Nédélec (first kind)

Click here to read what the information on this page means.

Alternative names	Whitney (triangle,tetrahedron), Nédélec, Q H(curl) (quadrilateral,hexahedron), Raviart–Thomas cubical H(curl) (quadrilateral), Nédélec cubical H(curl) (hexahedron), \(\mathcal{P}^-_k\Lambda^{1}(\Delta_d)\), \(\mathcal{Q}^-_k\Lambda^{1}(\square_d)\)
Abbreviated names	N1curl, NC, RTce (quadrilateral), Nce (hexahedron)
Orders	\(1\leqslant k\)
Reference elements	triangle, tetrahedron, quadrilateral, hexahedron, prism
Polynomial set	\(\mathcal{P}_{k-1}^d \oplus \mathcal{Z}^{(20)}_{k}\) (triangle, tetrahedron) \(\mathcal{Q}_{k-1}^d \oplus \mathcal{Z}^{(21)}_{k}\) (quadrilateral) \(\mathcal{Q}_{k}^d \oplus \mathcal{Z}^{(21)}_{k}\) (hexahedron) ↓ Show polynomial set definitions ↓↑ Hide polynomial set definitions ↑ \(\mathcal{P}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i\leqslant k\right\}\) \(\mathcal{Z}^{(20)}_k=\left\{\boldsymbol{p}\in\hat{\mathcal{P}}_{k}^d\middle\|\boldsymbol{p}(\boldsymbol{x})\cdot \boldsymbol{x}=0\right\}\) \(\hat{\mathcal{P}}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\sum_{i=1}^dp_i=k\right\}=\mathcal{P}_k\setminus\mathcal{P}_{k-1}\) \(\mathcal{Q}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\max_i(p_i)\leqslant k\right\}\) \(\mathcal{Z}^{(21)}_k=\left\{\boldsymbol{q}\in\hat{\mathcal{Q}}_{k}\middle\|\boldsymbol{q}(\boldsymbol{x})\cdot x_i\boldsymbol{e}_i\in\mathcal{Q}_{k}\text{ for }i=1,\dots,d\right\}\) \(\hat{\mathcal{Q}}_k=\operatorname{span}\left\{\prod_{i=1}^dx_i^{p_i}\middle\|\max_i(p_i)=k\right\}=\mathcal{Q}_k\setminus\mathcal{Q}_{k-1}\)
DOFs	On each edge: tangent integral moments with an order \(k-1\) Lagrange space On each face (triangle): integral moments with an order \(k-2\) vector Lagrange space On each face (quadrilateral): integral moments with an order \(k-1\) Q H(div) space On each volume (tetrahedron): integral moments with an order \(k-3\) vector Lagrange space On each volume (hexahedron): integral moments with an order \(k-1\) Q H(div) space
Number of DOFs	triangle: \(k(k+2)\) (A005563) tetrahedron: \(k(k+2)(k+3)/2\) (A005564) quadrilateral: \(2k(k+1)\) (A046092) hexahedron: \(3k(k+1)^2\) (A059986) prism: \(3k(k+2)(k+1)/2\)
Categories	Vector-valued elements, H(curl) conforming elements

Implementations

Symfem string	`"N1curl"` (triangle, tetrahedron) `"Qcurl"` (quadrilateral, hexahedron) `"Ncurl"` (prism) ↓ Show Symfem examples ↓↑ Hide Symfem examples ↑ Before trying this example, you must install Symfem: pip3 install symfem This element can then be created with the following lines of Python: import symfem # Create Nedelec (first kind) order 1 on a triangle element = symfem.create_element("triangle", "N1curl", 1) # Create Nedelec (first kind) order 2 on a triangle element = symfem.create_element("triangle", "N1curl", 2) # Create Nedelec (first kind) order 1 on a quadrilateral element = symfem.create_element("quadrilateral", "Qcurl", 1) # Create Nedelec (first kind) order 2 on a quadrilateral element = symfem.create_element("quadrilateral", "Qcurl", 2) # Create Nedelec (first kind) order 1 on a tetrahedron element = symfem.create_element("tetrahedron", "N1curl", 1) # Create Nedelec (first kind) order 2 on a tetrahedron element = symfem.create_element("tetrahedron", "N1curl", 2) # Create Nedelec (first kind) order 1 on a hexahedron element = symfem.create_element("hexahedron", "Qcurl", 1) # Create Nedelec (first kind) order 2 on a hexahedron element = symfem.create_element("hexahedron", "Qcurl", 2) # Create Nedelec (first kind) order 1 on a prism element = symfem.create_element("prism", "Ncurl", 1) # Create Nedelec (first kind) order 2 on a prism element = symfem.create_element("prism", "Ncurl", 2)
Basix string	`"Nedelec 1st kind H(curl)"` (triangle, tetrahedron, quadrilateral, hexahedron) ↓ Show Basix examples ↓↑ Hide Basix examples ↑ Before trying this example, you must install Basix. This element can then be created with the following lines of Python: import basix # Create Nedelec (first kind) order 1 on a triangle element = basix.create_element("Nedelec 1st kind H(curl)", "triangle", 1) # Create Nedelec (first kind) order 2 on a triangle element = basix.create_element("Nedelec 1st kind H(curl)", "triangle", 2) # Create Nedelec (first kind) order 1 on a quadrilateral element = basix.create_element("Nedelec 1st kind H(curl)", "quadrilateral", 1) # Create Nedelec (first kind) order 2 on a quadrilateral element = basix.create_element("Nedelec 1st kind H(curl)", "quadrilateral", 2) # Create Nedelec (first kind) order 1 on a tetrahedron element = basix.create_element("Nedelec 1st kind H(curl)", "tetrahedron", 1) # Create Nedelec (first kind) order 2 on a tetrahedron element = basix.create_element("Nedelec 1st kind H(curl)", "tetrahedron", 2) # Create Nedelec (first kind) order 1 on a hexahedron element = basix.create_element("Nedelec 1st kind H(curl)", "hexahedron", 1) # Create Nedelec (first kind) order 2 on a hexahedron element = basix.create_element("Nedelec 1st kind H(curl)", "hexahedron", 2)
UFL string	`"N1curl"` (triangle, tetrahedron) `"RTCE"` (quadrilateral) `"NCE"` (hexahedron) ↓ Show UFL examples ↓↑ Hide UFL examples ↑ Before trying this example, you must install UFL: pip3 install UFL This element can then be created with the following lines of Python: import ufl # Create Nedelec (first kind) order 1 on a triangle element = ufl.FiniteElement("N1curl", "triangle", 1) # Create Nedelec (first kind) order 2 on a triangle element = ufl.FiniteElement("N1curl", "triangle", 2) # Create Nedelec (first kind) order 1 on a quadrilateral element = ufl.FiniteElement("RTCE", "quadrilateral", 1) # Create Nedelec (first kind) order 2 on a quadrilateral element = ufl.FiniteElement("RTCE", "quadrilateral", 2) # Create Nedelec (first kind) order 1 on a tetrahedron element = ufl.FiniteElement("N1curl", "tetrahedron", 1) # Create Nedelec (first kind) order 2 on a tetrahedron element = ufl.FiniteElement("N1curl", "tetrahedron", 2) # Create Nedelec (first kind) order 1 on a hexahedron element = ufl.FiniteElement("NCE", "hexahedron", 1) # Create Nedelec (first kind) order 2 on a hexahedron element = ufl.FiniteElement("NCE", "hexahedron", 2)
Bempp string	`"SNC"` (triangle) ↓ Show Bempp examples ↓↑ Hide Bempp examples ↑ Before trying this example, you must install Bempp. This element can then be created with the following lines of Python: import bempp.api grid = bempp.api.shapes.regular_sphere(1)

