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

TL;DR

This paper analyzes the dimensions of divergence-free finite element spaces in 3D, comparing different mesh splits and polynomial degrees to understand their degrees of freedom and potential for de Rham complexes.

Contribution

It introduces a counting strategy to determine degrees of freedom for various finite element spaces on tetrahedral meshes with divergence constraints, providing new insights into their structure.

Findings

Bounds on degrees of freedom for different splits and polynomial degrees

Asymptotic behavior of degrees of freedom under mesh refinement

Insights into potential precursor spaces in de Rham complexes

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.