Exact smooth piecewise polynomial sequences on Alfeld splits
Provides foundational theoretical tools for finite element methods, benefiting researchers in numerical analysis and computational PDEs.
The paper develops exact piecewise polynomial sequences on Alfeld splits in any spatial dimension and any polynomial degree, showing enhanced smoothness in the kernel of the exterior derivative. This yields characterizations of divergence-free subspaces for Stokes problems, commutative projections, and dimension formulas for smooth polynomial spaces.
We develop exact piecewise polynomial sequences on Alfeld splits in any spatial dimension and any polynomial degree. An Alfeld split of a simplex is obtained by connecting the vertices of an $n$-simplex with its barycenter. We show that, on these triangulations, the kernel of the exterior derivative has enhanced smoothness. Byproducts of this theory include characterizations of discrete divergence-free subspaces for the Stokes problem, commutative projections, and simple formulas for the dimensions of smooth polynomial spaces.