Low-order preconditioning of the Stokes equations

TL;DR

This paper demonstrates that high-quality preconditioners for the Stokes equations can be developed by leveraging low-order discretizations, using geometric multigrid and local Fourier analysis to optimize convergence.

Contribution

It introduces a novel approach to preconditioning the Taylor-Hood discretization of the Stokes equations by using low-order analogs and multigrid techniques, with analysis and verification.

Findings

Effective multigrid preconditioners for high-order Stokes discretizations.

Optimized damping parameters for relaxation schemes via local Fourier analysis.

Robust convergence demonstrated through numerical experiments.

Abstract

A well-known strategy for building effective preconditioners for higher-order discretizations of some PDEs, such as Poisson's equation, is to leverage effective preconditioners for their low-order analogs. In this work, we show that high-quality preconditioners can also be derived for the Taylor-Hood discretization of the Stokes equations in much the same manner. In particular, we investigate the use of geometric multigrid based on the $\boldsymbol{ \mathbb{Q}}_1iso\boldsymbol{ \mathbb{Q}}_2/ \mathbb{Q}_1$ discretization of the Stokes operator as a preconditioner for the $\boldsymbol{ \mathbb{Q}}_2/\mathbb{Q}_1$ discretization of the Stokes system. We utilize local Fourier analysis to optimize the damping parameters for Vanka and Braess-Sarazin relaxation schemes and to achieve robust convergence. These results are then verified and compared against the measured multigrid performance. While geometric multigrid can be applied directly to the $\boldsymbol{ \mathbb{Q}}_2/\mathbb{Q}_1$ system, our ultimate motivation is to apply algebraic multigrid within solvers for $\boldsymbol{ \mathbb{Q}}_2/\mathbb{Q}_1$ systems via the $\boldsymbol{ \mathbb{Q}}_1iso\boldsymbol{ \mathbb{Q}}_2/ \mathbb{Q}_1$ discretization, which will be considered in a companion paper.