Hexahedral 𝐇(div) and 𝐇(curl) finite elements
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 1, p. 115-143

We study the approximation properties of some finite element subspaces of H(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral H(div;Ω) finite elements and for quadrilateral scalar finite element spaces. The finite element spaces we consider are constructed starting from a given finite dimensional space of vector fields on the reference cube, which is then transformed to a space of vector fields on a hexahedron using the appropriate transform (e.g., the Piola transform) associated to a trilinear isomorphism of the cube onto the hexahedron. After determining what vector fields are needed on the reference element to insure O(h) approximation in L2(Ω) and in H(div;Ω) and H(curl;Ω) on the physical element, we study the properties of the resulting finite element spaces.

DOI : https://doi.org/10.1051/m2an/2010034
Classification:  65N30
Keywords: hexahedral finite element
@article{M2AN_2011__45_1_115_0,
     author = {Falk, Richard S. and Gatto, Paolo and Monk, Peter},
     title = {Hexahedral $\mathbf {H}(\operatorname{div})$ and $\mathbf {H}(\operatorname{curl})$ finite elements},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {1},
     year = {2011},
     pages = {115-143},
     doi = {10.1051/m2an/2010034},
     zbl = {1270.65066},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2011__45_1_115_0}
}
Falk, Richard S.; Gatto, Paolo; Monk, Peter. Hexahedral $\mathbf {H}(\operatorname{div})$ and $\mathbf {H}(\operatorname{curl})$ finite elements. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 1, pp. 115-143. doi : 10.1051/m2an/2010034. http://www.numdam.org/item/M2AN_2011__45_1_115_0/

[1] D.N. Arnold, D. Boffi, R.S. Falk and L. Gastaldi, Finite element approximation on quadrilateral meshes. Comm. Num. Meth. Eng. 17 (2001) 805-812. | MR 1872639 | Zbl 0999.76073

[2] D.N. Arnold, D. Boffi and R.S. Falk, Approximation by quadrilateral finite elements. Math. Comp. 71 (2002) 909-922. | MR 1898739 | Zbl 0993.65125

[3] D.N. Arnold, D. Boffi and R.S. Falk, Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42 (2005) 2429-2451. | MR 2139400 | Zbl 1086.65105

[4] I. Babuška and A.K. Aziz, Survey lectures on the mathematical foundations of the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Proc. Sympos., Univ. Maryland, Baltimore, Md. Academic Press, New York (1972) 1-359. | MR 421106 | Zbl 0268.65052

[5] A. Bermúdez, P. Gamallo, M.R. Nogeiras and R. Rodríguez, Approximation properties of lowest-order hexahedral Raviart-Thomas elements. C. R. Acad. Sci. Paris, Sér. I 340 (2005) 687-692. | MR 2139278 | Zbl 1071.65148

[6] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer, New York (1994). | MR 1278258 | Zbl 1135.65042

[7] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer, New York (1991). | MR 1115205 | Zbl 0788.73002

[8] F. Dubois, Discrete vector potential representation of a divergence free vector field in three dimensional domains: Numerical analysis of a model problem. SINUM 27 (1990) 1103-1142. | MR 1061122 | Zbl 0717.65086

[9] T. Dupont and R. Scott, Polynomial Approximation of Functions in Sobolev Spaces. Math. Comp. 34 (1980) 441-463. | MR 559195 | Zbl 0423.65009

[10] P. Gatto, Elementi finiti su mesh di esaedri distorti per l'approssimazione di H(div) [Approximation of H(div) via finite elements over meshes of distorted hexahedra]. Master's Thesis, Dipartimento di Matematica, Università Pavia, Italy (2006).

[11] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics 5. Springer-Verlag, Berlin (1986). | MR 851383 | Zbl 0585.65077

[12] R.L. Naff, T.F. Russell and J.D. Wilson, Shape Functions for Velocity Interpolation in General Hexahedral Cells. Comput. Geosci. 6 (2002) 285-314. | MR 1956019 | Zbl 1094.76542

[13] T.F. Russell, C.I. Heberton, L.F. Konikow and G.Z. Hornberger, A finite-volume ELLAM for three-dimensional solute-transport modeling. Ground Water 41 (2003) 258-272.

[14] P. Šolín, K. Segeth and I. Doležel, Higher Order Finite Elements Methods, Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton (2004). | Zbl 1032.65132

[15] S. Zhang, On the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation. Numer. Math. 98 (2004) 559-579. | MR 2088927 | Zbl 1065.65135

[16] S. Zhang, Invertible Jacobian for hexahedral finite elements. Part 1. Bijectivity. Preprint available at http://www.math.udel.edu/ szhang/research/p/bijective1.ps (2005).

[17] S. Zhang, Invertible Jacobian for hexahedral finite elements. Part 2. Global positivity. Preprint available at http://www.math.udel.edu/ szhang/research/p/bijective2.ps (2005).

[18] S. Zhang, Subtetrahedral test for the positive Jacobian of hexahedral elements. Preprint available at http://www.math.udel.edu/ szhang/research/p/subtettest.pdf (2005).