Low-order preconditioning of the Stokes equations

Voronin, Alexey; He, Yunhui; MacLachlan, Scott; Olson, Luke N.; Tuminaro, Raymond

doi:10.1002/nla.2426

Title: Low-order preconditioning of the Stokes equations

Journal Article · Fri Dec 03 00:00:00 EST 2021 · Numerical Linear Algebra with Applications

DOI:https://doi.org/10.1002/nla.2426· OSTI ID:1834325

^[1]; He, Yunhui ^[2];

^[3]; Olson, Luke N. ^[1]; Tuminaro, Raymond ^[4]

Univ. of Illinois at Urbana-Champaign, IL (United States)
Univ. of Waterloo, ON (Canada)
Memorial Univ. of Newfoundland, Newfoundland and Labrador (Canada)
Sandia National Lab. (SNL-CA), Livermore, CA (United States)

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 Q₁ iso Q₂/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. Furthermore, 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 Q₁ iso Q₂/Q₁ discretization, which will be considered in a companion paper.

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Research Organization:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: NA0003525

OSTI ID:: 1834325

Alternate ID(s):: OSTI ID: 1833908

Report Number(s):: SAND-2021-14411J; 701513

Journal Information:: Numerical Linear Algebra with Applications, Vol. 29, Issue 3; ISSN 1070-5325

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

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