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Degree 2 Brezzi–Douglas–Fortin–Marini 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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x \left(x + y + z\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(x + y + z\right)\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(x + y + z\right)\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle y^{2}\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle x y\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x \left(x + y\right)\\\displaystyle 0\\\displaystyle x y\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{17}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 6 x \left(11 x + 9 y + 9 z - 8\right)\\\displaystyle 6 y \left(- x - 3 y - 3 z + 2\right)\\\displaystyle 6 z \left(- x - 3 y - 3 z + 2\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 6 x \left(11 x + 9 y + 9 z - 8\right)\\\displaystyle 6 y \left(- x - 3 y - 3 z + 2\right)\\\displaystyle 6 z \left(- x - 3 y - 3 z + 2\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 6 x \left(- 3 x - y - 3 z + 2\right)\\\displaystyle 6 y \left(9 x + 11 y + 9 z - 8\right)\\\displaystyle 6 z \left(- 3 x - y - 3 z + 2\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 6 x \left(- 3 x - y - 3 z + 2\right)\\\displaystyle 6 y \left(9 x + 11 y + 9 z - 8\right)\\\displaystyle 6 z \left(- 3 x - y - 3 z + 2\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 6 x \left(- 3 x - 3 y - z + 2\right)\\\displaystyle 6 y \left(- 3 x - 3 y - z + 2\right)\\\displaystyle 6 z \left(9 x + 9 y + 11 z - 8\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 6 x \left(- 3 x - 3 y - z + 2\right)\\\displaystyle 6 y \left(- 3 x - 3 y - z + 2\right)\\\displaystyle 6 z \left(9 x + 9 y + 11 z - 8\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 66 x^{2} + 84 x y + 84 x z - 84 x - 24 y - 24 z + 18\\\displaystyle 6 y \left(- x + 2 y + 2 z - 1\right)\\\displaystyle 6 z \left(- x + 2 y + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 66 x^{2} + 84 x y + 84 x z - 84 x - 24 y - 24 z + 18\\\displaystyle 6 y \left(- x + 2 y + 2 z - 1\right)\\\displaystyle 6 z \left(- x + 2 y + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle - \frac{162 x^{2}}{5} - 84 x y + \frac{324 x z}{5} + \frac{84 x}{5} + 24 y - 6\\\displaystyle \frac{6 y \left(45 x + 2 y + 6 z - 11\right)}{5}\\\displaystyle - 144 x y + \frac{54 x z}{5} + \frac{144 x}{5} - \frac{204 y z}{5} + \frac{144 y}{5} - 36 z^{2} + \frac{102 z}{5} - \frac{36}{5}\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle - \frac{162 x^{2}}{5} - 84 x y + \frac{324 x z}{5} + \frac{84 x}{5} + 24 y - 6\\\displaystyle \frac{6 y \left(45 x + 2 y + 6 z - 11\right)}{5}\\\displaystyle - 144 x y + \frac{54 x z}{5} + \frac{144 x}{5} - \frac{204 y z}{5} + \frac{144 y}{5} - 36 z^{2} + \frac{102 z}{5} - \frac{36}{5}\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - \frac{18 x^{2}}{5} - \frac{744 x z}{5} + \frac{156 x}{5} + 24 z - 6\\\displaystyle \frac{6 y \left(- 15 x - 12 y - 16 z + 11\right)}{5}\\\displaystyle 144 x y + \frac{126 x z}{5} - \frac{144 x}{5} + \frac{144 y z}{5} - \frac{144 y}{5} + 24 z^{2} - \frac{102 z}{5} + \frac{36}{5}\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - \frac{18 x^{2}}{5} - \frac{744 x z}{5} + \frac{156 x}{5} + 24 z - 6\\\displaystyle \frac{6 y \left(- 15 