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Degree 2 Brezzi–Douglas–Fortin–Marini 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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{20}\}\)
- 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 \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x z - 6 x + 6 y z - 6 y + 3 z^{2} - 10 z + 7\end{array}\right)\)
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 \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x z - 6 x + 6 y z - 6 y + 3 z^{2} - 10 z + 7\end{array}\right)\)
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 \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 6 x z + 6 x + 6 y z - 6 y + 3 z^{2} - 4 z + 1\end{array}\right)\)
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 \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 6 x z + 6 x + 6 y z - 6 y + 3 z^{2} - 4 z + 1\end{array}\right)\)
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 \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x z - 6 x - 6 y z + 6 y + 3 z^{2} - 4 z + 1\end{array}\right)\)
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 \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 x z - 6 x - 6 y z + 6 y + 3 z^{2} - 4 z + 1\end{array}\right)\)
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 \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 6 x y + 6 x - 3 y^{2} - 6 y z + 10 y + 6 z - 7\\\displaystyle 0\end{array}\right)\)
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 \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 6 x y + 6 x - 3 y^{2} - 6 y z + 10 y + 6 z - 7\\\displaystyle 0\end{array}\right)\)
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 \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x y - 6 x - 3 y^{2} - 6 y z + 4 y + 6 z - 1\\\displaystyle 0\end{array}\right)\)
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 \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x y - 6 x - 3 y^{2} - 6 y z + 4 y + 6 z - 1\\\displaystyle 0\end{array}\right)\)
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 \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 6 x y + 6 x - 3 y^{2} + 6 y z + 4 y - 6 z - 1\\\displaystyle 0\end{array}\right)\)
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 \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 6 x y + 6 x - 3 y^{2} + 6 y z + 4 y - 6 z - 1\\\displaystyle 0\end{array}\right)\)
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 \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 3 x^{2} + 6 x y + 6 x z - 10 x - 6 y - 6 z + 7\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 3 x^{2} + 6 x y + 6 x z - 10 x - 6 y - 6 z + 7\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 3 x^{2} - 6 x y + 6 x z - 4 x + 6 y - 6 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 3 x^{2} - 6 x y + 6 x z - 4 x + 6 y - 6 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 3 x^{2} + 6 x y - 6 x z - 4 x - 6 y + 6 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 3 x^{2} + 6 x y - 6 x z - 4 x - 6 y + 6 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle x \left(3 x - 6 y - 6 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle x \left(3 x - 6 y - 6 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle x \left(3 x + 6 y - 6 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle x \left(3 x + 6 y - 6 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle x \left(3 x - 6 y + 6 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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 \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle x \left(3 x - 6 y + 6 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{4}\).
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(6 x - 3 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{4}\).
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(6 x - 3 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{4}\).
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(- 6 x - 3 y + 6 z + 2\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{4}\).
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(- 6 x - 3 y + 6 z + 2\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{4}\)
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{4}\).
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(6 x - 3 y - 6 z + 2\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
where \(f_{4}\) is the 4th face;
\(\hat{\boldsymbol{n}}_{4}\) is the normal to facet 4;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{4}\).
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(6 x - 3 y - 6 z + 2\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(- s_{0} - s_{1} + 1)\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{5}\).
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(- 6 x - 6 y + 3 z + 4\right)\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{5}\).
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(- 6 x - 6 y + 3 z + 4\right)\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{5}\).
\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(6 x - 6 y + 3 z - 2\right)\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{5}\).
\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(6 x - 6 y + 3 z - 2\right)\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot(s_{1})\hat{\boldsymbol{n}}_{5}\)
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{5}\).
\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(- 6 x + 6 y + 3 z - 2\right)\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
where \(f_{5}\) is the 5th face;
\(\hat{\boldsymbol{n}}_{5}\) is the normal to facet 5;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{5}\).
\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(- 6 x + 6 y + 3 z - 2\right)\end{array}\right)\)
This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
where \(R\) is the reference element.
\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 6 x \left(1 - x\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 6 x \left(1 - x\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
where \(R\) is the reference element.
\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 y \left(1 - y\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 y \left(1 - y\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
where \(R\) is the reference element.
\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 z \left(1 - z\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 6 z \left(1 - z\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.