Quadratic serendipity finite elements on polygons using generalized barycentric coordinates

Abstract:

We introduce a finite element construction for use on the class of convex, planar polygons and show that it obtains a quadratic error convergence estimate. On a convex $n$-gon, 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. This 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 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.