A simple equilibration procedure leading to polynomial-degree-robust a posteriori error estimators for the curl-curl problem

TL;DR

This paper presents two reliable and efficient a posteriori error estimators for the curl-curl problem using Nédélec finite elements, which are robust with respect to polynomial degree and easy to implement, suitable for adaptive methods.

Contribution

The paper introduces novel polynomial-degree-robust a posteriori error estimators based on a new Prager-Synge identity and an equilibration procedure for the curl-curl problem.

Findings

Estimators are reliable and efficient across polynomial degrees.

Error bounds are fully computable for convex domains.

Estimators effectively guide adaptive refinement in numerical experiments.

Abstract

We introduce two a posteriori error estimators for N\'ed\'elec finite element discretizations of the curl-curl problem. These estimators pertain to a new Prager-Synge identity and an associated equilibration procedure. They are reliable and efficient, and the error estimates are polynomial-degree-robust. In addition, when the domain is convex, the reliability constants are fully computable. The proposed error estimators are also cheap and easy to implement, as they are computed by solving divergence-constrained minimization problems over edge patches. Numerical examples highlight our key findings, and show that both estimators are suited to drive adaptive refinement algorithms. Besides, these examples seem to indicate that guaranteed upper bounds can be achieved even in non-convex domains.