Low-order finite-element discretizations are well-known to provide effective preconditioners for the linear systems that arise from higher-order discretizations of the Poissonequation . 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}_1iso \mathbb{Q}_2/ Q_1 discretization of the Stokes operator as a preconditioner for the \mathbb{Q}_2/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/Q_1 system, our ultimate motivation is to apply algebraic multigrid within solvers for \mathbb{Q}_2/Q_1 systems via the \boldsymbol{ \mathbb{Q}}_1iso\boldsymbol{ \mathbb{Q}}_2/ \mathbb{Q}_1 discretization, which will be considered in a companion paper.
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}_1iso \mathbb{Q}_2/ Q_1 discretization of the Stokes operator as a preconditioner for the \mathbb{Q}_2/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/Q_1 system, our ultimate motivation is to apply algebraic multigrid within solvers for \mathbb{Q}_2/Q_1 systems via the \boldsymbol{ \mathbb{Q}}_1iso\boldsymbol{ \mathbb{Q}}_2/ \mathbb{Q}_1 discretization, which will be considered in a companion paper.
R
Low-order finite-element discretizations are
A
R
to provide
strategy for building
D
the linear systems that arise from
R
the Poissonequation
some PDEs, such as Poisson's equation, is to leverage effective preconditioners for their low-order analogs