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Raviart–Thomas

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Alternative namesRao–Wilton–Glisson, Nédélec (first kind) H(div)
De Rham complex families\(\left[S_{2,k}^\unicode{0x25FA}\right]_{d-1}\) / \(\mathcal{P}^-_{k}\Lambda^{d-1}(\Delta_d)\)
Abbreviated namesRT, RWG
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
Polynomial set\(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(26)}_{k}\)
↓ Show polynomial set definitions ↓
DOFsOn each facet: normal integral moments with an order \(k-1\) Lagrange space
On the interior of the reference element: integral moments with an order \(k-2\) vector Lagrange space
Number of DOFstriangle: \(k(k+2)\) (A005563)
tetrahedron: \(k(k+1)(k+3)/2\) (A077414)
Mappingcontravariant Piola
continuityComponents normal to facets are continuous
CategoriesVector-valued elements, H(div) conforming elements

Implementations

Basixbasix.ElementFamily.RT
↓ Show Basix examples ↓ This implementation is correct for all the examples below.
Basix.UFLbasix.ElementFamily.RT
↓ Show Basix.UFL examples ↓ This implementation is correct for all the examples below.
Bempp
↓ Show Bempp examples ↓
FIATFIAT.RaviartThomas(..., variant="integral")
↓ Show FIAT examples ↓ This implementation is correct for all the examples below that it supports.
Symfem"N1div", variant="legendre" (Legendre)
"N1div" (Lagrange)
↓ Show Symfem examples ↓ This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL"RT"
↓ 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)
tetrahedron
order 1
Lagrange variant

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

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

(click to view basis functions)

References

DefElement stats

Element added30 December 2020
Element last updated16 September 2023