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Degree 1 Tiniest tensor H(curl) on a hexahedron
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- \(R\) is the reference hexahedron. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 x z \left(z - 1\right)}{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle \frac{3 x y \left(1 - y\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 x z \left(- 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{3 x y \left(2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 z \left(- x z + x + z - 1\right)}{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle \frac{3 y \left(x y - x - y + 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{3 y \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y z \left(1 - z\right)}{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle \frac{3 x y \left(x - 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y z \left(2 x z - 2 x - z + 1\right)}{2}\\\displaystyle 0\\\displaystyle \frac{3 x y \left(- 2 x z + x + 2 z - 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 z \left(y z - y - z + 1\right)}{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle \frac{3 x \left(- x y + x + y - 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 z \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle 0\\\displaystyle \frac{3 x \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y z \left(y - 1\right)}{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 x z \left(1 - x\right)}{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y z \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle \frac{3 x z \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y \left(- y z + y + z - 1\right)}{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 x \left(x z - x - z + 1\right)}{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle \frac{3 y \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle \frac{3 x \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle 0\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{41}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 9 x y^{2} z - 9 x y^{2} + 9 x y z^{2} - 24 x y z + 15 x y - 9 x z^{2} + 15 x z - 6 x - \frac{15 y^{2} z}{2} + \frac{15 y^{2}}{2} - \frac{15 y z^{2}}{2} + 19 y z - \frac{23 y}{2} + \frac{15 z^{2}}{2} - \frac{23 z}{2} + 4\\\displaystyle \frac{9 x \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{9 x \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 9 x y^{2} z - 9 x y^{2} + 9 x y z^{2} - 24 x y z + 15 x y - 9 x z^{2} + 15 x z - 6 x - \frac{15 y^{2} z}{2} + \frac{15 y^{2}}{2} - \frac{15 y z^{2}}{2} + 19 y z - \frac{23 y}{2} + \frac{15 z^{2}}{2} - \frac{23 z}{2} + 4\\\displaystyle \frac{9 x \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{9 x \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle - 9 x y^{2} z + 9 x y^{2} - 9 x y z^{2} + 24 x y z - 15 x y + 9 x z^{2} - 15 x z + 6 x + \frac{3 y^{2} z}{2} - \frac{3 y^{2}}{2} + \frac{3 y z^{2}}{2} - 5 y z + \frac{7 y}{2} - \frac{3 z^{2}}{2} + \frac{7 z}{2} - 2\\\displaystyle \frac{9 x \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{9 x \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle - 9 x y^{2} z + 9 x y^{2} - 9 x y z^{2} + 24 x y z - 15 x y + 9 x z^{2} - 15 x z + 6 x + \frac{3 y^{2} z}{2} - \frac{3 y^{2}}{2} + \frac{3 y z^{2}}{2} - 5 y z + \frac{7 y}{2} - \frac{3 z^{2}}{2} + \frac{7 z}{2} - 2\\\displaystyle \frac{9 x \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{9 x \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle 9 x^{2} y z - 9 x^{2} y - \frac{15 x^{2} z}{2} + \frac{15 x^{2}}{2} + 9 x y z^{2} - 24 x y z + 15 x y - \frac{15 x z^{2}}{2} + 19 x z - \frac{23 x}{2} - 9 y z^{2} + 15 y z - 6 y + \frac{15 z^{2}}{2} - \frac{23 z}{2} + 4\\\displaystyle \frac{9 y \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle 9 x^{2} y z - 9 x^{2} y - \frac{15 x^{2} z}{2} + \frac{15 x^{2}}{2} + 9 x y z^{2} - 24 x y z + 15 x y - \frac{15 x z^{2}}{2} + 19 x z - \frac{23 x}{2} - 9 y z^{2} + 15 y z - 6 y + \frac{15 z^{2}}{2} - \frac{23 z}{2} + 4\\\displaystyle \frac{9 y \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle - 9 x^{2} y z + 9 x^{2} y + \frac{3 x^{2} z}{2} - \frac{3 x^{2}}{2} - 9 x y z^{2} + 24 x y z - 15 x y + \frac{3 x z^{2}}{2} - 5 x z + \frac{7 x}{2} + 9 y z^{2} - 15 y z + 6 y - \frac{3 z^{2}}{2} + \frac{7 z}{2} - 2\\\displaystyle \frac{9 y \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle - 9 x^{2} y z + 9 x^{2} y + \frac{3 x^{2} z}{2} - \frac{3 x^{2}}{2} - 9 x y z^{2} + 24 x y z - 15 x y + \frac{3 x z^{2}}{2} - 5 x z + \frac{7 x}{2} + 9 y z^{2} - 15 y z + 6 y - \frac{3 z^{2}}{2} + \frac{7 z}{2} - 2\\\displaystyle \frac{9 y \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{9 z \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle \frac{9 z \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle 9 x^{2} y z - \frac{15 x^{2} y}{2} - 9 x^{2} z + \frac{15 x^{2}}{2} + 9 x y^{2} z - \frac{15 x y^{2}}{2} - 24 x y z + 19 x y + 15 x z - \frac{23 x}{2} - 9 y^{2} z + \frac{15 y^{2}}{2} + 15 y z - \frac{23 y}{2} - 6 z + 4\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{9 z \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle \frac{9 z \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle 9 x^{2} y z - \frac{15 x^{2} y}{2} - 9 x^{2} z + \frac{15 x^{2}}{2} + 9 x y^{2} z - \frac{15 x y^{2}}{2} - 24 x y z + 19 x y + 15 x z - \frac{23 x}{2} - 9 y^{2} z + \frac{15 y^{2}}{2} + 15 y z - \frac{23 y}{2} - 6 z + 4\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \frac{9 z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle \frac{9 z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle - 9 x^{2} y z + \frac{3 x^{2} y}{2} + 9 x^{2} z - \frac{3 x^{2}}{2} - 9 x y^{2} z + \frac{3 x y^{2}}{2} + 24 x y z - 5 x y - 15 x z + \frac{7 x}{2} + 9 y^{2} z - \frac{3 y^{2}}{2} - 15 y z + \frac{7 y}{2} + 6 z - 2\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \frac{9 z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle \frac{9 z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle - 9 x^{2} y z + \frac{3 x^{2} y}{2} + 9 x^{2} z - \frac{3 x^{2}}{2} - 9 x y^{2} z + \frac{3 x y^{2}}{2} + 24 x y z - 5 x y - 15 x z + \frac{7 x}{2} + 9 y^{2} z - \frac{3 y^{2}}{2} - 15 y z + \frac{7 y}{2} + 6 z - 2\end{array}\right)\)
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 3rd edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{3}\).
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle \frac{x \left(18 x y z - 18 x y - 15 x z + 15 x - 18 y z^{2} + 12 y z + 6 y + 15 z^{2} - 8 z - 7\right)}{2}\\\displaystyle \frac{9 x y \left(2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 3 of the reference element.
where \(e_{3}\) is the 3rd edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{3}\).
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle \frac{x \left(18 x y z - 18 x y - 15 x z + 15 x - 18 y z^{2} + 12 y z + 6 y + 15 z^{2} - 8 z - 7\right)}{2}\\\displaystyle \frac{9 x y \left(2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)
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 3rd edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{3}\).
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle \frac{x \left(- 18 x y z + 18 x y + 3 x z - 3 x + 18 y z^{2} - 12 y z - 6 y - 3 z^{2} + 4 z - 1\right)}{2}\\\displaystyle \frac{9 x y \left(- 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 3 of the reference element.
where \(e_{3}\) is the 3rd edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{3}\).
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \frac{9 y \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle \frac{x \left(- 18 x y z + 18 x y + 3 x z - 3 x + 18 y z^{2} - 12 y z - 6 y - 3 z^{2} + 4 z - 1\right)}{2}\\\displaystyle \frac{9 x y \left(- 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
\(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{4}\).
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{9 z \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle \frac{9 x z \left(2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{x \left(18 x y z - 15 x y - 18 x z + 15 x - 18 y^{2} z + 15 y^{2} + 12 y z - 8 y + 6 z - 7\right)}{2}\end{array}\right)\)
This DOF is associated with edge 4 of the reference element.
where \(e_{4}\) is the 4th edge;
\(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{4}\).
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{9 z \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)}{2}\\\displaystyle \frac{9 x z \left(2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{x \left(18 x y z - 15 x y - 18 x z + 15 x - 18 y^{2} z + 15 y^{2} + 12 y z - 8 y + 6 z - 7\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{4}\).
