An equilibrated estimator for mixed finite element discretizations of the curl-curl problem

TL;DR

This paper introduces a new a posteriori error estimator for mixed finite element methods solving the curl-curl problem, offering guaranteed, efficient, and topology-independent error bounds suitable for adaptive mesh refinement.

Contribution

The paper develops a novel, fully guaranteed, polynomial-degree-robust error estimator based on a Prager--Synge inequality, applicable without domain topology assumptions.

Findings

Provides guaranteed upper bounds on error without topology assumptions

Demonstrates efficiency and robustness of the estimator

Numerical tests show suitability for mesh adaptivity

Abstract

We propose a new a posteriori error estimator for mixed finite element discretizations of the curl-curl problem. This estimator relies on a Prager--Synge inequality, and therefore leads to fully guaranteed constant-free upper bounds on the error. The estimator is also locally efficient and polynomial-degree-robust. The construction is based on patch-wise divergence-constrained minimization problems, leading to a cheap embarrassingly parallel algorithm. Crucially, the estimator operates without any assumption on the topology of the domain, and unconventional arguments are required to establish the reliability estimate. Numerical examples illustrate the key theoretical results, and suggest that the estimator is suited for mesh adaptivity purposes.