Examples

triangle
order 1 triangle
order 2 quadrilateral
order 1 quadrilateral
order 2 tetrahedron
order 1 tetrahedron
order 2 hexahedron
order 1 hexahedron
order 2 prism
order 1 prism
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:

\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0.

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

This DOF is associated with edge 0 of the reference element.

\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1.

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

This DOF is associated with edge 1 of the reference element.

\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2.

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

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:

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

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

This DOF is associated with edge 0 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 2 y \left(- 4 x - 4 y + 3\right)\\\displaystyle 8 x^{2} + 8 x y - 12 x - 6 y + 4\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 4 y \left(2 y - 1\right)\\\displaystyle - 8 x y + 2 x + 6 y - 2\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 8 x y - 6 x + 8 y^{2} - 12 y + 4\\\displaystyle 2 x \left(- 4 x - 4 y + 3\right)\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - 8 x y + 6 x + 2 y - 2\\\displaystyle 4 x \left(2 x - 1\right)\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.

\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
where \(R\) is 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.

\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
where \(R\) is 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 quadrilateral. The following numbering of the subentities of the reference is used:

\(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\)
\(\mathcal{L}=\{l_0,...,l_{3}\}\)
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.

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

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.

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

This DOF is associated with edge 1 of the reference element.

\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2.

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

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 3th edge;
and \(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3.

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\)

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

\(R\) is the reference quadrilateral. The following numbering of the subentities of the reference is used:

\(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y\end{array}\right)\)
\(\mathcal{L}=\{l_0,...,l_{11}\}\)
Functionals and basis functions:

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

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 18 x y^{2} - 24 x y + 6 x - 6 y^{2} + 8 y - 2\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 18 x^{2} y - 6 x^{2} - 24 x y + 8 x + 6 y - 2\end{array}\right)\)

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 \(e_{2}\).

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

This DOF is associated with edge 2 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x \left(9 x y - 3 x - 6 y + 2\right)\end{array}\right)\)

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 \(e_{3}\).

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

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 \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 2 y \left(9 x y - 6 x - 3 y + 2\right)\\\displaystyle 0\end{array}\right)\)

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

\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1 - s_{1}\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1})\) is a parametrisation of \(R(2)\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x \left(3 x y - 2 x - 3 y + 2\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 12 y \left(- 3 x y + 3 x + 2 y - 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 12 y \left(3 x y - 3 x - y + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x \left(- 3 x y + x + 3 y - 1\right)\end{array}\right)\)

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:

\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{t}}_{0}\) is the tangent to edge 0.

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

This DOF is associated with edge 0 of the reference element.

\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{t}}_{1}\) is the tangent to edge 1.

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

This DOF is associated with edge 1 of the reference element.

\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{t}}_{2}\) is the tangent to edge 2.

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

This DOF is associated with edge 2 of the reference element.

\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th edge;
and \(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 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.

\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{4}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{4}\)
where \(e_{4}\) is the 4th edge;
and \(\hat{\boldsymbol{t}}_{4}\) is the tangent to edge 4.

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

This DOF is associated with edge 4 of the reference element.

\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{5}\)
where \(e_{5}\) is the 5th edge;
and \(\hat{\boldsymbol{t}}_{5}\) is the tangent to edge 5.

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - y - z + 1\\\displaystyle x\\\displaystyle x\end{array}\right)\)

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:

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

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

This DOF is associated with edge 0 of the reference element.

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

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

This DOF is associated with edge 0 of the reference element.

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

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

This DOF is associated with edge 1 of the reference element.

