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

TL;DR

This paper introduces new polynomial-degree-robust a posteriori error estimates for various PDE problems using auxiliary space techniques, improving error control in finite element methods.

Contribution

It develops novel p-robust a posteriori error estimates based on auxiliary space decomposition for $H( m curl)$, $H( m div)$, and $H( m divdiv)$ problems, with guaranteed bounds.

Findings

Effective error estimators for Nédélec and H-H-J methods

Demonstrated p-robustness through numerical experiments

Provided guaranteed upper bounds under specific domain conditions

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 N\'ed\'elec edge element methods and the Hellan--Herrmann--Johnson methods.