Numerische Mathematik

, Volume 129, Issue 1, pp 1–20 | Cite as

Finite element differential forms on curvilinear cubic meshes and their approximation properties

  • Douglas N. Arnold
  • Daniele Boffi
  • Francesca Bonizzoni


We study the approximation properties of a wide class of finite element differential forms on curvilinear cubic meshes in \(n\) dimensions. Specifically, we consider meshes in which each element is the image of a cubical reference element under a diffeomorphism, and finite element spaces in which the shape functions and degrees of freedom are obtained from the reference element by pullback of differential forms. In the case where the diffeomorphisms from the reference element are all affine, i.e., mesh consists of parallelotopes, it is standard that the rate of convergence in \(L^2\) exceeds by one the degree of the largest full polynomial space contained in the reference space of shape functions. When the diffeomorphism is multilinear, the rate of convergence for the same space of reference shape function may degrade severely, the more so when the form degree is larger. The main result of the paper gives a sufficient condition on the reference shape functions to obtain a given rate of convergence.

Mathematics Subject Classification (2010)

65N30 41A25 41A10 41A15 41A63 


  1. 1.
    Arnold, D.N.: Differential complexes and numerical stability. In: Proceedings of the International Congress of Mathematicians, vol. I (Beijing, 2002), pp. 137–157. Higher Education Press, Beijing (2002)Google Scholar
  2. 2.
    Arnold, D.N.: Spaces of finite element differential forms. In: Gianazza, U., Brezzi, F., Colli Franzone, P., Gilardi G. (eds.) Analysis and Numerics of Partial Differential Equations, pp. 117–140. Springer, Berlin (2013)Google Scholar
  3. 3.
    Arnold, D.N., Awanou, G.: The serendipity family of finite elements. Found. Comput. Math. 11, 337–344 (2011)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Arnold, D.N., Awanou, G.: Finite element differential forms on cubical meshes. Math. Comput. 83, 1551–1570 (2014). doi:10.1090/S0025-5718-2013-02783-4
  5. 5.
    Arnold, D.N., Boffi, D., Falk, R.S.: Approximation by quadrilateral finite elements. Math. Comput. 71(239), 909–922 (2002, electronic). doi:10.1090/S0025-5718-02-01439-4
  6. 6.
    Arnold, D.N., Boffi, D., Falk, R.S.: Quadrilateral \(H\)(div) finite elements. SIAM J. Numer. Anal. 42(6), 2429–2451 (2005, electronic). doi:10.1137/S0036142903431924
  7. 7.
    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006). doi:10.1017/S0962492906210018
  8. 8.
    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. (N.S.) 47(2), 281–354 (2010). doi:10.1090/S0273-0979-10-01278-4
  9. 9.
    Bergot, M., Duruflé, M.: Approximation of \(H(div)\) with high-order optimal finite elements for pyramids, prisms and hexahedra. Commun. Comput. Phys. 14(5), 1372–1414 (2013)MathSciNetGoogle Scholar
  10. 10.
    Bergot, M., Duruflé, M.: High-order optimal edge elements for pyramids, prisms and hexahedra. J. Comput. Phys. 232, 189–213 (2013). doi:10.1016/
  11. 11.
    Christiansen, S.H.: Foundations of finite element methods for wave equations of Maxwell type. In: Quak, E., Soomere, T. (eds.) Applied Wave Mathematics, pp. 335–393. Springer, Berlin (2009). doi:10.1007/978-3-642-00585-5_17
  12. 12.
    Christiansen, S.H., Munthe-Kaas, H.Z., Owren, B.: Topics in structure-preserving discretization. Acta Numer. 20, 1–119 (2011). doi:10.1017/S096249291100002X
  13. 13.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, vol. 4. North-Holland Publishing Co., Amsterdam (1978)Google Scholar
  14. 14.
    Clément, P.: Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, RAIRO Analyse Numérique 9(2), 77–84 (1975)MATHGoogle Scholar
  15. 15.
    Durán, R.G.: On polynomial approximation in Sobolev spaces. SIAM J. Numer. Anal. 20(5), 985–988 (1983). doi:10.1137/0720068
  16. 16.
    Falk, R.S., Gatto, P., Monk, P.: Hexahedral \(H\)(div) and \(H\)(curl) finite elements. ESAIM Math. Model. Numer. Anal. 45, 115–143 (2011). doi: 10.1051/m2an/2010034 CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Hilton, P., Wylie, S.: Homology Theory: An Introduction to Algebraic Topology. Cambridge University Press, Cambridge (1967)MATHGoogle Scholar
  18. 18.
    Matthies, G.: Mapped finite elements on hexahedra. Necessary and sufficient conditions for optimal interpolation errors. Numer. Algorithms 27(4), 317–327 (2001). doi:10.1023/A:1013860707381
  19. 19.
    Naff, R.L., Russell, T.F., Wilson, J.D.: Shape functions for velocity interpolation in general hexahedral cells. Comput. Geosci. 6, 285–314 (2002)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Nédélec, J.C.: Mixed finite elements in \({\bf R}^{3}\). Numer. Math. 35, 315–341 (1980)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Verfürth, R.: A note on polynomial approximation in Sobolev spaces. M2AN Math. Model. Numer. Anal. 33(4), 715–719 (1999). doi:10.1051/m2an:1999159

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Douglas N. Arnold
    • 1
  • Daniele Boffi
    • 2
  • Francesca Bonizzoni
    • 3
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Dipartimento di Matematica “F. Casorati”Università di PaviaPaviaItaly
  3. 3.Faculty of MathematicsUniversity of ViennaWienAustria

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