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Nédélec (first kind)
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| Alternative names | Whitney, \(\mathcal{P}^-_k\Lambda^{1}(\Delta_d)\) |
| Abbreviated names | N1curl, NC |
| Orders | \(1\leqslant k\) |
| Reference elements | triangle, tetrahedron |
| Polynomial set | \(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(15)}_{k}\) ↓ Show polynomial set definitions ↓ |
| DOFs | On each edge: tangent integral moments with an order \(k-1\) Lagrange space On each face: integral moments with an order \(k-2\) vector Lagrange space On each volume: integral moments with an order \(k-3\) vector Lagrange space |
| Number of DOFs | triangle: \(k(k+2)\) (A005563) tetrahedron: \(k(k+2)(k+3)/2\) (A005564) |
| Categories | Vector-valued elements, H(curl) conforming elements |
Implementations
| Symfem string | "N1curl"↓ Show Symfem examples ↓ |
| Basix string | "Nedelec 1st kind H(curl)"↓ Show Basix examples ↓ |
Examples
triangleorder 1triangle
order 2tetrahedron
order 1tetrahedron
order 2
- \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle - x\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{2}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{0}\)
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - y\\\displaystyle x\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - y\\\displaystyle x\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{1}\)
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle y\\\displaystyle 1 - x\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle y\\\displaystyle 1 - x\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{2}\)
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 1 - y\\\displaystyle x\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 1 - y\\\displaystyle x\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
- \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle - x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle - x y\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{7}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{0}\)
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 2 y \left(1 - 4 x\right)\\\displaystyle 4 x \left(2 x - 1\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 2 y \left(1 - 4 x\right)\\\displaystyle 4 x \left(2 x - 1\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{0}\)
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 4 y \left(1 - 2 y\right)\\\displaystyle 2 x \left(4 y - 1\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 4 y \left(1 - 2 y\right)\\\displaystyle 2 x \left(4 y - 1\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{1}\)
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 2 y \left(- 4 x - 4 y + 3\right)\\\displaystyle 8 x^{2} + 8 x y - 12 x - 6 y + 4\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 2 y \left(- 4 x - 4 y + 3\right)\\\displaystyle 8 x^{2} + 8 x y - 12 x - 6 y + 4\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{1}\)
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 4 y \left(2 y - 1\right)\\\displaystyle - 8 x y + 2 x + 6 y - 2\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 4 y \left(2 y - 1\right)\\\displaystyle - 8 x y + 2 x + 6 y - 2\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{2}\)
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 8 x y - 6 x + 8 y^{2} - 12 y + 4\\\displaystyle 2 x \left(- 4 x - 4 y + 3\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 8 x y - 6 x + 8 y^{2} - 12 y + 4\\\displaystyle 2 x \left(- 4 x - 4 y + 3\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{2}\)
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - 8 x y + 6 x + 2 y - 2\\\displaystyle 4 x \left(2 x - 1\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - 8 x y + 6 x + 2 y - 2\\\displaystyle 4 x \left(2 x - 1\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 8 y \left(- x - 2 y + 2\right)\\\displaystyle 8 x \left(x + 2 y - 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 8 y \left(- x - 2 y + 2\right)\\\displaystyle 8 x \left(x + 2 y - 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 8 y \left(2 x + y - 1\right)\\\displaystyle 8 x \left(- 2 x - y + 2\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 8 y \left(2 x + y - 1\right)\\\displaystyle 8 x \left(- 2 x - y + 2\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
- \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle - x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle - y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle - x\\\displaystyle 0\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{5}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{0}\)
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - z\\\displaystyle y\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - z\\\displaystyle y\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{1}\)
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle - z\\\displaystyle 0\\\displaystyle x\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle - z\\\displaystyle 0\\\displaystyle x\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{2}\)
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle - y\\\displaystyle x\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle - y\\\displaystyle x\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{3}\)
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle z\\\displaystyle z\\\displaystyle - x - y + 1\end{array}\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle z\\\displaystyle z\\\displaystyle - x - y + 1\end{array}\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{4}\)
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle y\\\displaystyle - x - z + 1\\\displaystyle y\end{array}\right)\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle y\\\displaystyle - x - z + 1\\\displaystyle y\end{array}\right)\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{5}\)
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - y - z + 1\\\displaystyle x\\\displaystyle x\end{array}\right)\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - y - z + 1\\\displaystyle x\\\displaystyle x\end{array}\right)\)
This DOF is associated with edge 5 of the reference element.