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

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

This DOF is associated with edge 1 of the reference element.

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

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

This DOF is associated with edge 2 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 4 y \left(1 - 2 y\right)\\\displaystyle 2 x \left(4 y - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 2 z \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 2 z \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 8 x^{2} + 16 x y + 8 x z - 12 x + 8 y^{2} + 8 y z - 12 y - 6 z + 4\end{array}\right)\)

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

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

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 4 z \left(2 z - 1\right)\\\displaystyle 4 z \left(2 z - 1\right)\\\displaystyle - 8 x z + 2 x - 8 y z + 2 y + 6 z - 2\end{array}\right)\)

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

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

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 2 y \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 8 x^{2} + 8 x y + 16 x z - 12 x + 8 y z - 6 y + 8 z^{2} - 12 z + 4\\\displaystyle 2 y \left(- 4 x - 4 y - 4 z + 3\right)\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 4 y \left(2 y - 1\right)\\\displaystyle - 8 x y + 2 x - 8 y z + 6 y + 2 z - 2\\\displaystyle 4 y \left(2 y - 1\right)\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 8 x y + 8 x z - 6 x + 8 y^{2} + 16 y z - 12 y + 8 z^{2} - 12 z + 4\\\displaystyle 2 x \left(- 4 x - 4 y - 4 z + 3\right)\\\displaystyle 2 x \left(- 4 x - 4 y - 4 z + 3\right)\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle - 8 x y - 8 x z + 6 x + 2 y + 2 z - 2\\\displaystyle 4 x \left(2 x - 1\right)\\\displaystyle 4 x \left(2 x - 1\right)\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.

\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}- \frac{\sqrt{3}}{3}\\\frac{\sqrt{3}}{3}\\0\end{array}\right)\)
where \(f_{0}\) is the 0th face.

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle - 8 y z\\\displaystyle 16 x z\\\displaystyle - 8 x y\end{array}\right)\)

This DOF is associated with face 0 of the reference element.

\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}- \frac{\sqrt{3}}{3}\\0\\\frac{\sqrt{3}}{3}\end{array}\right)\)
where \(f_{0}\) is the 0th face.

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle - 8 y z\\\displaystyle - 8 x z\\\displaystyle 16 x y\end{array}\right)\)

This DOF is associated with face 0 of the reference element.

\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
where \(f_{1}\) is the 1st face.

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 8 y z\\\displaystyle 8 z \left(- 2 x - y - 2 z + 2\right)\\\displaystyle 8 y \left(x + y + 2 z - 1\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.

\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
where \(f_{1}\) is the 1st face.

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 8 y z\\\displaystyle 8 z \left(x + 2 y + z - 1\right)\\\displaystyle 8 y \left(- 2 x - 2 y - z + 2\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.

\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
where \(f_{2}\) is the 2nd face.

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 8 z \left(- x - 2 y - 2 z + 2\right)\\\displaystyle 8 x z\\\displaystyle 8 x \left(x + y + 2 z - 1\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.

\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)
where \(f_{2}\) is the 2nd face.

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 8 z \left(2 x + y + z - 1\right)\\\displaystyle 8 x z\\\displaystyle 8 x \left(- 2 x - 2 y - z + 2\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.

\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
where \(f_{3}\) is the 3th face.

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 8 y \left(- x - 2 y - 2 z + 2\right)\\\displaystyle 8 x \left(x + 2 y + z - 1\right)\\\displaystyle 8 x y\end{array}\right)\)

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

\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
where \(f_{3}\) is the 3th face.

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 8 y \left(2 x + y + z - 1\right)\\\displaystyle 8 x \left(- 2 x - y - 2 z + 2\right)\\\displaystyle 8 x y\end{array}\right)\)

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

\(R\) 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.

\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{1}\)
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.

\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{2}\)
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.

\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{3}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{3}\)
where \(e_{3}\) is the 3th 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.

\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{5}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{5}\)
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.

\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{6}\)
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.

\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{7}\)
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.

\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{8}\)
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.

\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{9}\)
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.

\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{10}\)
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.

\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{t}}_{11}\)
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.

\(R\) is the reference hexahedron. The following numbering of the subentities of the reference is used:

\(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2} z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2} z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{2} z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{2} z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2} z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y^{2} z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y^{2} z\end{array}\right)\)
\(\mathcal{L}=\{l_0,...,l_{53}\}\)
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 \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 54 x y^{2} z^{2} + 72 x y^{2} z - 18 x y^{2} + 72 x y z^{2} - 96 x y z + 24 x y - 18 x z^{2} + 24 x z - 6 x + 36 y^{2} z^{2} - 48 y^{2} z + 12 y^{2} - 48 y z^{2} + 64 y z - 16 y + 12 z^{2} - 16 z + 4\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 54 x y^{2} z^{2} - 72 x y^{2} z + 18 x y^{2} - 72 x y z^{2} + 96 x y z - 24 x y + 18 x z^{2} - 24 x z + 6 x - 18 y^{2} z^{2} + 24 y^{2} z - 6 y^{2} + 24 y z^{2} - 32 y z + 8 y - 6 z^{2} + 8 z - 2\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

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 \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 54 x^{2} y z^{2} + 72 x^{2} y z - 18 x^{2} y + 36 x^{2} z^{2} - 48 x^{2} z + 12 x^{2} + 72 x y z^{2} - 96 x y z + 24 x y - 48 x z^{2} + 64 x z - 16 x - 18 y z^{2} + 24 y z - 6 y + 12 z^{2} - 16 z + 4\\\displaystyle 0\end{array}\right)\)

