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Nédélec (second kind)

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Exterior calculus names\(\mathcal{P}_{k}\Lambda^{1}(\Delta_d)\)
Cockburn–fu names\(\left[S_{1,k}^\unicode{0x25FA}\right]_{1}\)
Abbreviated namesN2curl
Orders\(1\leqslant k\)
Reference elementstriangle, tetrahedron
Polynomial set\(\mathcal{P}_{k}^d\)
↓ Show polynomial set definitions ↓
DOFsOn each edge: tangent integral moments with an order \(k\) Lagrange space
On each face: integral moments with an order \(k-1\) Raviart–Thomas space
On each volume: integral moments with an order \(k-2\) Raviart–Thomas space
Number of DOFsinterval: \(k+1\) (A000027)
triangle: \((k+1)(k+2)\) (A002378)
tetrahedron: \((k+1)(k+2)(k+3)/2\) (A027480)
CategoriesVector-valued elements, H(curl) conforming elements

Implementations

Symfem"N2curl"
↓ Show Symfem examples ↓
Basixbasix.ElementFamily.N2E
↓ Show Basix examples ↓
UFL"N2curl"
↓ Show UFL examples ↓

Examples

triangle
order 1
triangle
order 2
tetrahedron
order 1
tetrahedron
order 2
  • \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{5}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 2 y\\\displaystyle 4 x\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle - 4 y\\\displaystyle - 2 x\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle - 2 y\\\displaystyle - 4 x - 6 y + 4\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 4 y\\\displaystyle 2 x + 6 y - 2\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle - 6 x - 4 y + 4\\\displaystyle - 2 x\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 6 x + 2 y - 2\\\displaystyle 4 x\end{array}\right)\)

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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{11}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 3 y \left(4 x - 1\right)\\\displaystyle 9 x \left(2 x - 1\right)\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 9 y \left(1 - 2 y\right)\\\displaystyle 3 x \left(1 - 4 y\right)\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(- 6 x - y + 2\right)}{2}\\\displaystyle \frac{3 x \left(x + 6 y - 2\right)}{2}\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 3 y \left(4 x + 4 y - 3\right)\\\displaystyle 18 x^{2} + 48 x y - 27 x + 30 y^{2} - 36 y + 9\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 9 y \left(2 y - 1\right)\\\displaystyle 12 x y - 3 x + 30 y^{2} - 24 y + 3\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(- 6 x - 5 y + 4\right)}{2}\\\displaystyle \frac{3 x^{2}}{2} - 15 x y - 15 y^{2} + 15 y - \frac{3}{2}\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 48 x y - 36 x + 18 y^{2} - 27 y + 9\\\displaystyle 3 x \left(4 x + 4 y - 3\right)\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 12 x y - 24 x - 3 y + 3\\\displaystyle 9 x \left(2 x - 1\right)\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle - 15 x^{2} - 15 x y + 15 x + \frac{3 y^{2}}{2} - \frac{3}{2}\\\displaystyle \frac{3 x \left(- 5 x - 6 y + 4\right)}{2}\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{0}\\- s_{1}\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 12 y \left(- 4 x - y + 1\right)\\\displaystyle 12 x \left(- x - 4 y + 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}s_{0} - 1\\s_{1}\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 12 y \left(4 x + 3 y - 3\right)\\\displaystyle 12 x \left(x + 2 y - 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{0}\\1 - s_{1}\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 12 y \left(- 2 x - y + 1\right)\\\displaystyle 12 x \left(- 3 x - 4 y + 3\right)\end{array}\right)\)

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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{11}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 z\\\displaystyle 4 y\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 4 z\\\displaystyle - 2 y\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 2 z\\\displaystyle 0\\\displaystyle 4 x\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - 4 z\\\displaystyle 0\\\displaystyle - 2 x\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 2 y\\\displaystyle 4 x\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - 4 y\\\displaystyle - 2 x\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle - 2 z\\\displaystyle - 2 z\\\displaystyle - 4 x - 4 y - 6 z + 4\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 4 z\\\displaystyle 4 z\\\displaystyle 2 x + 2 y + 6 z - 2\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
\(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4;
and \(s_{0}\) is a parametrisation of \(e_{4}\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle - 2 y\\\displaystyle - 4 x - 6 y - 4 z + 4\\\displaystyle - 2 y\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
\(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4;
and \(s_{0}\) is a parametrisation of \(e_{4}\).

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 4 y\\\displaystyle 2 x + 6 y + 2 z - 2\\\displaystyle 4 y\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
\(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5;
and \(s_{0}\) is a parametrisation of \(e_{5}\).

