| Alternative names | Whitney (triangle,tetrahedron), Nédélec, Q H(curl) (quadrilateral,hexahedron), Raviart–Thomas cubical H(curl) (quadrilateral), Nédélec cubical H(curl) (hexahedron) |
| Exterior calculus names | \(\mathcal{P}^-_{k}\Lambda^{1}(\Delta_d)\), \(\mathcal{Q}^-_{k}\Lambda^{1}(\square_d)\) |
| Cockburn–fu names | \(\left[S_{2,k}^\unicode{0x25FA}\right]_{1}\), \(\left[S_{4,k}^\square\right]_{1}\) |
| Abbreviated names | N1curl, NC, RTce (quadrilateral), Nce (hexahedron) |
| Orders | \(1\leqslant k\) |
| Reference elements | triangle, tetrahedron, quadrilateral, hexahedron, prism |
| Polynomial set | \(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(21)}_{k}\) (triangle, tetrahedron)
\(\mathcal{Q}_{k-1}^d \oplus \mathcal{Z}^{(22)}_{k}\) (quadrilateral, hexahedron)
↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑\(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|\sum_{i=1}^dp_i\leqslant k\right\}\)
\(\mathcal{Z}^{(21)}_k=\left\{\boldsymbol{p}\in\hat{\mathcal{P}}_{k}^d\middle|\boldsymbol{p}(\boldsymbol{x})\cdot \boldsymbol{x}=0\right\}\)
\(\hat{\mathcal{P}}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|\sum_{i=1}^dp_i=k\right\}=\mathcal{P}_k\setminus\mathcal{P}_{k-1}\)
\(\mathcal{Q}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|\max_i(p_i)\leqslant k\right\}\)
\(\mathcal{Z}^{(22)}_k=\left\{\boldsymbol{q}\in\hat{\mathcal{Q}}_{k}\middle|\boldsymbol{q}(\boldsymbol{x})\cdot x_i\boldsymbol{e}_i\in\mathcal{Q}_{k}\text{ for }i=1,\dots,d\right\}\)
\(\hat{\mathcal{Q}}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle|\max_i(p_i)=k\right\}=\mathcal{Q}_k\setminus\mathcal{Q}_{k-1}\) |
| DOFs | On each edge: tangent integral moments with an order \(k-1\) Lagrange space
On each face (triangle): integral moments with an order \(k-2\) vector Lagrange space
On each face (quadrilateral): integral moments with an order \(k-1\) Q H(div) space
On each volume (tetrahedron): integral moments with an order \(k-3\) vector Lagrange space
On each volume (hexahedron): integral moments with an order \(k-1\) Q H(div) space |
| Number of DOFs | triangle: \(k(k+2)\) (A005563)
tetrahedron: \(k(k+2)(k+3)/2\) (A005564)
quadrilateral: \(2k(k+1)\) (A046092)
hexahedron: \(3k(k+1)^2\) (A059986)
prism: \(3k(k+2)(k+1)/2\) |
| Mapping | covariant Piola |
| continuity | Components tandential to facets are continuous |
| Categories | Vector-valued elements, H(curl) conforming elements |
| Basix | basix.ElementFamily.N1E (triangle, tetrahedron, quadrilateral, hexahedron)
↓ Show Basix examples ↓↑ Hide Basix examples ↑Before trying this example, you must install Basix: pip3 install git+https://github.com/fenics/basix.git This element can then be created with the following lines of Python: import basix
# Create Nedelec (first kind) order 1 on a triangle
element = basix.create_element(basix.ElementFamily.N1E, basix.CellType.triangle, 1)
# Create Nedelec (first kind) order 2 on a triangle
element = basix.create_element(basix.ElementFamily.N1E, basix.CellType.triangle, 2)
# Create Nedelec (first kind) order 1 on a quadrilateral
element = basix.create_element(basix.ElementFamily.N1E, basix.CellType.quadrilateral, 1)
# Create Nedelec (first kind) order 2 on a quadrilateral
element = basix.create_element(basix.ElementFamily.N1E, basix.CellType.quadrilateral, 2)
