  In this paper, we present a geometric multigrid methodology for the solution
of matrix systems associated with isogeometric compatible discretizations of
the generalized Stokes and Oseen problems. The methodology provably yields a
pointwise divergence-free velocity field independent of the number of
pre-smoothing steps, post-smoothing steps, grid levels, or cycles in a V-cycle
implementation. The methodology relies upon Scwharz-style smoothers in
conjunction with specially defined overlapping subdomains that respect the
underlying topological structure of the generalized Stokes and Oseen problems.
Numerical results in both two- and three-dimensions demonstrate the robustness
of the methodology through the invariance of convergence rates with respect to
grid resolution and flow parameters for the generalized Stokes problem as well
as the generalized Oseen problem provided it is not advection-dominated.
