Quadratic Serendipity Finite Elements on Polygons Using Generalized Barycentric Coordinates

TL;DR

This paper presents a new finite element method for convex polygons using generalized barycentric coordinates, achieving quadratic convergence and extending serendipity elements beyond quadrilaterals.

Contribution

It introduces a novel construction of quadratic serendipity finite elements on convex polygons, broadening their applicability with theoretical error estimates and numerical validation.

Findings

Achieves quadratic error convergence on convex polygons.

Extends serendipity elements to general convex polygons.

Provides numerical evidence on trapezoidal meshes.

Abstract

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.