The Serendipity Family of Finite Elements

Douglas N. Arnold, Gerard Awanou

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We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s-r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r. The degrees of freedom are given by moments of degree at most r-2d on each face of dimension d. We establish unisolvence and a geometric decomposition of the space.

Original languageEnglish (US)
Pages (from-to)337-344
Number of pages8
JournalFoundations of Computational Mathematics
Issue number3
StatePublished - Jun 2011

Bibliographical note

Funding Information:
The work of G. Awanou was supported in part by NSF grant DMS-0811052 and the Sloan Foundation.

Funding Information:
The work of D.N. Arnold was supported in part by NSF grant DMS-0713568.


  • Finite element
  • Serendipity
  • Unisolvence


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