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Tangential-Normal Decompositions of Finite Element Differential Forms

Long Chen, Xuehai Huang

TL;DR

This work addresses the challenge of constructing bases for finite element differential forms that are dual to degrees of freedom in FEEC. It introduces a tangential-normal ($t$-$n$) decomposition on simplices, provides explicit dual bases for the second family of polynomial differential forms, and couples these with the Lagrange element to enable efficient stiffness assembly, interpolation, and integration. A geometric decomposition using bubble polynomial forms is developed, and the approach is extended to full polynomial differential forms, de Rham complexes, and Lagrange-type bases. The resulting framework yields dimension-independent local bases on simplicial meshes, with robust trace properties and exactness in the discrete de Rham sequence, offering practical pathways for high-order FEEC implementations.

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.

Tangential-Normal Decompositions of Finite Element Differential Forms

TL;DR

This work addresses the challenge of constructing bases for finite element differential forms that are dual to degrees of freedom in FEEC. It introduces a tangential-normal (-) decomposition on simplices, provides explicit dual bases for the second family of polynomial differential forms, and couples these with the Lagrange element to enable efficient stiffness assembly, interpolation, and integration. A geometric decomposition using bubble polynomial forms is developed, and the approach is extended to full polynomial differential forms, de Rham complexes, and Lagrange-type bases. The resulting framework yields dimension-independent local bases on simplicial meshes, with robust trace properties and exactness in the discrete de Rham sequence, offering practical pathways for high-order FEEC implementations.

Abstract

This paper introduces a novel tangential-normal (-) 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 - 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.

Paper Structure

This paper contains 27 sections, 18 theorems, 115 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Simplex and sub-simplexes
  4. Barycentric coordinates and Bernstein polynomials
  5. Tangential-Normal ($t$-$n$) bases
  6. Normal bases of an edge on a face
  7. Background of differential forms
  8. Increasing sequence
  9. Differential forms
  10. Proxy vector
  11. Hodge star
  12. Inner product
  13. Differential Forms on Simplexes
  14. Extension
  15. Traces of differential forms
  16. ...and 12 more sections

Key Result

Lemma 2.1

For $f \in \Delta_{\ell}(T)$ with $0\leq \ell \leq d-1,$ the tangential-normal basis of $\mathscr{N}^f$ is the scaled dual of the face normal basis $\{\nabla\lambda_i \mid i \in f^*\}$.

Figures (3)

  • Figure 1: Face normal basis and tangential-normal basis of a vertex and an edge in a tetrahedron. For the face normal basis functions, a negative sign is added to improve the clarity of the illustration.
  • Figure 2: A $t$--$n$ decomposition of $\mathscr{T}^f$ based on a sub-simplex $e\subset f$.
  • Figure 3: A grid on the $(e,f)$ plane, $f\in \Delta_{\ell}(T), e\in \Delta_s(f)$, $\ell = k, \ldots, d, s = \ell -k, \ldots, \ell$, for the $t$-$n$ decomposition of polynomial differential forms $\mathbb{P}_r\Lambda^{k}(T)$ on a simplex $T$. Each dot represents the space $\mathbb{P}_{r - (s+1)}(e) \otimes b_e \star_f \operatorname{Alt}^{\ell - k}(\mathscr{T}^e)$. Summing vertically gives the geometric decomposition of $\operatorname{Alt}^k(\mathscr{T}^T)$, and summing horizontally gives the bubble polynomial form space $\mathbb{B}_k\Lambda^k(f)$. Summing over all entries yields the full space $\mathbb{P}_r\Lambda^k(T)$.

Theorems & Definitions (36)

  • Lemma 2.1
  • proof
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • ...and 26 more