Low-order preconditioning of the Stokes equations

Low-order finite-element discretizations are well-known to provide effective preconditioners for the linear systems that arise from higher-order discretizations of the Poisson equation. 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 $\boldsymbol{ \mathbb{Q}}_1iso\boldsymbol{ \mathbb{Q}}_2/ \mathbb{Q}_1$ discretization of the Stokes operator as a preconditioner for the $\boldsymbol{ \mathbb{Q}}_2/\mathbb{Q}_1$ 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 $\boldsymbol{ \mathbb{Q}}_2/\mathbb{Q}_1$ system, our ultimate motivation is to apply algebraic multigrid within solvers for $\boldsymbol{ \mathbb{Q}}_2/\mathbb{Q}_1$ systems via the $\boldsymbol{ \mathbb{Q}}_1iso\boldsymbol{ \mathbb{Q}}_2/ \mathbb{Q}_1$ discretization, which will be considered in a companion paper.