Exact smooth piecewise polynomial sequences on Alfeld splits
Guosheng Fu, Johnny Guzmán, Michael Neilan
Abstract
We construct local 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -simplex with its barycenter. We show that, on these splits, the kernel of the exterior derivative has enhanced smoothness. Byproducts of this result include characterizations of discrete divergence-free subspaces and simple formulas for the dimensions of smooth polynomial spaces. In addition, we construct analogous global exact sequences and commuting projections in three-dimensions with varying levels of smoothness.