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.

翻译：我们引入了 N\'ed\'elec 等离子元素对曲线- 曲线问题的后端误差估计值。 这些估计值涉及一个新的 Prager- Synge 身份和相关平衡程序。 它们可靠有效, 误差估计值是多度- 紫外线。 此外, 当域为连接时, 可靠性常数完全可以计算 。 提议的误差估计值也是廉价和容易执行的, 因为它们是通过解决边缘补丁上的差异限制最小化问题来计算的。 数字示例突显了我们的主要发现, 并表明两个估计值都适合驱动适应性精细算法。 此外, 这些示例似乎表明即使在非节点域中, 也能够实现保证的上限 。