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \frac{9 z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle \frac{9 x z \left(- 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{x \left(- 18 x y z + 3 x y + 18 x z - 3 x + 18 y^{2} z - 3 y^{2} - 12 y z + 4 y - 6 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 4 of the reference element.
where \(e_{4}\) is the 4th edge;
\(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{4}\).
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle \frac{9 z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)}{2}\\\displaystyle \frac{9 x z \left(- 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{x \left(- 18 x y z + 3 x y + 18 x z - 3 x + 18 y^{2} z - 3 y^{2} - 12 y z + 4 y - 6 z - 1\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{5}\).
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle \frac{y \left(18 x y z - 18 x y - 18 x z^{2} + 12 x z + 6 x - 15 y z + 15 y + 15 z^{2} - 8 z - 7\right)}{2}\\\displaystyle \frac{9 x \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{9 x y \left(2 x z - x - 2 z + 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 5 of the reference element.
where \(e_{5}\) is the 5th edge;
\(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{5}\).
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle \frac{y \left(18 x y z - 18 x y - 18 x z^{2} + 12 x z + 6 x - 15 y z + 15 y + 15 z^{2} - 8 z - 7\right)}{2}\\\displaystyle \frac{9 x \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{9 x y \left(2 x z - x - 2 z + 1\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{5}\).
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{y \left(- 18 x y z + 18 x y + 18 x z^{2} - 12 x z - 6 x + 3 y z - 3 y - 3 z^{2} + 4 z - 1\right)}{2}\\\displaystyle \frac{9 x \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{9 x y \left(- 2 x z + x + 2 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 5 of the reference element.
where \(e_{5}\) is the 5th edge;
\(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{5}\).
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{y \left(- 18 x y z + 18 x y + 18 x z^{2} - 12 x z - 6 x + 3 y z - 3 y - 3 z^{2} + 4 z - 1\right)}{2}\\\displaystyle \frac{9 x \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{9 x y \left(- 2 x z + x + 2 z - 1\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{6}\).
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(2 x z - 2 x - z + 1\right)}{2}\\\displaystyle \frac{9 z \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{y \left(- 18 x^{2} z + 15 x^{2} + 18 x y z - 15 x y + 12 x z - 8 x - 18 y z + 15 y + 6 z - 7\right)}{2}\end{array}\right)\)
This DOF is associated with edge 6 of the reference element.
where \(e_{6}\) is the 6th edge;
\(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{6}\).
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(2 x z - 2 x - z + 1\right)}{2}\\\displaystyle \frac{9 z \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{y \left(- 18 x^{2} z + 15 x^{2} + 18 x y z - 15 x y + 12 x z - 8 x - 18 y z + 15 y + 6 z - 7\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{6}\).
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(- 2 x z + 2 x + z - 1\right)}{2}\\\displaystyle \frac{9 z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{y \left(18 x^{2} z - 3 x^{2} - 18 x y z + 3 x y - 12 x z + 4 x + 18 y z - 3 y - 6 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 6 of the reference element.
where \(e_{6}\) is the 6th edge;
\(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{6}\).
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(- 2 x z + 2 x + z - 1\right)}{2}\\\displaystyle \frac{9 z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{y \left(18 x^{2} z - 3 x^{2} - 18 x y z + 3 x y - 12 x z + 4 x + 18 y z - 3 y - 6 z - 1\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{7}\).
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(2 x z - 2 x - z + 1\right)}{2}\\\displaystyle \frac{9 x z \left(2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{x y \left(- 18 x z + 15 x - 18 y z + 15 y + 24 z - 22\right)}{2}\end{array}\right)\)
This DOF is associated with edge 7 of the reference element.
where \(e_{7}\) is the 7th edge;
\(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{7}\).
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(2 x z - 2 x - z + 1\right)}{2}\\\displaystyle \frac{9 x z \left(2 y z - 2 y - z + 1\right)}{2}\\\displaystyle \frac{x y \left(- 18 x z + 15 x - 18 y z + 15 y + 24 z - 22\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{7}\).
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(- 2 x z + 2 x + z - 1\right)}{2}\\\displaystyle \frac{9 x z \left(- 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{x y \left(18 x z - 3 x + 18 y z - 3 y - 24 z + 2\right)}{2}\end{array}\right)\)
This DOF is associated with edge 7 of the reference element.
where \(e_{7}\) is the 7th edge;
\(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{7}\).
