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Finite Element Complexes in Two Dimensions

Long Chen, Xuehai Huang

TL;DR

The paper presents a systematic, two-dimensional framework to construct finite element complexes with multiple smoothness levels, unifying de Rham, curl–div, elasticity, and div–div sequences via a geometric, lattice-based element design and the Bernstein basis. It leverages a non-overlapping decomposition of the simplicial lattice and the Bernstein–Gelfand–Gelfand (BGG) construction to derive discrete elasticity and div–div complexes from discretized de Rham complexes, with rigorous DoF definitions, dimension counts, and unisolvence arguments. The approach yields conforming finite element spaces that form exact sequences, including discrete Stokes, curl–div, elasticity, and div–div complexes, and provides explicit DoF prescriptions and bubble-space analyses. While focused on 2D, the framework identifies key challenges for extending to 3D (notably continuity mismatches and edge-type elements) and outlines directions for systematic BGG-based FE complexes in higher dimensions. Overall, the work offers a principled, modular pathway to design stable, exact FE complexes in two dimensions and sets the stage for broader applicability and 3D generalizations.

Abstract

In this study, two-dimensional finite element complexes with various levels of smoothness, including the de Rham complex, the curldiv complex, the elasticity complex, and the divdiv complex, are systematically constructed. Smooth scalar finite elements in two dimensions are developed based on a non-overlapping decomposition of the simplicial lattice and the Bernstein basis of the polynomial space, with the order of differentiability at vertices being greater than twice that at edges. Finite element de Rham complexes with different levels of smoothness are devised using smooth finite elements with smoothness parameters that satisfy certain relations. Finally, finite element elasticity complexes and finite element divdiv complexes are derived from finite element de Rham complexes by using the Bernstein-Gelfand-Gelfand (BGG) framework. This study is the first work to construct finite element complexes in a systematic way. Moreover, the novel tools developed in this work, such as the non-overlapping decomposition of the simplicial lattice and the discrete BGG construction, can be useful for further research in this field.

Finite Element Complexes in Two Dimensions

TL;DR

The paper presents a systematic, two-dimensional framework to construct finite element complexes with multiple smoothness levels, unifying de Rham, curl–div, elasticity, and div–div sequences via a geometric, lattice-based element design and the Bernstein basis. It leverages a non-overlapping decomposition of the simplicial lattice and the Bernstein–Gelfand–Gelfand (BGG) construction to derive discrete elasticity and div–div complexes from discretized de Rham complexes, with rigorous DoF definitions, dimension counts, and unisolvence arguments. The approach yields conforming finite element spaces that form exact sequences, including discrete Stokes, curl–div, elasticity, and div–div complexes, and provides explicit DoF prescriptions and bubble-space analyses. While focused on 2D, the framework identifies key challenges for extending to 3D (notably continuity mismatches and edge-type elements) and outlines directions for systematic BGG-based FE complexes in higher dimensions. Overall, the work offers a principled, modular pathway to design stable, exact FE complexes in two dimensions and sets the stage for broader applicability and 3D generalizations.

Abstract

In this study, two-dimensional finite element complexes with various levels of smoothness, including the de Rham complex, the curldiv complex, the elasticity complex, and the divdiv complex, are systematically constructed. Smooth scalar finite elements in two dimensions are developed based on a non-overlapping decomposition of the simplicial lattice and the Bernstein basis of the polynomial space, with the order of differentiability at vertices being greater than twice that at edges. Finite element de Rham complexes with different levels of smoothness are devised using smooth finite elements with smoothness parameters that satisfy certain relations. Finally, finite element elasticity complexes and finite element divdiv complexes are derived from finite element de Rham complexes by using the Bernstein-Gelfand-Gelfand (BGG) framework. This study is the first work to construct finite element complexes in a systematic way. Moreover, the novel tools developed in this work, such as the non-overlapping decomposition of the simplicial lattice and the discrete BGG construction, can be useful for further research in this field.
Paper Structure (24 sections, 24 theorems, 150 equations, 3 figures, 2 tables)

This paper contains 24 sections, 24 theorems, 150 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on Hilbert complexes
  3. Notation
  4. Hilbert complex and exact sequence
  5. The de Rham complex
  6. Bernstein-Gelfand-Gelfand construction
  7. Smooth Finite Elements in Two Dimensions
  8. Simplicial lattice
  9. Bernstein basis
  10. Sub-simplicial lattices and distance
  11. Derivative and distance
  12. Derivatives at vertices
  13. Normal derivatives on edges
  14. Geometric decompositions of the simplicial lattice
  15. Smooth finite elements in two dimensions
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let be a complex, where $\mathcal{V}_i$ are finite-dimensional linear spaces for $i=0,\ldots,3$. Assume $\mathcal{V}_0=\mathcal{V}_1\cap\ker(\,{\rm d}_1)$, and If either $\,{\rm d}_1\mathcal{V}_1 = \mathcal{V}_2\cap \ker(\,{\rm d}_2)$ or $\,{\rm d}_2 \mathcal{V}_2 = \mathcal{V}_3$, then complex eq:shortexactsequence is exact.

Figures (3)

  • Figure 1: Two embedding of the simplicial lattice $\mathbb T_8^2$ in two dimensions.
  • Figure 2: Comparison of the geometric decompositions of a two-dimensional Hermite element and a $C^1$-conforming element.
  • Figure 3: Explanation of the smooth finite element de Rham complex with increased smoothness in pressure.

Theorems & Definitions (54)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 44 more