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

This study investigates high-order face and edge elements in finite elementmethods, with a focus on their geometric attributes, indexing management, andpractical application. The exposition begins by a geometric decomposition ofLagrange finite elements, setting the foundation for further analysis. Thediscussion then extends to H(div)-conforming andH(curl)-conforming finite element spaces, adopting variable framesacross differing sub-simplices. The imposition of tangential or normalcontinuity is achieved through the strategic selection of corresponding bases.The paper concludes with a focus on efficient indexing management strategiesfor degrees of freedom, offering practical guidance to researchers andengineers. It serves as a comprehensive resource that bridges the gap betweentheory and practice.