  While the use of finite element methods for the numerical approximation of
eigenvalues is a well-studied problem, the use of serendipity elements for this
purpose has received little attention in the literature. We show by numerical
experiments that serendipity elements, which are defined on a square reference
geometry, can attain the same order of accuracy as their tensor product
counterparts while using dramatically fewer degrees of freedom. In some cases,
the serendipity method uses only half as many basis functions as the tensor
product method while still producing the same numerical approximation of an
eigenvalue. To encourage the further use and study of serendipity elements, we
provide a table of serendipity basis functions for low order cases and a
Mathematica file that can be used to generate the basis functions for higher
order cases.
