Tangential-Normal Decompositions of Finite Element Differential Forms

TL;DR

This paper introduces a new tangential-normal decomposition for finite element differential forms, creating a dual basis framework that improves computational efficiency and practical implementation in finite element exterior calculus.

Contribution

It develops a novel $t$-$n$ basis with explicitly dual degrees of freedom and shape functions, facilitating easier assembly and integration in finite element methods.

Findings

Dual $t$-$n$ basis simplifies stiffness matrix assembly

Supports practical implementation with Lagrange element basis

Includes geometric decomposition with bubble polynomial forms

Abstract

This paper introduces a novel tangential-normal ($t$-$n$) decomposition for finite element differential forms, presenting a new framework for constructing bases in finite element exterior calculus. The main contribution is the development of a $t$-$n$ basis where degrees of freedom and shape functions are explicitly dual, a property that streamlines stiffness matrix assembly and enhances the efficiency of interpolation and numerical integration. Additionally, the integration of the well-documented Lagrange element basis supports practical implementation of finite element differential forms in applications. A geometric decomposition using newly defined bubble polynomial forms is also presented.