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(- 2 x z + 2 x + z - 1\right)}{2}\\\displaystyle \frac{9 x z \left(- 2 y z + 2 y + z - 1\right)}{2}\\\displaystyle \frac{x y \left(18 x z - 3 x + 18 y z - 3 y - 24 z + 2\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{8}\).
\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle \frac{z \left(- 18 x y^{2} + 18 x y z + 12 x y - 18 x z + 6 x + 15 y^{2} - 15 y z - 8 y + 15 z - 7\right)}{2}\\\displaystyle \frac{9 x z \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle \frac{9 x \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 8 of the reference element.
where \(e_{8}\) is the 8th edge;
\(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{8}\).
\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle \frac{z \left(- 18 x y^{2} + 18 x y z + 12 x y - 18 x z + 6 x + 15 y^{2} - 15 y z - 8 y + 15 z - 7\right)}{2}\\\displaystyle \frac{9 x z \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle \frac{9 x \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{8}\).
\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle \frac{z \left(18 x y^{2} - 18 x y z - 12 x y + 18 x z - 6 x - 3 y^{2} + 3 y z + 4 y - 3 z - 1\right)}{2}\\\displaystyle \frac{9 x z \left(- 2 x y + x + 2 y - 1\right)}{2}\\\displaystyle \frac{9 x \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 8 of the reference element.
where \(e_{8}\) is the 8th edge;
\(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{8}\).
\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle \frac{z \left(18 x y^{2} - 18 x y z - 12 x y + 18 x z - 6 x - 3 y^{2} + 3 y z + 4 y - 3 z - 1\right)}{2}\\\displaystyle \frac{9 x z \left(- 2 x y + x + 2 y - 1\right)}{2}\\\displaystyle \frac{9 x \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{9}\) is the tangent to edge 9;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{9}\).
\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(2 x y - 2 x - y + 1\right)}{2}\\\displaystyle \frac{z \left(- 18 x^{2} y + 15 x^{2} + 18 x y z + 12 x y - 15 x z - 8 x - 18 y z + 6 y + 15 z - 7\right)}{2}\\\displaystyle \frac{9 y \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 9 of the reference element.
where \(e_{9}\) is the 9th edge;
\(\hat{\boldsymbol{t}}_{9}\) is the tangent to edge 9;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{9}\).
\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(2 x y - 2 x - y + 1\right)}{2}\\\displaystyle \frac{z \left(- 18 x^{2} y + 15 x^{2} + 18 x y z + 12 x y - 15 x z - 8 x - 18 y z + 6 y + 15 z - 7\right)}{2}\\\displaystyle \frac{9 y \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{9}\) is the tangent to edge 9;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{9}\).
\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle \frac{z \left(18 x^{2} y - 3 x^{2} - 18 x y z - 12 x y + 3 x z + 4 x + 18 y z - 6 y - 3 z - 1\right)}{2}\\\displaystyle \frac{9 y \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 9 of the reference element.
where \(e_{9}\) is the 9th edge;
\(\hat{\boldsymbol{t}}_{9}\) is the tangent to edge 9;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{9}\).
\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle \frac{z \left(18 x^{2} y - 3 x^{2} - 18 x y z - 12 x y + 3 x z + 4 x + 18 y z - 6 y - 3 z - 1\right)}{2}\\\displaystyle \frac{9 y \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{10}\) is the tangent to edge 10;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{10}\).
\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(2 x y - 2 x - y + 1\right)}{2}\\\displaystyle \frac{x z \left(- 18 x y + 15 x - 18 y z + 24 y + 15 z - 22\right)}{2}\\\displaystyle \frac{9 x y \left(2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 10 of the reference element.
where \(e_{10}\) is the 10th edge;
\(\hat{\boldsymbol{t}}_{10}\) is the tangent to edge 10;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{10}\).
\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(2 x y - 2 x - y + 1\right)}{2}\\\displaystyle \frac{x z \left(- 18 x y + 15 x - 18 y z + 24 y + 15 z - 22\right)}{2}\\\displaystyle \frac{9 x y \left(2 y z - y - 2 z + 1\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{10}\) is the tangent to edge 10;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{10}\).