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 \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 54 x^{2} y z^{2} - 72 x^{2} y z + 18 x^{2} y - 18 x^{2} z^{2} + 24 x^{2} z - 6 x^{2} - 72 x y z^{2} + 96 x y z - 24 x y + 24 x z^{2} - 32 x z + 8 x + 18 y z^{2} - 24 y z + 6 y - 6 z^{2} + 8 z - 2\\\displaystyle 0\end{array}\right)\)

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 \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 54 x^{2} y^{2} z + 36 x^{2} y^{2} + 72 x^{2} y z - 48 x^{2} y - 18 x^{2} z + 12 x^{2} + 72 x y^{2} z - 48 x y^{2} - 96 x y z + 64 x y + 24 x z - 16 x - 18 y^{2} z + 12 y^{2} + 24 y z - 16 y - 6 z + 4\end{array}\right)\)

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 \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 54 x^{2} y^{2} z - 18 x^{2} y^{2} - 72 x^{2} y z + 24 x^{2} y + 18 x^{2} z - 6 x^{2} - 72 x y^{2} z + 24 x y^{2} + 96 x y z - 32 x y - 24 x z + 8 x + 18 y^{2} z - 6 y^{2} - 24 y z + 8 y + 6 z - 2\end{array}\right)\)

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 \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x \left(- 27 x y z^{2} + 36 x y z - 9 x y + 18 x z^{2} - 24 x z + 6 x + 18 y z^{2} - 24 y z + 6 y - 12 z^{2} + 16 z - 4\right)\\\displaystyle 0\end{array}\right)\)

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 \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x \left(27 x y z^{2} - 36 x y z + 9 x y - 9 x z^{2} + 12 x z - 3 x - 18 y z^{2} + 24 y z - 6 y + 6 z^{2} - 8 z + 2\right)\\\displaystyle 0\end{array}\right)\)

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 \(e_{4}\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x \left(- 27 x y^{2} z + 18 x y^{2} + 36 x y z - 24 x y - 9 x z + 6 x + 18 y^{2} z - 12 y^{2} - 24 y z + 16 y + 6 z - 4\right)\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x \left(27 x y^{2} z - 9 x y^{2} - 36 x y z + 12 x y + 9 x z - 3 x - 18 y^{2} z + 6 y^{2} + 24 y z - 8 y - 6 z + 2\right)\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 2 y \left(- 27 x y z^{2} + 36 x y z - 9 x y + 18 x z^{2} - 24 x z + 6 x + 18 y z^{2} - 24 y z + 6 y - 12 z^{2} + 16 z - 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

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 \(e_{5}\).

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 2 y \left(27 x y z^{2} - 36 x y z + 9 x y - 18 x z^{2} + 24 x z - 6 x - 9 y z^{2} + 12 y z - 3 y + 6 z^{2} - 8 z + 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.

\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{6}\)
where \(e_{6}\) is the 6th edge;
\(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6;
and \(s_{0}\) is a parametrisation of \(e_{6}\).

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 y \left(- 27 x^{2} y z + 18 x^{2} y + 18 x^{2} z - 12 x^{2} + 36 x y z - 24 x y - 24 x z + 16 x - 9 y z + 6 y + 6 z - 4\right)\end{array}\right)\)

This DOF is associated with edge 6 of the reference element.

\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{6}\)
where \(e_{6}\) is the 6th edge;
\(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6;
and \(s_{0}\) is a parametrisation of \(e_{6}\).

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 y \left(27 x^{2} y z - 9 x^{2} y - 18 x^{2} z + 6 x^{2} - 36 x y z + 12 x y + 24 x z - 8 x + 9 y z - 3 y - 6 z + 2\right)\end{array}\right)\)

This DOF is associated with edge 6 of the reference element.

\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{7}\)
where \(e_{7}\) is the 7th edge;
\(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7;
and \(s_{0}\) is a parametrisation of \(e_{7}\).

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x y \left(- 27 x y z + 18 x y + 18 x z - 12 x + 18 y z - 12 y - 12 z + 8\right)\end{array}\right)\)

This DOF is associated with edge 7 of the reference element.

\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{7}\)
where \(e_{7}\) is the 7th edge;
\(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7;
and \(s_{0}\) is a parametrisation of \(e_{7}\).

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x y \left(27 x y z - 9 x y - 18 x z + 6 x - 18 y z + 6 y + 12 z - 4\right)\end{array}\right)\)

This DOF is associated with edge 7 of the reference element.

\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{8}\)
where \(e_{8}\) is the 8th edge;
\(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8;
and \(s_{0}\) is a parametrisation of \(e_{8}\).

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 2 z \left(- 27 x y^{2} z + 18 x y^{2} + 36 x y z - 24 x y - 9 x z + 6 x + 18 y^{2} z - 12 y^{2} - 24 y z + 16 y + 6 z - 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 8 of the reference element.

\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{8}\)
where \(e_{8}\) is the 8th edge;
\(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8;
and \(s_{0}\) is a parametrisation of \(e_{8}\).

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 2 z \left(27 x y^{2} z - 18 x y^{2} - 36 x y z + 24 x y + 9 x z - 6 x - 9 y^{2} z + 6 y^{2} + 12 y z - 8 y - 3 z + 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 8 of the reference element.

\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{9}\)
where \(e_{9}\) is the 9th edge;
\(\hat{\boldsymbol{t}}_{9}\) is the tangent to edge 9;
and \(s_{0}\) is a parametrisation of \(e_{9}\).

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 z \left(- 27 x^{2} y z + 18 x^{2} y + 18 x^{2} z - 12 x^{2} + 36 x y z - 24 x y - 24 x z + 16 x - 9 y z + 6 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 9 of the reference element.