x - 12 y - 16 z + 11\right)}{5}\\\displaystyle 144 x y + \frac{126 x z}{5} - \frac{144 x}{5} + \frac{144 y z}{5} - \frac{144 y}{5} + 24 z^{2} - \frac{102 z}{5} + \frac{36}{5}\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 6 x \left(- 2 x + y - 2 z + 1\right)\\\displaystyle - 84 x y + 24 x - 66 y^{2} - 84 y z + 84 y + 24 z - 18\\\displaystyle 6 z \left(- 2 x + y - 2 z + 1\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 6 x \left(- 2 x + y - 2 z + 1\right)\\\displaystyle - 84 x y + 24 x - 66 y^{2} - 84 y z + 84 y + 24 z - 18\\\displaystyle 6 z \left(- 2 x + y - 2 z + 1\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \frac{6 x \left(- 2 x - 45 y + 54 z - 1\right)}{5}\\\displaystyle 84 x y - 24 x + \frac{162 y^{2}}{5} + \frac{36 y z}{5} - \frac{156 y}{5} + 6\\\displaystyle - 144 x y + \frac{204 x z}{5} + \frac{144 x}{5} - \frac{54 y z}{5} + \frac{144 y}{5} - 36 z^{2} + \frac{42 z}{5} - \frac{36}{5}\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \frac{6 x \left(- 2 x - 45 y + 54 z - 1\right)}{5}\\\displaystyle 84 x y - 24 x + \frac{162 y^{2}}{5} + \frac{36 y z}{5} - \frac{156 y}{5} + 6\\\displaystyle - 144 x y + \frac{204 x z}{5} + \frac{144 x}{5} - \frac{54 y z}{5} + \frac{144 y}{5} - 36 z^{2} + \frac{42 z}{5} - \frac{36}{5}\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{6 x \left(12 x + 15 y - 44 z + 1\right)}{5}\\\displaystyle \frac{18 y^{2}}{5} + \frac{384 y z}{5} - \frac{84 y}{5} - 24 z + 6\\\displaystyle 144 x y - \frac{144 x z}{5} - \frac{144 x}{5} - \frac{126 y z}{5} - \frac{144 y}{5} + 48 z^{2} - \frac{42 z}{5} + \frac{36}{5}\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{6 x \left(12 x + 15 y - 44 z + 1\right)}{5}\\\displaystyle \frac{18 y^{2}}{5} + \frac{384 y z}{5} - \frac{84 y}{5} - 24 z + 6\\\displaystyle 144 x y - \frac{144 x z}{5} - \frac{144 x}{5} - \frac{126 y z}{5} - \frac{144 y}{5} + 48 z^{2} - \frac{42 z}{5} + \frac{36}{5}\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 6 x \left(2 x + 2 y - z - 1\right)\\\displaystyle 6 y \left(2 x + 2 y - z - 1\right)\\\displaystyle 84 x z - 24 x + 84 y z - 24 y + 66 z^{2} - 84 z + 18\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 6 x \left(2 x + 2 y - z - 1\right)\\\displaystyle 6 y \left(2 x + 2 y - z - 1\right)\\\displaystyle 84 x z - 24 x + 84 y z - 24 y + 66 z^{2} - 84 z + 18\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle \frac{66 x \left(- 2 x + 9 z - 1\right)}{5}\\\displaystyle \frac{6 y \left(- 10 x + 12 y - 9 z - 1\right)}{5}\\\displaystyle - 144 x y - \frac{276 x z}{5} + \frac{264 x}{5} - \frac{144 y z}{5} + \frac{144 y}{5} - 54 z^{2} + \frac{192 z}{5} - \frac{66}{5}\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle \frac{66 x \left(- 2 x + 9 z - 1\right)}{5}\\\displaystyle \frac{6 y \left(- 10 x + 12 y - 9 z - 1\right)}{5}\\\displaystyle - 144 x y - \frac{276 x z}{5} + \frac{264 x}{5} - \frac{144 y z}{5} + \frac{144 y}{5} - 54 z^{2} + \frac{192 z}{5} - \frac{66}{5}\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{6 x \left(12 x - 10 y - 69 z + 11\right)}{5}\\\displaystyle \frac{6 y \left(- 22 y + 39 z + 1\right)}{5}\\\displaystyle 144 x y - \frac{144 x z}{5} - \frac{144 x}{5} - \frac{276 y z}{5} - \frac{24 y}{5} + 18 z^{2} + \frac{48 z}{5} + \frac{6}{5}\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face;