\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle \frac{x z \left(18 x y - 3 x + 18 y z - 24 y - 3 z + 2\right)}{2}\\\displaystyle \frac{9 x y \left(- 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 10 of the reference element.
where \(e_{10}\) is the 10th edge;
\(\hat{\boldsymbol{t}}_{10}\) is the tangent to edge 10;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{10}\).
\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle \frac{9 y z \left(- 2 x y + 2 x + y - 1\right)}{2}\\\displaystyle \frac{x z \left(18 x y - 3 x + 18 y z - 24 y - 3 z + 2\right)}{2}\\\displaystyle \frac{9 x y \left(- 2 y z + y + 2 z - 1\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{11}\) is the tangent to edge 11;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{11}\).
\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle \frac{y z \left(- 18 x y - 18 x z + 24 x + 15 y + 15 z - 22\right)}{2}\\\displaystyle \frac{9 x z \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle \frac{9 x y \left(2 x z - x - 2 z + 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 11 of the reference element.
where \(e_{11}\) is the 11th edge;
\(\hat{\boldsymbol{t}}_{11}\) is the tangent to edge 11;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{11}\).
\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle \frac{y z \left(- 18 x y - 18 x z + 24 x + 15 y + 15 z - 22\right)}{2}\\\displaystyle \frac{9 x z \left(2 x y - x - 2 y + 1\right)}{2}\\\displaystyle \frac{9 x y \left(2 x z - x - 2 z + 1\right)}{2}\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{11}\) is the tangent to edge 11;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{11}\).
\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle \frac{y z \left(18 x y + 18 x z - 24 x - 3 y - 3 z + 2\right)}{2}\\\displaystyle \frac{9 x z \left(- 2 x y + x + 2 y - 1\right)}{2}\\\displaystyle \frac{9 x y \left(- 2 x z + x + 2 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 11 of the reference element.
where \(e_{11}\) is the 11th edge;
\(\hat{\boldsymbol{t}}_{11}\) is the tangent to edge 11;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{11}\).
\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle \frac{y z \left(18 x y + 18 x z - 24 x - 3 y - 3 z + 2\right)}{2}\\\displaystyle \frac{9 x z \left(- 2 x y + x + 2 y - 1\right)}{2}\\\displaystyle \frac{9 x y \left(- 2 x z + x + 2 z - 1\right)}{2}\end{array}\right)\)
This DOF is associated with edge 11 of the reference element.
\(\displaystyle l_{24}:\mathbf{v}\mapsto\displaystyle\int_{f_{0}}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{0}\) is the 0th face.
\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 3 y \left(- 6 x y z + 6 x y + 6 x z - 6 x + 5 y z - 5 y - 5 z + 5\right)\\\displaystyle 9 x \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face.
\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 3 y \left(- 6 x y z + 6 x y + 6 x z - 6 x + 5 y z - 5 y - 5 z + 5\right)\\\displaystyle 9 x \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{25}:\mathbf{v}\mapsto\displaystyle\int_{f_{0}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{0}\) is the 0th face.
\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 9 y \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)\\\displaystyle 3 x \left(6 x y z - 6 x y - 5 x z + 5 x - 6 y z + 6 y + 5 z - 5\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face.
\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 9 y \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)\\\displaystyle 3 x \left(6 x y z - 6 x y - 5 x z + 5 x - 6 y z + 6 y + 5 z - 5\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{26}:\mathbf{v}\mapsto\displaystyle\int_{f_{0}}\left(\begin{array}{c}\displaystyle s_{0}\\\displaystyle - s_{1}\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{0}\) is the 0th face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 18 y \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)\\\displaystyle 18 x \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 18 y \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)\\\displaystyle 18 x \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{27}:\mathbf{v}\mapsto\displaystyle\int_{f_{1}}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{1}\) is the 1st face.
\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 3 z \left(- 6 x y z + 6 x y + 6 x z - 6 x + 5 y z - 5 y - 5 z + 5\right)\\\displaystyle 0\\\displaystyle 9 x \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face.
\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 3 z \left(- 6 x y z + 6 x y + 6 x z - 6 x + 5 y z - 5 y - 5 z + 5\right)\\\displaystyle 0\\\displaystyle 9 x \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{28}:\mathbf{v}\mapsto\displaystyle\int_{f_{1}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle -1\end{array}\right)\cdot\mathbf{v}\)
where \(f_{1}\) is the 1st face.
