Monolithic Multigrid Preconditioners for High-Order Discretizations of Stokes Equations

TL;DR

This paper presents a monolithic $ph$MG multigrid framework for high-order discretizations of Stokes equations, improving efficiency by coarsening in both approximation order and mesh resolution, outperforming traditional methods in many scenarios.

Contribution

It introduces a novel $ph$MG multigrid approach that coarsens in both $p$ and $h$, demonstrating significant efficiency gains for high-order Stokes discretizations.

Findings

$ph$MG outperforms $h$MG in setup and solve times for Taylor-Hood elements.

$ph$MG achieves low iteration counts and competitive timings for Scott-Vogelius discretizations.

Setup costs are higher for monolithic $ph$MG compared to FBF in Scott-Vogelius cases.

Abstract

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.