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Raviart–Thomas

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Alternative namesRao–Wilton–Glisson, Nédélec (first kind) H(div), Raviart–Thomas cubical H(div) (quadrilateral), Nédélec cubical H(div) (hexahedron), Q H(div) (quadrilateral, hexahedron)
De Rham complex families\(\left[S_{4,k}^\square\right]_{d-1}\) or \(\mathcal{Q}^-_{k}\Lambda^{d-1}(\square_d)\)
Abbreviated namesRT, RWG, RTcf (quadrilateral), Ncf (hexahedron)
VariantsLegendre: Integral moments are taken against orthonormal polynomials
Lagrange: Integral moments are taken against Lagrange basis functions
Degrees\(1\leqslant k\)
where \(k\) is the Lagrange superdegree
Polynomial subdegree\(k-1\)
Polynomial superdegreetriangle: \(k\)
tetrahedron: \(k\)
quadrilateral: \(dk - d + 1\)
hexahedron: \(dk - d + 1\)
Lagrange subdegree\(k-1\)
Lagrange superdegree\(k\)
Reference elementstriangle, tetrahedron, quadrilateral, hexahedron
Polynomial set\(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(24)}_{k}\) (triangle, tetrahedron)
\(\mathcal{Q}_{k-1}^d \oplus \mathcal{Z}^{(25)}_{k}\) (quadrilateral)
\(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(25)}_{k}\) (hexahedron)
↓ Show polynomial set definitions ↓
DOFsOn each facet: normal integral moments with an degree \(k-1\) Lagrange space
On the interior of the reference element (triangle): integral moments with an degree \(k-2\) vector Lagrange space
On the interior of the reference element (tetrahedron): integral moments with an degree \(k-2\) vector Lagrange space
On the interior of the reference element (quadrilateral): integral moments with an degree \(k-1\) Nédélec (first kind) space
On the interior of the reference element (hexahedron): integral moments with an degree \(k-1\) Nédélec (first kind) space
Number of DOFstriangle: \(k(k+2)\) (A005563)
tetrahedron: \(k(k+1)(k+3)/2\) (A077414)
quadrilateral: \(2k(k+1)\) (A046092)
hexahedron: \(3k^2(k+1)\) (A270205)
Mappingcontravariant Piola
continuityComponents normal to facets are continuous
CategoriesVector-valued elements, H(div) conforming elements

Implementations

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

Examples

triangle
degree 1
Lagrange variant

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

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

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

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

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

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

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

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

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

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

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

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

(click to view basis functions)
tetrahedron
degree 2
Legendre variant

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

(click to view basis functions)
hexahedron
degree 2
Legendre variant

(click to view basis functions)

References

DefElement stats

Element added30 December 2020
Element last updated27 November 2024