A new family of a posteriori error estimates for non-conforming finite element methods leading to stabilization-free error bounds

TL;DR

This paper introduces new a posteriori error estimators for non-conforming finite element methods that do not require stabilization, improving efficiency and robustness across various discretization schemes.

Contribution

It presents novel reformulations of the Prager-Synge identity, enabling stabilization-free, polynomial-degree-robust error bounds for second-order elliptic PDE discretizations.

Findings

Residual-based estimator with optimal polynomial degree scaling

Two equilibrated estimators that are polynomial-degree-robust

One estimator achieves asymptotically constant-free error bounds

Abstract

We propose new a posteriori error estimators for non-conforming finite element discretizations of second-order elliptic PDE problems. These estimators are based on novel reformulations of the standard Prager-Synge identity, and enable to prove efficiency estimates without extra stabilization terms in the error measure for a large class of discretization schemes. We propose a residual-based estimator for which the efficiency constant scales optimally in polynomial degree, as well as two equilibrated estimators that are polynomial-degree-robust. One of the two estimators further leads to asymptotically constant-free error bounds.