an encyclopedia of finite element definitions

# Raviart–Thomas

 Alternative names Rao–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 names RT, RWG Variants Legendre: Integral moments are taken against orthonormal polynomialsLagrange: Integral moments are taken against (Lagrange)[element:lagrange] basis functions Orders $$1\leqslant k$$ Reference elements triangle, tetrahedron Polynomial set $$\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(26)}_{k}$$↓ Show polynomial set definitions ↓ DOFs On 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 DOFs triangle: $$k(k+2)$$ (A005563)tetrahedron: $$k(k+1)(k+3)/2$$ (A077414) Mapping contravariant Piola continuity Components normal to facets are continuous Categories Vector-valued elements, H(div) conforming elements

## Implementations

 Basix basix.ElementFamily.RT↓ Show Basix examples ↓ This implementation is correct for all the examples below. Basix.UFL basix.ElementFamily.RT↓ Show Basix.UFL examples ↓ This implementation is correct for all the examples below. Bempp ↓ Show Bempp examples ↓ FIAT FIAT.RaviartThomas(..., variant="integral")↓ Show FIAT examples ↓ This implementation is correct for all the examples below that it supports.↓ Show more ↓ 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

 triangleorder 1Lagrange variant (click to view basis functions) triangleorder 2Lagrange variant (click to view basis functions) tetrahedronorder 1Lagrange variant (click to view basis functions) tetrahedronorder 2Lagrange variant (click to view basis functions) triangleorder 1Legendre variant (click to view basis functions) triangleorder 2Legendre variant (click to view basis functions) tetrahedronorder 1Legendre variant (click to view basis functions) tetrahedronorder 2Legendre variant (click to view basis functions)
• Raviart, Pierre-Arnaud and Thomas, Jean-Marie. A mixed finite element method for 2nd order elliptic problems, in Mathematical aspects of finite element methods (eds: Galligani, Ilio and Magenes, Enrico), 1977. [BibTeX]
• Nédélec, Jean-Claude. Mixed finite elements in $$\mathbb{R}^3$$, Numerische Mathematik 35(3), 315–341, 1980. [DOI: 10.1007/BF01396415] [BibTeX]
• Rao, S. S. M., Wilton, Donald R., and Glisson, Allen W. Electromagnetic scattering by surfaces of arbitrary shape, IEEE transactions on antennas and propagation 30, 409–418, 1982. [DOI: 10.1109/TAP.1982.1142818] [BibTeX]
• Arnold, Douglas N. and Logg, Anders. Periodic table of the finite elements, SIAM News 47, 2014. [sinews.siam.org/Details-Page/periodic-table-of-the-finite-elements] [BibTeX]
• Cockburn, Bernardo and Fu, Guosheng. A systematic construction of finite element commuting exact sequences, SIAM journal on numerical analysis 55, 1650–1688, 2017. [DOI: 10.1137/16M1073352] [BibTeX]

## DefElement stats

 Element added 30 December 2020 Element last updated 16 September 2023