\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 9 z \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)\\\displaystyle 0\\\displaystyle 3 x \left(6 x y z - 5 x y - 6 x z + 5 x - 6 y z + 5 y + 6 z - 5\right)\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face.
\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 9 z \left(- 2 x y z + 2 x y + 2 x z - 2 x + y z - y - z + 1\right)\\\displaystyle 0\\\displaystyle 3 x \left(6 x y z - 5 x y - 6 x z + 5 x - 6 y z + 5 y + 6 z - 5\right)\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{29}:\mathbf{v}\mapsto\displaystyle\int_{f_{1}}\left(\begin{array}{c}\displaystyle s_{0}\\\displaystyle 0\\\displaystyle - s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(f_{1}\) is the 1st face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 18 z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)\\\displaystyle 0\\\displaystyle 18 x \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 18 z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)\\\displaystyle 0\\\displaystyle 18 x \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{30}:\mathbf{v}\mapsto\displaystyle\int_{f_{2}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{2}\) is the 2nd face.
\(\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 z \left(- 6 x y z + 6 x y + 5 x z - 5 x + 6 y z - 6 y - 5 z + 5\right)\\\displaystyle 9 y \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face.
\(\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 z \left(- 6 x y z + 6 x y + 5 x z - 5 x + 6 y z - 6 y - 5 z + 5\right)\\\displaystyle 9 y \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{31}:\mathbf{v}\mapsto\displaystyle\int_{f_{2}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle -1\end{array}\right)\cdot\mathbf{v}\)
where \(f_{2}\) is the 2nd face.
\(\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 9 z \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)\\\displaystyle 3 y \left(6 x y z - 5 x y - 6 x z + 5 x - 6 y z + 5 y + 6 z - 5\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face.
\(\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 9 z \left(- 2 x y z + 2 x y + x z - x + 2 y z - 2 y - z + 1\right)\\\displaystyle 3 y \left(6 x y z - 5 x y - 6 x z + 5 x - 6 y z + 5 y + 6 z - 5\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{32}:\mathbf{v}\mapsto\displaystyle\int_{f_{2}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle s_{0}\\\displaystyle - s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(f_{2}\) is the 2nd face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 18 z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)\\\displaystyle 18 y \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 18 z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)\\\displaystyle 18 y \left(- 2 x y z + x y + 2 x z - x + 2 y z - y - 2 z + 1\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{33}:\mathbf{v}\mapsto\displaystyle\int_{f_{3}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{3}\) is the 3rd face.
\(\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 x z \left(6 y z - 6 y - 5 z + 5\right)\\\displaystyle 9 x y \left(- 2 y z + y + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face.
\(\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 x z \left(6 y z - 6 y - 5 z + 5\right)\\\displaystyle 9 x y \left(- 2 y z + y + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{34}:\mathbf{v}\mapsto\displaystyle\int_{f_{3}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle -1\end{array}\right)\cdot\mathbf{v}\)
where \(f_{3}\) is the 3rd face.
\(\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 9 x z \left(2 y z - 2 y - z + 1\right)\\\displaystyle 3 x y \left(- 6 y z + 5 y + 6 z - 5\right)\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face.
\(\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 9 x z \left(2 y z - 2 y - z + 1\right)\\\displaystyle 3 x y \left(- 6 y z + 5 y + 6 z - 5\right)\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{35}:\mathbf{v}\mapsto\displaystyle\int_{f_{3}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle s_{0}\\\displaystyle - s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(f_{3}\) is the 3rd face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 18 x z \left(- 2 y z + 2 y + z - 1\right)\\\displaystyle 18 x y \left(2 y z - y - 2 z + 1\right)\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 18 x z \left(- 2 y z + 2 y + z - 1\right)\\\displaystyle 18 x y \left(2 y z - y - 2 z + 1\right)\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{36}:\mathbf{v}\mapsto\displaystyle\int_{f_{4}}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{4}\) is the 4th face.
\(\displaystyle \boldsymbol{\phi}_{36} = \left(\begin{array}{c}\displaystyle 3 y z \left(6 x z - 6 x - 5 z + 5\right)\\\displaystyle 0\\\displaystyle 9 x y \left(- 2 x z + x + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
where \(f_{4}\) is the 4th face.
