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Degree 2 Nédélec (second kind) on a tetrahedron

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In this example:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 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.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0.

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

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0.

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

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 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.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1.

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

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1.

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \frac{3 z \left(- 6 x - z + 2\right)}{2}\\\displaystyle 0\\\displaystyle \frac{3 x \left(x + 6 z - 2\right)}{2}\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 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.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2.

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 9 y \left(1 - 2 y\right)\\\displaystyle 3 x \left(1 - 4 y\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2.

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(- 6 x - y + 2\right)}{2}\\\displaystyle \frac{3 x \left(x + 6 y - 2\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
and \(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 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.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
and \(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3.

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 9 z \left(2 z - 1\right)\\\displaystyle 9 z \left(2 z - 1\right)\\\displaystyle 12 x z - 3 x + 12 y z - 3 y + 30 z^{2} - 24 z + 3\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
and \(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3.

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{3 z \left(- 6 x - 6 y - 5 z + 4\right)}{2}\\\displaystyle \frac{3 z \left(- 6 x - 6 y - 5 z + 4\right)}{2}\\\displaystyle \frac{3 x^{2}}{2} + 3 x y - 15 x z + \frac{3 y^{2}}{2} - 15 y z - 15 z^{2} + 15 z - \frac{3}{2}\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
and \(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 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.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
and \(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4.

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 9 y \left(2 y - 1\right)\\\displaystyle 12 x y - 3 x + 30 y^{2} + 12 y z - 24 y - 3 z + 3\\\displaystyle 9 y \left(2 y - 1\right)\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
and \(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4.

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(- 6 x - 5 y - 6 z + 4\right)}{2}\\\displaystyle \frac{3 x^{2}}{2} - 15 x y + 3 x z - 15 y^{2} - 15 y z + 15 y + \frac{3 z^{2}}{2} - \frac{3}{2}\\\displaystyle \frac{3 y \left(- 6 x - 5 y - 6 z + 4\right)}{2}\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
and \(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 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.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(s_{0} \cdot \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
and \(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5.

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 12 x y + 12 x z - 24 x - 3 y - 3 z + 3\\\displaystyle 9 x \left(2 x - 1\right)\\\displaystyle 9 x \left(2 x - 1\right)\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(4 s_{0} \cdot \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
and \(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5.

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle - 15 x^{2} - 15 x y - 15 x z + 15 x + \frac{3 y^{2}}{2} + 3 y z + \frac{3 z^{2}}{2} - \frac{3}{2}\\\displaystyle \frac{3 x \left(- 5 x - 6 y - 6 z + 4\right)}{2}\\\displaystyle \frac{3 x \left(- 5 x - 6 y - 6 z + 4\right)}{2}\end{array}\right)\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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