Homogeneous Multigrid for Hybrid Discretizations: Application to HHO Methods

Core Concepts

Uniform convergence of geometric multigrid V-cycle for hybrid high-order methods.

Abstract

This article discusses the convergence analysis of geometric multigrid methods for hybrid high-order (HHO) and other discontinuous skeletal methods. It generalizes results for HDG methods, focusing on the fast solution of condensed systems from hybrid discretizations. The paper outlines a framework for multigrid convergence with a V-cycle, including HHO methods in its scope. The theoretical results are supported by numerical experiments. The content is structured as follows: Introduction to hybrid discretization methods. Problem formulation and notation. Discontinuous skeletal methods. Multigrid algorithm and abstract convergence results. Convergence analysis with Lemmas and Theorems. Verification of assumptions for HHO. Key Highlights: Proposal of a generalized framework for multigrid convergence analysis. Inclusion of HHO methods in the application scope. Proof of uniform convergence using V-cycle multigrid method.

Stats

Various multigrid solvers designed over the last decade focus on HDG and HHO methods [CDGT14, SA17, WMBT19, MBTS20, LRK21, LRK22a]. Modern hybrid discretization methods handle polyhedral elements but are restricted to simplicial meshes.

Quotes

Key Insights Distilled From

Homogeneous multigrid for hybrid discretizations

by Daniele A. D... at arxiv.org 03-26-2024

https://arxiv.org/pdf/2403.15858.pdf

Deeper Inquiries

How do the assumptions made in this article impact the practical implementation of multigrid algorithms

The assumptions made in the article play a crucial role in determining the practical implementation of multigrid algorithms. For instance, the assumption of boundedness and conformity with linear finite elements for injection operators impacts how efficiently information is transferred between different levels of the multigrid hierarchy. These assumptions ensure that the interpolation and restriction operations maintain stability and accuracy throughout the algorithm. Additionally, assumptions related to regularity of approximation and consistency with standard methods help guarantee that the multigrid algorithm converges effectively towards a solution.

What are the potential limitations or challenges faced when applying homogeneous multigrid to complex domains

When applying homogeneous multigrid to complex domains, there are potential limitations or challenges that may arise. One significant challenge is related to handling irregular geometries or domains with re-entrant corners. The weak version of elliptic regularity assumed in the article may not fully capture all complexities present in such domains, leading to difficulties in ensuring convergence across all parts of the domain. Additionally, restrictions to simplicial meshes can limit applicability to more general mesh structures commonly found in complex engineering problems.

How can the findings in this article be extended to other types of discretization schemes beyond HHO methods

The findings presented in this article regarding homogeneous multigrid for hybrid discretizations can be extended beyond HHO methods to other types of discretization schemes by adapting similar principles and techniques. By incorporating appropriate local solvers, stabilization terms, and interpolation operators tailored to specific discretization methods like HDG or LDG-H, one can apply homogeneous multigrid concepts effectively across various numerical schemes used for solving partial differential equations. Extending these findings requires careful consideration of each method's unique characteristics while maintaining key properties such as boundedness, consistency with standard methods, and efficient transfer operations between mesh levels.