\(\displaystyle \boldsymbol{\phi}_{36} = \left(\begin{array}{c}\displaystyle 3 y z \left(6 x z - 6 x - 5 z + 5\right)\\\displaystyle 0\\\displaystyle 9 x y \left(- 2 x z + x + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{37}:\mathbf{v}\mapsto\displaystyle\int_{f_{4}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle -1\end{array}\right)\cdot\mathbf{v}\)
where \(f_{4}\) is the 4th face.
\(\displaystyle \boldsymbol{\phi}_{37} = \left(\begin{array}{c}\displaystyle 9 y z \left(2 x z - 2 x - z + 1\right)\\\displaystyle 0\\\displaystyle 3 x y \left(- 6 x z + 5 x + 6 z - 5\right)\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
where \(f_{4}\) is the 4th face.
\(\displaystyle \boldsymbol{\phi}_{37} = \left(\begin{array}{c}\displaystyle 9 y z \left(2 x z - 2 x - z + 1\right)\\\displaystyle 0\\\displaystyle 3 x y \left(- 6 x z + 5 x + 6 z - 5\right)\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{38}:\mathbf{v}\mapsto\displaystyle\int_{f_{4}}\left(\begin{array}{c}\displaystyle s_{0}\\\displaystyle 0\\\displaystyle - s_{1}\end{array}\right)\cdot\mathbf{v}\)
where \(f_{4}\) is the 4th face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{4}\).
\(\displaystyle \boldsymbol{\phi}_{38} = \left(\begin{array}{c}\displaystyle 18 y z \left(- 2 x z + 2 x + z - 1\right)\\\displaystyle 0\\\displaystyle 18 x y \left(2 x z - x - 2 z + 1\right)\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
where \(f_{4}\) is the 4th face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{4}\).
\(\displaystyle \boldsymbol{\phi}_{38} = \left(\begin{array}{c}\displaystyle 18 y z \left(- 2 x z + 2 x + z - 1\right)\\\displaystyle 0\\\displaystyle 18 x y \left(2 x z - x - 2 z + 1\right)\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{39}:\mathbf{v}\mapsto\displaystyle\int_{f_{5}}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{5}\) is the 5th face.
\(\displaystyle \boldsymbol{\phi}_{39} = \left(\begin{array}{c}\displaystyle 3 y z \left(6 x y - 6 x - 5 y + 5\right)\\\displaystyle 9 x z \left(- 2 x y + x + 2 y - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
where \(f_{5}\) is the 5th face.
\(\displaystyle \boldsymbol{\phi}_{39} = \left(\begin{array}{c}\displaystyle 3 y z \left(6 x y - 6 x - 5 y + 5\right)\\\displaystyle 9 x z \left(- 2 x y + x + 2 y - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{40}:\mathbf{v}\mapsto\displaystyle\int_{f_{5}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{5}\) is the 5th face.
\(\displaystyle \boldsymbol{\phi}_{40} = \left(\begin{array}{c}\displaystyle 9 y z \left(2 x y - 2 x - y + 1\right)\\\displaystyle 3 x z \left(- 6 x y + 5 x + 6 y - 5\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
where \(f_{5}\) is the 5th face.
\(\displaystyle \boldsymbol{\phi}_{40} = \left(\begin{array}{c}\displaystyle 9 y z \left(2 x y - 2 x - y + 1\right)\\\displaystyle 3 x z \left(- 6 x y + 5 x + 6 y - 5\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{41}:\mathbf{v}\mapsto\displaystyle\int_{f_{5}}\left(\begin{array}{c}\displaystyle s_{0}\\\displaystyle - s_{1}\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{5}\) is the 5th face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{5}\).
\(\displaystyle \boldsymbol{\phi}_{41} = \left(\begin{array}{c}\displaystyle 18 y z \left(- 2 x y + 2 x + y - 1\right)\\\displaystyle 18 x z \left(2 x y - x - 2 y + 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
where \(f_{5}\) is the 5th face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{5}\).
\(\displaystyle \boldsymbol{\phi}_{41} = \left(\begin{array}{c}\displaystyle 18 y z \left(- 2 x y + 2 x + y - 1\right)\\\displaystyle 18 x z \left(2 x y - x - 2 y + 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 5 of the reference element.