an encyclopedia of finite element definitions

# Raviart–Thomas

 Alternative names Rao–Wilton–Glisson, Nédélec (first kind) H(div) Exterior calculus names $$\mathcal{P}^-_{k}\Lambda^{d-1}(\Delta_d)$$ Cockburn–fu names $$\left[S_{2,k}^\unicode{0x25FA}\right]_{d-1}$$ Abbreviated names RT, RWG Orders $$1\leqslant k$$ Reference elements triangle, tetrahedron Polynomial set $$\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(15)}_{k}$$↓ 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 ↓ Bempp "RWG" (triangle)↓ Show Bempp examples ↓ Symfem "N1div"↓ Show Symfem examples ↓ UFL "RT"↓ Show UFL examples ↓

## Examples

 triangleorder 1 (click to view basis functions) triangleorder 2 (click to view basis functions) tetrahedronorder 1 (click to view basis functions) tetrahedronorder 2 (click to view basis functions)

## References

• 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 02 August 2022