Dimensions of Exactly Divergence-Free Finite Element Spaces in 3D

SIAM Journal on Scientific Computing, Volume 46, Issue 2, Page A1102-A1131, April 2024.
Abstract. We examine the dimensions of various inf-sup stable mixed finite element spaces on tetrahedral meshes in three dimensions 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 only a 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 the Stokes complex for finite element methods on the Worsey–Farin split. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://git-ce.rwth-aachen.de/pub/meshquantities/ and in the supplementary materials (meshquantities-main.zip [3.13KB]).