Trimmed Serendipity Finite Element Differential Forms
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(288) 2014] 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(3) 2016], namely, a minimal compatible finite element system on squares or cubes containing order r-1 polynomial differential forms.
Cite
@article{arxiv.1607.00571,
title = {Trimmed Serendipity Finite Element Differential Forms},
author = {Andrew Gillette and Tyler Kloefkorn},
journal= {arXiv preprint arXiv:1607.00571},
year = {2018}
}
Comments
Improved results, detailed comparison to prior and contemporary work, and further explanation of computational benefits have been added since the original version. This version has been accepted for publication in Mathematics of Computation