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle - 6 x - 4 y - 4 z + 4\\\displaystyle - 2 x\\\displaystyle - 2 x\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
\(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5;
and \(s_{0}\) is a parametrisation of \(e_{5}\).

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 6 x + 2 y + 2 z - 2\\\displaystyle 4 x\\\displaystyle 4 x\end{array}\right)\)

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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{29}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 z \left(4 y - 1\right)\\\displaystyle 9 y \left(2 y - 1\right)\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 9 z \left(1 - 2 z\right)\\\displaystyle 3 y \left(1 - 4 z\right)\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0;
and \(s_{0}\) is a parametrisation of \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle \frac{3 z \left(- 6 y - z + 2\right)}{2}\\\displaystyle \frac{3 y \left(y + 6 z - 2\right)}{2}\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 3 z \left(4 x - 1\right)\\\displaystyle 0\\\displaystyle 9 x \left(2 x - 1\right)\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 9 z \left(1 - 2 z\right)\\\displaystyle 0\\\displaystyle 3 x \left(1 - 4 z\right)\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1;
and \(s_{0}\) is a parametrisation of \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \frac{3 z \left(- 6 x - z + 2\right)}{2}\\\displaystyle 0\\\displaystyle \frac{3 x \left(x + 6 z - 2\right)}{2}\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 3 y \left(4 x - 1\right)\\\displaystyle 9 x \left(2 x - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 9 y \left(1 - 2 y\right)\\\displaystyle 3 x \left(1 - 4 y\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2;
and \(s_{0}\) is a parametrisation of \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(- 6 x - y + 2\right)}{2}\\\displaystyle \frac{3 x \left(x + 6 y - 2\right)}{2}\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 3 z \left(4 x + 4 y + 4 z - 3\right)\\\displaystyle 3 z \left(4 x + 4 y + 4 z - 3\right)\\\displaystyle 18 x^{2} + 36 x y + 48 x z - 27 x + 18 y^{2} + 48 y z - 27 y + 30 z^{2} - 36 z + 9\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 9 z \left(2 z - 1\right)\\\displaystyle 9 z \left(2 z - 1\right)\\\displaystyle 12 x z - 3 x + 12 y z - 3 y + 30 z^{2} - 24 z + 3\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
\(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3;
and \(s_{0}\) is a parametrisation of \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle \frac{3 z \left(- 6 x - 6 y - 5 z + 4\right)}{2}\\\displaystyle \frac{3 z \left(- 6 x - 6 y - 5 z + 4\right)}{2}\\\displaystyle \frac{3 x^{2}}{2} + 3 x y - 15 x z + \frac{3 y^{2}}{2} - 15 y z - 15 z^{2} + 15 z - \frac{3}{2}\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
\(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4;
and \(s_{0}\) is a parametrisation of \(e_{4}\).

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 3 y \left(4 x + 4 y + 4 z - 3\right)\\\displaystyle 18 x^{2} + 48 x y + 36 x z - 27 x + 30 y^{2} + 48 y z - 36 y + 18 z^{2} - 27 z + 9\\\displaystyle 3 y \left(4 x + 4 y + 4 z - 3\right)\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
\(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4;
and \(s_{0}\) is a parametrisation of \(e_{4}\).

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 9 y \left(2 y - 1\right)\\\displaystyle 12 x y - 3 x + 30 y^{2} + 12 y z - 24 y - 3 z + 3\\\displaystyle 9 y \left(2 y - 1\right)\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
\(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4;
and \(s_{0}\) is a parametrisation of \(e_{4}\).

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle \frac{3 y \left(- 6 x - 5 y - 6 z + 4\right)}{2}\\\displaystyle \frac{3 x^{2}}{2} - 15 x y + 3 x z - 15 y^{2} - 15 y z + 15 y + \frac{3 z^{2}}{2} - \frac{3}{2}\\\displaystyle \frac{3 y \left(- 6 x - 5 y - 6 z + 4\right)}{2}\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(2 s_{0}^{2} - 3 s_{0} + 1)\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
\(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5;
and \(s_{0}\) is a parametrisation of \(e_{5}\).

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 48 x y + 48 x z - 36 x + 18 y^{2} + 36 y z - 27 y + 18 z^{2} - 27 z + 9\\\displaystyle 3 x \left(4 x + 4 y + 4 z - 3\right)\\\displaystyle 3 x \left(4 x + 4 y + 4 z - 3\right)\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(s_{0} \left(2 s_{0} - 1\right))\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
\(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5;
and \(s_{0}\) is a parametrisation of \(e_{5}\).

