an encyclopedia of finite element definitions

# Nédélec (second kind)

 Abbreviated names N2curl Orders $$1\leqslant k$$ Reference elements triangle, tetrahedron Polynomial set $$\mathcal{P}_{k}^d$$↓ Show polynomial set definitions ↓ DOFs On each edge: tangent integral moments with an order $$k$$ Lagrange space On each face: integral moments with an order $$k-1$$ Raviart–Thomas space On each volume: integral moments with an order $$k-2$$ 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) Categories Vector-valued elements, H(curl) conforming elements

## Implementations

 Symfem string "N2curl"↓ Show Symfem examples ↓ Basix string "Nedelec 2nd kind H(curl)"↓ Show Basix examples ↓ UFL string "N2curl"↓ Show UFL examples ↓

## Examples

triangle
order 1
triangle
order 2
tetrahedron
order 1
tetrahedron
order 2
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{5}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 2 y\\\displaystyle 4 x\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle - 4 y\\\displaystyle - 2 x\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle - 2 y\\\displaystyle - 4 x - 6 y + 4\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 4 y\\\displaystyle 2 x + 6 y - 2\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle - 6 x - 4 y + 4\\\displaystyle - 2 x\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 6 x + 2 y - 2\\\displaystyle 4 x\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{11}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(2 t_{0}^{2} - 3 t_{0} + 1)\hat{\boldsymbol{t}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 3 y \left(4 x - 1\right)\\\displaystyle 9 x \left(2 x - 1\right)\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(4 t_{0} \left(1 - t_{0}\right))\hat{\boldsymbol{t}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(- 6 x - y + 2\right)}{2}\\\displaystyle \frac{3 x \left(x + 6 y - 2\right)}{2}\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{t}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 9 y \left(1 - 2 y\right)\\\displaystyle 3 x \left(1 - 4 y\right)\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(2 t_{0}^{2} - 3 t_{0} + 1)\hat{\boldsymbol{t}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 3 y \left(4 x + 4 y - 3\right)\\\displaystyle 18 x^{2} + 48 x y - 27 x + 30 y^{2} - 36 y + 9\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(4 t_{0} \left(1 - t_{0}\right))\hat{\boldsymbol{t}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(- 6 x - 5 y + 4\right)}{2}\\\displaystyle \frac{3 x^{2}}{2} - 15 x y - 15 y^{2} + 15 y - \frac{3}{2}\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{t}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 9 y \left(2 y - 1\right)\\\displaystyle 12 x y - 3 x + 30 y^{2} - 24 y + 3\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(2 t_{0}^{2} - 3 t_{0} + 1)\hat{\boldsymbol{t}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 48 x y - 36 x + 18 y^{2} - 27 y + 9\\\displaystyle 3 x \left(4 x + 4 y - 3\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(4 t_{0} \left(1 - t_{0}\right))\hat{\boldsymbol{t}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle - 15 x^{2} - 15 x y + 15 x + \frac{3 y^{2}}{2} - \frac{3}{2}\\\displaystyle \frac{3 x \left(- 5 x - 6 y + 4\right)}{2}\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{t}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 12 x y - 24 x - 3 y + 3\\\displaystyle 9 x \left(2 x - 1\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}- t_{0}\\- t_{1}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 12 y \left(- 4 x - y + 1\right)\\\displaystyle 12 x \left(- x - 4 y + 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}t_{0} - 1\\t_{1}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 12 y \left(4 x + 3 y - 3\right)\\\displaystyle 12 x \left(x + 2 y - 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}- t_{0}\\1 - t_{1}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 12 y \left(- 2 x - y + 1\right)\\\displaystyle 12 x \left(- 3 x - 4 y + 3\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
• $$R$$ is the reference tetrahedron. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{11}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 z\\\displaystyle 4 y\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 4 z\\\displaystyle - 2 y\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 2 z\\\displaystyle 0\\\displaystyle 4 x\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - 4 z\\\displaystyle 0\\\displaystyle - 2 x\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 2 y\\\displaystyle 4 x\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - 4 y\\\displaystyle - 2 x\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle - 2 z\\\displaystyle - 2 z\\\displaystyle - 4 x - 4 y - 6 z + 4\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 4 z\\\displaystyle 4 z\\\displaystyle 2 x + 2 y + 6 z - 2\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle - 2 y\\\displaystyle - 4 x - 6 y - 4 z + 4\\\displaystyle - 2 y\end{array}\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 4 y\\\displaystyle 2 x + 6 y + 2 z - 2\\\displaystyle 4 y\end{array}\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle - 6 x - 4 y - 4 z + 4\\\displaystyle - 2 x\\\displaystyle - 2 x\end{array}\right)$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 6 x + 2 y + 2 z - 2\\\displaystyle 4 x\\\displaystyle 4 x\end{array}\right)$$

