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