Quadratic serendipity finite elements on polygons using generalized barycentric coordinates

Alexander Rand, Andrew Gillette, Chandrajit Bajaj

Research output: Contribution to journalArticle

43 Citations (Scopus)

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
StatePublished - 2014

Fingerprint

Generalized Polygon
Barycentric Coordinates
Finite Element
Polygon
Basis Functions
Adaptive Meshing
Interior angle
Quadrilateral Mesh
n-gon
Convergence Estimates
A Priori Error Estimates
Quadratic Convergence
Midpoint
Aspect ratio
Regularity Conditions
Lagrange
Aspect Ratio
Error Estimates
Mesh
Vertex of a graph

Keywords

  • Barycentric coordinates
  • Finite element
  • Serendipity

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics
  • Computational Mathematics

Cite this

Quadratic serendipity finite elements on polygons using generalized barycentric coordinates. / Rand, Alexander; Gillette, Andrew; Bajaj, Chandrajit.

In: Mathematics of Computation, Vol. 83, No. 290, 2014, p. 2691-2716.

Research output: Contribution to journalArticle

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