This DOF is associated with edge 5 of the reference element.
• $$R$$ is the reference tetrahedron. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{29}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(2 t_{0}^{2} - 3 t_{0} + 1)\hat{\boldsymbol{t}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 z \left(4 y - 1\right)\\\displaystyle 9 y \left(2 y - 1\right)\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(4 t_{0} \left(1 - t_{0}\right))\hat{\boldsymbol{t}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 z \left(- 6 y - z + 2\right)}{2}\\\displaystyle \frac{3 y \left(y + 6 z - 2\right)}{2}\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{t}}_{0}$$

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 9 z \left(1 - 2 z\right)\\\displaystyle 3 y \left(1 - 4 z\right)\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(2 t_{0}^{2} - 3 t_{0} + 1)\hat{\boldsymbol{t}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 3 z \left(4 x - 1\right)\\\displaystyle 0\\\displaystyle 9 x \left(2 x - 1\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(4 t_{0} \left(1 - t_{0}\right))\hat{\boldsymbol{t}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{3 z \left(- 6 x - z + 2\right)}{2}\\\displaystyle 0\\\displaystyle \frac{3 x \left(x + 6 z - 2\right)}{2}\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{t}}_{1}$$

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 9 z \left(1 - 2 z\right)\\\displaystyle 0\\\displaystyle 3 x \left(1 - 4 z\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(2 t_{0}^{2} - 3 t_{0} + 1)\hat{\boldsymbol{t}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 3 y \left(4 x - 1\right)\\\displaystyle 9 x \left(2 x - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(4 t_{0} \left(1 - t_{0}\right))\hat{\boldsymbol{t}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(- 6 x - y + 2\right)}{2}\\\displaystyle \frac{3 x \left(x + 6 y - 2\right)}{2}\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{t}}_{2}$$

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 9 y \left(1 - 2 y\right)\\\displaystyle 3 x \left(1 - 4 y\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(2 t_{0}^{2} - 3 t_{0} + 1)\hat{\boldsymbol{t}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 3 z \left(4 x + 4 y + 4 z - 3\right)\\\displaystyle 3 z \left(4 x + 4 y + 4 z - 3\right)\\\displaystyle 18 x^{2} + 36 x y + 48 x z - 27 x + 18 y^{2} + 48 y z - 27 y + 30 z^{2} - 36 z + 9\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(4 t_{0} \left(1 - t_{0}\right))\hat{\boldsymbol{t}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle \frac{3 z \left(- 6 x - 6 y - 5 z + 4\right)}{2}\\\displaystyle \frac{3 z \left(- 6 x - 6 y - 5 z + 4\right)}{2}\\\displaystyle \frac{3 x^{2}}{2} + 3 x y - 15 x z + \frac{3 y^{2}}{2} - 15 y z - 15 z^{2} + 15 z - \frac{3}{2}\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{t}}_{3}$$

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 9 z \left(2 z - 1\right)\\\displaystyle 9 z \left(2 z - 1\right)\\\displaystyle 12 x z - 3 x + 12 y z - 3 y + 30 z^{2} - 24 z + 3\end{array}\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(2 t_{0}^{2} - 3 t_{0} + 1)\hat{\boldsymbol{t}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 3 y \left(4 x + 4 y + 4 z - 3\right)\\\displaystyle 18 x^{2} + 48 x y + 36 x z - 27 x + 30 y^{2} + 48 y z - 36 y + 18 z^{2} - 27 z + 9\\\displaystyle 3 y \left(4 x + 4 y + 4 z - 3\right)\end{array}\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(4 t_{0} \left(1 - t_{0}\right))\hat{\boldsymbol{t}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(- 6 x - 5 y - 6 z + 4\right)}{2}\\\displaystyle \frac{3 x^{2}}{2} - 15 x y + 3 x z - 15 y^{2} - 15 y z + 15 y + \frac{3 z^{2}}{2} - \frac{3}{2}\\\displaystyle \frac{3 y \left(- 6 x - 5 y - 6 z + 4\right)}{2}\end{array}\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{t}}_{4}$$

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 9 y \left(2 y - 1\right)\\\displaystyle 12 x y - 3 x + 30 y^{2} + 12 y z - 24 y - 3 z + 3\\\displaystyle 9 y \left(2 y - 1\right)\end{array}\right)$$

