  A nonlinear multigrid solver for two-phase flow and transport in a mixed
fractional-flow velocity-pressure-saturation formulation is proposed. The
solver, which is under the framework of the full approximation scheme (FAS),
extends our previous work on nonlinear multigrid for heterogeneous diffusion
problems. The coarse spaces in the multigrid hierarchy are constructed by first
aggregating degrees of freedom, and then solving some local flow problems. The
mixed formulation and the choice of coarse spaces allow us to assemble the
coarse problems without visiting finer levels during the solving phase, which
is crucial for the scalability of multigrid methods. Specifically, a natural
generalization of the upwind flux can be evaluated directly on coarse levels
using the precomputed coarse flux basis vectors. The resulting solver is
applicable to problems discretized on general unstructured grids. The
performance of the proposed nonlinear multigrid solver in comparison with the
standard single level Newton's method is demonstrated through challenging
numerical examples. It is observed that the proposed solver is robust for
highly nonlinear problems and clearly outperforms Newton's method in the case
of high Courant-Friedrichs-Lewy (CFL) numbers.
