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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr


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Volume 16, Issue 4

Issues

Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes

Andrew Gillette
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  • Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, Tucson, AZ, United States of America
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/ Alexander Rand / Chandrajit Bajaj
  • Department of Computer Science, Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX, United States of America
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Published Online: 2016-05-18 | DOI: https://doi.org/10.1515/cmam-2016-0019

Abstract

We combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic convex polytope meshes in dimensions 2 and 3. Our construction recovers well-known bases for the lowest order Nédélec, Raviart–Thomas, and Brezzi–Douglas–Marini elements on simplicial meshes and generalizes the notion of Whitney forms to non-simplicial convex polygons and polyhedra. We show that our basis functions lie in the correct function space with regards to global continuity and that they reproduce the requisite polynomial differential forms described by finite element exterior calculus. We present a method to count the number of basis functions required to ensure these two key properties.

Keywords: Generalized Barycentric Coordinates; Polygonal Finite Element Methods; Finite Element Exterior Calculus

MSC 2010: 65N30; 41A30; 65D05; 41A25

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About the article

Received: 2016-02-20

Revised: 2016-04-23

Accepted: 2016-04-25

Published Online: 2016-05-18

Published in Print: 2016-10-01


Funding Source: NSF Office of the Director

Award identifier / Grant number: 1522289

Funding Source: NIH Office of the Director

Award identifier / Grant number: R01-GM117594

Funding Source: Sandia National Laboratories

Award identifier / Grant number: BD-4485

AG was supported in part by NSF Award 1522289 and a J. Tinsley Oden Fellowship. CB was supported in part by a grant from NIH (R01-GM117594) and contract (BD-4485) from Sandia National Labs.


Citation Information: Computational Methods in Applied Mathematics, Volume 16, Issue 4, Pages 667–683, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2016-0019.

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