Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

IMPACT FACTOR 2017: 0.658

CiteScore 2017: 1.05

SCImago Journal Rank (SJR) 2017: 1.291
Source Normalized Impact per Paper (SNIP) 2017: 0.893

Mathematical Citation Quotient (MCQ) 2017: 0.76

See all formats and pricing
More options …
Volume 16, Issue 4


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

Andrew Gillette
  • Corresponding author
  • Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, Tucson, AZ, United States of America
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ 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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-05-18 | DOI: https://doi.org/10.1515/cmam-2016-0019


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


  • [1]

    Abraham R., Marsden J. E. and Ratiu T., Manifolds, Tensor Analysis, and Applications, 2nd ed. Appl. Math. Sci. 75, Springer, New York, 1988. Google Scholar

  • [2]

    Alnæs M., Blechta J., Hake J., Johansson A., Kehlet B., Logg A., Richardson C., Ring J., Rognes M. E. and Wells G. N., The FEniCS Project version 1.5, Arch. Numer. Softw. 3 (2015), Paper No. 100. Google Scholar

  • [3]

    Arnold D., Falk R. and Winther R., Finite element exterior calculus, homological techniques, and applications, Act. Numer. 15 (2006), 1–155. Google Scholar

  • [4]

    Arnold D., Falk R. and Winther R., Geometric decompositions and local bases for spaces of finite element differential forms, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 21–26, 1660–1672. Google Scholar

  • [5]

    Arnold D., Falk R. and Winther R., Finite element exterior calculus: From Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2, 281–354. Google Scholar

  • [6]

    Beirão da Veiga L., Brezzi F., Cangiani A., Manzini G., Marini L. D. and Russo A., Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013), no. 1, 199–214. Google Scholar

  • [7]

    Bossavit A., Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism, IEE Proc. A 135 (1988), no. 8, 493–500. Google Scholar

  • [8]

    Bossavit A., A uniform rationale for Whitney forms on various supporting shapes, Math. Comput. Simulation 80 (2010), no. 8, 1567–1577. Google Scholar

  • [9]

    Brezzi F., Douglas, Jr. J., Durán R. and Fortin M., Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, 237–250. Google Scholar

  • [10]

    Brezzi F., Douglas, Jr. J. and Marini L. D., Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. Google Scholar

  • [11]

    Chen W. and Wang Y., Minimal degree H(curl) and H(div) conforming finite elements on polytopal meshes, preprint 2015, http://arxiv.org/abs/1502.01553.

  • [12]

    Christiansen S. H., A construction of spaces of compatible differential forms on cellular complexes, Math. Models Methods Appl. Sci. 18 (2008), no. 5, 739–757. Google Scholar

  • [13]

    Christiansen S. H. and Winther R., Smoothed projections into finite element exterior calculus, Math. Comp. 77 (2008), no. 262, 813–829. Google Scholar

  • [14]

    Clément P., Approximation by finite element functions using local regularization, Rev. Franc. Automat. Inform. Rech. Operat. 9 (1975), no. R-2, 77–84. Google Scholar

  • [15]

    Ern A. and Guermond J.-L., Theory and Practice of Finite Elements, Appl. Math. Sci. 159, Springer, New York, 2004. Google Scholar

  • [16]

    Euler T., Schuhmann R. and Weiland T., Polygonal finite elements, IEEE Trans. Magn. 42 (2006), no. 4, 675–678. Google Scholar

  • [17]

    Farin G., Surfaces over Dirichlet tessellations, Comput. Aided Geom. Design 7 (1990), no. 1–4, 281–292. Google Scholar

  • [18]

    Floater M., Mean value coordinates, Comput. Aided Geom. Design 20 (2003), no. 1, 19–27. Google Scholar

  • [19]

    Floater M., Gillette A. and Sukumar N., Gradient bounds for Wachspress coordinates on polytopes, SIAM J. Numer. Anal. 52 (2014), no. 1, 515–532. Google Scholar

  • [20]

    Floater M., Hormann K. and Kós G., A general construction of barycentric coordinates over convex polygons, Adv. Comput. Math. 24 (2006), no. 1, 311–331. Google Scholar

  • [21]

    Floater M., Kós G. and Reimers M., Mean value coordinates in 3D, Comput. Aided Geom. Design 22 (2005), no. 7, 623–631. Google Scholar

  • [22]

    Gillette A. and Bajaj C., A generalization for stable mixed finite elements, Proceedings of the 14th ACM Symposium on Solid and Physical Modeling (SPM ’10), ACM, New York (2010), 41–50. Google Scholar

  • [23]

    Gillette A. and Bajaj C., Dual formulations of mixed finite element methods with applications, Comput. Aided Des. 43 (2011), no. 10, 1213–1221. Google Scholar

  • [24]

    Gillette A., Rand A. and Bajaj C., Error estimates for generalized barycentric coordinates, Adv. Comput. Math. 37 (2012), no. 3, 417–439. Google Scholar

  • [25]

    Gradinaru V., Whitney elements on sparse grids, Ph.D. thesis, Universität Tübingen, Tübingen, 2002. Google Scholar

  • [26]

    Gradinaru V. and Hiptmair R., Whitney elements on pyramids, Electron. Trans. Numer. Anal. 8 (1999), 154–168. Google Scholar

  • [27]

    Hirani A. N., Discrete exterior calculus, Dissertation, California Institute of Technology, Pasedena, 2003. Google Scholar

  • [28]

    Hormann K. and Sukumar N., Maximum entropy coordinates for arbitrary polytopes, Comp. Graph. Forum 27 (2008), no. 5, 1513–1520. Google Scholar

  • [29]

    Joshi P., Meyer M., DeRose T., Green B. and Sanocki T., Harmonic coordinates for character articulation, ACM Trans. Graph. 26 (2007), Article ID 71. CrossrefGoogle Scholar

