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Q H(div)

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Alternative namesRaviart–Thomas cubical H(div) (quadrilateral), Nédélec cubical H(div) (hexahedron)
Exterior calculus names\(\mathcal{Q}^-_{k}\Lambda^{d-1}(\square_d)\)
Cockburn–fu names\(\left[S_{4,k}^\square\right]_{d-1}\)
Abbreviated namesRTcf (quadrilateral), Ncf (hexahedron)
Orders\(1\leqslant k\)
Reference elementsquadrilateral, hexahedron
Polynomial set\(\mathcal{Q}_{k-1}^d \oplus \mathcal{Z}^{(2)}_{k}\) (quadrilateral)
\(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(2)}_{k}\) (hexahedron)
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DOFsOn each facet: normal integral moments with an order \(k-1\) Lagrange space
On the interior of the reference element: integral moments with an order \(k-1\) Nédélec (first kind) space
Number of DOFsquadrilateral: \(2k(k+1)\) (A046092)
hexahedron: \(3k^2(k+1)\) (A270205)
Mappingcontravariant Piola
continuityComponents normal to facets are continuous
CategoriesVector-valued elements, H(div) conforming elements

Implementations

Symfem"Qdiv"
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UFL"RTCF" (quadrilateral)
"NCF" (hexahedron)
↓ Show UFL examples ↓

Examples

quadrilateral
order 1

(click to view basis functions)
quadrilateral
order 2

(click to view basis functions)
hexahedron
order 1

(click to view basis functions)
hexahedron
order 2

(click to view basis functions)

References

DefElement stats

Element added31 December 2020
Element last updated02 August 2022