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

TL;DR

This paper presents a geometric decomposition approach for high-order face and edge finite elements, emphasizing efficient implementation and practical indexing strategies to enhance their application in computational methods.

Contribution

It introduces a geometric decomposition framework for high-order finite elements and proposes efficient indexing management techniques for degrees of freedom.

Findings

Provides a geometric decomposition of Lagrange finite elements

Develops strategies for efficient indexing of degrees of freedom

Bridges theoretical concepts with practical implementation guidance

Abstract

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.