- \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle - x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle - x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle - x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle - y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle - y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle - x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle - x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle - x y\\\displaystyle 0\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{19}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{0}\)
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 z \left(1 - 4 y\right)\\\displaystyle 4 y \left(2 y - 1\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 z \left(1 - 4 y\right)\\\displaystyle 4 y \left(2 y - 1\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{0}\)
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 z \left(1 - 2 z\right)\\\displaystyle 2 y \left(4 z - 1\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 z \left(1 - 2 z\right)\\\displaystyle 2 y \left(4 z - 1\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{1}\)
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 2 z \left(1 - 4 x\right)\\\displaystyle 0\\\displaystyle 4 x \left(2 x - 1\right)\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 2 z \left(1 - 4 x\right)\\\displaystyle 0\\\displaystyle 4 x \left(2 x - 1\right)\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{1}\)
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 4 z \left(1 - 2 z\right)\\\displaystyle 0\\\displaystyle 2 x \left(4 z - 1\right)\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 4 z \left(1 - 2 z\right)\\\displaystyle 0\\\displaystyle 2 x \left(4 z - 1\right)\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{2}\)
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 2 y \left(1 - 4 x\right)\\\displaystyle 4 x \left(2 x - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 2 y \left(1 - 4 x\right)\\\displaystyle 4 x \left(2 x - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{2}\)
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 4 y \left(1 - 2 y\right)\\\displaystyle 2 x \left(4 y - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 4 y \left(1 - 2 y\right)\\\displaystyle 2 x \left(4 y - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{3}\)
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 2 z \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 2 z \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 8 x^{2} + 16 x y + 8 x z - 12 x + 8 y^{2} + 8 y z - 12 y - 6 z + 4\end{array}\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 2 z \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 2 z \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 8 x^{2} + 16 x y + 8 x z - 12 x + 8 y^{2} + 8 y z - 12 y - 6 z + 4\end{array}\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{3}\)
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 4 z \left(2 z - 1\right)\\\displaystyle 4 z \left(2 z - 1\right)\\\displaystyle - 8 x z + 2 x - 8 y z + 2 y + 6 z - 2\end{array}\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 4 z \left(2 z - 1\right)\\\displaystyle 4 z \left(2 z - 1\right)\\\displaystyle - 8 x z + 2 x - 8 y z + 2 y + 6 z - 2\end{array}\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{4}\)
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 2 y \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 8 x^{2} + 8 x y + 16 x z - 12 x + 8 y z - 6 y + 8 z^{2} - 12 z + 4\\\displaystyle 2 y \left(- 4 x - 4 y - 4 z + 3\right)\end{array}\right)\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 2 y \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 8 x^{2} + 8 x y + 16 x z - 12 x + 8 y z - 6 y + 8 z^{2} - 12 z + 4\\\displaystyle 2 y \left(- 4 x - 4 y - 4 z + 3\right)\end{array}\right)\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{4}\)