\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{9}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{9}\)
where \(e_{9}\) is the 9th edge;
\(\hat{\boldsymbol{t}}_{9}\) is the tangent to edge 9;
and \(s_{0}\) is a parametrisation of \(e_{9}\).

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 z \left(27 x^{2} y z - 18 x^{2} y - 9 x^{2} z + 6 x^{2} - 36 x y z + 24 x y + 12 x z - 8 x + 9 y z - 6 y - 3 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 9 of the reference element.

\(\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{10}\)
where \(e_{10}\) is the 10th edge;
\(\hat{\boldsymbol{t}}_{10}\) is the tangent to edge 10;
and \(s_{0}\) is a parametrisation of \(e_{10}\).

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x z \left(- 27 x y z + 18 x y + 18 x z - 12 x + 18 y z - 12 y - 12 z + 8\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 10 of the reference element.

\(\displaystyle l_{21}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{10}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{10}\)
where \(e_{10}\) is the 10th edge;
\(\hat{\boldsymbol{t}}_{10}\) is the tangent to edge 10;
and \(s_{0}\) is a parametrisation of \(e_{10}\).

\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x z \left(27 x y z - 18 x y - 9 x z + 6 x - 18 y z + 12 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 10 of the reference element.

\(\displaystyle l_{22}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{11}\)
where \(e_{11}\) is the 11th edge;
\(\hat{\boldsymbol{t}}_{11}\) is the tangent to edge 11;
and \(s_{0}\) is a parametrisation of \(e_{11}\).

\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 2 y z \left(- 27 x y z + 18 x y + 18 x z - 12 x + 18 y z - 12 y - 12 z + 8\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 11 of the reference element.

\(\displaystyle l_{23}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{11}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{11}\)
where \(e_{11}\) is the 11th edge;
\(\hat{\boldsymbol{t}}_{11}\) is the tangent to edge 11;
and \(s_{0}\) is a parametrisation of \(e_{11}\).

\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle 2 y z \left(27 x y z - 18 x y - 18 x z + 12 x - 9 y z + 6 y + 6 z - 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 11 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x \left(9 x y z^{2} - 12 x y z + 3 x y - 6 x z^{2} + 8 x z - 2 x - 9 y z^{2} + 12 y z - 3 y + 6 z^{2} - 8 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 12 y \left(- 9 x y z^{2} + 12 x y z - 3 x y + 9 x z^{2} - 12 x z + 3 x + 6 y z^{2} - 8 y z + 2 y - 6 z^{2} + 8 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 12 y \left(9 x y z^{2} - 12 x y z + 3 x y - 9 x z^{2} + 12 x z - 3 x - 3 y z^{2} + 4 y z - y + 3 z^{2} - 4 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x \left(- 9 x y z^{2} + 12 x y z - 3 x y + 3 x z^{2} - 4 x z + x + 9 y z^{2} - 12 y z + 3 y - 3 z^{2} + 4 z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x \left(9 x y^{2} z - 6 x y^{2} - 12 x y z + 8 x y + 3 x z - 2 x - 9 y^{2} z + 6 y^{2} + 12 y z - 8 y - 3 z + 2\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 12 z \left(- 9 x y^{2} z + 9 x y^{2} + 12 x y z - 12 x y - 3 x z + 3 x + 6 y^{2} z - 6 y^{2} - 8 y z + 8 y + 2 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.

\(\displaystyle l_{30}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{0}\\0\\0\end{array}\right)\)
where \(f_{1}\) is the 1st face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 12 z \left(9 x y^{2} z - 9 x y^{2} - 12 x y z + 12 x y + 3 x z - 3 x - 3 y^{2} z + 3 y^{2} + 4 y z - 4 y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.

\(\displaystyle l_{31}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\s_{1}\end{array}\right)\)
where \(f_{1}\) is the 1st face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x \left(- 9 x y^{2} z + 3 x y^{2} + 12 x y z - 4 x y - 3 x z + x + 9 y^{2} z - 3 y^{2} - 12 y z + 4 y + 3 z - 1\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.

\(\displaystyle l_{32}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1 - s_{1}\end{array}\right)\)
where \(f_{2}\) is the 2nd face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 y \left(9 x^{2} y z - 6 x^{2} y - 9 x^{2} z + 6 x^{2} - 12 x y z + 8 x y + 12 x z - 8 x + 3 y z - 2 y - 3 z + 2\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.

\(\displaystyle l_{33}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\s_{0} - 1\\0\end{array}\right)\)
where \(f_{2}\) is the 2nd face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 z \left(- 9 x^{2} y z + 9 x^{2} y + 6 x^{2} z - 6 x^{2} + 12 x y z - 12 x y - 8 x z + 8 x - 3 y z + 3 y + 2 z - 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.

\(\displaystyle l_{34}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\- s_{0}\\0\end{array}\right)\)
where \(f_{2}\) is the 2nd face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 z \left(9 x^{2} y z - 9 x^{2} y - 3 x^{2} z + 3 x^{2} - 12 x y z + 12 x y + 4 x z - 4 x + 3 y z - 3 y - z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.

\(\displaystyle l_{35}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\s_{1}\end{array}\right)\)
where \(f_{2}\) is the 2nd face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 y \left(- 9 x^{2} y z + 3 x^{2} y + 9 x^{2} z - 3 x^{2} + 12 x y z - 4 x y - 12 x z + 4 x - 3 y z + y + 3 z - 1\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.

