In this work, N\'ed\'elec elements on locally refined meshes with hanging nodes are considered. A crucial aspect is the orientation of the hanging edges and faces. For non-orientable meshes, no solution or implementation has been available to date. The problem statement and corresponding algorithms are described in great detail. As a model problem, the time-harmonic Maxwell's equations are adopted because N\'ed\'elec elements constitute their natural discretization. The algorithms and implementation are demonstrated through two numerical examples on different uniformly and adaptively refined meshes. The implementation is performed within the finite element library deal.II.

翻译：本文研究了含悬挂节点的局部加密网格上的Nédélec单元。悬挂边与悬挂面的定向问题是关键所在。对于不可定向网格，迄今尚无可用解决方案或实现方法。本文详细阐述了问题描述及相应算法。采用时谐麦克斯韦方程组作为模型问题，因为Nédélec单元是其自然离散化方案。通过在不同均匀加密与自适应加密网格上的两个数值算例，验证了算法与实现的可行性。该实现基于有限元库deal.II完成。