id: 01368731
dt: j
an: 01368731
au: Knoll, D.A.; Rider, W.J.
ti: A multigrid preconditioned Newton-Krylov method.
so: SIAM J. Sci. Comput. 21, No.2, 691-710 (1999).
py: 1999
pu: Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA
la: EN
cc: 
ut: Newton-Krylov methods; multigrid preconditioning; nonlinear partial
    differential equations; incompressible Navier-Stokes equations; GMRES;
    diffusion-convection problem; convergence; numerical examples; Burgers
    equations
ci: Zbl 0373.65054
li: doi:10.1137/S1064827598332709
ab: The authors consider the realization of Newton’s method for the numerical
    solution of discretized boundary value problems for nonlinear partial
    differential equations (like the incompressible Navier-Stokes
    equations). The proposal is to use (on a sequence of grids) a
    Krylov-method (GMRES) for the solution of the linear equations arising
    from the Newton method in a way which avoids the formation of the
    Jacobi matrix, and to precondition these linear equations using a crude
    approximation of the Jacobi matrix on the coarse grids. The authors
    experimentally show that this preconditioning limits the growth of the
    number of Krylov iterations per Newton iterations, and that, in case of
    a diffusion-convection problem, a pure diffusion preconditioning works,
    or for a high-order approximation of such a problem, a low-order
    preconditioning is efficient in the sense that the convergence rates of
    inexact Newton methods are preserved. Their numerical examples include
    1D and 2D Burgers equations and 2D Navier-Stokes equations and show
    that the approach is better than single-grid preconditioning or the
    classical Newton method with multigrid solution of the linear
    equations. What becomes not clear is a comparison with the full
    approximation scheme of {\it A. Brandt} [Math. Comput. 31, 333-390
    (1977; Zbl 0373.65054)], but the proposed method seems to be
    competitive.
rv: Gisbert Stoyan (Budapest)