  We introduce a finite element construction for use on the class of convex,
planar polygons and show it obtains a quadratic error convergence estimate. On
a convex n-gon satisfying simple geometric criteria, our construction produces
2n basis functions, associated in a Lagrange-like fashion to each vertex and
each edge midpoint, by transforming and combining a set of n(n+1)/2 basis
functions known to obtain quadratic convergence. The technique broadens the
scope of the so-called `serendipity' elements, previously studied only for
quadrilateral and regular hexahedral meshes, by employing the theory of
generalized barycentric coordinates. Uniform `a priori' error estimates are
established over the class of convex quadrilaterals with bounded aspect ratio
as well as over the class of generic convex planar polygons satisfying
additional shape regularity conditions to exclude large interior angles and
short edges. Numerical evidence is provided on a trapezoidal quadrilateral
mesh, previously not amenable to serendipity constructions, and applications to
adaptive meshing are discussed.
