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

TL;DR

This work proposes the first optimal geometric multigrid solver for hybrid high-order discretizations that can handle arbitrary polytopal agglomeration hierarchies in both two and three dimensions, and proves robust convergence with respect to the mesh size and the number of levels.

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.

Figures (2)

• Figure 1: Schematic representation of the two star-patch smoothers considered: on the left, two interface-star patches (in red and blue); on the right, a single vertex-star patch (in green). Generating interfaces are represented as thick colored lines. For the vertex-interface-star patch the generating vertex is represented as a colored square. Note that only skeletal dof on the generating (colored) interfaces are active.
• Figure 2: Illustration of the considered families of meshes in 2D: Rep-tile meshes (top row), and agglomerated Voronoi meshes (bottom row). Each row shows consecutive levels in the mesh hierarchy, with finer meshes on the left and coarser meshes on the right. Note that interfaces between coarse cells retain all vertices from the agglomeration process.

Theorems & Definitions (50)

• Remark 2.1
• Definition 3.1
• Theorem 4.4
• proof
• Theorem 4.5
• proof
• Lemma 4.9
• proof
• Definition 4.10: Consistency error
• Theorem 4.11