Achieving $h$- and $p$-robust monolithic multigrid solvers for the Stokes equations

TL;DR

This paper develops and analyzes multigrid preconditioners for higher-order discretizations of the Stokes equations, achieving robustness in iterative solutions across polynomial orders.

Contribution

It introduces variants of Vanka relaxation schemes that serve as effective multigrid preconditioners for both conforming and non-conforming discretizations of the Stokes equations.

Findings

Robust FGMRES iteration counts across polynomial orders.

Effective preconditioning for conforming Taylor-Hood discretizations.

Open questions on stopping tolerances at high polynomial orders.

Abstract

The numerical analysis of higher-order mixed finite-element discretizations for saddle-point problems, such as the Stokes equations, has been well-studied in recent years. While the theory and practice of such discretizations is now well-understood, the same cannot be said for efficient preconditioners for solving the resulting linear (or linearized) systems of equations. In this work, we propose and study variants of the well-known Vanka relaxation scheme that lead to effective geometric multigrid preconditioners for both the conforming Taylor-Hood discretizations and non-conforming ${\bf H}(\text{div})$-$L^2$ discretizations of the Stokes equations. Numerical results demonstrate robust performance with respect to FGMRES iteration counts for increasing polynomial order for some of the considered discretizations, and expose open questions about stopping tolerances for effectively preconditioned iterations at high polynomial order.