Low‐order preconditioning of the Stokes equations

• Published in: Numerical linear algebra with applications 2022-05, Vol.29 (3), p.n/a
• Main Authors: Voronin, Alexey, He, Yunhui, MacLachlan, Scott, Olson, Luke N., Tuminaro, Raymond
• Format: Article
• Language: English
• Subjects: additive Vanka; Braess–Sarazin; Damping; Discretization; Fourier analysis; local Fourier analysis; Mathematical analysis; monolithic multigrid; Navier-Stokes equations; Poisson equation; Preconditioning; Stokes equations
• ISSN: 1070-5325; 1099-1506
• DOI: 10.1002/nla.2426

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 ℚ1isoℚ2/ℚ1 discretization of the Stokes operator as a preconditioner for the ℚ2/ℚ1 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 ℚ2/ℚ1 system, our ultimate motivation is to apply algebraic multigrid within solvers for ℚ2/ℚ1 systems via the ℚ1isoℚ2/ℚ1 discretization, which will be considered in a companion paper.