Hexahedral \(\mathbf H(\operatorname{div})\) and \(\mathbf H(\operatorname{curl})\) finite elements (Q2836470)

The authors study the approximation properties of some finite element subspaces of \(\mathbf H(\operatorname{div};\Omega)\) and \(\mathbf H(\operatorname{curl};\Omega)\) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral \(\mathbf H(\operatorname{div};\Omega)\) finite elements and for quadrilateral scalar finite element spaces. The considered finite element spaces 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 ensure \(\mathcal O(h)\) approximation in \(L^2(\Omega)\) and in \(\mathbf H(\operatorname{div};\Omega)\) and \(\mathbf H(\operatorname{curl};\Omega)\) on the physical element, the authors study the properties of the resulting finite element spaces.