This study investigates high-order face and edge elements in finite element methods, with a focus on their geometric attributes, indexing management, and practical application. The exposition begins by a geometric decomposition of Lagrange finite elements, setting the foundation for further analysis. The discussion then extends to $H(\rm{div})$-conforming and $H(\rm{curl})$-conforming finite element spaces, adopting variable frames across differing sub-simplices. The imposition of tangential or normal continuity is achieved through the strategic selection of corresponding bases. The paper concludes with a focus on efficient indexing management strategies for degrees of freedom, offering practical guidance to researchers and engineers. It serves as a comprehensive resource that bridges the gap between theory and practice.

翻译：本研究探讨有限元方法中的高阶面单元与边单元，重点关注其几何属性、索引管理及实际应用。首先通过对拉格朗日有限元进行几何分解，为后续分析奠定基础。随后讨论$H(\rm{div})$相容与$H(\rm{curl})$相容的有限元空间，在不同子单纯形上采用可变标架。通过策略性选择对应基函数，实现切向或法向连续性的施加。本文最后重点介绍自由度的有效索引管理策略，为研究人员与工程师提供实用指导。该研究作为一套综合性资源，弥合了理论与实践的差距。