  • [30]

    Ju T., Schaefer S., Warren J. D. and Desbrun M., A geometric construction of coordinates for convex polyhedra using polar duals, Proceedings of the Third Eurographics Symposium on Geometry Processing (SGP ’05), Eurographics Association, Aire-la-Ville (2015), 181–186. Google Scholar

  • [31]

    Klausen R., Rasmussen A. and Stephansen A., Velocity interpolation and streamline tracing on irregular geometries, Comput. Geosci. 16 (2011), no. 2, 1–16. Google Scholar

  • [32]

    Lipnikov K., Manzini G. and Shashkov M., Mimetic finite difference method, J. Comput. Phys. 257B (2014), 1163–1227. Google Scholar

  • [33]

    Manson J. and Schaefer S., Moving least squares coordinates, Comp. Graph. Forum 29 (2010), no. 5, 1517–1524. Google Scholar

  • [34]

    Manzini G., Russo A. and Sukumar N., New perspectives on polygonal and polyhedral finite element methods, Math. Models Methods Appl. Sci. 24 (2014), no. 8, 1665–1699. Google Scholar

  • [35]

    Martin S., Kaufmann P., Botsch M., Wicke M. and Gross M., Polyhedral finite elements using harmonic basis functions, Comp. Graph. Forum 27 (2008), no. 5, 1521–1529. Google Scholar

  • [36]

    Milbradt P. and Pick T., Polytope finite elements, Internat. J. Numer. Methods Engrg. 73 (2008), no. 12, 1811–1835. Google Scholar

  • [37]

    Nédélec J.-C., Mixed finite elements in 3, Numer. Math. 35 (1980), no. 3, 315–341. Google Scholar

  • [38]

    Nédélec J.-C., A new family of mixed finite elements in 3, Numer. Math. 50 (1986), no. 1, 57–81. Google Scholar

  • [39]

    Rand A., Average interpolation under the maximum angle condition, SIAM J. Numer. Anal. 50 (2012), no. 5, 2538–2559. Google Scholar

  • [40]

    Rand A., Gillette A. and Bajaj C., Interpolation error estimates for mean value coordinates, Adv. Comput. Math. 39 (2013), 327–347. Google Scholar

  • [41]

    Rand A., Gillette A. and Bajaj C., Quadratic serendipity finite elements on polygons using generalized barycentric coordinates, Math. Comp. 83 (2014), no. 290, 2691–2716. Google Scholar

  • [42]

    Rashid M. and Selimotic M., A three-dimensional finite element method with arbitrary polyhedral elements, Internat. J. Numer. Methods Engrg. 67 (2006), no. 2, 226–252. Google Scholar

  • [43]

    Raviart P.-A. and Thomas J. M., A mixed finite element method for 2nd order elliptic problems, Mathematical Aspects of Finite Element Methods (Rome 1975), Lecture Notes in Math. 606, Springer, Berlin (1977), 292–315. Google Scholar

  • [44]

    Scott L. and Zhang S., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. Google Scholar

  • [45]

    Sibson R., A vector identity for the Dirichlet tessellation, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 1, 151–155. Google Scholar

  • [46]

    Sukumar N., Construction of polygonal interpolants: A maximum entropy approach, Internat. J. Numer. Methods Engrg. 61 (2004), no. 12, 2159–2181. Google Scholar

  • [47]

    Sukumar N. and Malsch E. A., Recent advances in the construction of polygonal finite element interpolants, Arch. Comput. Methods Eng. 13 (2006), no. 1, 129–163. Google Scholar

  • [48]

    Sukumar N. and Tabarraei A., Conforming polygonal finite elements, Internat. J. Numer. Methods Engrg. 61 (2004), no. 12, 2045–2066. Google Scholar

  • [49]

    Wachspress E., A Rational Finite Element Basis, Math. Sci. Eng. 114, Academic Press, New York, 1975. Google Scholar

  • [50]

    Wachspress E., Barycentric coordinates for polytopes, Comput. Math. Appl. 61 (2011), no. 11, 3319–3321. Google Scholar

  • [51]

    Warren J., Barycentric coordinates for convex polytopes, Adv. Comput. Math. 6 (1996), no. 1, 97–108. Google Scholar

  • [52]

    Warren J., Schaefer S., Hirani A. N. and Desbrun M., Barycentric coordinates for convex sets, Adv. Comput. Math. 27 (2007), no. 3, 319–338. Google Scholar

  • [53]

    Whitney H., Geometric Integration Theory, Princeton University Press, Princeton, 1957. Google Scholar

  • [54]

    Wicke M., Botsch M. and Gross M., A finite element method on convex polyhedra, Comp. Graph. Forum 26 (2007), no. 3, 355–364. Google Scholar

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.

Export Citation

© 2016 by De Gruyter.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

B. Emek Abali and Felix A. Reich
Continuum Mechanics and Thermodynamics, 2018
Jiangguo Liu, Simon Tavener, and Zhuoran Wang
SIAM Journal on Scientific Computing, 2018, Volume 40, Number 5, Page B1229
Akash Anand, Jeffrey S. Ovall, and Steffen Weißer
Computers & Mathematics with Applications, 2018
Ali Vaziri Astaneh, Federico Fuentes, Jaime Mora, and Leszek Demkowicz
Computer Methods in Applied Mechanics and Engineering, 2017
B. Emek Abali and Felix A. Reich
Computer Methods in Applied Mechanics and Engineering, 2017, Volume 319, Page 567
Max Budninskiy, Beibei Liu, Yiying Tong, and Mathieu Desbrun
ACM Transactions on Graphics, 2016, Volume 35, Number 6, Page 1

Comments (0)

Please log in or register to comment.
Log in