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Quadratic serendipity finite elements on polygons using generalized barycentric coordinates


Authors: Alexander Rand, Andrew Gillette and Chandrajit Bajaj
Journal: Math. Comp. 83 (2014), 2691-2716
MSC (2010): Primary 65D05, 65N30, 41A30, 41A25
DOI: https://doi.org/10.1090/S0025-5718-2014-02807-X
Published electronically: February 20, 2014
MathSciNet review: 3246806
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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.


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Additional Information

Alexander Rand
Affiliation: CD-adapco, 10800 Pecan Park Blvd, Austin, Texas 78750
Email: alexprand@gmaiil.com

Andrew Gillette
Affiliation: The University of Arizona, 617 N. Santa Rita Avenue, Tucson, Arizona 85721
Email: agillette@math.arizona.edu

Chandrajit Bajaj
Affiliation: Department of Computer Science, The University of Texas at Austin, 2317 Speedway, Stop D9500, Austin, Texas 78712
Email: bajaj@cs.utexas.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02807-X
Keywords: Finite element, barycentric coordinates, serendipity
Received by editor(s): August 31, 2011
Received by editor(s) in revised form: July 19, 2012, and February 13, 2013
Published electronically: February 20, 2014
Additional Notes: This research was supported in part by NIH contracts R01-EB00487, R01-GM074258, and a grant from the UT-Portugal CoLab project.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.