Trimmed serendipity finite element differential forms

Andrew Gillette, Tyler Kloefkorn

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)583-606
Number of pages24
JournalMathematics of Computation
Issue number316
StatePublished - 2019


  • Cubes
  • Cubical meshes
  • Finite element differential forms
  • Finite element exterior calculus
  • Serendipity elements

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics


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