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.
@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},
volume = {45},
year = {2011},
pages = {115-143},
doi = {10.1051/m2an/2010034},
zbl = {1270.65066},
language = {en},
url = {http://dml.mathdoc.fr/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, Tome 45 (2011) pp. 115-143. doi : 10.1051/m2an/2010034. http://gdmltest.u-ga.fr/item/M2AN_2011__45_1_115_0/
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