 
Summary: Found Comput Math
DOI 10.1007/s1020801190873
The Serendipity Family of Finite Elements
Douglas N. Arnold · Gerard Awanou
Received: 23 February 2010 / Accepted: 19 November 2010
© SFoCM 2011
Abstract We give a new, simple, dimensionindependent definition of the serendip
ity 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.
Keywords Serendipity · Finite element · Unisolvence
Mathematics Subject Classification (2000) Primary 65N30
Communicated by Philippe Ciarlet.
The work of D.N. Arnold was supported in part by NSF grant DMS0713568.
The work of G. Awanou was supported in part by NSF grant DMS0811052 and the Sloan
Foundation.
D.N. Arnold ( )
