Homogeneous multigrid for hybrid discretizations: application to HHO methods

TL;DR

This work addresses fast solvers for condensed systems arising from hybrid discretizations, notably HHO, by developing a generalized homogeneous multigrid framework that preserves the skeleton discretization across mesh levels. The authors prove uniform convergence of a symmetric V-cycle under mild elliptic regularity assumptions and verify the framework for HHO by establishing LS1–LS9 and constructing three injection operators. Numerical experiments in 2D and 3D demonstrate uniform convergence and illustrate how injection choice affects robustness, highlighting the practical value of the higher-order reconstruction-based injection. The results provide rigorous, scalable guidance for applying geometric multigrid to skeletal methods, enabling robust solvers on complex domains without changing the discretization scheme.

Abstract

We prove the uniform convergence of the geometric multigrid V-cycle for hybrid high-order (HHO) and other discontinuous skeletal methods. Our results generalize previously established results for HDG methods, and our multigrid method uses standard smoothers and local solvers that are bounded, convergent, and consistent. We use a weak version of elliptic regularity in our proofs. Numerical experiments confirm our theoretical results.

Theorems & Definitions (31)

• Remark 2.1: More general boundary conditions
• Remark 3.1: Possible choices of hybrid methods
• Remark 3.2: HDG methods
• Lemma 3.3: Quasi-orthogonality
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Lemma 5.1