\(\displaystyle l_{36}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1 - s_{1}\end{array}\right)\)
where \(f_{3}\) is the 3th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{36} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x y \left(9 x y z - 6 x y - 9 x z + 6 x - 6 y z + 4 y + 6 z - 4\right)\end{array}\right)\)

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

\(\displaystyle l_{37}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\s_{0} - 1\\0\end{array}\right)\)
where \(f_{3}\) is the 3th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{37} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x z \left(- 9 x y z + 9 x y + 6 x z - 6 x + 6 y z - 6 y - 4 z + 4\right)\\\displaystyle 0\end{array}\right)\)

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

\(\displaystyle l_{38}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\- s_{0}\\0\end{array}\right)\)
where \(f_{3}\) is the 3th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{38} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x z \left(9 x y z - 9 x y - 3 x z + 3 x - 6 y z + 6 y + 2 z - 2\right)\\\displaystyle 0\end{array}\right)\)

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

\(\displaystyle l_{39}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\s_{1}\end{array}\right)\)
where \(f_{3}\) is the 3th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{39} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x y \left(- 9 x y z + 3 x y + 9 x z - 3 x + 6 y z - 2 y - 6 z + 2\right)\end{array}\right)\)

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

\(\displaystyle l_{40}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1 - s_{1}\end{array}\right)\)
where \(f_{4}\) is the 4th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{40} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x y \left(9 x y z - 6 x y - 6 x z + 4 x - 9 y z + 6 y + 6 z - 4\right)\end{array}\right)\)

This DOF is associated with face 4 of the reference element.

\(\displaystyle l_{41}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot\left(\begin{array}{c}s_{0} - 1\\0\\0\end{array}\right)\)
where \(f_{4}\) is the 4th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{41} = \left(\begin{array}{c}\displaystyle 12 y z \left(- 9 x y z + 9 x y + 6 x z - 6 x + 6 y z - 6 y - 4 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.

\(\displaystyle l_{42}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{0}\\0\\0\end{array}\right)\)
where \(f_{4}\) is the 4th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{42} = \left(\begin{array}{c}\displaystyle 12 y z \left(9 x y z - 9 x y - 6 x z + 6 x - 3 y z + 3 y + 2 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.

\(\displaystyle l_{43}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\s_{1}\end{array}\right)\)
where \(f_{4}\) is the 4th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{4}\).

\(\displaystyle \boldsymbol{\phi}_{43} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x y \left(- 9 x y z + 3 x y + 6 x z - 2 x + 9 y z - 3 y - 6 z + 2\right)\end{array}\right)\)

This DOF is associated with face 4 of the reference element.

\(\displaystyle l_{44}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1 - s_{1}\\0\end{array}\right)\)
where \(f_{5}\) is the 5th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{44} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x z \left(9 x y z - 6 x y - 6 x z + 4 x - 9 y z + 6 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 5 of the reference element.

\(\displaystyle l_{45}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot\left(\begin{array}{c}s_{0} - 1\\0\\0\end{array}\right)\)
where \(f_{5}\) is the 5th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{45} = \left(\begin{array}{c}\displaystyle 12 y z \left(- 9 x y z + 6 x y + 9 x z - 6 x + 6 y z - 4 y - 6 z + 4\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 5 of the reference element.

\(\displaystyle l_{46}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{0}\\0\\0\end{array}\right)\)
where \(f_{5}\) is the 5th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{46} = \left(\begin{array}{c}\displaystyle 12 y z \left(9 x y z - 6 x y - 9 x z + 6 x - 3 y z + 2 y + 3 z - 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 5 of the reference element.

\(\displaystyle l_{47}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{5}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\s_{1}\\0\end{array}\right)\)
where \(f_{5}\) is the 5th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{5}\).

\(\displaystyle \boldsymbol{\phi}_{47} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 12 x z \left(- 9 x y z + 6 x y + 3 x z - 2 x + 9 y z - 6 y - 3 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 5 of the reference element.

\(\displaystyle l_{48}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1 - s_{2}\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1},s_{2})\) is a parametrisation of \(R(3)\).

\(\displaystyle \boldsymbol{\phi}_{48} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 72 x y \left(- 3 x y z + 2 x y + 3 x z - 2 x + 3 y z - 2 y - 3 z + 2\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.

\(\displaystyle l_{49}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\s_{1} - 1\\0\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1},s_{2})\) is a parametrisation of \(R(3)\).

\(\displaystyle \boldsymbol{\phi}_{49} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 72 x z \left(3 x y z - 3 x y - 2 x z + 2 x - 3 y z + 3 y + 2 z - 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.

\(\displaystyle l_{50}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}1 - s_{0}\\0\\0\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1},s_{2})\) is a parametrisation of \(R(3)\).

\(\displaystyle \boldsymbol{\phi}_{50} = \left(\begin{array}{c}\displaystyle 72 y z \left(- 3 x y z + 3 x y + 3 x z - 3 x + 2 y z - 2 y - 2 z + 2\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.

\(\displaystyle l_{51}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}s_{0}\\0\\0\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1},s_{2})\) is a parametrisation of \(R(3)\).

\(\displaystyle \boldsymbol{\phi}_{51} = \left(\begin{array}{c}\displaystyle 72 y z \left(3 x y z - 3 x y - 3 x z + 3 x - y z + y + z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.

\(\displaystyle l_{52}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\- s_{1}\\0\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1},s_{2})\) is a parametrisation of \(R(3)\).

\(\displaystyle \boldsymbol{\phi}_{52} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 72 x z \left(- 3 x y z + 3 x y + x z - x + 3 y z - 3 y - z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.

\(\displaystyle l_{53}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\s_{2}\end{array}\right)\)
where \(R\) is the reference element;
and \((s_{0},s_{1},s_{2})\) is a parametrisation of \(R(3)\).

