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Nédélec (first kind)

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Alternative namesWhitney (triangle,tetrahedron), Nédélec, Q H(curl) (quadrilateral,hexahedron), Raviart–Thomas cubical H(curl) (quadrilateral), Nédélec cubical H(curl) (hexahedron)
Exterior calculus names\(\mathcal{P}^-_{k}\Lambda^{1}(\Delta_d)\), \(\mathcal{Q}^-_{k}\Lambda^{1}(\square_d)\)
Cockburn–fu names\(\left[S_{2,k}^\unicode{0x25FA}\right]_{1}\), \(\left[S_{4,k}^\square\right]_{1}\)
Abbreviated namesN1curl, NC, RTce (quadrilateral), Nce (hexahedron)
Orders\(1\leqslant k\)
Reference elementstriangle, tetrahedron, quadrilateral, hexahedron, prism
Polynomial set\(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(35)}_{k}\) (triangle, tetrahedron)
\(\mathcal{Q}_{k-1}^d \oplus \mathcal{Z}^{(36)}_{k}\) (quadrilateral, hexahedron)
↓ Show polynomial set definitions ↓
DOFsOn each edge: tangent integral moments with an order \(k-1\) Lagrange space
On each face (triangle): integral moments with an order \(k-2\) vector Lagrange space
On each face (quadrilateral): integral moments with an order \(k-1\) Q H(div) space
On each volume (tetrahedron): integral moments with an order \(k-3\) vector Lagrange space
On each volume (hexahedron): integral moments with an order \(k-1\) Q H(div) space
Number of DOFstriangle: \(k(k+2)\) (A005563)
tetrahedron: \(k(k+2)(k+3)/2\) (A005564)
quadrilateral: \(2k(k+1)\) (A046092)
hexahedron: \(3k(k+1)^2\) (A059986)
prism: \(3k(k+2)(k+1)/2\)
Mappingcovariant Piola
continuityComponents tandential to facets are continuous
CategoriesVector-valued elements, H(curl) conforming elements

Implementations

Basixbasix.ElementFamily.N1E (triangle, tetrahedron, quadrilateral, hexahedron)
↓ Show Basix examples ↓
Bempp"SNC" (triangle)
↓ Show Bempp examples ↓
Symfem"N1curl" (triangle, tetrahedron)
"Qcurl" (quadrilateral, hexahedron)
"Ncurl" (prism)
↓ Show Symfem examples ↓
UFL"N1curl" (triangle, tetrahedron)
"RTCE" (quadrilateral)
"NCE" (hexahedron)
↓ Show UFL examples ↓

Examples

triangle
order 1

(click to view basis functions)
triangle
order 2

(click to view basis functions)
quadrilateral
order 1

(click to view basis functions)
quadrilateral
order 2

(click to view basis functions)
tetrahedron
order 1

(click to view basis functions)
tetrahedron
order 2

(click to view basis functions)
hexahedron
order 1

(click to view basis functions)
hexahedron
order 2

(click to view basis functions)
prism
order 1

(click to view basis functions)
prism
order 2

(click to view basis functions)

References

DefElement stats

Element added30 December 2020
Element last updated02 August 2022