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.