 
Summary: THE SERENDIPITY FAMILY OF FINITE ELEMENTS
DOUGLAS N. ARNOLD AND GERARD AWANOU
Abstract. We give a new, simple, dimensionindependent definition of the seren
dipity 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 r2d on each face of dimension d. We establish
unisolvence and a geometric decomposition of the space.
1. Introduction
The serendipity family of finite element spaces are among the most popular fi
nite element spaces for parallelepiped meshes in two, and, to a lesser extent, three
dimensions. For each such mesh and each degree r 1 they provide a finite ele
ment subspace with C0
continuity which has significantly smaller dimension than the
more obvious alternative, the tensor product Lagrange element family. However, the
serendipity elements are rarely studied systematically, particularly in 3D. Usually
only the lowest degree examples are discussed, with the pattern for higher degrees
not evident. In this paper, we give a simple, but apparently new, definition of the
serendipity elements, by specifying in a dimensionindependent fashion the space of
