Monolithic Multigrid Preconditioners for High-Order Discretizations of Stokes Equations
Abstract
This work introduces and assesses the efficiency of a novel monolithic $ph\text{MG}$ multigrid method, specifically designed for high-order discretizations of stationary Stokes systems using Taylor-Hood and Scott-Vogelius elements. The $ph\text{MG}$ approach integrates approximation order ($p$) and spatial ($h$) coarsening to address the computational and memory efficiency challenges that are often encountered in conventional high-order numerical simulations. Our comparative analysis reveals that $ph\text{MG}$ offers significant improvements over traditional spatial-coarsening-only multigrid ($h\text{MG}$) techniques for problems discretized with Taylor-Hood elements across a variety of problem sizes and discretization orders. In particular, the $ph\text{MG}$ 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 $ph\text{MG}$ 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 ($\text{FBF}$) preconditioner where
$ph\text{MG}$ is employed only for the velocity unknowns. This is primarily due to
the setup costs of the larger mixed-field relaxation patches with monolithic
$ph\text{MG}$ versus the patch setup costs with a single unknown type for $\text{FBF}$.