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Discrete Poincaré and Trace Inequalities for the Hybridizable Discontinuous Galerkin Method

Yukun Yue

TL;DR

The paper addresses the stability analysis of HDG discretizations for second-order elliptic problems by deriving discrete Poincaré and trace inequalities tailored to hybridizable spaces. It leverages the Crouzeix–Raviart lifting as a bridge between HDG spaces and classical nonconforming discrete tools, achieving a discrete Poincaré inequality with an O(h) jump term and a discrete trace inequality of order O(1). These tools enable stability results for HDG formulations under minimal regularity assumptions, without relying on a translation argument. The results are extended to an HDG Poisson problem via a boundary lifting operator and a mixed formulation, yielding mesh-independent energy estimates that depend only on the data and the domain. Overall, the work provides robust, CR-based analytical tools that enhance the stability analysis and applicability of HDG methods for elliptic problems.

Abstract

In this paper, we derive discrete Poincaré and trace inequalities for the hybridizable discontinuous Galerkin (HDG) method. We employ the Crouzeix-Raviart space as a bridge, connecting classical discrete functional tools from Brenner's foundational work \cite{brenner2003poincare} with hybridizable finite element spaces comprised of piecewise polynomial functions defined both within the element interiors and on the mesh skeleton. This approach yields custom-tailored inequalities that underpin the stability analysis of HDG discretizations. The resulting framework is then used to demonstrate the well-posedness and robustness of HDG-based numerical schemes for second-order elliptic problems, even under minimal regularity assumptions on the source term and boundary data.

Discrete Poincaré and Trace Inequalities for the Hybridizable Discontinuous Galerkin Method

TL;DR

The paper addresses the stability analysis of HDG discretizations for second-order elliptic problems by deriving discrete Poincaré and trace inequalities tailored to hybridizable spaces. It leverages the Crouzeix–Raviart lifting as a bridge between HDG spaces and classical nonconforming discrete tools, achieving a discrete Poincaré inequality with an O(h) jump term and a discrete trace inequality of order O(1). These tools enable stability results for HDG formulations under minimal regularity assumptions, without relying on a translation argument. The results are extended to an HDG Poisson problem via a boundary lifting operator and a mixed formulation, yielding mesh-independent energy estimates that depend only on the data and the domain. Overall, the work provides robust, CR-based analytical tools that enhance the stability analysis and applicability of HDG methods for elliptic problems.

Abstract

In this paper, we derive discrete Poincaré and trace inequalities for the hybridizable discontinuous Galerkin (HDG) method. We employ the Crouzeix-Raviart space as a bridge, connecting classical discrete functional tools from Brenner's foundational work \cite{brenner2003poincare} with hybridizable finite element spaces comprised of piecewise polynomial functions defined both within the element interiors and on the mesh skeleton. This approach yields custom-tailored inequalities that underpin the stability analysis of HDG discretizations. The resulting framework is then used to demonstrate the well-posedness and robustness of HDG-based numerical schemes for second-order elliptic problems, even under minimal regularity assumptions on the source term and boundary data.
Paper Structure (19 sections, 19 theorems, 134 equations, 1 figure)

This paper contains 19 sections, 19 theorems, 134 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notations and Assumptions
  4. Space Discretization
  5. Function Spaces
  6. Other Notations
  7. Mesh Assumptions
  8. Technical Lemmas
  9. Crouzeix–Raviart Space
  10. Discrete Poincaré Inequality
  11. Discrete Poincaré Inequality for Piecewise $H^1$ Functions
  12. CR Lifting Operator
  13. Poincaré Inequalities in $\mathcal{X}_h^k$
  14. Discrete Trace Inequality
  15. Application to HDG formulation
  16. ...and 4 more sections

Key Result

Lemma 2.3

Consider a simplex $K$ in $\mathbb{R}^{d}$, with $e$ representing one of its faces. For any function $f$ belonging to $\mathcal{P}^k(K)$, the following inequality holds true:

Figures (1)

  • Figure 1: Comparison of Spaces: Nonconforming Space for DG Method (Left) versus Hybridizable Space for HDG or HHO Method (Right)

Theorems & Definitions (40)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 3.1
  • proof
  • ...and 30 more