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 $\boldsymbol{ \mathbb{Q}}_1iso\boldsymbol{ \mathbb{Q}}_2/ \mathbb{Q}_1$ 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.

翻译：用于为某些 PDE 的更高顺序离散建立有效先决条件的著名战略, 如 Posson 方程式, 是用来为它们的低序类比提供有效的先决条件。 在此工作中, 我们显示也可以以同样的方式为 Stokes 方程式的泰勒- Hood 离散提供高质量的先决条件。 特别是, 我们根据 $\ boldsymbol {\ mathbbb_ 1 iso\ boldsymbol {\ mathbb2/2/\ boldsymbol {\ mathbbb2\ block_ block_ black_ block_ block_ block_\ block_ block_ block_\ block_ block_ block_\ block_ block_ block_ blickr\\ bal_ blickr\ blickr\\\ b=mabbbbbbl_ br_ bld_ br_ al_ alth salth systems, 我们 roupal_ b_ b_ brb_ b_ lib_ lib_ bl_ brb_ lib_mab_mab_ bh_ bh_mabsldsldrodrodsld) rod_ b) y_ b) rod_ b) ystemstystemldmdsmldst ystemldmldstystemld roddsmdsmdsmdddddddddsxxxxxxxxx。