Raviart–Thomas

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Alternative names	Rao–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 names	RT, RWG, RTcf (quadrilateral), Ncf (hexahedron)
Variants	Legendre: 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 superdegree	triangle: \(k\) tetrahedron: \(k\) quadrilateral: \(dk - d + 1\) hexahedron: \(dk - d + 1\)
Lagrange subdegree	\(k-1\)
Lagrange superdegree	\(k\)
Reference elements	triangle, 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 ↓↑ Hide polynomial set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Z}^{(24)}_k=\left\{\left(\begin{array}{c}px_1\\\vdots\\px_d\end{array}\right)\middle\|p\in\hat{\mathcal{P}}_{k-1}\right\}\) \(\hat{\mathcal{P}}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i=k\right\}=\mathcal{P}_k\setminus\mathcal{P}_{k-1}\) \(\mathcal{Q}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\max_i(p_i)\leqslant k\right\}\) \(\mathcal{Z}^{(25)}_k=\left\{\boldsymbol{q}\in\hat{\mathcal{Q}}_{k}\middle\|\boldsymbol{q}(\boldsymbol{x})\cdot x_i\boldsymbol{e}_j\in\begin{cases}\hat{\mathcal{Q}}_{k+1}&i=j\\\hat{\mathcal{Q}}_{k}&i\not=j\end{cases}\text{ for }i,j=1,\dots,d\right\}\) \(\hat{\mathcal{Q}}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\max_i(p_i)=k\right\}=\mathcal{Q}_k\setminus\mathcal{Q}_{k-1}\)
DOFs	On 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 DOFs	triangle: \(k(k+2)\) (A005563) tetrahedron: \(k(k+1)(k+3)/2\) (A077414) quadrilateral: \(2k(k+1)\) (A046092) hexahedron: \(3k^2(k+1)\) (A270205)
Mapping	contravariant Piola
continuity	Components normal to facets are continuous
Categories	Vector-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 ↓

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Basix	`basix.ElementFamily.RT` ↓ Show Basix examples ↓↑ Hide Basix examples ↑ Before running this example, you must install Basix: pip3 install fenics-basix This element can then be created with the following lines of Python: import basix # Create Raviart-Thomas (Lagrange variant) degree 1 on a triangle element = basix.create_element(basix.ElementFamily.RT, basix.CellType.triangle, 1) # Create Raviart-Thomas (Lagrange variant) degree 2 on a triangle element = basix.create_element(basix.ElementFamily.RT, basix.CellType.triangle, 2) # Create Raviart-Thomas (Lagrange variant) degree 1 on a quadrilateral element = basix.create_element(basix.ElementFamily.RT, basix.CellType.quadrilateral, 1) # Create Raviart-Thomas (Lagrange variant) degree 2 on a quadrilateral element = basix.create_element(basix.ElementFamily.RT, basix.CellType.quadrilateral, 2) # Create Raviart-Thomas (Lagrange variant) degree 1 on a tetrahedron element = basix.create_element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 1) # Create Raviart-Thomas (Lagrange variant) degree 2 on a tetrahedron element = basix.create_element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 2) # Create Raviart-Thomas (Lagrange variant) degree 1 on a hexahedron element = basix.create_element(basix.ElementFamily.RT, basix.CellType.hexahedron, 1) # Create Raviart-Thomas (Lagrange variant) degree 2 on a hexahedron element = basix.create_element(basix.ElementFamily.RT, basix.CellType.hexahedron, 2) # Create Raviart-Thomas (Legendre variant) degree 1 on a triangle element = basix.create_element(basix.ElementFamily.RT, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.legendre) # Create Raviart-Thomas (Legendre variant) degree 2 on a triangle element = basix.create_element(basix.ElementFamily.RT, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.legendre) # Create Raviart-Thomas (Legendre variant) degree 1 on a quadrilateral element = basix.create_element(basix.ElementFamily.RT, basix.CellType.quadrilateral, 1, lagrange_variant=basix.LagrangeVariant.legendre) # Create Raviart-Thomas (Legendre variant) degree 2 on a quadrilateral element = basix.create_element(basix.ElementFamily.RT, basix.CellType.quadrilateral, 2, lagrange_variant=basix.LagrangeVariant.legendre) # Create Raviart-Thomas (Legendre variant) degree 1 on a tetrahedron element = basix.create_element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.legendre) # Create Raviart-Thomas (Legendre variant) degree 2 on a tetrahedron element = basix.create_element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.legendre) # Create Raviart-Thomas (Legendre variant) degree 1 on a hexahedron element = basix.create_element(basix.ElementFamily.RT, basix.CellType.hexahedron, 1, lagrange_variant=basix.LagrangeVariant.legendre) # Create Raviart-Thomas (Legendre variant) degree 2 on a hexahedron element = basix.create_element(basix.ElementFamily.RT, basix.CellType.hexahedron, 2, lagrange_variant=basix.LagrangeVariant.legendre) This implementation is correct for all the examples below.
