  An efficient nonlinear multigrid method for a mixed finite element method of
the Darcy-Forchheimer model is constructed in this paper. A Peaceman-Rachford
type iteration is used as a smoother to decouple the nonlinearity from the
divergence constraint. The nonlinear equation can be solved element-wise with a
closed formulae. The linear saddle point system for the constraint is reduced
into a symmetric positive definite system of Poisson type. Furthermore an
empirical choice of the parameter used in the splitting is proposed and the
resulting multigrid method is robust to the so-called Forchheimer number which
controls the strength of the nonlinearity. By comparing the number of
iterations and CPU time of different solvers in several numerical experiments,
our multigrid method is shown to convergent with a rate independent of the mesh
size and the Forchheimer number and with a nearly linear computational cost.
