title: 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 it obtains a quadratic error convergence estimate. On a convexn-gon, our construction produces 2nbasis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set ofn(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. Uniforma priorierror 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.
authors.affiliation: Institute for Computational Engineering and Sciences, The University of Texas at Austin.
authors.forename: Alexander
authors.initials: A
authors.lastname: Rand
authors.affiliation: Department of Mathematics, The University of Texas at Austin.
authors.forename: Andrew
authors.initials: A
authors.lastname: Gillette
authors.affiliation: Department of Computer Science, The University of Texas at Austin.
authors.forename: Chandrajit
authors.initials: C
authors.lastname: Bajaj
keywords: barycentric coordinates; finite element; serendipity