A geometric multigrid method for isogeometric compatible discretizations of the generalized Stokes and Oseen problems
It provides a provably robust solver for incompressible flow problems in isogeometric analysis, addressing a computational bottleneck for practitioners.
The paper presents a geometric multigrid method for solving matrix systems from isogeometric discretizations of the generalized Stokes and Oseen problems, achieving pointwise divergence-free velocity fields and robust convergence rates independent of grid resolution and flow parameters (for Stokes and non-advection-dominated Oseen).
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.