This DOF is associated with edge 4 of the reference element.
$$\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(2 t_{0}^{2} - 3 t_{0} + 1)\hat{\boldsymbol{t}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 48 x y + 48 x z - 36 x + 18 y^{2} + 36 y z - 27 y + 18 z^{2} - 27 z + 9\\\displaystyle 3 x \left(4 x + 4 y + 4 z - 3\right)\\\displaystyle 3 x \left(4 x + 4 y + 4 z - 3\right)\end{array}\right)$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(4 t_{0} \left(1 - t_{0}\right))\hat{\boldsymbol{t}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle - 15 x^{2} - 15 x y - 15 x z + 15 x + \frac{3 y^{2}}{2} + 3 y z + \frac{3 z^{2}}{2} - \frac{3}{2}\\\displaystyle \frac{3 x \left(- 5 x - 6 y - 6 z + 4\right)}{2}\\\displaystyle \frac{3 x \left(- 5 x - 6 y - 6 z + 4\right)}{2}\end{array}\right)$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(t_{0} \left(2 t_{0} - 1\right))\hat{\boldsymbol{t}}_{5}$$

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 12 x y + 12 x z - 24 x - 3 y - 3 z + 3\\\displaystyle 9 x \left(2 x - 1\right)\\\displaystyle 9 x \left(2 x - 1\right)\end{array}\right)$$

This DOF is associated with edge 5 of the reference element.
$$\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}\frac{\sqrt{3} \left(t_{0} + t_{1}\right)}{3}\\- \frac{\sqrt{3} t_{0}}{3}\\- \frac{\sqrt{3} t_{1}}{3}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 36 y z\\\displaystyle 12 x z\\\displaystyle 12 x y\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}\frac{\sqrt{3} \left(- t_{0} - t_{1} + 1\right)}{3}\\\frac{\sqrt{3} \left(t_{0} - 1\right)}{3}\\\frac{\sqrt{3} t_{1}}{3}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle - 12 y z\\\displaystyle - 36 x z\\\displaystyle - 12 x y\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}\frac{\sqrt{3} \left(t_{0} + t_{1} - 1\right)}{3}\\- \frac{\sqrt{3} t_{0}}{3}\\\frac{\sqrt{3} \left(1 - t_{1}\right)}{3}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 12 y z\\\displaystyle 12 x z\\\displaystyle 36 x y\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\- t_{0}\\- t_{1}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle - 36 y z\\\displaystyle 12 z \left(- x - 4 y - z + 1\right)\\\displaystyle 12 y \left(- x - y - 4 z + 1\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\t_{0} - 1\\t_{1}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 12 y z\\\displaystyle 12 z \left(3 x + 4 y + 3 z - 3\right)\\\displaystyle 12 y \left(x + y + 2 z - 1\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\- t_{0}\\1 - t_{1}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle - 12 y z\\\displaystyle 12 z \left(- x - 2 y - z + 1\right)\\\displaystyle 12 y \left(- 3 x - 3 y - 4 z + 3\right)\end{array}\right)$$

This DOF is associated with face 1 of the reference element.
$$\displaystyle l_{24}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}- t_{0}\\0\\- t_{1}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 12 z \left(- 4 x - y - z + 1\right)\\\displaystyle - 36 x z\\\displaystyle 12 x \left(- x - y - 4 z + 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{25}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}t_{0} - 1\\0\\t_{1}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 12 z \left(4 x + 3 y + 3 z - 3\right)\\\displaystyle 12 x z\\\displaystyle 12 x \left(x + y + 2 z - 1\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{26}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}- t_{0}\\0\\1 - t_{1}\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 12 z \left(- 2 x - y - z + 1\right)\\\displaystyle - 12 x z\\\displaystyle 12 x \left(- 3 x - 3 y - 4 z + 3\right)\end{array}\right)$$

This DOF is associated with face 2 of the reference element.
$$\displaystyle l_{27}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}- t_{0}\\- t_{1}\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 12 y \left(- 4 x - y - z + 1\right)\\\displaystyle 12 x \left(- x - 4 y - z + 1\right)\\\displaystyle - 36 x y\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{28}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}t_{0} - 1\\t_{1}\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 12 y \left(4 x + 3 y + 3 z - 3\right)\\\displaystyle 12 x \left(x + 2 y + z - 1\right)\\\displaystyle 12 x y\end{array}\right)$$

This DOF is associated with face 3 of the reference element.
$$\displaystyle l_{29}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}- t_{0}\\1 - t_{1}\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 12 y \left(- 2 x - y - z + 1\right)\\\displaystyle 12 x \left(- 3 x - 4 y - 3 z + 3\right)\\\displaystyle - 12 x y\end{array}\right)$$

This DOF is associated with face 3 of the reference element.

## References

• Nédélec, J. C. A new family of mixed finite elements in $$\mathbb{R}^3$$, Numerische Mathematik 50(1), 57–81, 1986. [DOI: 10.1007/BF01389668] [BibTeX]