Nédélec (second kind)

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De Rham complex families	\(\left[S_{1,k}^\unicode{0x25FA}\right]_{1}\) or \(\mathcal{P}_{k}\Lambda^{1}(\Delta_d)\)
Abbreviated names	N2curl
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 Polynomial superdegree
Polynomial subdegree	\(k\)
Polynomial superdegree	\(k\)
Reference elements	triangle, tetrahedron
Polynomial set	\(\mathcal{P}_{k}^d\) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\)
DOFs	On each edge: tangent integral moments with an degree \(k\) Lagrange space On each face: integral moments with an degree \(k-1\) Raviart–Thomas space On each volume: integral moments with an degree \(k-2\) Raviart–Thomas space
Number of DOFs	interval: \(k+1\) (A000027) triangle: \((k+1)(k+2)\) (A002378) tetrahedron: \((k+1)(k+2)(k+3)/2\) (A027480)
Mapping	covariant Piola
continuity	Components tangential to facets are continuous
Categories	Vector-valued elements, H(curl) conforming elements

Implementations

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

↑ Hide implementation detail ↑

Basix	`basix.ElementFamily.N2E` ↓ 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 Nedelec (second kind) (Lagrange variant) degree 1 on a triangle element = basix.create_element(basix.ElementFamily.N2E, basix.CellType.triangle, 1) # Create Nedelec (second kind) (Lagrange variant) degree 2 on a triangle element = basix.create_element(basix.ElementFamily.N2E, basix.CellType.triangle, 2) # Create Nedelec (second kind) (Lagrange variant) degree 1 on a tetrahedron element = basix.create_element(basix.ElementFamily.N2E, basix.CellType.tetrahedron, 1) # Create Nedelec (second kind) (Lagrange variant) degree 2 on a tetrahedron element = basix.create_element(basix.ElementFamily.N2E, basix.CellType.tetrahedron, 2) # Create Nedelec (second kind) (Legendre variant) degree 1 on a triangle element = basix.create_element(basix.ElementFamily.N2E, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.legendre) # Create Nedelec (second kind) (Legendre variant) degree 2 on a triangle element = basix.create_element(basix.ElementFamily.N2E, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.legendre) # Create Nedelec (second kind) (Legendre variant) degree 1 on a tetrahedron element = basix.create_element(basix.ElementFamily.N2E, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.legendre) # Create Nedelec (second kind) (Legendre variant) degree 2 on a tetrahedron element = basix.create_element(basix.ElementFamily.N2E, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.legendre) This implementation is correct for all the examples below.
Basix.UFL	`basix.ElementFamily.N2E` ↓ 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 Nedelec (second kind) (Lagrange variant) degree 1 on a triangle element = basix.ufl.element(basix.ElementFamily.N2E, basix.CellType.triangle, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Nedelec (second kind) (Lagrange variant) degree 2 on a triangle element = basix.ufl.element(basix.ElementFamily.N2E, basix.CellType.triangle, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Nedelec (second kind) (Lagrange variant) degree 1 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.N2E, basix.CellType.tetrahedron, 1, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Nedelec (second kind) (Lagrange variant) degree 2 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.N2E, basix.CellType.tetrahedron, 2, lagrange_variant=basix.LagrangeVariant.equispaced) # Create Nedelec (second kind) (Legendre variant) degree 1 on a triangle element = basix.ufl.element(basix.ElementFamily.N2E, basix.CellType.triangle, 1) # Create Nedelec (second kind) (Legendre variant) degree 2 on a triangle element = basix.ufl.element(basix.ElementFamily.N2E, basix.CellType.triangle, 2) # Create Nedelec (second kind) (Legendre variant) degree 1 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.N2E, basix.CellType.tetrahedron, 1) # Create Nedelec (second kind) (Legendre variant) degree 2 on a tetrahedron element = basix.ufl.element(basix.ElementFamily.N2E, basix.CellType.tetrahedron, 2) This implementation is correct for all the examples below.
FIAT	`FIAT.NedelecSecondKind(..., 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 Nedelec (second kind) (Legendre variant) degree 1 element = FIAT.NedelecSecondKind(FIAT.ufc_cell("triangle"), 1, variant="integral") # Create Nedelec (second kind) (Legendre variant) degree 2 element = FIAT.NedelecSecondKind(FIAT.ufc_cell("triangle"), 2, variant="integral") # Create Nedelec (second kind) (Legendre variant) degree 1 element = FIAT.NedelecSecondKind(FIAT.ufc_cell("tetrahedron"), 1, variant="integral") # Create Nedelec (second kind) (Legendre variant) degree 2 element = FIAT.NedelecSecondKind(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; tetrahedron,1,lagrange; tetrahedron,2,lagrange
Symfem	`"N2curl", variant="legendre"` (Legendre) `"N2curl"` (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 Nedelec (second kind) (Lagrange variant) degree 1 on a triangle element = symfem.create_element("triangle", "N2curl", 1) # Create Nedelec (second kind) (Lagrange variant) degree 2 on a triangle element = symfem.create_element("triangle", "N2curl", 2) # Create Nedelec (second kind) (Lagrange variant) degree 1 on a tetrahedron element = symfem.create_element("tetrahedron", "N2curl", 1) # Create Nedelec (second kind) (Lagrange variant) degree 2 on a tetrahedron element = symfem.create_element("tetrahedron", "N2curl", 2) # Create Nedelec (second kind) (Legendre variant) degree 1 on a triangle element = symfem.create_element("triangle", "N2curl", 1, variant="legendre") # Create Nedelec (second kind) (Legendre variant) degree 2 on a triangle element = symfem.create_element("triangle", "N2curl", 2, variant="legendre") # Create Nedelec (second kind) (Legendre variant) degree 1 on a tetrahedron element = symfem.create_element("tetrahedron", "N2curl", 1, variant="legendre") # Create Nedelec (second kind) (Legendre variant) degree 2 on a tetrahedron element = symfem.create_element("tetrahedron", "N2curl", 2, variant="legendre") This implementation is used to compute the examples below and verify other implementations.
(legacy) UFL	`"N2curl"` ↓ 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 Nedelec (second kind) (Lagrange variant) degree 1 on a triangle element = ufl_legacy.FiniteElement("N2curl", "triangle", 1) # Create Nedelec (second kind) (Lagrange variant) degree 2 on a triangle element = ufl_legacy.FiniteElement("N2curl", "triangle", 2) # Create Nedelec (second kind) (Lagrange variant) degree 1 on a tetrahedron element = ufl_legacy.FiniteElement("N2curl", "tetrahedron", 1) # Create Nedelec (second kind) (Lagrange variant) degree 2 on a tetrahedron element = ufl_legacy.FiniteElement("N2curl", "tetrahedron", 2)

Examples

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

References

Nédélec, Jean-Claude. A new family of mixed finite elements in \(\mathbb{R}^3\), Numerische Mathematik 50(1), 57–81, 1986. [DOI: 10.1007/BF01389668] [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	28 December 2020
Element last updated	27 September 2024