Geometric Multigrid solvers for Hybrid High-Order methods on polytopal meshes

TL;DR

This paper introduces an optimal geometric multigrid solver for hybrid high-order methods on polytopal meshes, effectively handling complex mesh hierarchies in 2D and 3D with proven convergence.

Contribution

It presents the first multigrid solver for hybrid high-order discretizations that accommodates arbitrary polytopal agglomerations, using modified skeleton spaces.

Findings

Proves robust convergence with respect to mesh size and levels.

Validates effectiveness through numerical experiments.

Extends naturally to other hybrid discretizations.

Abstract

We propose the first optimal geometric multigrid solver for hybrid high-order discretizations that can handle arbitrary polytopal agglomeration hierarchies in both two and three dimensions. The key ingredient is the use of modified skeleton spaces, which naturally accommodate non-planar interfaces arising during coarsening while reducing the number of degrees of freedom. We prove robust convergence with respect to the mesh size and the number of levels, and we validate our results numerically on a range of agglomeration-based mesh hierarchies. The approach extends naturally to other hybrid discretizations such as hybridizable discontinuous Galerkin and Weak Galerkin methods.