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

Chunyu Chen, Long Chen, Xuehai Huang, Huayi Wei

TL;DR

This paper delves into the world of high-order curl and div elements within finite element methods, providing valuable insights into their geometric properties, indexing management, and practical implementation considerations, and concludes with a focus on efficientindexing management strategies for degrees of freedom.

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.

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

TL;DR

This paper delves into the world of high-order curl and div elements within finite element methods, providing valuable insights into their geometric properties, indexing management, and practical implementation considerations, and concludes with a focus on efficientindexing management strategies for degrees of freedom.

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 -conforming and -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.
Paper Structure (26 sections, 16 theorems, 102 equations, 6 figures)

This paper contains 26 sections, 16 theorems, 102 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial lattice
  4. Interpolation points
  5. Sub-simplexes and sub-simplicial lattices
  6. Triangulation
  7. Geometric Decompositions of Lagrange Elements
  8. Geometric decomposition
  9. Lagrange interpolation basis functions
  10. Geometric Decompositions of Face Elements
  11. Geometric decomposition
  12. A nodal basis for the BDM element
  13. Geometric Decompositions of Edge Elements
  14. Differential operator and its trace
  15. Polynomial bubble space
  16. ...and 11 more sections

Key Result

Theorem 3.1

For the polynomial space $\mathbb P_k(T)$ with $k\geq 1$ on an $n$-dimensional simplex $T$, we have the following decomposition The function $u\in \mathbb P_k(T)$ is uniquely determined by DoFs

Figures (6)

  • Figure 1: The interpolation points of $\mathbb T^2_4$ and their linear indices and multi-indexs.
  • Figure 2: The left figure shows $\{\boldsymbol e_0, \boldsymbol e_1\}$ at each interpolation point, the right figure shows $\{\hat{ \boldsymbol e}_0, \hat{ \boldsymbol e}_1\}$ at each interpolation point.
  • Figure 3: The left figure shows $\{\boldsymbol e_0, \boldsymbol e_1, \boldsymbol e_2\}$ at each interpolation point, the right figure shows $\{\hat{ \boldsymbol e}_0, \hat{ \boldsymbol e}_1, \hat{\boldsymbol e}_2\}$ at each interpolation point.
  • Figure 4: Local indexing (Left) and global indexing(Right) of DoFs for a face of tetrahedron, where the local vertex order is and global vertex order is . Due to the different ordering of vertices in local and global representation of the face, the ordering of the local indexing and the global indexing is different.
  • Figure 5: Errors $\|\boldsymbol u - \boldsymbol u_h\|_0$ and $\|p - p_h\|_0$ of finite element method \ref{['exm2eq1']} and \ref{['exm2eq2']} on uniformly refined mesh with $k = 2, 3, 4$.
  • ...and 1 more figures

Theorems & Definitions (30)

  • Theorem 3.1: Geometric decomposition of Lagrange element
  • Lemma 3.2: Lagrange interpolation basis functions nicolaides1972class
  • proof
  • Theorem 3.3: DoFs of Lagrange finite element on $\mathcal{T}_h$
  • proof
  • Corollary 3.4
  • proof
  • Lemma 4.1
  • Theorem 4.2
  • proof
  • ...and 20 more