Geodesic
Hexahedral 𝐇(div) and 𝐇(curl) finite elements
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 1, pp. 115-143

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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 : 10.1051/m2an/2010034
Classification : 65N30
Keywords: hexahedral finite element 
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     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 },
     pages = {115--143},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {1},
     year = {2011},
     doi = {10.1051/m2an/2010034},
     zbl = {1270.65066},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2010034/}
}
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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 , Tome 45 (2011) no. 1, pp. 115-143. doi: 10.1051/m2an/2010034

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