Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dimensions of exactly divergence-free finite element spaces in 3D

L. Ridgway Scott, Tabea Tscherpel

TL;DR

This work develops a mesh-quantity counting framework to compare the dimensions of 3D inf-sup stable mixed finite element spaces with exact divergence constraints on tetrahedral meshes, across the original mesh and split variants (Alfeld and Worsey–Farin). By expressing mesh quantities in terms of the vertex count $V$, average edge count per vertex $\bar{e}$, and topological constants, it derives asymptotic DOF formulas for velocity and pressure spaces and assesses their relative sizes under refinement. The key finding is that, in 3D, most low-order split-methods have larger DOFs than the Scott–Vogelius pair on the original mesh, except for the very first-order Worsey–Farin case, while higher-order Scott–Vogelius with $k=4$ often offers the best efficiency among exact-divergence constructions. The study also provides insights into discrete Stokes complexes and precursor spaces, suggesting directions for further theoretical and computational development of exact-divergence schemes on split meshes and their potential advantages in handling singularities and pressure robustness.

Abstract

We examine the dimensions of various inf-sup stable mixed finite element spaces on tetrahedral meshes in 3D with exact divergence constraints. More precisely, we compare the standard Scott-Vogelius elements of higher polynomial degree and low order methods on split meshes, the Alfeld and the Worsey-Farin split. The main tool is a counting strategy to express the degrees of freedom for given polynomial degree and given split in terms of few mesh quantities, for which bounds and asymptotic behavior under mesh refinement is investigated. Furthermore, this is used to obtain insights on potential precursor spaces in full de Rham complexes for finite element methods on the Worsey-Farin split.

Dimensions of exactly divergence-free finite element spaces in 3D

TL;DR

This work develops a mesh-quantity counting framework to compare the dimensions of 3D inf-sup stable mixed finite element spaces with exact divergence constraints on tetrahedral meshes, across the original mesh and split variants (Alfeld and Worsey–Farin). By expressing mesh quantities in terms of the vertex count , average edge count per vertex , and topological constants, it derives asymptotic DOF formulas for velocity and pressure spaces and assesses their relative sizes under refinement. The key finding is that, in 3D, most low-order split-methods have larger DOFs than the Scott–Vogelius pair on the original mesh, except for the very first-order Worsey–Farin case, while higher-order Scott–Vogelius with often offers the best efficiency among exact-divergence constructions. The study also provides insights into discrete Stokes complexes and precursor spaces, suggesting directions for further theoretical and computational development of exact-divergence schemes on split meshes and their potential advantages in handling singularities and pressure robustness.

Abstract

We examine the dimensions of various inf-sup stable mixed finite element spaces on tetrahedral meshes in 3D with exact divergence constraints. More precisely, we compare the standard Scott-Vogelius elements of higher polynomial degree and low order methods on split meshes, the Alfeld and the Worsey-Farin split. The main tool is a counting strategy to express the degrees of freedom for given polynomial degree and given split in terms of few mesh quantities, for which bounds and asymptotic behavior under mesh refinement is investigated. Furthermore, this is used to obtain insights on potential precursor spaces in full de Rham complexes for finite element methods on the Worsey-Farin split.
Paper Structure (19 sections, 8 theorems, 76 equations, 5 figures, 3 tables)

This paper contains 19 sections, 8 theorems, 76 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Split meshes
  3. Splits in 2D
  4. Splits in 3D
  5. Mesh quantities in 3D
  6. Average number of edges
  7. Lower bounds
  8. Upper bounds
  9. Piecewise polynomials on split meshes
  10. Original mesh
  11. Alfeld split
  12. Worsey--Farin split
  13. Compare and contrast
  14. Alternative mixed methods
  15. Approximately divergence-free methods
  16. ...and 4 more sections

Key Result

Lemma 1

Let ${\mathcal{T}}$ be a conforming simplicial triangulation of a polyhedral bounded domain $\Omega \subset \mathbb{R}^2$ with $V$ vertices, $E$ edges, and $T$ triangles. Let ${\mathcal{T}}_{{A}}$ be the Alfeld split and ${\mathcal{T}}_{{P}}$ the Powell--Sabin split of ${\mathcal{T}}$. Then we have

Figures (5)

  • Figure 1: Alfeld split into $3$ new triangles (left) and Powell--Sabin split split into $6$ new triangles each (right) in 2D with the new edges (dashed) and new vertices in gray.
  • Figure 2: Alfeld split into $4$ new tetrahedra (left) and Worsey–Farin split into $3$ new tetrahedra each (right) in 3D with the new edges (dashed) and new vertices in gray. For simpler visualization the $4$ tetrahedra resulting from the Alfeld split are detached in the right-hand figure.
  • Figure 3: Regular triangulation of a cube and combination of those to form the triangulation of a rectangular domain.
  • Figure 4: Refinement of a single tetrahedron.
  • Figure 5: Initial meshes for experiment in FEniCS, generated with tikzplotlib.

Theorems & Definitions (27)

  • Lemma 1: mesh quantities for 2D split meshes
  • Lemma 2: mesh quantities for 3D split meshes
  • Remark 3: 2D case
  • Proposition 4
  • proof
  • Remark 5
  • Example 6: star meshes
  • Example 7: regular meshes
  • Remark 8: alternative quantities
  • Example 9
  • ...and 17 more