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

## TL;DR

This paper introduces a geometric multigrid method for efficiently solving isogeometric discretizations of the generalized Stokes and Oseen problems, ensuring divergence-free velocity fields and robustness across grid resolutions.

## Contribution

The paper develops a multigrid solver with Schwarz-style smoothers that guarantees divergence-free solutions and maintains convergence rates regardless of grid size or flow parameters.

## Key findings

- Convergence rates are invariant to grid resolution.
- Method produces divergence-free velocity fields.
- Robust for non-advection-dominated flows.

## Abstract

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.