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Exact smooth piecewise polynomial sequences on Alfeld splits


Authors: Guosheng Fu, Johnny Guzmán and Michael Neilan
Journal: Math. Comp. 89 (2020), 1059-1091
MSC (2010): Primary 65N30
DOI: https://doi.org/10.1090/mcom/3520
Published electronically: January 13, 2020
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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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Additional Information

Guosheng Fu
Affiliation: Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, Indiana 46556
Email: gfu@nd.edu

Johnny Guzmán
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: johnny_guzman@brown.edu

Michael Neilan
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: neilan@pitt.edu

DOI: https://doi.org/10.1090/mcom/3520
Received by editor(s): May 31, 2019
Received by editor(s) in revised form: December 10, 2019
Published electronically: January 13, 2020
Article copyright: © Copyright 2020 American Mathematical Society