Numerische Mathematik

, Volume 142, Issue 1, pp 1–32 | Cite as

Construction of \(H({\mathrm{div}})\)-conforming mixed finite elements on cuboidal hexahedra

  • Todd ArbogastEmail author
  • Zhen Tao


We generalize the two dimensional mixed finite elements of Arbogast and Correa (SIAM J Numer Anal 54:3332–3356, 2016) defined on quadrilaterals to three dimensional cuboidal hexahedra. The construction is similar in that polynomials are used directly on the element and supplemented with functions defined on a reference element and mapped to the hexahedron using the Piola transform. The main contribution is providing a systematic procedure for defining supplemental functions that are divergence-free and have any prescribed polynomial normal flux. General procedures are also presented for determining which supplemental normal fluxes are required to define the finite element space. Both full and reduced \(H({\mathrm{div}})\)-approximation spaces may be defined, so the scalar variable, vector variable, and vector divergence are approximated optimally. The spaces can be constructed to be of minimal local dimension, if desired.

Mathematics Subject Classification

65N30 65N12 41A10 

Supplementary material


  1. 1.
    Arbogast, T., Correa, M.R.: Two families of H(div) mixed finite elements on quadrilaterals of minimal dimension. SIAM J. Numer. Anal. 54(6), 3332–3356 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arnold, D.N., Awanou, G.: Finite element differential forms on cubical meshes. Math. Comp. 83, 1551–1570 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arnold, D.N., Boffi, D., Bonizzoni, F.: Finite element differential forms on curvilinear cubic meshes and their approximation properties. Numer. Math. 129(1), 1–20 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arnold, D.N., Boffi, D., Falk, R.S.: Quadrilateral H(div) finite elements. SIAM. J. Numer. Anal. 42(6), 2429–2451 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19, 7–32 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bergot, M., Duruflé, M.: Approximation of H(div) with high-order optimal finite elements for pyramids, prisms and hexahedra. Communications in Computational Physics 14(5), 1372–1414 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brezzi, F., Douglas Jr., J., Duràn, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51, 237–250 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brezzi, F., Douglas Jr., J., Marini, L.D.: Two families of mixed elements for second order elliptic problems. Numer. Math. 47, 217–235 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer-Verlag, New York (1991)CrossRefzbMATHGoogle Scholar
  10. 10.
    Cockburn, B., Fu, G.: Superconvergence by M-decompositions. Part III: Construction of three-dimensional finite elements. ESAIM: Mathematical Modelling and Numerical Analysis 51(1), 365–398 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Huo-Yuan Duan, H.-Y., Liang, G.-P.: Nonconforming elements in least-squares mixed finite element methods Math. Comp. 73(245), 1–18 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Falk, R.S., Gatto, P., Monk, P.: Hexahedral H(div) and H(curl) finite elements. ESAIM Math. Model. Numer. Anal. 45(1), 115–143 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin (1986)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kwak, D.Y., Pyo, H.C.: Mixed finite element methods for general quadrilateral grids. Applied Mathematics and Computation 217, 6556–6565 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ponting, D.K.: Corner Point Geometry in Reservoir Simulation. In: P.R. King (eds.) Proc. of the 1st European Conf. on the Mathematics of Oil Recovery, pp. 45–65, Cambridge (1989)Google Scholar
  16. 16.
    Raviart, R.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: I. Galligani, E. Magenes (eds.) Mathematical Aspects of Finite Element Methods, no. 606 in Lecture Notes in Math., pp. 292–315. Springer-Verlag, New York (1977)Google Scholar
  17. 17.
    Shen, J.: Mixed finite element methods on distorted rectangular grids. Tech. Rep. ISC-94-13-MATH, Institute for Scientific Computation, Texas A&M University, College Station, Texas (1994)Google Scholar
  18. 18.
    Silvester, J.R.: Determinants of block matrices. The Mathematical Gazette, 84 (2000)Google Scholar
  19. 19.
    Thomas, J.M.: Sur l’analyse numerique des methodes d’elements finis hybrides et mixtes. Ph.D. thesis, Sciences Mathematiques, à l’Universite Pierre et Marie Curie (1977)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics C1200University of TexasAustinUSA
  2. 2.Institute for Computational Engineering and Sciences C0200University of TexasAustinUSA

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