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. ArnoldEmail author
  • 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 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Douglas N. Arnold
    • 1
    Email author
  • 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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