With cedram.org version française Home Presentation Advanced Search All online articles Latest articles Search for an article Table of contents for this volume | Next article Andrew Gillette; Tyler Kloefkorn; Victoria SandersComputational Serendipity and Tensor Product Finite Element Differential FormsSMAI-Journal of computational mathematics, 5 (2019), p. 1-21, doi: 10.5802/smai-jcm.41 Article PDF Class. Math.: 65N30Keywords: Finite element differential forms, finite element exterior calculus, serendipity elements, cubical meshes, cubes AbstractMany conforming finite elements on squares and cubes are elegantly classified into families by the language of finite element exterior calculus and presented in the Periodic Table of the Finite Elements. Use of these elements varies, based principally on the ease or difficulty in finding a “computational basis” of shape functions for element families. The tensor product family, $\mathcal{Q}^-_r\Lambda ^k$, is most commonly used because computational basis functions are easy to state and implement. The trimmed and non-trimmed serendipity families, $\mathcal{S}^-_r\Lambda ^k$ and $\mathcal{S}_r\Lambda ^k$ respectively, are used less frequently because they are newer to the community and, until now, lacked a straightforward technique for computational basis construction. This represents a missed opportunity for computational efficiency as the serendipity elements in general have fewer degrees of freedom than elements of equivalent accuracy from the tensor product family. Accordingly, in pursuit of easy adoption of the serendipity families, we present complete lists of computational bases for both serendipity families, for any order $r\ge 1$ and for any differential form order $0\le k\le n$, for problems in dimension $n=2$ or $3$. 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Zaglmayr, High Order Finite Element Methods for Electromagnetic Field Computation, Ph. D. Thesis, Johannes Kepler Universität, 2006 © 2015SMAI Journal of Computational Mathematics ISSN 2426-8399 Papers are published under the licence Creative Commons CC BY-NC-ND 3.0