Basix.UFL	`basix.ElementFamily.RT` ↓ Show Basix.UFL examples ↓↑ Hide Basix.UFL examples ↑ Before running this example, you must install Basix.UFL: pip3 install fenics-basix fenics-ufl This element can then be created with the following lines of Python: import basix import basix.ufl # Create Raviart-Thomas (Lagrange variant) degree 1 on a triangle element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Raviart-Thomas (Lagrange variant) degree 2 on a triangle element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Raviart-Thomas (Lagrange variant) degree 1 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.quadrilateral, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Raviart-Thomas (Lagrange variant) degree 2 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.quadrilateral, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Raviart-Thomas (Lagrange variant) degree 1 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Raviart-Thomas (Lagrange variant) degree 2 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Raviart-Thomas (Lagrange variant) degree 1 on a hexahedron element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.hexahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Raviart-Thomas (Lagrange variant) degree 2 on a hexahedron element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.hexahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Raviart-Thomas (Legendre variant) degree 1 on a triangle element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.triangle, 1) # Create Raviart-Thomas (Legendre variant) degree 2 on a triangle element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.triangle, 2) # Create Raviart-Thomas (Legendre variant) degree 1 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.quadrilateral, 1) # Create Raviart-Thomas (Legendre variant) degree 2 on a quadrilateral element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.quadrilateral, 2) # Create Raviart-Thomas (Legendre variant) degree 1 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 1) # Create Raviart-Thomas (Legendre variant) degree 2 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.tetrahedron, 2) # Create Raviart-Thomas (Legendre variant) degree 1 on a hexahedron element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.hexahedron, 1) # Create Raviart-Thomas (Legendre variant) degree 2 on a hexahedron element = basix.ufl.element(basix.ElementFamily.RT, basix.CellType.hexahedron, 2) This implementation is correct for all the examples below.
Bempp	`"RWG"` ↓ Show Bempp examples ↓↑ Hide Bempp examples ↑ Before running this example, you must install Bempp: pip3 install numba scipy meshio pip3 install bempp-cl This element can then be created with the following lines of Python: import bempp.api grid = bempp.api.shapes.regular_sphere(1) # Create Raviart-Thomas degree 1 element = bempp.api.function_space(grid, "RWG", 0)
FIAT	`FIAT.RaviartThomas(..., variant="integral")` ↓ Show FIAT examples ↓↑ Hide FIAT examples ↑ Before running this example, you must install FIAT: pip3 install git+https://github.com/firedrakeproject/fiat.git This element can then be created with the following lines of Python: import FIAT # Create Raviart-Thomas (Legendre variant) degree 1 element = FIAT.RaviartThomas(FIAT.ufc_cell("triangle"), 1, variant="integral") # Create Raviart-Thomas (Legendre variant) degree 2 element = FIAT.RaviartThomas(FIAT.ufc_cell("triangle"), 2, variant="integral") # Create Raviart-Thomas (Legendre variant) degree 1 element = FIAT.RaviartThomas(FIAT.ufc_cell("tetrahedron"), 1, variant="integral") # Create Raviart-Thomas (Legendre variant) degree 2 element = FIAT.RaviartThomas(FIAT.ufc_cell("tetrahedron"), 2, variant="integral") This implementation is correct for all the examples below that it supports. ↓ Show more ↓ ↑ Hide ↑ Correct: triangle,1,legendre; triangle,2,legendre; tetrahedron,1,legendre; tetrahedron,2,legendre Not implemented: triangle,1,lagrange; triangle,2,lagrange; quadrilateral,1,lagrange; quadrilateral,2,lagrange; tetrahedron,1,lagrange; tetrahedron,2,lagrange; hexahedron,1,lagrange; hexahedron,2,lagrange; quadrilateral,1,legendre; quadrilateral,2,legendre; hexahedron,1,legendre; hexahedron,2,legendre
NDElement	`Family.RaviartThomas, DEGREES=` ↓ Show NDElement examples ↓↑ Hide NDElement examples ↑ Before running this example, you must install NDElement: pip3 install ndelement This element can then be created with the following lines of Python: from ndelement.ciarlet import Family, create_family from ndelement.reference_cell import ReferenceCellType # Create Raviart-Thomas (Lagrange variant) degree 1 on a triangle family = create_family(Family.RaviartThomas, 1) element = family.element(ReferenceCellType.Triangle) This implementation is correct for all the examples below that it supports. ↓ Show more ↓ ↑ Hide ↑ Correct: triangle,1,lagrange Not implemented: triangle,2,lagrange; quadrilateral,1,lagrange; quadrilateral,2,lagrange; tetrahedron,1,lagrange; tetrahedron,2,lagrange; hexahedron,1,lagrange; hexahedron,2,lagrange; triangle,1,legendre; triangle,2,legendre; quadrilateral,1,legendre; quadrilateral,2,legendre; tetrahedron,1,legendre; tetrahedron,2,legendre; hexahedron,1,legendre; hexahedron,2,legendre
