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Some p-robust a posteriori error estimates based on auxiliary spaces

Yuwen Li

TL;DR

The paper develops polynomial-degree-robust equilibrated a posteriori error estimators for $H({\rm curl})$, $H({\rm div})$, and $H({\rm divdiv})$ problems by leveraging an $H^1$ auxiliary-space decomposition and regular decompositions. Through an auxiliary-space preconditioning framework and equilibrated flux concepts, residuals in $H^{-1}$ are bounded by $H^1$-type estimators, yielding simple, robust, and guaranteed upper bounds for both vector- and tensor-valued problems, including Nédélec and HHJ discretizations. The approach extends to the Hodge–Laplace equation and to stress variables in mixed FEMs (Poisson and biharmonic problems), with provable p-robustness and practical local-solve implementations on vertex patches. Numerical experiments confirm near-optimal adaptive convergence and robustness across polynomial degrees, highlighting the framework’s potential for hp-adaptive methods and high-order simulations in electromagnetics and elasticity.

Abstract

This work develops polynomial-degree-robust (p-robust) equilibrated a posteriori error estimates for $H(\rm curl)$, $H(\rm div)$ and $H(\rm divdiv)$ problems, based on $H^1$ auxiliary space decomposition. The proposed framework employs auxiliary space preconditioning and regular decompositions to decompose the finite element residual into $H^{-1}$ residuals that are further controlled by classical p-robust equilibrated a posteriori error analysis. As a result, we obtain novel and simple p-robust a posteriori error estimates of $H(\rm curl)$/$H(\rm div)$ conforming methods and mixed methods for the biharmonic equation. In addition, we prove guaranteed a posteriori upper error bounds under convex domains or certain boundary conditions. Numerical experiments demonstrate the effectiveness and p-robustness of the proposed error estimators for the Nédélec edge element methods and the Hellan--Herrmann--Johnson methods.

Some p-robust a posteriori error estimates based on auxiliary spaces

TL;DR

The paper develops polynomial-degree-robust equilibrated a posteriori error estimators for , , and problems by leveraging an auxiliary-space decomposition and regular decompositions. Through an auxiliary-space preconditioning framework and equilibrated flux concepts, residuals in are bounded by -type estimators, yielding simple, robust, and guaranteed upper bounds for both vector- and tensor-valued problems, including Nédélec and HHJ discretizations. The approach extends to the Hodge–Laplace equation and to stress variables in mixed FEMs (Poisson and biharmonic problems), with provable p-robustness and practical local-solve implementations on vertex patches. Numerical experiments confirm near-optimal adaptive convergence and robustness across polynomial degrees, highlighting the framework’s potential for hp-adaptive methods and high-order simulations in electromagnetics and elasticity.

Abstract

This work develops polynomial-degree-robust (p-robust) equilibrated a posteriori error estimates for , and problems, based on auxiliary space decomposition. The proposed framework employs auxiliary space preconditioning and regular decompositions to decompose the finite element residual into residuals that are further controlled by classical p-robust equilibrated a posteriori error analysis. As a result, we obtain novel and simple p-robust a posteriori error estimates of / conforming methods and mixed methods for the biharmonic equation. In addition, we prove guaranteed a posteriori upper error bounds under convex domains or certain boundary conditions. Numerical experiments demonstrate the effectiveness and p-robustness of the proposed error estimators for the Nédélec edge element methods and the Hellan--Herrmann--Johnson methods.

Paper Structure

This paper contains 15 sections, 14 theorems, 115 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equilibration in H(grad)
  4. Operator Preconditioning
  5. Natural norm a posteriori error estimates
  6. H(curl) Elliptic Equation
  7. H(div) Elliptic Equation
  8. Hodge--Laplace equation
  9. A Posteriori Error Estimates of Stress Variables
  10. Mixed FEM for the Laplacian
  11. Hellan--Herrmann--Johnson Method
  12. Numerical Experiments
  13. H(curl) Conforming Method
  14. HHJ Method
  15. Concluding remarks

Key Result

Lemma 2.2

Let $R\in H^{-1}(\Omega)$ satisfy $\langle R,\phi_i\rangle=0$ for each vertex $a_i\not\in\partial\Omega$ and admit a piecewise polynomial representation. Assume that Problem prob:min_sigmaRT admits a solution. Then

Figures (4)

  • Figure 5.1: A posteriori error estimates and $H(\rm curl)$ norm error of FEMs for the $H(\rm curl)$ elliptic equation.
  • Figure 5.2: A posteriori error estimates for the curl-curl equation.
  • Figure 5.3: A posteriori error estimates based on the local $\mathcal{RT}_p\times\mathcal{P}_p^{\rm dG}$ mixed problem for the HHJ method under simply supported (left) and clamped (right) boundary conditions.
  • Figure 5.4: A posteriori error estimates based on the local $\mathcal{RT}_{p+1}\times\mathcal{P}_{p+1}^{\rm dG}$ mixed problem for the HHJ method under simply supported (left) and clamped (right) boundary conditions.

Theorems & Definitions (25)

  • Lemma 2.2: Equilibrated residual in $H^{-1}(\Omega)$
  • proof
  • Lemma 2.3: Fictitious Space Lemma
  • Lemma 3.1: Regular Decomposition in H(curl)
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Corollary 3.4
  • proof
  • Lemma 3.5
  • ...and 15 more