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Degree 2 trimmed serendipity 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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle - y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - z^{2}\\\displaystyle 0\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle - x z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle - y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - y z\\\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle - x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - x z\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle - x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z^{2}\\\displaystyle x z^{2}\\\displaystyle 2 x y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2} z\\\displaystyle 2 x y z\\\displaystyle x y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 2 x y z\\\displaystyle x^{2} z\\\displaystyle x^{2} y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 2 x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 2 y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 y z\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle x z\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 2 x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 2 x y\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 2 x z\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle - x z^{2}\\\displaystyle x y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - y z^{2}\\\displaystyle 0\\\displaystyle x y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - y^{2} z\\\displaystyle x y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle - x y z\\\displaystyle x y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - x y z\\\displaystyle 0\\\displaystyle x^{2} y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - x y z\\\displaystyle x^{2} z\\\displaystyle 0\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{35}\}\)
- 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 \(e_{0}\).
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 6 x y z + 6 x y + 6 x z - 6 x - 3 y^{2} z + 3 y^{2} - 3 y z^{2} + 10 y z - 7 y + 3 z^{2} - 7 z + 4\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{0}\).
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 6 x y z + 6 x y + 6 x z - 6 x - 3 y^{2} z + 3 y^{2} - 3 y z^{2} + 10 y z - 7 y + 3 z^{2} - 7 z + 4\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{0}\).
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 6 x y z - 6 x y - 6 x z + 6 x - 3 y^{2} z + 3 y^{2} - 3 y z^{2} + 4 y z - y + 3 z^{2} - z - 2\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{0}\).
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 6 x y z - 6 x y - 6 x z + 6 x - 3 y^{2} z + 3 y^{2} - 3 y z^{2} + 4 y z - y + 3 z^{2} - z - 2\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{1}\).
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 3 x^{2} z + 3 x^{2} - 6 x y z + 6 x y - 3 x z^{2} + 10 x z - 7 x + 6 y z - 6 y + 3 z^{2} - 7 z + 4\\\displaystyle 0\end{array}\right)\)
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 \(e_{1}\).
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 3 x^{2} z + 3 x^{2} - 6 x y z + 6 x y - 3 x z^{2} + 10 x z - 7 x + 6 y z - 6 y + 3 z^{2} - 7 z + 4\\\displaystyle 0\end{array}\right)\)
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 \(e_{1}\).
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 3 x^{2} z + 3 x^{2} + 6 x y z - 6 x y - 3 x z^{2} + 4 x z - x - 6 y z + 6 y + 3 z^{2} - z - 2\\\displaystyle 0\end{array}\right)\)
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 \(e_{1}\).
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 3 x^{2} z + 3 x^{2} + 6 x y z - 6 x y - 3 x z^{2} + 4 x z - x - 6 y z + 6 y + 3 z^{2} - z - 2\\\displaystyle 0\end{array}\right)\)
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 \(e_{2}\).
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 3 x^{2} y + 3 x^{2} - 3 x y^{2} - 6 x y z + 10 x y + 6 x z - 7 x + 3 y^{2} + 6 y z - 7 y - 6 z + 4\end{array}\right)\)
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 \(e_{2}\).
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 3 x^{2} y + 3 x^{2} - 3 x y^{2} - 6 x y z + 10 x y + 6 x z - 7 x + 3 y^{2} + 6 y z - 7 y - 6 z + 4\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 3 x^{2} y + 3 x^{2} - 3 x y^{2} + 6 x y z + 4 x y - 6 x z - x + 3 y^{2} - 6 y z - y + 6 z - 2\end{array}\right)\)
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 \(e_{2}\).
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 3 x^{2} y + 3 x^{2} - 3 x y^{2} + 6 x y z + 4 x y - 6 x z - x + 3 y^{2} - 6 y z - y + 6 z - 2\end{array}\right)\)
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 3rd edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{3}\).
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(- 3 x z + 3 x + 6 y z - 6 y + 3 z^{2} - 4 z + 1\right)\\\displaystyle 0\end{array}\right)\)
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 \(e_{3}\).
