Quadratic serendipity finite elements on polygons using generalized barycentric coordinates

Alexander Rand, Andrew Gillette, Chandrajit Bajaj

Research output: Contribution to journalArticle

51 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)2691-2716
Number of pages26
JournalMathematics of Computation
Volume83
Issue number290
DOIs
StatePublished - Jan 1 2014

Keywords

  • Barycentric coordinates
  • Finite element
  • Serendipity

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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