We introduce a finite element construction for use on the class of convex,planar polygons and show it obtains a quadratic error convergence estimate. Ona convex n-gon satisfying simple geometric criteria, our construction produces2n basis functions, associated in a Lagrange-like fashion to each vertex andeach edge midpoint, by transforming and combining a set of n(n+1)/2 basisfunctions known to obtain quadratic convergence. The technique broadens thescope of the so-called `serendipity' elements, previously studied only forquadrilateral and regular hexahedral meshes, by employing the theory ofgeneralized barycentric coordinates. Uniform `a priori' error estimates areestablished over the class of convex quadrilaterals with bounded aspect ratioas well as over the class of generic convex planar polygons satisfyingadditional shape regularity conditions to exclude large interior angles andshort edges. Numerical evidence is provided on a trapezoidal quadrilateralmesh, previously not amenable to serendipity constructions, and applications toadaptive meshing are discussed.
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