\(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{6 x \left(12 x - 10 y - 69 z + 11\right)}{5}\\\displaystyle \frac{6 y \left(- 22 y + 39 z + 1\right)}{5}\\\displaystyle 144 x y - \frac{144 x z}{5} - \frac{144 x}{5} - \frac{276 y z}{5} - \frac{24 y}{5} + 18 z^{2} + \frac{48 z}{5} + \frac{6}{5}\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\- s_{2}\\s_{1}\end{array}\right)\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 12 x \left(- 6 x - 5 y + 32 z - 3\right)\\\displaystyle 12 y \left(y - 22 z + 2\right)\\\displaystyle - 720 x y + 144 x z + 144 x + 156 y z + 144 y - 120 z^{2} + 12 z - 36\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 12 x \left(- 6 x - 5 y + 32 z - 3\right)\\\displaystyle 12 y \left(y - 22 z + 2\right)\\\displaystyle - 720 x y + 144 x z + 144 x + 156 y z + 144 y - 120 z^{2} + 12 z - 36\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{2}\\0\\s_{0}\end{array}\right)\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 12 x \left(x - 52 z + 8\right)\\\displaystyle 12 y \left(- 5 x - 6 y + 2 z + 3\right)\\\displaystyle 720 x y + 156 x z - 144 x + 144 y z - 144 y + 240 z^{2} - 132 z + 36\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 12 x \left(x - 52 z + 8\right)\\\displaystyle 12 y \left(- 5 x - 6 y + 2 z + 3\right)\\\displaystyle 720 x y + 156 x z - 144 x + 144 y z - 144 y + 240 z^{2} - 132 z + 36\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{1}\\s_{0}\\0\end{array}\right)\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 12 x \left(- 11 x - 25 y + 27 z + 2\right)\\\displaystyle 12 y \left(25 x + 11 y + 3 z - 8\right)\\\displaystyle - 720 x y + 84 x z + 144 x - 84 y z + 144 y - 180 z^{2} + 72 z - 36\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 12 x \left(- 11 x - 25 y + 27 z + 2\right)\\\displaystyle 12 y \left(25 x + 11 y + 3 z - 8\right)\\\displaystyle - 720 x y + 84 x z + 144 x - 84 y z + 144 y - 180 z^{2} + 72 z - 36\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}s_{2}\\s_{2}\\- s_{0} - s_{1} + 1\end{array}\right)\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 60 x \left(x + y + 2 z - 1\right)\\\displaystyle 60 y \left(x + y + 2 z - 1\right)\\\displaystyle 60 z \left(- 5 x - 5 y - 4 z + 4\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 60 x \left(x + y + 2 z - 1\right)\\\displaystyle 60 y \left(x + y + 2 z - 1\right)\\\displaystyle 60 z \left(- 5 x - 5 y - 4 z + 4\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}s_{1}\\- s_{0} - s_{2} + 1\\s_{1}\end{array}\right)\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 60 x \left(x + 2 y + z - 1\right)\\\displaystyle 60 y \left(- 5 x - 4 y - 5 z + 4\right)\\\displaystyle 60 z \left(x + 2 y + z - 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 60 x \left(x + 2 y + z - 1\right)\\\displaystyle 60 y \left(- 5 x - 4 y - 5 z + 4\right)\\\displaystyle 60 z \left(x + 2 y + z - 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{1} - s_{2} + 1\\s_{0}\\s_{0}\end{array}\right)\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 60 x \left(- 4 x - 5 y - 5 z + 4\right)\\\displaystyle 60 y \left(2 x + y + z - 1\right)\\\displaystyle 60 z \left(2 x + y + z - 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 60 x \left(- 4 x - 5 y - 5 z + 4\right)\\\displaystyle 60 y \left(2 x + y + z - 1\right)\\\displaystyle 60 z \left(2 x + y + z - 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.