  We extend previously developed two-level coarsening procedures for graph
Laplacian problems written in a mixed saddle point form to the fully recursive
multilevel case. The resulting hierarchy of discretizations gives rise to a
hierarchy of upscaled models, in the sense that they provide approximation in
the natural norms (in the mixed setting). This property enables us to utilize
them in three applications: (i) as an accurate reduced model, (ii) as a tool in
multilevel Monte Carlo simulations (in application to finite volume
discretizations), and (iii) for providing a sequence of nonlinear operators in
FAS (full approximation scheme) for solving nonlinear pressure equations
discretized by the conservative two-point flux approximation. We illustrate the
potential of the proposed multilevel technique in all three applications on a
number of popular benchmark problems used in reservoir simulation.
