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$hp$-a posteriori error estimates for hybrid high-order methods applied to biharmonic problems

Zhaonan Dong, Alexandre Ern, Tanvi Wadhawan

TL;DR

This work develops hp-adaptive, residual-based a posteriori error estimators for hybrid high-order methods applied to the biharmonic equation on Lipschitz polytopal domains in 2D and 3D. A key novelty is bounding the nonconforming error through a local $C^1$ partition of unity built on Alfeld-split subcells and a stable local Helmholtz decomposition on vertex patches, avoiding dependence on domain topology. Two conforming-error estimators are derived from distinct interpolation operators: a canonical CH-based one yielding an $H^2$-elliptic projection and, separately, a Babuška–Suri-based operator; each provides hp-optimal or near hp-optimal bounds with different indicator terms. The main result combines these ingredients into hp-upper bounds that control the energy error by stabilization, residuals, and data oscillation, with numerical experiments confirming efficiency and adaptive refinement capability for handling singularities.

Abstract

We derive a residual-based $hp$-a posteriori error estimator for hybrid high-order (HHO) methods on simplicial meshes applied to the biharmonic problem posed on two- and three-dimensional polytopal Lipschitz domains. The a posteriori error estimator hinges on an error decomposition into conforming and nonconforming components. To bound the nonconforming error, we use a $C^1$-partition of unity constructed via Alfeld splittings, combined with local Helmholtz decompositions on vertex stars. For the conforming error, we design two residual-based estimators, each associated with a specific interpolation operator. In the first setting, the upper bound for the conforming error involves only the stabilization term and the data oscillation. In the second setting, the bound additionally incorporates bulk residuals, normal flux jumps, and tangential jumps. Numerical experiments confirm the theoretical findings and demonstrate the efficiency of the proposed estimators.

$hp$-a posteriori error estimates for hybrid high-order methods applied to biharmonic problems

TL;DR

This work develops hp-adaptive, residual-based a posteriori error estimators for hybrid high-order methods applied to the biharmonic equation on Lipschitz polytopal domains in 2D and 3D. A key novelty is bounding the nonconforming error through a local partition of unity built on Alfeld-split subcells and a stable local Helmholtz decomposition on vertex patches, avoiding dependence on domain topology. Two conforming-error estimators are derived from distinct interpolation operators: a canonical CH-based one yielding an -elliptic projection and, separately, a Babuška–Suri-based operator; each provides hp-optimal or near hp-optimal bounds with different indicator terms. The main result combines these ingredients into hp-upper bounds that control the energy error by stabilization, residuals, and data oscillation, with numerical experiments confirming efficiency and adaptive refinement capability for handling singularities.

Abstract

We derive a residual-based -a posteriori error estimator for hybrid high-order (HHO) methods on simplicial meshes applied to the biharmonic problem posed on two- and three-dimensional polytopal Lipschitz domains. The a posteriori error estimator hinges on an error decomposition into conforming and nonconforming components. To bound the nonconforming error, we use a -partition of unity constructed via Alfeld splittings, combined with local Helmholtz decompositions on vertex stars. For the conforming error, we design two residual-based estimators, each associated with a specific interpolation operator. In the first setting, the upper bound for the conforming error involves only the stabilization term and the data oscillation. In the second setting, the bound additionally incorporates bulk residuals, normal flux jumps, and tangential jumps. Numerical experiments confirm the theoretical findings and demonstrate the efficiency of the proposed estimators.
Paper Structure (25 sections, 12 theorems, 124 equations, 4 figures)

This paper contains 25 sections, 12 theorems, 124 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Analysis tools
  3. Basic notation
  4. Weak formulation
  5. Mesh
  6. Analysis tools
  7. Hybrid high-order discretization
  8. Local reconstruction and stabilization
  9. Discrete problem
  10. Local interpolation operators
  11. Local Helmholtz decomposition
  12. $hp$-a posteriori error analysis
  13. Abstract error bound
  14. Bound on dual residual norm
  15. Bound on nonconforming error
  16. ...and 10 more sections

Key Result

Lemma 2.1

For all $T \in \mathcal{T}_h$, all $v \in \mathbb{P}^p(T)$ with $p \ge 0$, and all $F\in\mathcal{F}_{\partial T}$, the following holds:

Figures (4)

  • Figure 1: Convergence of the interpolation error $\| \partial_{n} (v- \mathcal{C}_T^{k+2} (v) ) \|_{\partial T}$ as a function of the polynomial degree $(k+2)$. Left panel: $d=2$, $k \in \{0,\ldots,20\}$, and $\alpha \in\{ 1.01, 1.51\}$. Right panel: $d=3$, $k \in\{0,\ldots,13\}$, and $\alpha \in\{1.01, 1.51\}$.
  • Figure 2: Example 1: Energy error and a posteriori error estimator for $k\in\{0,1,2,3\}$ as a function of DoFs (left panel) and effectivity index as a function of DoFs (right panel).
  • Figure 3: Example 2: Energy error and a posteriori error estimator as a function of DoFs for $k\in\{0,1,2,3\}$.
  • Figure 4: Example 2: Effectivity index as a function of DoFs for $k\in\{0, 1,2,3\}$ (left panel). Effectivity index as a function of $k\in\{0, \ldots,12\}$ on a mesh composed of 96 cells (right panel).

Theorems & Definitions (28)

  • Lemma 2.1: $hp$-discrete trace inequality
  • proof
  • Lemma 2.2: $hp$-inverse inequality
  • proof
  • Lemma 2.3: Global Babuška--Suri $hp$-interpolation operator
  • proof
  • Corollary 2.4: Modified Babuška--Suri $hp$-interpolation operator
  • proof
  • Lemma 3.1: Useful property
  • proof
  • ...and 18 more