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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\(\left[S_{2,k}^\unicode{0x25FA}\right]_{1}\) or \(\mathcal{P}^-_{k}\Lambda^{1}(\Delta_d)\), \(\left[S_{4,k}^\square\right]_{1}\) or \(\mathcal{Q}^-_{k}\Lambda^{1}(\square_d)\)
Abbreviated namesN1curl, NC, RTce (quadrilateral), Nce (hexahedron)
VariantsLegendre: Integral moments are taken against orthonormal polynomials
Lagrange: Integral moments are taken against Lagrange basis functions
Degrees\(0\leqslant k\)
where \(k\) is the Polynomial subdegree
Polynomial subdegree\(k\)
Polynomial superdegreetriangle: \(k+1\)
tetrahedron: \(k+1\)
prism: \(2k+2\)
quadrilateral: \(dk+d-1\)
hexahedron: \(dk+d-1\)
Lagrange subdegree\(k\)
Lagrange superdegree\(k+1\)
Reference elementstriangle, tetrahedron, quadrilateral, hexahedron, prism
Polynomial set\(\mathcal{P}_{k}^d \oplus \mathcal{Z}^{(19)}_{k+1}\) (triangle, tetrahedron)
\(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(20)}_{k+1}\) (quadrilateral, hexahedron)
↓ Show polynomial set definitions ↓
DOFsOn each edge: tangent integral moments with a degree \(k\) Lagrange space
On each face (triangle): integral moments with a degree \(k-1\) vector Lagrange space
On each face (quadrilateral): integral moments with a degree \(k-1\) Raviart–Thomas space
On each volume (tetrahedron): integral moments with a degree \(k-2\) vector Lagrange space
On each volume (hexahedron): integral moments with a degree \(k-1\) Raviart–Thomas space
Number of DOFstriangle: \((k+1)(k+3)\) (A005563)
tetrahedron: \((k+1)(k+3)(k+4)/2\) (A005564)
quadrilateral: \(2(k+1)(k+2)\) (A046092)
hexahedron: \(3(k+1)(k+2)^2\) (A059986)
prism: \(3(k+1)(k+2)(k+3)/2\)
Mappingcovariant Piola
continuityComponents tangential to facets are continuous
CategoriesVector-valued elements, H(curl) conforming elements

Implementations

This element is implemented in Basix , Basix.UFL , Bempp-cl, FIAT , NDElement , Symfem , and (legacy) UFL.↓ Show implementation detail ↓

Examples

triangle
degree 0
Lagrange variant

(click to view basis functions)
triangle
degree 1
Lagrange variant

(click to view basis functions)
quadrilateral
degree 0
Lagrange variant

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

(click to view basis functions)
tetrahedron
degree 0
Lagrange variant

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

(click to view basis functions)
hexahedron
degree 0
Lagrange variant

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

(click to view basis functions)
prism
degree 0
Lagrange variant

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

(click to view basis functions)
triangle
degree 0
Legendre variant

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

(click to view basis functions)
quadrilateral
degree 0
Legendre variant

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

(click to view basis functions)

References

DefElement stats

Element added24 March 2021
Element last updated31 March 2025