Low-order divergence-free approximations for the Stokes problem on Worsey–Farin and Powell–Sabin splits

Abstract

We derive low-order, inf–sup stable and divergence-free finite element approximations for the Stokes problem using Worsey–Farin splits in three dimensions and Powell–Sabin splits in two dimensions. The velocity space simply consists of continuous, piecewise linear polynomials, whereas the pressure space is a subspace of piecewise constants with weak continuity properties at singular edges (3D) and singular vertices (2D). We discuss implementation aspects that arise when coding the pressure space, and in particular, show that the pressure constraints can be enforced at an algebraic level.