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Tiniest tensor H(curl)

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Alternative namesTNT H(curl)
Cockburn–fu names\(\left[S_{3,k}^\square\right]_{1}\)
Orders\(1\leqslant k\)
Reference elementsquadrilateral, hexahedron
Polynomial set\(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(44)}_{k}\) (quadrilateral)
\(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(45)}_{k} \oplus \mathcal{Z}^{(46)}_{k} \oplus \mathcal{Z}^{(47)}_{k}\) (hexahedron)
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DOFsOn each edge: tangent integral moments with an order \(k\) Lagrange space
On each face: integral moments with \(\nabla f\times \boldsymbol{n}\) (where \(\boldsymbol{n}\) is a unit vector normal to the face) for each \(f\) in an order \(k\) Lagrange space, and integral moments with \(\nabla\times\boldsymbol{f}\times \boldsymbol{n}\) (where \(\boldsymbol{n}\) is a unit vector normal to the face) for each \(\boldsymbol{f}\) in an order \(k\) vector Lagrange space such that \(\nabla\cdot\boldsymbol{f}=0\) and the normal trace of \(\boldsymbol{f}\) on the edges of the face is 0
On each volume: integral moments with \(\nabla\times\boldsymbol{f}\) for each \(\boldsymbol{f}\) in an order \(k\) vector Lagrange space such that the tangential trace of \(\boldsymbol{f}\) on the faces of the volume is 0, and integral moments with \(\nabla f\) for each \(f\) in an order \(k\) Lagrange space such that the trace of \(f\) on the faces of the volume is 0
Number of DOFsquadrilateral: \(2(k+1)^2 + 3\)
hexahedron: \(3(k+1)^3 + 18\)
Mappingcovariant Piola
continuityComponents tandential to facets are continuous
CategoriesVector-valued elements, H(curl) conforming elements

Implementations

Symfem"TNTcurl" (quadrilateral, hexahedron)
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Examples

quadrilateral
order 1

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quadrilateral
order 2

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quadrilateral
order 3

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hexahedron
order 1

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References

DefElement stats

Element added24 October 2021
Element last updated02 August 2022