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Degree 1 serendipity H(curl) on a hexahedron

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

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

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

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

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

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

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

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

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

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 3 z \left(- y z + y + z - 1\right)\\\displaystyle 3 z \left(- x z + x + z - 1\right)\\\displaystyle - 6 x y z + 4 x y + 6 x z - 4 x + 6 y z - 4 y - 6 z + 4\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2.

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 3 z \left(y z - y - z + 1\right)\\\displaystyle 3 z \left(x z - x - z + 1\right)\\\displaystyle 6 x y z - 2 x y - 6 x z + 2 x - 6 y z + 2 y + 6 z - 2\end{array}\right)\)

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

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

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
and \(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3.

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

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
and \(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4.

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 3 z \left(y z - y - z + 1\right)\\\displaystyle 3 x z \left(z - 1\right)\\\displaystyle 2 x \left(3 y z - 2 y - 3 z + 2\right)\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
and \(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4.

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

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
and \(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5.

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

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
and \(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5.

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

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{6}\)
where \(e_{6}\) is the 6th edge;
and \(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6.

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

This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{6}\)
where \(e_{6}\) is the 6th edge;
and \(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6.

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

This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{7}\)
where \(e_{7}\) is the 7th edge;
and \(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7.

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 3 y z \left(1 - z\right)\\\displaystyle 3 x z \left(1 - z\right)\\\displaystyle 2 x y \left(2 - 3 z\right)\end{array}\right)\)

This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{7}\)
where \(e_{7}\) is the 7th edge;
and \(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7.

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 3 y z \left(z - 1\right)\\\displaystyle 3 x z \left(z - 1\right)\\\displaystyle 2 x y \left(3 z - 1\right)\end{array}\right)\)

This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{8}\)
where \(e_{8}\) is the 8th edge;
and \(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8.

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

This DOF is associated with edge 8 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{8}\)
where \(e_{8}\) is the 8th edge;
and \(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8.

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

This DOF is associated with edge 8 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{9}\)
where \(e_{9}\) is the 9th edge;
and \(\hat{\boldsymbol{t}}_{9}\) is the tangent to edge 9.

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 3 y z \left(y - 1\right)\\\displaystyle 2 z \left(3 x y - 2 x - 3 y + 2\right)\\\displaystyle 3 y \left(x y - x - y + 1\right)\end{array}\right)\)

This DOF is associated with edge 9 of the reference element.
\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{9}\)
where \(e_{9}\) is the 9th edge;
and \(\hat{\boldsymbol{t}}_{9}\) is the tangent to edge 9.

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 3 y z \left(1 - y\right)\\\displaystyle 2 z \left(- 3 x y + x + 3 y - 1\right)\\\displaystyle 3 y \left(- x y + x + y - 1\right)\end{array}\right)\)

This DOF is associated with edge 9 of the reference element.
\(\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{10}\)
where \(e_{10}\) is the 10th edge;
and \(\hat{\boldsymbol{t}}_{10}\) is the tangent to edge 10.

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 3 y z \left(1 - y\right)\\\displaystyle 2 x z \left(2 - 3 y\right)\\\displaystyle 3 x y \left(1 - y\right)\end{array}\right)\)

This DOF is associated with edge 10 of the reference element.
\(\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{10}\)
where \(e_{10}\) is the 10th edge;
and \(\hat{\boldsymbol{t}}_{10}\) is the tangent to edge 10.

\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 3 y z \left(y - 1\right)\\\displaystyle 2 x z \left(3 y - 1\right)\\\displaystyle 3 x y \left(y - 1\right)\end{array}\right)\)

This DOF is associated with edge 10 of the reference element.
\(\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{11}\)
where \(e_{11}\) is the 11th edge;
and \(\hat{\boldsymbol{t}}_{11}\) is the tangent to edge 11.

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

This DOF is associated with edge 11 of the reference element.
\(\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{11}\)
where \(e_{11}\) is the 11th edge;
and \(\hat{\boldsymbol{t}}_{11}\) is the tangent to edge 11.

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

This DOF is associated with edge 11 of the reference element.