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

Summary 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, postsmoothing steps, grid levels, or cycles in a V‐cycle implementation, provided that the initial velocity guess is also divergence free. 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 and the generalized Oseen problem, provided that it is not advection dominated.