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

A geometric multigrid method for isogeometric compatible discretizations of the generalized Stokes and Oseen problems