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Degree 1 Nédélec (first kind) 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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{11}\}\)
- 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.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle y z - y - z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle y z - y - z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1.
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x z - x - z + 1\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x z - x - z + 1\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2.
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y - x - y + 1\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y - x - y + 1\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3rd edge;
and \(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3.
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(1 - z\right)\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(1 - z\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
and \(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4.
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x \left(1 - y\right)\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x \left(1 - y\right)\end{array}\right)\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
and \(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5.
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle y \left(1 - z\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 5 of the reference element.
where \(e_{5}\) is the 5th edge;
and \(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5.
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle y \left(1 - z\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{6}\)
where \(e_{6}\) is the 6th edge;
and \(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y \left(1 - x\right)\end{array}\right)\)
This DOF is associated with edge 6 of the reference element.
where \(e_{6}\) is the 6th edge;
and \(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y \left(1 - x\right)\end{array}\right)\)
This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{7}\)
where \(e_{7}\) is the 7th edge;
and \(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7.
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)\)
This DOF is associated with edge 7 of the reference element.
where \(e_{7}\) is the 7th edge;
and \(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7.
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)\)
This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{8}\)
where \(e_{8}\) is the 8th edge;
and \(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8.
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle z \left(1 - y\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 8 of the reference element.
where \(e_{8}\) is the 8th edge;
and \(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8.
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle z \left(1 - y\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 8 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{9}\)
where \(e_{9}\) is the 9th edge;
and \(\hat{\boldsymbol{t}}_{9}\) is the tangent to edge 9.
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(1 - x\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 9 of the reference element.
where \(e_{9}\) is the 9th edge;
and \(\hat{\boldsymbol{t}}_{9}\) is the tangent to edge 9.
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(1 - x\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 9 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{10}\)
where \(e_{10}\) is the 10th edge;
and \(\hat{\boldsymbol{t}}_{10}\) is the tangent to edge 10.
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 10 of the reference element.
where \(e_{10}\) is the 10th edge;
and \(\hat{\boldsymbol{t}}_{10}\) is the tangent to edge 10.
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 10 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{11}\)
where \(e_{11}\) is the 11th edge;
and \(\hat{\boldsymbol{t}}_{11}\) is the tangent to edge 11.
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 11 of the reference element.
where \(e_{11}\) is the 11th edge;
and \(\hat{\boldsymbol{t}}_{11}\) is the tangent to edge 11.
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 11 of the reference element.