Geometric Decomposition and Efficient Implementation of High Order Face and Edge Elements

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 $H(\mathrm{d}\mathrm{i}\mathrm{v})$-conforming and $H(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l})$-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.