  This work introduces and assesses the efficiency of a monolithic $ph$MG
multigrid framework designed for high-order discretizations of stationary
Stokes systems using Taylor-Hood and Scott-Vogelius elements. The proposed
approach integrates coarsening in both approximation order ($p$) and mesh
resolution ($h$), to address the computational and memory efficiency challenges
that are often encountered in conventional high-order numerical simulations.
Our numerical results reveal that $ph$MG offers significant improvements over
traditional spatial-coarsening-only multigrid ($h$MG) techniques for problems
discretized with Taylor-Hood elements across a variety of problem sizes and
discretization orders. In particular, the $ph$MG method exhibits superior
performance in reducing setup and solve times, particularly when dealing with
higher discretization orders and unstructured problem domains. For
Scott-Vogelius discretizations, while monolithic $ph$MG delivers low iteration
counts and competitive solve phase timings, it exhibits a discernibly slower
setup phase when compared to a multilevel (non-monolithic)
full-block-factorization (FBF) preconditioner where $ph$MG is employed only for
the velocity unknowns. This is primarily due to the setup costs of the larger
mixed-field relaxation patches with monolithic $ph$MG versus the patch setup
costs with a single unknown type for FBF.