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(- 3 x z + 3 x + 6 y z - 6 y + 3 z^{2} - 4 z + 1\right)\\\displaystyle 0\end{array}\right)\)
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 \(e_{3}\).
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(- 3 x z + 3 x - 6 y z + 6 y + 3 z^{2} + 2 z - 5\right)\\\displaystyle 0\end{array}\right)\)
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 \(e_{3}\).
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(- 3 x z + 3 x - 6 y z + 6 y + 3 z^{2} + 2 z - 5\right)\\\displaystyle 0\end{array}\right)\)
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 \(e_{4}\).
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x \left(- 3 x y + 3 x + 3 y^{2} + 6 y z - 4 y - 6 z + 1\right)\end{array}\right)\)
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 \(e_{4}\).
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x \left(- 3 x y + 3 x + 3 y^{2} + 6 y z - 4 y - 6 z + 1\right)\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{4}\).
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x \left(- 3 x y + 3 x + 3 y^{2} - 6 y z + 2 y + 6 z - 5\right)\end{array}\right)\)
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 \(e_{4}\).
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x \left(- 3 x y + 3 x + 3 y^{2} - 6 y z + 2 y + 6 z - 5\right)\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{5}\).
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle y \left(6 x z - 6 x - 3 y z + 3 y + 3 z^{2} - 4 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{5}\).
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle y \left(6 x z - 6 x - 3 y z + 3 y + 3 z^{2} - 4 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{5}\).
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle y \left(- 6 x z + 6 x - 3 y z + 3 y + 3 z^{2} + 2 z - 5\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{5}\).
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle y \left(- 6 x z + 6 x - 3 y z + 3 y + 3 z^{2} + 2 z - 5\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{6}\).
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y \left(3 x^{2} - 3 x y + 6 x z - 4 x + 3 y - 6 z + 1\right)\end{array}\right)\)
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 \(e_{6}\).
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y \left(3 x^{2} - 3 x y + 6 x z - 4 x + 3 y - 6 z + 1\right)\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{6}\).
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y \left(3 x^{2} - 3 x y - 6 x z + 2 x + 3 y + 6 z - 5\right)\end{array}\right)\)
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 \(e_{6}\).
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y \left(3 x^{2} - 3 x y - 6 x z + 2 x + 3 y + 6 z - 5\right)\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{7}\).
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y \left(3 x + 3 y - 6 z - 2\right)\end{array}\right)\)
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 \(e_{7}\).
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y \left(3 x + 3 y - 6 z - 2\right)\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{7}\).
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y \left(3 x + 3 y + 6 z - 8\right)\end{array}\right)\)
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 \(e_{7}\).
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y \left(3 x + 3 y + 6 z - 8\right)\end{array}\right)\)
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;
\(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{8}\).
\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle z \left(6 x y - 6 x + 3 y^{2} - 3 y z - 4 y + 3 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{8}\).
\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle z \left(6 x y - 6 x + 3 y^{2} - 3 y z - 4 y + 3 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{8}\).
\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle z \left(- 6 x y + 6 x + 3 y^{2} - 3 y z + 2 y + 3 z - 5\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{8}\).
\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle z \left(- 6 x y + 6 x + 3 y^{2} - 3 y z + 2 y + 3 z - 5\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{9}\).
\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(3 x^{2} + 6 x y - 3 x z - 4 x - 6 y + 3 z + 1\right)\\\displaystyle 0\end{array}\right)\)
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 \(e_{9}\).
\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(3 x^{2} + 6 x y - 3 x z - 4 x - 6 y + 3 z + 1\right)\\\displaystyle 0\end{array}\right)\)
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 \(e_{9}\).
\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(3 x^{2} - 6 x y - 3 x z + 2 x + 6 y + 3 z - 5\right)\\\displaystyle 0\end{array}\right)\)
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 \(e_{9}\).
\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(3 x^{2} - 6 x y - 3 x z + 2 x + 6 y + 3 z - 5\right)\\\displaystyle 0\end{array}\right)\)
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 \(e_{10}\).
