Matrix-free Monolithic Multigrid Methods for Stokes and Generalized Stokes Problems

TL;DR

This paper introduces matrix-free monolithic geometric multigrid methods with Chebyshev-Jacobi smoothers for efficiently solving Stokes and generalized Stokes systems discretized with Taylor-Hood elements in 2D and 3D.

Contribution

It develops and analyzes a novel matrix-free multigrid solver tailored for Taylor-Hood finite element discretizations of Stokes problems, enhancing computational efficiency.

Findings

Effective multigrid solver for Stokes systems

Numerical results demonstrate solver efficiency

Applicable to 2D and 3D benchmark problems

Abstract

We consider the widely used continuous $\mathcal{Q}_{k}$-$\mathcal{Q}_{k-1}$ quadrilateral or hexahedral Taylor-Hood elements for the finite element discretization of the Stokes and generalized Stokes systems in two and three spatial dimensions. For the fast solution of the corresponding symmetric, but indefinite system of finite element equations, we propose and analyze matrix-free monolithic geometric multigrid solvers that are based on appropriately scaled Chebyshev-Jacobi smoothers. The analysis is based on results by Sch\"oberl and Zulehner (2003). We present and discuss several numerical results for typical benchmark problems.