Analysis of a two-level algorithm for HDG methods for diffusion problems

TL;DR

This paper provides a unified analysis of a two-level algorithm for HDG and WG methods for diffusion problems, deriving sharp convergence estimates with minimal regularity and flexible mesh assumptions, supported by numerical validation.

Contribution

It introduces a novel analysis framework using the extended X-Z identity that applies to both HDG and WG methods under minimal regularity and non-quasi-uniform meshes.

Findings

Sharp convergence rate estimates derived

Analysis applies to both HDG and WG methods

Numerical experiments confirm theoretical predictions

Abstract

This paper analyzes an abstract two-level algorithm for hybridizable discontinuous Galerkin (HDG) methods in a unified fashion. We use an extended version of the Xu-Zikatanov (X-Z) identity to derive a sharp estimate of the convergence rate of the algorithm, and show that the theoretical results also apply to weak Galerkin (WG) methods. The main features of our analysis are twofold: one is that we only need the minimal regularity of the model problem; the other is that we do not require the triangulations to be quasi-uniform. Numerical experiments are provided to confirm the theoretical results.