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 4 y \left(2 y - 1\right)\\\displaystyle - 8 x y + 2 x - 8 y z + 6 y + 2 z - 2\\\displaystyle 4 y \left(2 y - 1\right)\end{array}\right)\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 4 y \left(2 y - 1\right)\\\displaystyle - 8 x y + 2 x - 8 y z + 6 y + 2 z - 2\\\displaystyle 4 y \left(2 y - 1\right)\end{array}\right)\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(1 - t_{0})\hat{\boldsymbol{t}}_{5}\)
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 8 x y + 8 x z - 6 x + 8 y^{2} + 16 y z - 12 y + 8 z^{2} - 12 z + 4\\\displaystyle 2 x \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 2 x \left(- 4 x - 4 y - 4 z + 3\right)\end{array}\right)\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 8 x y + 8 x z - 6 x + 8 y^{2} + 16 y z - 12 y + 8 z^{2} - 12 z + 4\\\displaystyle 2 x \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 2 x \left(- 4 x - 4 y - 4 z + 3\right)\end{array}\right)\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(t_{0})\hat{\boldsymbol{t}}_{5}\)
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle - 8 x y - 8 x z + 6 x + 2 y + 2 z - 2\\\displaystyle 4 x \left(2 x - 1\right)\\\displaystyle 4 x \left(2 x - 1\right)\end{array}\right)\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle - 8 x y - 8 x z + 6 x + 2 y + 2 z - 2\\\displaystyle 4 x \left(2 x - 1\right)\\\displaystyle 4 x \left(2 x - 1\right)\end{array}\right)\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}- \frac{\sqrt{3}}{3}\\\frac{\sqrt{3}}{3}\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle - 8 y z\\\displaystyle 16 x z\\\displaystyle - 8 x y\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle - 8 y z\\\displaystyle 16 x z\\\displaystyle - 8 x y\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}- \frac{\sqrt{3}}{3}\\0\\\frac{\sqrt{3}}{3}\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle - 8 y z\\\displaystyle - 8 x z\\\displaystyle 16 x y\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle - 8 y z\\\displaystyle - 8 x z\\\displaystyle 16 x y\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 8 y z\\\displaystyle 8 z \left(- 2 x - y - 2 z + 2\right)\\\displaystyle 8 y \left(x + y + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 8 y z\\\displaystyle 8 z \left(- 2 x - y - 2 z + 2\right)\\\displaystyle 8 y \left(x + y + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 8 y z\\\displaystyle 8 z \left(x + 2 y + z - 1\right)\\\displaystyle 8 y \left(- 2 x - 2 y - z + 2\right)\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 8 y z\\\displaystyle 8 z \left(x + 2 y + z - 1\right)\\\displaystyle 8 y \left(- 2 x - 2 y - z + 2\right)\end{array}\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 8 z \left(- x - 2 y - 2 z + 2\right)\\\displaystyle 8 x z\\\displaystyle 8 x \left(x + y + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 8 z \left(- x - 2 y - 2 z + 2\right)\\\displaystyle 8 x z\\\displaystyle 8 x \left(x + y + 2 z - 1\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 8 z \left(2 x + y + z - 1\right)\\\displaystyle 8 x z\\\displaystyle 8 x \left(- 2 x - 2 y - z + 2\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 8 z \left(2 x + y + z - 1\right)\\\displaystyle 8 x z\\\displaystyle 8 x \left(- 2 x - 2 y - z + 2\right)\end{array}\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 8 y \left(- x - 2 y - 2 z + 2\right)\\\displaystyle 8 x \left(x + 2 y + z - 1\right)\\\displaystyle 8 x y\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 8 y \left(- x - 2 y - 2 z + 2\right)\\\displaystyle 8 x \left(x + 2 y + z - 1\right)\\\displaystyle 8 x y\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 8 y \left(2 x + y + z - 1\right)\\\displaystyle 8 x \left(- 2 x - y - 2 z + 2\right)\\\displaystyle 8 x y\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 8 y \left(2 x + y + z - 1\right)\\\displaystyle 8 x \left(- 2 x - y - 2 z + 2\right)\\\displaystyle 8 x y\end{array}\right)\)
This DOF is associated with face 3 of the reference element.
References
- Nédélec, J. C. Mixed finite elements in \(\mathbb{R}^3\), Numerische Mathematik 35(3), 315–341, 1980. [DOI: 10.1007/BF01396415] [BibTeX]