, Volume 11, Issue 3, pp 337-344
Date: 01 Mar 2011

The Serendipity Family of Finite Elements

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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 sr 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.

Communicated by Philippe Ciarlet.
The work of D.N. Arnold was supported in part by NSF grant DMS-0713568.
The work of G. Awanou was supported in part by NSF grant DMS-0811052 and the Sloan Foundation.