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 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.
Authors
- Alexey Voronin (voronin2@illinois.edu)
- Yunhui He
- Scott MacLachlan
- Luke N. Olson
- Raymond Tuminaro
The published paper can be found at
This git reposetory contains code described in above mention publications.
Running the code
To be able to use this code, you will need
The code can be found in sysmg directory and all the relevant data collection scripts are in the data_collection folders.