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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)
De Rham complex families\(\left[S_{2,k}^\unicode{0x25FA}\right]_{1}\) / \(\mathcal{P}^-_{k}\Lambda^{1}(\Delta_d)\), \(\left[S_{4,k}^\square\right]_{1}\) / \(\mathcal{Q}^-_{k}\Lambda^{1}(\square_d)\)
Abbreviated namesN1curl, NC, RTce (quadrilateral), Nce (hexahedron)
VariantsLegendre: Integral moments are taken against orthonormal polynomials
Lagrange: Integral moments are taken against (Lagrange)[element:lagrange] basis functions
Orders\(1\leqslant k\)
Reference elementstriangle, tetrahedron, quadrilateral, hexahedron, prism
Polynomial set\(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(20)}_{k}\) (triangle, tetrahedron)
\(\mathcal{Q}_{k-1}^d \oplus \mathcal{Z}^{(21)}_{k}\) (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 tangential to facets are continuous
CategoriesVector-valued elements, H(curl) conforming elements

Implementations

Basixbasix.ElementFamily.N1E
↓ Show Basix examples ↓ This implementation is correct for all the examples below that it supports.
Basix.UFLbasix.ElementFamily.N1E
↓ Show Basix.UFL examples ↓ This implementation is correct for all the examples below that it supports.
Bempp"SNC"
↓ Show Bempp examples ↓
FIATFIAT.Nedelec(..., variant="integral")
↓ Show FIAT examples ↓ This implementation is correct for all the examples below that it supports.
Symfem"N1curl", variant="legendre" (triangle, Legendre; tetrahedron, Legendre)
"Qcurl", variant="legendre" (quadrilateral, Legendre; hexahedron, Legendre)
"Ncurl", variant="legendre" (prism, Legendre)
"N1curl" (triangle, Lagrange; tetrahedron, Lagrange)
"Qcurl" (quadrilateral, Lagrange; hexahedron, Lagrange)
"Ncurl" (prism, Lagrange)
↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL"N1curl" (triangle, Lagrange; tetrahedron, Lagrange)
"RTCE" (quadrilateral, Lagrange)
"NCE" (hexahedron, Lagrange)
↓ Show (legacy) UFL examples ↓

Examples

triangle
order 1
Lagrange variant

(click to view basis functions)
triangle
order 2
Lagrange variant

(click to view basis functions)
quadrilateral
order 1
Lagrange variant

(click to view basis functions)
quadrilateral
order 2
Lagrange variant

(click to view basis functions)
tetrahedron
order 1
Lagrange variant

(click to view basis functions)
tetrahedron
order 2
Lagrange variant

(click to view basis functions)
hexahedron
order 1
Lagrange variant

(click to view basis functions)
hexahedron
order 2
Lagrange variant

(click to view basis functions)
prism
order 1
Lagrange variant

(click to view basis functions)
prism
order 2
Lagrange variant

(click to view basis functions)
triangle
order 1
Legendre variant

(click to view basis functions)
triangle
order 2
Legendre variant

(click to view basis functions)
quadrilateral
order 1
Legendre variant

(click to view basis functions)
quadrilateral
order 2
Legendre variant

(click to view basis functions)

References

DefElement stats

Element added31 December 2020
Element last updated16 September 2023