\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x z \left(3 x - 6 y + 3 z - 2\right)\\\displaystyle 0\end{array}\right)\)
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 \(e_{10}\).
\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x z \left(3 x - 6 y + 3 z - 2\right)\\\displaystyle 0\end{array}\right)\)
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 \(e_{10}\).
\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x z \left(3 x + 6 y + 3 z - 8\right)\\\displaystyle 0\end{array}\right)\)
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 \(e_{10}\).
\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x z \left(3 x + 6 y + 3 z - 8\right)\\\displaystyle 0\end{array}\right)\)
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 \(e_{11}\).
\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle y z \left(- 6 x + 3 y + 3 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{11}\).
\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle y z \left(- 6 x + 3 y + 3 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{11}\).
\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle y z \left(6 x + 3 y + 3 z - 8\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(e_{11}\).
\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle y z \left(6 x + 3 y + 3 z - 8\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 11 of the reference element.
\(\displaystyle l_{24}:\mathbf{v}\mapsto\displaystyle\int_{f_{0}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{0}\) is the 0th face.
\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x \left(- x z + x + z - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face.
\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x \left(- x z + x + z - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{25}:\mathbf{v}\mapsto\displaystyle\int_{f_{0}}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{0}\) is the 0th face.
\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 6 y \left(y z - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face.
\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 6 y \left(y z - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{26}:\mathbf{v}\mapsto\displaystyle\int_{f_{1}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle -1\end{array}\right)\cdot\mathbf{v}\)
where \(f_{1}\) is the 1st face.
\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x \left(- x y + x + y - 1\right)\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face.
\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x \left(- x y + x + y - 1\right)\end{array}\right)\)
This DOF is associated with face 1 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.
\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 6 z \left(y z - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face.
\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 6 z \left(y z - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{28}:\mathbf{v}\mapsto\displaystyle\int_{f_{2}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle -1\end{array}\right)\cdot\mathbf{v}\)
where \(f_{2}\) is the 2nd face.
\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 y \left(- x y + x + y - 1\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face.
\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 y \left(- x y + x + y - 1\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{29}:\mathbf{v}\mapsto\displaystyle\int_{f_{2}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{2}\) is the 2nd face.
\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 z \left(x z - x - z + 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face.
\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 z \left(x z - x - z + 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{30}:\mathbf{v}\mapsto\displaystyle\int_{f_{3}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle -1\end{array}\right)\cdot\mathbf{v}\)
where \(f_{3}\) is the 3rd face.
\(\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x y \left(y - 1\right)\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face.
\(\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x y \left(y - 1\right)\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{31}:\mathbf{v}\mapsto\displaystyle\int_{f_{3}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{3}\) is the 3rd face.
\(\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x z \left(1 - z\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face.
\(\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x z \left(1 - z\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{32}:\mathbf{v}\mapsto\displaystyle\int_{f_{4}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle -1\end{array}\right)\cdot\mathbf{v}\)
where \(f_{4}\) is the 4th face.
\(\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x y \left(x - 1\right)\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
where \(f_{4}\) is the 4th face.
\(\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x y \left(x - 1\right)\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{33}:\mathbf{v}\mapsto\displaystyle\int_{f_{4}}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{4}\) is the 4th face.
\(\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 6 y z \left(1 - z\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
where \(f_{4}\) is the 4th face.
\(\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 6 y z \left(1 - z\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{34}:\mathbf{v}\mapsto\displaystyle\int_{f_{5}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle -1\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{5}\) is the 5th face.
\(\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x z \left(x - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
where \(f_{5}\) is the 5th face.
\(\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x z \left(x - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{35}:\mathbf{v}\mapsto\displaystyle\int_{f_{5}}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\cdot\mathbf{v}\)
where \(f_{5}\) is the 5th face.
\(\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle 6 y z \left(1 - y\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
where \(f_{5}\) is the 5th face.
\(\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle 6 y z \left(1 - y\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 5 of the reference element.