a Newton-Krylov scheme, whether it is used as a coarse grid solver or as a standalone solver. As a standalone solver, Newton-Krylov schemes with multigrid preconditioning can considerably enhance the robustness of available smoothers.
Classical fixed point methods for solving the incompressible steady-state Navier Stokes equations, such as SIMPLE and SIMPLER, can be used as multigrid smoothers, and are also effective preconditioners in a Newton-Krylov scheme. In this latter context, they have the additional advantage of requiring no explicit linearization in order to be effective. While this presents an opportunity to leverage prior software investments, some care must be taken in the way these fixed point methods are deployed in the context of a hybrid solver.
Even within this framework, numerous questions still need to be addressed. What is the best multigrid cycling strategy to use as a preconditioner? How accurately should the FAS coarse grid problem be solved? What modifications of the fixed point single grid solver are needed to enhance its effectiveness as a smoother? Where in the grid hierarchy is the Newton-Krylov method most effective? What software components are needed, and how should they be organized, to facilitate exploration of these issues? Few analytic results are available to point to the most effective strategy, but numerical experiments help to identify fruitful avenues to investigate.
-- Michael Pernice Center for High Performance Computing firstname.lastname@example.org http://www.chpc.utah.edu/~pernice