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Semi-robust equal-order hybridized discontinuous methods

Xiaoqi Ma, Jin Zhang

TL;DR

The paper addresses stable, high-order discretizations for incompressible flow by developing a unified equal-order HDG/EDG/E-HDG framework for the Oseen equations. It employs a symmetric pressure stabilization to bypass the inf-sup constraint and derives semi-robust error estimates: an energy-norm error of O(h^k) with constants independent of ν^{-1}, along with optimal O(h^{k+1}) velocity convergence in L^2 and O(h^k) pressure convergence. Theoretical analysis is complemented by interpolation and stability results and validated through numerical experiments across ν=1 and ν=0.1, confirming robustness in convection-dominated regimes. The work offers a practical, reduced-DOF, high-order approach for steady/incompressible flow simulations with reliable performance at small viscosity.

Abstract

This paper introduces a unified analysis framework of equal-order hybridized discontinuous finite element (HDG) methods. The general framework covers standard HDG, embedded discontinuous finite element, and embedded-hybridized discontinuous finite element methods.

Semi-robust equal-order hybridized discontinuous methods

TL;DR

The paper addresses stable, high-order discretizations for incompressible flow by developing a unified equal-order HDG/EDG/E-HDG framework for the Oseen equations. It employs a symmetric pressure stabilization to bypass the inf-sup constraint and derives semi-robust error estimates: an energy-norm error of O(h^k) with constants independent of ν^{-1}, along with optimal O(h^{k+1}) velocity convergence in L^2 and O(h^k) pressure convergence. Theoretical analysis is complemented by interpolation and stability results and validated through numerical experiments across ν=1 and ν=0.1, confirming robustness in convection-dominated regimes. The work offers a practical, reduced-DOF, high-order approach for steady/incompressible flow simulations with reliable performance at small viscosity.

Abstract

This paper introduces a unified analysis framework of equal-order hybridized discontinuous finite element (HDG) methods. The general framework covers standard HDG, embedded discontinuous finite element, and embedded-hybridized discontinuous finite element methods.

Paper Structure

This paper contains 20 sections, 6 theorems, 77 equations, 3 figures, 12 tables.

Table of Contents

  1. Introduction
  2. The Oseen problem
  3. The equal-order HDG methods
  4. Mesh-related notation
  5. Fnite element spaces and norms
  6. Weak formulation
  7. Stability of the method
  8. Interpolation and interpolation estimates
  9. Interpolation operators and approximation properties
  10. Interpolation error estimate
  11. Error estimates
  12. Numerical experiment
  13. Experimental Setup
  14. Stabilization Parameter Selection
  15. Observed convergence rates
  16. ...and 5 more sections

Key Result

lemma thmcounterlemma

(Consistency) Let $(u, p) \in ([H_{0}^{1}(\Omega)]^{d} \cap [H^{2}(\Omega)]^{d}) \times (L^{2}_{0}(\Omega) \cap H^{1}(\Omega))$ and ${\bm{u}} = (u, \zeta(u))$ and ${\bm{p}} = (p, \zeta(p))$, where $(u, p)$ is the solution of eq:SSSS-1,

Figures (3)

  • Figure 1: $\Vert u-u_{h}\Vert$ and pressure penalty parameter $\alpha$ for E-HDG ($\nu=1$, $k=1$, $\eta=4$)
  • Figure 2: $\Vert p-p_{h}\Vert$ and pressure penalty parameter $\alpha$ for E-HDG ($\nu=1$, $k=1$, $\eta=4$)
  • Figure 3: $|||(\mathbf{e}_{u}, \mathbf{e}_{p})|||$ and pressure penalty parameter $\alpha$ for E-HDG ($\nu=1$, $k=1$, $\eta=4$)

Theorems & Definitions (13)

  • remark thmcounterremark
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • theorem 1
  • proof
  • theorem 2
  • ...and 3 more