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.

翻译：本文提出并评估了一种用于Taylor-Hood和Scott-Vogelius单元离散定常Stokes方程组的整体$ph$MG多重网格框架。该方法结合了近似阶数（$p$）和网格分辨率（$h$）的双重粗化策略，以应对传统高阶数值模拟中常见的计算与内存效率挑战。数值结果表明，对于采用Taylor-Hood单元离散的不同规模与离散阶数问题，$ph$MG相比传统仅空间粗化的多重网格（$h$MG）技术具有显著优势。特别是在处理高离散阶数及非结构化求解域时，$ph$MG方法在降低建立与求解时间方面表现出更优越的性能。对于Scott-Vogelius离散格式，虽然整体$ph$MG能够实现较低的迭代次数和具有竞争力的求解阶段耗时，但其建立阶段相较于采用非整体多层全块分解（FBF）预处理器（其中$ph$MG仅用于速度未知量）时明显更慢。这主要源于整体$ph$MG中较大混合场松弛块与FBF中单一未知量类型块在建立成本上的差异。