In this note we continue our investigation [1] of multigrid methods as preconditioners to a Jacobian-free Newton-Krylov method [2,3]. We consider two different options for the formation of the coarse grid operators required in the multigrid preconditioner. The first option (Method 1) involves restricting the dependent variables down through a series of grids, rediscritizing, and forming the coarse grid matrices. In the second option, considered (Method 2), the coarse grid matrices are formed from the fine grid matrix using a Galerkin [4] procedure. Both methods use low-complexity piecewise constant restriction and prolongation. Additionally, we consider the option of using either a coupled or a distributed (segregated) [4,5] approach in our preconditioner. In the more standard coupled approach, the multigrid smoother works directly on the coupled system of equations. In the distributed approach, each equation in the system is treated individually in the preconditioner and approximately inverted using a scalar multigrid approach. We use the standard driven cavity problem [6] and the natural convection problem [7] as our test problems. Other research in this area includes the work of Pernice [8], who is studying combinations of SIMPLE [9], nonlinear multigrid, and Newton-Krylov methods.
Newton's method requires the solution of the linear system
where J is the Jacobian matrix, F(u) is the nonlinear system of equations (the discretized partial differential equations), u is the state vector, δu is the Newton update vector, d is an adaptively chosen damping scalar, and n is the Newton iteration level. The Generalized Minimal RESidual (GMRES) algorithm [10] is used to solve Eq. ( 1). The GMRES algorithm requires the action of the Jacobian (J) only in the form of Jacobian-vector products, which may be approximated by a first-order Taylor series expansion [2, 3],
Jv ≈ [F(u + v) -F(u)]/ , (2) 262