  In this paper, we present a monolithic multigrid method for the efficient
solution of flow problems in fractured porous media. Specifically, we consider
a mixed-dimensional model which couples Darcy flow in the porous matrix with
Forchheimer flow within the fractures. A suitable finite volume discretization
permits to reduce the coupled problem to a system of nonlinear equations with a
saddle point structure. In order to solve this system, we propose a full
approximation scheme (FAS) multigrid solver that appropriately deals with the
mixed-dimensional nature of the problem by using mixed-dimensional smoothing
and inter-grid transfer operators. Remarkably, the nonlinearity is localized in
the fractures, and no coupling between the porous matrix and the fracture
unknowns is needed in the smoothing procedure. Numerical experiments show that
the proposed multigrid method is robust with respect to the fracture
permeability, the Forchheimer coefficient and the mesh size.
