Exact smooth piecewise polynomial sequences on Alfeld splits

Fu, Guosheng; Guzmán, Johnny; Neilan, Michael

doi:10.1090/mcom/3520

Exact smooth piecewise polynomial sequences on Alfeld splits
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Math. Comp. 89 (2020), 1059-1091 Request permission

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 $n$-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.

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