Multigrid methods for Hdiv-conforming discontinuous Galerkin methods for the Stokes equations

TL;DR

This paper introduces a multigrid solver for the Stokes equations discretized with an Hdiv-conforming discontinuous Galerkin method, achieving mesh-independent convergence without Schur complement approximation.

Contribution

It develops a multigrid method that operates directly on velocity and pressure spaces, utilizing Schwarz smoothers and nested divergence-free subspaces, with proven convergence.

Findings

Convergence rates are mesh-independent.

The method does not require Schur complement approximation.

Numerical experiments confirm small, stable convergence rates.

Abstract

A multigrid method for the Stokes system discretized with an Hdiv-conforming discontinuous Galerkin method is presented. It acts on the combined velocity and pressure spaces and thus does not need a Schur complement approximation. The smoothers used are of overlapping Schwarz type and employ a local Helmholtz decomposition. Additionally, we use the fact that the discretization provides nested divergence free subspaces. We present the convergence analysis and numerical evidence that convergence rates are not only independent of mesh size, but also reasonably small.