\(\displaystyle \boldsymbol{\phi}_{53} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 72 x y \left(3 x y z - x y - 3 x z + x - 3 y z + y + 3 z - 1\right)\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.

\(R\) is the reference prism. The following numbering of the subentities of the reference is used:

\(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle - x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle - x z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\)
\(\mathcal{L}=\{l_0,...,l_{8}\}\)
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.

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

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.

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

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.

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

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 3th edge;
and \(\hat{\boldsymbol{t}}_{3}\) is the tangent to edge 3.

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

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.

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

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.

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

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.

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

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.

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

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.

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle - y z\\\displaystyle x z\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 8 of the reference element.

\(R\) is the reference prism. The following numbering of the subentities of the reference is used:

\(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle - x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y z\\\displaystyle - x^{2} z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y z^{2}\\\displaystyle - x^{2} z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle - x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2} z\\\displaystyle - x y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2} z^{2}\\\displaystyle - x y z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2} z\end{array}\right)\)
\(\mathcal{L}=\{l_0,...,l_{35}\}\)
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 \(e_{0}\).

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 24 x y z^{2} - 32 x y z + 8 x y - 18 x z^{2} + 24 x z - 6 x + 24 y^{2} z^{2} - 32 y^{2} z + 8 y^{2} - 36 y z^{2} + 48 y z - 12 y + 12 z^{2} - 16 z + 4\\\displaystyle 2 x \left(- 12 x z^{2} + 16 x z - 4 x - 12 y z^{2} + 16 y z - 4 y + 9 z^{2} - 12 z + 3\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle - 24 x y z^{2} + 32 x y z - 8 x y + 18 x z^{2} - 24 x z + 6 x + 6 y z^{2} - 8 y z + 2 y - 6 z^{2} + 8 z - 2\\\displaystyle 4 x \left(6 x z^{2} - 8 x z + 2 x - 3 z^{2} + 4 z - 1\right)\\\displaystyle 0\end{array}\right)\)

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 \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 2 y \left(- 12 x z^{2} + 16 x z - 4 x - 12 y z^{2} + 16 y z - 4 y + 9 z^{2} - 12 z + 3\right)\\\displaystyle 24 x^{2} z^{2} - 32 x^{2} z + 8 x^{2} + 24 x y z^{2} - 32 x y z + 8 x y - 36 x z^{2} + 48 x z - 12 x - 18 y z^{2} + 24 y z - 6 y + 12 z^{2} - 16 z + 4\\\displaystyle 0\end{array}\right)\)

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 \(e_{1}\).

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 4 y \left(6 y z^{2} - 8 y z + 2 y - 3 z^{2} + 4 z - 1\right)\\\displaystyle - 24 x y z^{2} + 32 x y z - 8 x y + 6 x z^{2} - 8 x z + 2 x + 18 y z^{2} - 24 y z + 6 y - 6 z^{2} + 8 z - 2\\\displaystyle 0\end{array}\right)\)

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 \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 18 x^{2} z + 12 x^{2} - 36 x y z + 24 x y + 24 x z - 16 x - 18 y^{2} z + 12 y^{2} + 24 y z - 16 y - 6 z + 4\end{array}\right)\)

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 \(e_{2}\).

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 18 x^{2} z - 6 x^{2} + 36 x y z - 12 x y - 24 x z + 8 x + 18 y^{2} z - 6 y^{2} - 24 y z + 8 y + 6 z - 2\end{array}\right)\)

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 \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 2 y \left(- 12 x z^{2} + 16 x z - 4 x + 3 z^{2} - 4 z + 1\right)\\\displaystyle 4 x \left(6 x z^{2} - 8 x z + 2 x - 3 z^{2} + 4 z - 1\right)\\\displaystyle 0\end{array}\right)\)

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 \(e_{3}\).

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 4 y \left(- 6 y z^{2} + 8 y z - 2 y + 3 z^{2} - 4 z + 1\right)\\\displaystyle 2 x \left(12 y z^{2} - 16 y z + 4 y - 3 z^{2} + 4 z - 1\right)\\\displaystyle 0\end{array}\right)\)

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 \(e_{4}\).

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x \left(- 9 x z + 6 x + 6 z - 4\right)\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.

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

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

This DOF is associated with edge 4 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 y \left(- 9 y z + 6 y + 6 z - 4\right)\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 y \left(9 y z - 3 y - 6 z + 2\right)\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.

\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{6}\)
where \(e_{6}\) is the 6th edge;
\(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6;
and \(s_{0}\) is a parametrisation of \(e_{6}\).

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 2 z \left(12 x y z - 8 x y - 9 x z + 6 x + 12 y^{2} z - 8 y^{2} - 18 y z + 12 y + 6 z - 4\right)\\\displaystyle 2 x z \left(- 12 x z + 8 x - 12 y z + 8 y + 9 z - 6\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 6 of the reference element.

\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{6}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{6}\)
where \(e_{6}\) is the 6th edge;
\(\hat{\boldsymbol{t}}_{6}\) is the tangent to edge 6;
and \(s_{0}\) is a parametrisation of \(e_{6}\).

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 2 z \left(- 12 x y z + 8 x y + 9 x z - 6 x + 3 y z - 2 y - 3 z + 2\right)\\\displaystyle 4 x z \left(6 x z - 4 x - 3 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 6 of the reference element.

\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{7}\)
where \(e_{7}\) is the 7th edge;
\(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7;
and \(s_{0}\) is a parametrisation of \(e_{7}\).

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 2 y z \left(- 12 x z + 8 x - 12 y z + 8 y + 9 z - 6\right)\\\displaystyle 2 z \left(12 x^{2} z - 8 x^{2} + 12 x y z - 8 x y - 18 x z + 12 x - 9 y z + 6 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 7 of the reference element.

