Finite Element Discretizations for Incompressible Flow on Split Meshes

We construct families of divergence-free finite element methods for the Stokes/Navier-Stokes problem. Borrowing tools from the theory of multi-variate splines, we define these spaces on splits of a simplicial triangulation. We show that the divergence-free Stokes finite element pairs are connected to smooth piecewise polynomial spaces via a smooth discrete de Rham complex. Byproducts of this construction include characterizations of discrete divergence-free subspaces for the Stokes problem, commutative projections, and simple formulas for the dimensions of smooth polynomial spaces.