Low‐order preconditioning of the Stokes equations

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 Taylo...

Full description

Saved in:
Bibliographic Details
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
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:1070-5325
1099-1506
DOI:10.1002/nla.2426