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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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle y z\\\displaystyle z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle y^{2}\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle x y\\\displaystyle x z\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{14}\}\)
- 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 \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 6 x \left(5 x - 2\right)\\\displaystyle 6 y \left(5 x - 1\right)\\\displaystyle 6 z \left(5 x - 1\right)\end{array}\right)\)
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 \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 6 x \left(5 x - 2\right)\\\displaystyle 6 y \left(5 x - 1\right)\\\displaystyle 6 z \left(5 x - 1\right)\end{array}\right)\)
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 \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 6 x \left(5 y - 1\right)\\\displaystyle 6 y \left(5 y - 2\right)\\\displaystyle 6 z \left(5 y - 1\right)\end{array}\right)\)
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 \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 6 x \left(5 y - 1\right)\\\displaystyle 6 y \left(5 y - 2\right)\\\displaystyle 6 z \left(5 y - 1\right)\end{array}\right)\)
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 \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 6 x \left(5 z - 1\right)\\\displaystyle 6 y \left(5 z - 1\right)\\\displaystyle 6 z \left(5 z - 2\right)\end{array}\right)\)
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 \(f_{0}\).
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 6 x \left(5 z - 1\right)\\\displaystyle 6 y \left(5 z - 1\right)\\\displaystyle 6 z \left(5 z - 2\right)\end{array}\right)\)
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 \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 30 x y + 30 x z - 48 x - 24 y - 24 z + 18\\\displaystyle 6 y \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle 6 z \left(5 x + 5 y + 5 z - 4\right)\end{array}\right)\)
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 \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 30 x y + 30 x z - 48 x - 24 y - 24 z + 18\\\displaystyle 6 y \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle 6 z \left(5 x + 5 y + 5 z - 4\right)\end{array}\right)\)
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 \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle - 30 x y + 6 x + 24 y - 6\\\displaystyle 6 y \left(2 - 5 y\right)\\\displaystyle 6 z \left(1 - 5 y\right)\end{array}\right)\)
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 \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle - 30 x y + 6 x + 24 y - 6\\\displaystyle 6 y \left(2 - 5 y\right)\\\displaystyle 6 z \left(1 - 5 y\right)\end{array}\right)\)
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 \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - 30 x z + 6 x + 24 z - 6\\\displaystyle 6 y \left(1 - 5 z\right)\\\displaystyle 6 z \left(2 - 5 z\right)\end{array}\right)\)
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 \(f_{1}\).
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - 30 x z + 6 x + 24 z - 6\\\displaystyle 6 y \left(1 - 5 z\right)\\\displaystyle 6 z \left(2 - 5 z\right)\end{array}\right)\)
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 \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 6 x \left(- 5 x - 5 y - 5 z + 4\right)\\\displaystyle - 30 x y + 24 x - 30 y^{2} - 30 y z + 48 y + 24 z - 18\\\displaystyle 6 z \left(- 5 x - 5 y - 5 z + 4\right)\end{array}\right)\)
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 \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 6 x \left(- 5 x - 5 y - 5 z + 4\right)\\\displaystyle - 30 x y + 24 x - 30 y^{2} - 30 y z + 48 y + 24 z - 18\\\displaystyle 6 z \left(- 5 x - 5 y - 5 z + 4\right)\end{array}\right)\)
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 \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 6 x \left(5 x - 2\right)\\\displaystyle 30 x y - 24 x - 6 y + 6\\\displaystyle 6 z \left(5 x - 1\right)\end{array}\right)\)
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 \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 6 x \left(5 x - 2\right)\\\displaystyle 30 x y - 24 x - 6 y + 6\\\displaystyle 6 z \left(5 x - 1\right)\end{array}\right)\)
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 \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 6 x \left(5 z - 1\right)\\\displaystyle 30 y z - 6 y - 24 z + 6\\\displaystyle 6 z \left(5 z - 2\right)\end{array}\right)\)
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 \(f_{2}\).
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 6 x \left(5 z - 1\right)\\\displaystyle 30 y z - 6 y - 24 z + 6\\\displaystyle 6 z \left(5 z - 2\right)\end{array}\right)\)
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 \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 6 x \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle 6 y \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle 30 x z - 24 x + 30 y z - 24 y + 30 z^{2} - 48 z + 18\end{array}\right)\)
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 \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 6 x \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle 6 y \left(5 x + 5 y + 5 z - 4\right)\\\displaystyle 30 x z - 24 x + 30 y z - 24 y + 30 z^{2} - 48 z + 18\end{array}\right)\)
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 \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 6 x \left(2 - 5 x\right)\\\displaystyle 6 y \left(1 - 5 x\right)\\\displaystyle - 30 x z + 24 x + 6 z - 6\end{array}\right)\)
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 \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 6 x \left(2 - 5 x\right)\\\displaystyle 6 y \left(1 - 5 x\right)\\\displaystyle - 30 x z + 24 x + 6 z - 6\end{array}\right)\)
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 \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 6 x \left(1 - 5 y\right)\\\displaystyle 6 y \left(2 - 5 y\right)\\\displaystyle - 30 y z + 24 y + 6 z - 6\end{array}\right)\)
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 \(f_{3}\).
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 6 x \left(1 - 5 y\right)\\\displaystyle 6 y \left(2 - 5 y\right)\\\displaystyle - 30 y z + 24 y + 6 z - 6\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
where \(R\) is the reference element.
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 30 x \left(- 2 x - y - z + 2\right)\\\displaystyle 30 y \left(- 2 x - y - z + 1\right)\\\displaystyle 30 z \left(- 2 x - y - z + 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 30 x \left(- 2 x - y - z + 2\right)\\\displaystyle 30 y \left(- 2 x - y - z + 1\right)\\\displaystyle 30 z \left(- 2 x - y - z + 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
where \(R\) is the reference element.
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 30 x \left(- x - 2 y - z + 1\right)\\\displaystyle 30 y \left(- x - 2 y - z + 2\right)\\\displaystyle 30 z \left(- x - 2 y - z + 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 30 x \left(- x - 2 y - z + 1\right)\\\displaystyle 30 y \left(- x - 2 y - z + 2\right)\\\displaystyle 30 z \left(- x - 2 y - z + 1\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
where \(R\) is the reference element.
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 30 x \left(- x - y - 2 z + 1\right)\\\displaystyle 30 y \left(- x - y - 2 z + 1\right)\\\displaystyle 30 z \left(- x - y - 2 z + 2\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 30 x \left(- x - y - 2 z + 1\right)\\\displaystyle 30 y \left(- x - y - 2 z + 1\right)\\\displaystyle 30 z \left(- x - y - 2 z + 2\right)\end{array}\right)\)
This DOF is associated with volume 0 of the reference element.