# Create Nedelec (first kind) order 1 on a tetrahedron
element = basix.create_element(basix.ElementFamily.N1E, basix.CellType.tetrahedron, 1)
# Create Nedelec (first kind) order 2 on a tetrahedron
element = basix.create_element(basix.ElementFamily.N1E, basix.CellType.tetrahedron, 2)
# Create Nedelec (first kind) order 1 on a hexahedron
element = basix.create_element(basix.ElementFamily.N1E, basix.CellType.hexahedron, 1)
# Create Nedelec (first kind) order 2 on a hexahedron
element = basix.create_element(basix.ElementFamily.N1E, basix.CellType.hexahedron, 2) |
| Bempp | "SNC" (triangle)
↓ Show Bempp examples ↓↑ Hide Bempp examples ↑Before trying this example, you must install Bempp: 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) |
| Symfem | "N1curl" (triangle, tetrahedron)
"Qcurl" (quadrilateral, hexahedron)
"Ncurl" (prism)
↓ Show Symfem examples ↓↑ Hide Symfem examples ↑Before trying 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 (first kind) order 1 on a triangle
element = symfem.create_element("triangle", "N1curl", 1)
# Create Nedelec (first kind) order 2 on a triangle
element = symfem.create_element("triangle", "N1curl", 2)
# Create Nedelec (first kind) order 1 on a quadrilateral
element = symfem.create_element("quadrilateral", "Qcurl", 1)
# Create Nedelec (first kind) order 2 on a quadrilateral
element = symfem.create_element("quadrilateral", "Qcurl", 2)
# Create Nedelec (first kind) order 1 on a tetrahedron
element = symfem.create_element("tetrahedron", "N1curl", 1)
# Create Nedelec (first kind) order 2 on a tetrahedron
element = symfem.create_element("tetrahedron", "N1curl", 2)
# Create Nedelec (first kind) order 1 on a hexahedron
element = symfem.create_element("hexahedron", "Qcurl", 1)
# Create Nedelec (first kind) order 2 on a hexahedron
element = symfem.create_element("hexahedron", "Qcurl", 2)
# Create Nedelec (first kind) order 1 on a prism
element = symfem.create_element("prism", "Ncurl", 1)
# Create Nedelec (first kind) order 2 on a prism
element = symfem.create_element("prism", "Ncurl", 2) |
| UFL | "N1curl" (triangle, tetrahedron)
"RTCE" (quadrilateral)
"NCE" (hexahedron)
↓ Show UFL examples ↓↑ Hide UFL examples ↑Before trying this example, you must install UFL: pip3 install UFL This element can then be created with the following lines of Python: import ufl
# Create Nedelec (first kind) order 1 on a triangle
element = ufl.FiniteElement("N1curl", "triangle", 1)
# Create Nedelec (first kind) order 2 on a triangle
element = ufl.FiniteElement("N1curl", "triangle", 2)
# Create Nedelec (first kind) order 1 on a quadrilateral
element = ufl.FiniteElement("RTCE", "quadrilateral", 1)
# Create Nedelec (first kind) order 2 on a quadrilateral
element = ufl.FiniteElement("RTCE", "quadrilateral", 2)
# Create Nedelec (first kind) order 1 on a tetrahedron
element = ufl.FiniteElement("N1curl", "tetrahedron", 1)
# Create Nedelec (first kind) order 2 on a tetrahedron
element = ufl.FiniteElement("N1curl", "tetrahedron", 2)
# Create Nedelec (first kind) order 1 on a hexahedron
element = ufl.FiniteElement("NCE", "hexahedron", 1)
# Create Nedelec (first kind) order 2 on a hexahedron
element = ufl.FiniteElement("NCE", "hexahedron", 2) |
