We present a local construction of H(curl)-conforming piecewise polynomials satisfying a prescribed curl constraint. We start from a piecewise polynomial not contained in the H(curl) space but satisfying a suitable orthogonality property. The procedure employs minimizations in vertex patches and the outcome is, up to a generic constant independent of the underlying polynomial degree, as accurate as the best-approximations over the entire local versions of H(curl). This allows to design guaranteed, fully computable, constant-free, and polynomial-degree-robust a posteriori error estimates of Prager-Synge type for N\'ed\'elec finite element approximations of the curl-curl problem. A divergence-free decomposition of a divergence-free H(div)-conforming piecewise polynomial, relying on over-constrained minimizations in Raviart-Thomas spaces, is the key ingredient. Numerical results illustrate the theoretical developments.

翻译：我们展示了一种符合H( curl) 的局部成像片状多面体的构造, 满足了指定的卷曲限制。 我们从一个不包含在 H( curl) 空间中但满足了合适正数属性的片状多面体开始。 程序在顶端补丁中采用最小值, 结果是, 与基本多面度不相干, 与整个本地版本H( curl) 的最佳匹配度一样准确。 这样可以设计一个有保障的、 完全可比较的、 恒定的和 多元度- 度- robust, 一种用于 N\'ed\\' elec 曲线- curnation 问题的边际元素近似的边际- Synge 的事后误差估计值。 一个无差异的无差异的 H( div) 相配成的片状多面体, 依赖Raviart- Thomas 空间中过度受限制的最小值, 是关键成份。 。 Numerical 的结果说明了理论发展。