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

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.