A new family of a posteriori error estimates for non-conforming finite element methods leading to stabilization-free error bounds
T. Chaumont-Frelet
TL;DR
This work develops hp-robust a posteriori error estimators for non-conforming finite element discretizations of second-order elliptic PDEs by reformulating the Prager--Synge identity. It introduces a residual-based estimator and two fully equilibrated estimators, including a curl-free reconstruction approach, to bound the error in the natural energy norm without additional stabilization terms. The framework applies to interior penalty DG and Raviart–Thomas mixed methods, with additional alternative equilibrations based on edge patches. The resulting estimators are provably reliable and efficient, with polynomial-degree-robust constants and, in one variant, asymptotically constant-free bounds, enabling robust adaptive refinement on general three-dimensional domains.
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.