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$H(\textrm{div})$-conforming Finite Element Tensors with Constraints

Long Chen, Xuehai Huang

TL;DR

The article develops a unified construction of $H(\operatorname{div})$-conforming finite element tensors, including vector, symmetric, and traceless types, by integrating a geometric decomposition of Lagrange elements with tangent-normal decompositions on sub-simplices. It introduces a constraint tensor space $\mathbb X = \ker(s^{k,n-1})$ within a differential-form framework, provides intrinsic bases and projection formulae, and builds explicit $H(\operatorname{div})$-conforming spaces with robust DoFs and discrete inf-sup stability. The framework encompasses classical elements (BDM, Stenberg, CHH) as special cases and supports arbitrary dimensions, aided by face redistribution to achieve stable trace and divergence properties. This yields a systematic, basis-explicit method for stable, constrained tensor discretizations applicable to elasticity, relativity, and related PDEs, with clear pathways for extending to various symmetry and constraint patterns. Overall, the work offers a cohesive, dimension-agnostic toolkit for designing and analyzing $H(\operatorname{div})$-conforming tensor finite elements with intrinsic bases and provable stability.

Abstract

A unified construction of $H(\textrm{div})$-conforming finite element tensors, including vector element, symmetric matrix element, traceless matrix element, and, in general, tensors with linear constraints, is developed in this work. It is based on the geometric decomposition of Lagrange elements into bubble functions on each sub-simplex. Each tensor at a sub-simplex is decomposed into tangential and normal components. The tangential component forms the bubble function space, while the normal component characterizes the trace. Some degrees of freedom can be redistributed to $(n-1)$-dimensional faces. The developed finite element spaces are $H(\textrm{div})$-conforming and satisfy the discrete inf-sup condition. Intrinsic bases of the constraint tensor space are also established.

$H(\textrm{div})$-conforming Finite Element Tensors with Constraints

TL;DR

The article develops a unified construction of -conforming finite element tensors, including vector, symmetric, and traceless types, by integrating a geometric decomposition of Lagrange elements with tangent-normal decompositions on sub-simplices. It introduces a constraint tensor space within a differential-form framework, provides intrinsic bases and projection formulae, and builds explicit -conforming spaces with robust DoFs and discrete inf-sup stability. The framework encompasses classical elements (BDM, Stenberg, CHH) as special cases and supports arbitrary dimensions, aided by face redistribution to achieve stable trace and divergence properties. This yields a systematic, basis-explicit method for stable, constrained tensor discretizations applicable to elasticity, relativity, and related PDEs, with clear pathways for extending to various symmetry and constraint patterns. Overall, the work offers a cohesive, dimension-agnostic toolkit for designing and analyzing -conforming tensor finite elements with intrinsic bases and provable stability.

Abstract

A unified construction of -conforming finite element tensors, including vector element, symmetric matrix element, traceless matrix element, and, in general, tensors with linear constraints, is developed in this work. It is based on the geometric decomposition of Lagrange elements into bubble functions on each sub-simplex. Each tensor at a sub-simplex is decomposed into tangential and normal components. The tangential component forms the bubble function space, while the normal component characterizes the trace. Some degrees of freedom can be redistributed to -dimensional faces. The developed finite element spaces are -conforming and satisfy the discrete inf-sup condition. Intrinsic bases of the constraint tensor space are also established.
Paper Structure (36 sections, 38 theorems, 247 equations, 9 figures)

This paper contains 36 sections, 38 theorems, 247 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Simplex and sub-simplices
  4. $t$-$n$ bases
  5. Barycentric coordinates and Bernstein polynomials
  6. Geometric decomposition of Lagrange elements
  7. Geometric Decompositions of Vector Face Elements
  8. Examples in three dimensions
  9. Face elements in arbitrary dimension
  10. Bubble spaces
  11. Brezzi-Douglas-Marini element
  12. Stenberg's type element
  13. Discrete inf-sup condition
  14. Geometric Decompositions of Matrix Face Elements
  15. Traceless matrix elements
  16. ...and 21 more sections

Key Result

Lemma 2.1

For $f\in \Delta_{\ell}(T)$, the rescaled tangential normal basis $\{\boldsymbol n_{f\cup\{i\}}^f/(\boldsymbol n_{f\cup\{i\}}^f\cdot\boldsymbol n_{F_i}), i\in f^*\}$ of $\mathscr N^f$ is dual to the face normal basis $\{\boldsymbol n_{F_i}, i\in f^*\}$.

Figures (9)

  • Figure 1: Face normal basis and tangential normal basis of a vertex and an edge in a tetrahedron.
  • Figure 2: Classical face elements can be obtained by a special $t$-$n$ decomposition and face-wise redistribution of normal components.
  • Figure 3: Different $H(\operatorname{div})$-conforming finite elements can be obtained by different $t$-$n$ decompositions.
  • Figure 4: Several $t$-$n$ decompositions for $\mathbb S$ in $\mathbb R^3$. Blocks with the same symbol (circle, square, or diamond) are in the same constraint sequence and the white block is used as the pair index. Color of the block represents: Green: free rows and free blocks; Blue: all free indices not in free rows; Red: bubble functions. Blue or green blocks are free indices in $\mathscr N^f(\mathbb S)$. All white blocks are pair indices and the corresponding coefficients are determined by the free variables through the constraints.
  • Figure 5: Sub-simplices and their tangential and normal vectors.
  • ...and 4 more figures

Theorems & Definitions (84)

  • Lemma 2.1
  • proof
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5: Geometric decomposition of Lagrange element, (2.6) in ArnoldFalkWinther2009
  • proof
  • Remark 2.6
  • Example 3.1: Nédélec element/BDM element
  • ...and 74 more