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 30 x^{2} + 12 x y + 12 x z - 24 x - 3 y - 3 z + 3\\\displaystyle 9 x \left(2 x - 1\right)\\\displaystyle 9 x \left(2 x - 1\right)\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(4 s_{0} \left(1 - s_{0}\right))\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
\(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5;
and \(s_{0}\) is a parametrisation of \(e_{5}\).

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle - 15 x^{2} - 15 x y - 15 x z + 15 x + \frac{3 y^{2}}{2} + 3 y z + \frac{3 z^{2}}{2} - \frac{3}{2}\\\displaystyle \frac{3 x \left(- 5 x - 6 y - 6 z + 4\right)}{2}\\\displaystyle \frac{3 x \left(- 5 x - 6 y - 6 z + 4\right)}{2}\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}\frac{\sqrt{3} \left(s_{0} + s_{1}\right)}{3}\\- \frac{\sqrt{3} s_{0}}{3}\\- \frac{\sqrt{3} s_{1}}{3}\end{array}\right)\)
where \(f_{0}\) is the 0th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 36 y z\\\displaystyle 12 x z\\\displaystyle 12 x y\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}\frac{\sqrt{3} \left(- s_{0} - s_{1} + 1\right)}{3}\\\frac{\sqrt{3} \left(s_{0} - 1\right)}{3}\\\frac{\sqrt{3} s_{1}}{3}\end{array}\right)\)
where \(f_{0}\) is the 0th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle - 12 y z\\\displaystyle - 36 x z\\\displaystyle - 12 x y\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}\frac{\sqrt{3} \left(s_{0} + s_{1} - 1\right)}{3}\\- \frac{\sqrt{3} s_{0}}{3}\\\frac{\sqrt{3} \left(1 - s_{1}\right)}{3}\end{array}\right)\)
where \(f_{0}\) is the 0th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{0}\).

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 12 y z\\\displaystyle 12 x z\\\displaystyle 36 x y\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\- s_{0}\\- s_{1}\end{array}\right)\)
where \(f_{1}\) is the 1st face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle - 36 y z\\\displaystyle 12 z \left(- x - 4 y - z + 1\right)\\\displaystyle 12 y \left(- x - y - 4 z + 1\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\s_{0} - 1\\s_{1}\end{array}\right)\)
where \(f_{1}\) is the 1st face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 12 y z\\\displaystyle 12 z \left(3 x + 4 y + 3 z - 3\right)\\\displaystyle 12 y \left(x + y + 2 z - 1\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\- s_{0}\\1 - s_{1}\end{array}\right)\)
where \(f_{1}\) is the 1st face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle - 12 y z\\\displaystyle 12 z \left(- x - 2 y - z + 1\right)\\\displaystyle 12 y \left(- 3 x - 3 y - 4 z + 3\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{24}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{0}\\0\\- s_{1}\end{array}\right)\)
where \(f_{2}\) is the 2nd face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 12 z \left(- 4 x - y - z + 1\right)\\\displaystyle - 36 x z\\\displaystyle 12 x \left(- x - y - 4 z + 1\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{25}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}s_{0} - 1\\0\\s_{1}\end{array}\right)\)
where \(f_{2}\) is the 2nd face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 12 z \left(4 x + 3 y + 3 z - 3\right)\\\displaystyle 12 x z\\\displaystyle 12 x \left(x + y + 2 z - 1\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{26}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{0}\\0\\1 - s_{1}\end{array}\right)\)
where \(f_{2}\) is the 2nd face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 12 z \left(- 2 x - y - z + 1\right)\\\displaystyle - 12 x z\\\displaystyle 12 x \left(- 3 x - 3 y - 4 z + 3\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{27}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{0}\\- s_{1}\\0\end{array}\right)\)
where \(f_{3}\) is the 3th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 12 y \left(- 4 x - y - z + 1\right)\\\displaystyle 12 x \left(- x - 4 y - z + 1\right)\\\displaystyle - 36 x y\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{28}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}s_{0} - 1\\s_{1}\\0\end{array}\right)\)
where \(f_{3}\) is the 3th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 12 y \left(4 x + 3 y + 3 z - 3\right)\\\displaystyle 12 x \left(x + 2 y + z - 1\right)\\\displaystyle 12 x y\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{29}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{0}\\1 - s_{1}\\0\end{array}\right)\)
where \(f_{3}\) is the 3th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 12 y \left(- 2 x - y - z + 1\right)\\\displaystyle 12 x \left(- 3 x - 4 y - 3 z + 3\right)\\\displaystyle - 12 x y\end{array}\right)\)

This DOF is associated with face 3 of the reference element.

References

DefElement stats

Element added28 December 2020
Element last updated09 August 2021