Low-order preconditioning of the Stokes equations

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 ${\mathbb{Q}}_{1}iso{\mathbb{Q}}_2/{\mathbb{Q}}_1$ discretization of the Stokes operator as a preconditioner for the ${\mathbb{Q}}_2/{\mathbb{Q}}_1$ 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 ${\mathbb{Q}}_2/{\mathbb{Q}}_1$ system, our ultimate motivation is to apply algebraic multigrid within solvers for ${\mathbb{Q}}_2/{\mathbb{Q}}_1$ systems via the ${\mathbb{Q}}_{1}iso{\mathbb{Q}}_2/{\mathbb{Q}}_1$ discretization, which will be considered in a companion paper.

arXiv commentIn the process of being to submitted to NLA@Wiley

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Cited by in corpus (2)

• A closed-form multigrid smoothing factor for an additive Vanka-type smoother applied to the Poisson equation
• Novel mass-based multigrid relaxation schemes for the Stokes equations