Monolithic Algebraic Multigrid Preconditioners for the Stokes Equations

TL;DR

This paper introduces a new algebraic multigrid preconditioner for the Stokes equations that improves efficiency and robustness across various 2D and 3D problems, especially in challenging geometries.

Contribution

It develops a monolithic AMG preconditioner using a novel defect-correction approach with specialized relaxation strategies for Stokes discretizations.

Findings

Demonstrates robust performance on 2D and 3D Stokes problems

Reduces storage and computational costs with a new block factorization

Often outperforms traditional inexact block-triangular preconditioners

Abstract

We investigate a novel monolithic algebraic multigrid (AMG) preconditioner for the Taylor-Hood ($\pmb{\mathbb{P}}_2/\mathbb{P}_1$) and Scott-Vogelius ($\pmb{\mathbb{P}}_2/\mathbb{P}_1^{disc}$) discretizations of the Stokes equations. The algorithm is based on the use of the lower-order $\pmb{\mathbb{P}}_1\text{iso}\kern1pt\pmb{\mathbb{P}}_2/\mathbb{P}_1$ operator within a defect-correction setting, in combination with AMG construction of interpolation operators for velocities and pressures. The preconditioning framework is primarily algebraic, though the $\pmb{\mathbb{P}}_1\text{iso}\kern1pt\pmb{\mathbb{P}}_2/\mathbb{P}_1$ operator must be provided. We investigate two relaxation strategies in this setting. Specifically, a novel block factorization approach is devised for Vanka patch systems, which significantly reduces storage requirements and computational overhead, and a Chebyshev adaptation of the LSC-DGS relaxation is developed to improve parallelism. The preconditioner demonstrates robust performance across a variety of 2D and 3D Stokes problems, often matching or exceeding the effectiveness of an inexact block-triangular (or Uzawa) preconditioner, especially in challenging scenarios such as elongated-domain problems.