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Trimmed serendipity finite element differential forms

Authors: Andrew Gillette and Tyler Kloefkorn
Journal: Math. Comp. 88 (2019), 583-606
MSC (2010): Primary 65N30
Published electronically: May 18, 2018
MathSciNet review: 3882277
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Abstract: We introduce the family of trimmed serendipity finite element differential form spaces, defined on cubical meshes in any number of dimensions, for any polynomial degree, and for any form order. The relation between the trimmed serendipity family and the (non-trimmed) serendipity family developed by Arnold and Awanou [Math. Comp. 83 (2014), pp. 1551–1570] is analogous to the relation between the trimmed and (non-trimmed) polynomial finite element differential form families on simplicial meshes from finite element exterior calculus. We provide degrees of freedom in the general setting and prove that they are unisolvent for the trimmed serendipity spaces. The sequence of trimmed serendipity spaces with a fixed polynomial order $r$ provides an explicit example of a system described by Christiansen and Gillette [ESAIM:M2AN 50 (2016), pp. 883–850], namely, a minimal compatible finite element system on squares or cubes containing order $r-1$ polynomial differential forms.

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Additional Information

Andrew Gillette
Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
MR Author ID: 991138

Tyler Kloefkorn
Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
MR Author ID: 1086521

Keywords: Finite element differential forms, finite element exterior calculus, serendipity elements, cubical meshes, cubes
Received by editor(s): July 2, 2016
Received by editor(s) in revised form: March 24, 2017, October 26, 2017, October 28, 2017, and December 28, 2017
Published electronically: May 18, 2018
Additional Notes: Both authors were supported in part by NSF Award DMS-1522289.
Article copyright: © Copyright 2018 American Mathematical Society