  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.
