A Multigrid-Preconditioned Newton--Krylov Method for the Incompressible Navier--Stokes Equations

Abstract

Globalized inexact Newton methods are well suited for solving large-scale systems of nonlinear equations. When combined with a Krylov iterative method, an explicit Jacobian is never needed, and the resulting matrix-free Newton--Krylov method greatly simplifies application of the method to complex problems. Despite asymptotically superlinear rates of convergence, the overall efficiency of a Newton--Krylov solver is determined by the preconditioner. High-quality preconditioners can be constructed from methods that incorporate problem-specific information, and for the incompressible Navier--Stokes equations, classical pressure-correction methods such as SIMPLE and SIMPLER fulfill this requirement. A preconditioner is constructed by using these pressure-correction methods as smoothers in a linear multigrid procedure. The effectiveness of the resulting Newton--Krylov-multigrid method is demonstrated on benchmark incompressible flow problems.