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Stabilization-free HHO a posteriori error control

Fleurianne Bertrand, Carsten Carstensen, Benedikt Gräßle, Ngoc Tien Tran

TL;DR

The paper develops stabilization-free a posteriori error control for hybrid high-order methods on simplicial meshes by introducing two explicit estimators, $\eta_{\rm res}$ and $\eta_{\mathrm{eq},p}$, and establishing guaranteed upper bounds (GUB) for the true error without relying on the stabilization term. It proves reliability and efficiency of these estimators up to data oscillations, and constructs equilibrated flux reconstructions $Q_p$ to realize the GUB through a post-processing framework that includes nodal averaging and $H(\mathrm{div})$-conforming corrections. The results are supported by detailed theoretical foundations (a Helmholtz-type decomposition and explicit constants) and numerical benchmarks on the Poisson problem, showing that adaptive refinement driven by the stabilization-free estimators achieves optimal convergence and accurate error control. The work broadens the HHO error-analysis toolkit by enabling stabilization-free, data-oscillation-aware guarantees, with practical impact for reliable adaptive simulations on simplicial meshes.

Abstract

The known a posteriori error analysis of hybrid high-order methods (HHO) treats the stabilization contribution as part of the error and as part of the error estimator for an efficient and reliable error control. This paper circumvents the stabilization contribution on simplicial meshes and arrives at a stabilization-free error analysis with an explicit residual-based a posteriori error estimator for adaptive mesh-refining as well as an equilibrium-based guaranteed upper error bound (GUB). Numerical evidence in a Poisson model problem supports that the GUB leads to realistic upper bounds for the displacement error in the piecewise energy norm. The adaptive mesh-refining algorithm associated to the explicit residual-based a posteriori error estimator recovers the optimal convergence rates in computational benchmarks.

Stabilization-free HHO a posteriori error control

TL;DR

The paper develops stabilization-free a posteriori error control for hybrid high-order methods on simplicial meshes by introducing two explicit estimators, and , and establishing guaranteed upper bounds (GUB) for the true error without relying on the stabilization term. It proves reliability and efficiency of these estimators up to data oscillations, and constructs equilibrated flux reconstructions to realize the GUB through a post-processing framework that includes nodal averaging and -conforming corrections. The results are supported by detailed theoretical foundations (a Helmholtz-type decomposition and explicit constants) and numerical benchmarks on the Poisson problem, showing that adaptive refinement driven by the stabilization-free estimators achieves optimal convergence and accurate error control. The work broadens the HHO error-analysis toolkit by enabling stabilization-free, data-oscillation-aware guarantees, with practical impact for reliable adaptive simulations on simplicial meshes.

Abstract

The known a posteriori error analysis of hybrid high-order methods (HHO) treats the stabilization contribution as part of the error and as part of the error estimator for an efficient and reliable error control. This paper circumvents the stabilization contribution on simplicial meshes and arrives at a stabilization-free error analysis with an explicit residual-based a posteriori error estimator for adaptive mesh-refining as well as an equilibrium-based guaranteed upper error bound (GUB). Numerical evidence in a Poisson model problem supports that the GUB leads to realistic upper bounds for the displacement error in the piecewise energy norm. The adaptive mesh-refining algorithm associated to the explicit residual-based a posteriori error estimator recovers the optimal convergence rates in computational benchmarks.
Paper Structure (6 sections, 6 theorems, 40 equations, 1 table)

This paper contains 6 sections, 6 theorems, 40 equations, 1 table.

Table of Contents

  1. Introduction
  2. Known a posteriori error estimator
  3. Stabilization-free a posteriori error control
  4. Further contributions and outline
  5. Overall notation
  6. Foundations of the a posteriori error analysis

Key Result

theorem 1

Suppose that ${G}\in H^1(\mathcal{T};\mathbb R^n)$ and $f\in L^2(\Omega)$ satisfy eq:solution_property--eq:div_0_property. Then the error estimator $\eta(\mathcal{T}, {G})$ from eq:eta is a GUB of the error $\|\nabla u-{G}\|$ for the solution $u\in V$ to eq:poisson. The constants $C_{\rm 1}, C_{2}, C_{\rm H}$ exclusively depend on $\Omega$ and the shape-regularity of $\mathcal{T}$.

Theorems & Definitions (12)

  • theorem 1: residual-based GUB
  • lemma thmcounterlemma: decomposition
  • proof
  • lemma thmcounterlemma: Helmholtz-decomposition
  • proof
  • lemma thmcounterlemma: trace inequality
  • proof
  • lemma thmcounterlemma: equilibrium
  • proof
  • lemma thmcounterlemma: conformity
  • ...and 2 more