Symfem	`"N1div", variant="legendre"` (triangle, Legendre; tetrahedron, Legendre) `"Qdiv", variant="legendre"` (quadrilateral, Legendre; hexahedron, Legendre) `"N1div"` (triangle, Lagrange; tetrahedron, Lagrange) `"Qdiv"` (quadrilateral, Lagrange; hexahedron, Lagrange) ↓ Show Symfem examples ↓↑ Hide Symfem examples ↑ Before running this example, you must install Symfem: pip3 install symfem This element can then be created with the following lines of Python: import symfem # Create Raviart-Thomas (Lagrange variant) degree 1 on a triangle element = symfem.create_element("triangle", "N1div", 1) # Create Raviart-Thomas (Lagrange variant) degree 2 on a triangle element = symfem.create_element("triangle", "N1div", 2) # Create Raviart-Thomas (Lagrange variant) degree 1 on a quadrilateral element = symfem.create_element("quadrilateral", "Qdiv", 1) # Create Raviart-Thomas (Lagrange variant) degree 2 on a quadrilateral element = symfem.create_element("quadrilateral", "Qdiv", 2) # Create Raviart-Thomas (Lagrange variant) degree 1 on a tetrahedron element = symfem.create_element("tetrahedron", "N1div", 1) # Create Raviart-Thomas (Lagrange variant) degree 2 on a tetrahedron element = symfem.create_element("tetrahedron", "N1div", 2) # Create Raviart-Thomas (Lagrange variant) degree 1 on a hexahedron element = symfem.create_element("hexahedron", "Qdiv", 1) # Create Raviart-Thomas (Lagrange variant) degree 2 on a hexahedron element = symfem.create_element("hexahedron", "Qdiv", 2) # Create Raviart-Thomas (Legendre variant) degree 1 on a triangle element = symfem.create_element("triangle", "N1div", 1, variant="legendre") # Create Raviart-Thomas (Legendre variant) degree 2 on a triangle element = symfem.create_element("triangle", "N1div", 2, variant="legendre") # Create Raviart-Thomas (Legendre variant) degree 1 on a quadrilateral element = symfem.create_element("quadrilateral", "Qdiv", 1, variant="legendre") # Create Raviart-Thomas (Legendre variant) degree 2 on a quadrilateral element = symfem.create_element("quadrilateral", "Qdiv", 2, variant="legendre") # Create Raviart-Thomas (Legendre variant) degree 1 on a tetrahedron element = symfem.create_element("tetrahedron", "N1div", 1, variant="legendre") # Create Raviart-Thomas (Legendre variant) degree 2 on a tetrahedron element = symfem.create_element("tetrahedron", "N1div", 2, variant="legendre") # Create Raviart-Thomas (Legendre variant) degree 1 on a hexahedron element = symfem.create_element("hexahedron", "Qdiv", 1, variant="legendre") # Create Raviart-Thomas (Legendre variant) degree 2 on a hexahedron element = symfem.create_element("hexahedron", "Qdiv", 2, variant="legendre") This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL	`"RT"` (triangle, Lagrange; tetrahedron, Lagrange) `"RTCF"` (quadrilateral, Lagrange) `"NCF"` (hexahedron, Lagrange) ↓ Show (legacy) UFL examples ↓↑ Hide (legacy) UFL examples ↑ Before running this example, you must install (legacy) UFL: pip3 install setuptools pip3 install fenics-ufl-legacy This element can then be created with the following lines of Python: import ufl_legacy # Create Raviart-Thomas (Lagrange variant) degree 1 on a triangle element = ufl_legacy.FiniteElement("RT", "triangle", 1) # Create Raviart-Thomas (Lagrange variant) degree 2 on a triangle element = ufl_legacy.FiniteElement("RT", "triangle", 2) # Create Raviart-Thomas (Lagrange variant) degree 1 on a quadrilateral element = ufl_legacy.FiniteElement("RTCF", "quadrilateral", 1) # Create Raviart-Thomas (Lagrange variant) degree 2 on a quadrilateral element = ufl_legacy.FiniteElement("RTCF", "quadrilateral", 2) # Create Raviart-Thomas (Lagrange variant) degree 1 on a tetrahedron element = ufl_legacy.FiniteElement("RT", "tetrahedron", 1) # Create Raviart-Thomas (Lagrange variant) degree 2 on a tetrahedron element = ufl_legacy.FiniteElement("RT", "tetrahedron", 2) # Create Raviart-Thomas (Lagrange variant) degree 1 on a hexahedron element = ufl_legacy.FiniteElement("NCF", "hexahedron", 1) # Create Raviart-Thomas (Lagrange variant) degree 2 on a hexahedron element = ufl_legacy.FiniteElement("NCF", "hexahedron", 2)

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

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. [www.siam.org/publications/siam-news/issues/volume-47-number-09-november-2014/] [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 October 2024