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Hexahedral $$\mathbf H(\operatorname{div})$$ and $$\mathbf H(\operatorname{curl})$$ finite elements. (English) Zbl 1270.65066
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.

MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
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