\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{7}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{7}\)
where \(e_{7}\) is the 7th edge;
\(\hat{\boldsymbol{t}}_{7}\) is the tangent to edge 7;
and \(s_{0}\) is a parametrisation of \(e_{7}\).

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 4 y z \left(6 y z - 4 y - 3 z + 2\right)\\\displaystyle 2 z \left(- 12 x y z + 8 x y + 3 x z - 2 x + 9 y z - 6 y - 3 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 7 of the reference element.

\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(1 - s_{0})\hat{\boldsymbol{t}}_{8}\)
where \(e_{8}\) is the 8th edge;
\(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8;
and \(s_{0}\) is a parametrisation of \(e_{8}\).

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 2 y z \left(- 12 x z + 8 x + 3 z - 2\right)\\\displaystyle 4 x z \left(6 x z - 4 x - 3 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 8 of the reference element.

\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{8}}\boldsymbol{v}\cdot(s_{0})\hat{\boldsymbol{t}}_{8}\)
where \(e_{8}\) is the 8th edge;
\(\hat{\boldsymbol{t}}_{8}\) is the tangent to edge 8;
and \(s_{0}\) is a parametrisation of \(e_{8}\).

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 4 y z \left(- 6 y z + 4 y + 3 z - 2\right)\\\displaystyle 2 x z \left(12 y z - 8 y - 3 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 8 of the reference element.

\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
where \(f_{0}\) is the 0th face.

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle 8 y \left(- 3 x z^{2} + 4 x z - x - 6 y z^{2} + 8 y z - 2 y + 6 z^{2} - 8 z + 2\right)\\\displaystyle 8 x \left(3 x z^{2} - 4 x z + x + 6 y z^{2} - 8 y z + 2 y - 3 z^{2} + 4 z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.

\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
where \(f_{0}\) is the 0th face.

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 8 y \left(6 x z^{2} - 8 x z + 2 x + 3 y z^{2} - 4 y z + y - 3 z^{2} + 4 z - 1\right)\\\displaystyle 8 x \left(- 6 x z^{2} + 8 x z - 2 x - 3 y z^{2} + 4 y z - y + 6 z^{2} - 8 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.

\(\displaystyle l_{20}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\1 - s_{1}\end{array}\right)\)
where \(f_{1}\) is the 1st face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{1}\).

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x \left(3 x z - 2 x + 3 y z - 2 y - 3 z + 2\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle 12 z \left(4 x y z - 4 x y - 3 x z + 3 x + 4 y^{2} z - 4 y^{2} - 6 y z + 6 y + 2 z - 2\right)\\\displaystyle 12 x z \left(- 4 x z + 4 x - 4 y z + 4 y + 3 z - 3\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 12 z \left(- 4 x y z + 4 x y + 3 x z - 3 x + y z - y - z + 1\right)\\\displaystyle 24 x z \left(2 x z - 2 x - z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x \left(- 3 x z + x - 3 y z + y + 3 z - 1\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 y \left(3 x z - 2 x + 3 y z - 2 y - 3 z + 2\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 12 y z \left(- 4 x z + 4 x - 4 y z + 4 y + 3 z - 3\right)\\\displaystyle 12 z \left(4 x^{2} z - 4 x^{2} + 4 x y z - 4 x y - 6 x z + 6 x - 3 y z + 3 y + 2 z - 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.

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

\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 24 y z \left(2 y z - 2 y - z + 1\right)\\\displaystyle 12 z \left(- 4 x y z + 4 x y + x z - x + 3 y z - 3 y - z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.

\(\displaystyle l_{27}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\s_{1}\end{array}\right)\)
where \(f_{2}\) is the 2nd face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{2}\).

\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 y \left(- 3 x z + x - 3 y z + y + 3 z - 1\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.

\(\displaystyle l_{28}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\\frac{\sqrt{2} \left(1 - s_{1}\right)}{2}\end{array}\right)\)
where \(f_{3}\) is the 3th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x y \left(2 - 3 z\right)\end{array}\right)\)

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

\(\displaystyle l_{29}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}\frac{\sqrt{2} \left(1 - s_{0}\right)}{2}\\\frac{\sqrt{2} \left(s_{0} - 1\right)}{2}\\0\end{array}\right)\)
where \(f_{3}\) is the 3th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 12 y z \left(- 4 x z + 4 x + z - 1\right)\\\displaystyle 24 x z \left(2 x z - 2 x - z + 1\right)\\\displaystyle 0\end{array}\right)\)

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

\(\displaystyle l_{30}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}\frac{\sqrt{2} s_{0}}{2}\\- \frac{\sqrt{2} s_{0}}{2}\\0\end{array}\right)\)
where \(f_{3}\) is the 3th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 24 y z \left(- 2 y z + 2 y + z - 1\right)\\\displaystyle 12 x z \left(4 y z - 4 y - z + 1\right)\\\displaystyle 0\end{array}\right)\)

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

\(\displaystyle l_{31}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\0\\\frac{\sqrt{2} s_{1}}{2}\end{array}\right)\)
where \(f_{3}\) is the 3th face;
and \((s_{0},s_{1})\) is a parametrisation of \(f_{3}\).

\(\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 12 x y \left(3 z - 1\right)\end{array}\right)\)

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

\(\displaystyle l_{32}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)
where \(f_{4}\) is the 4th face.

\(\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle 8 y z \left(- 3 x z + 2 x - 6 y z + 4 y + 6 z - 4\right)\\\displaystyle 8 x z \left(3 x z - 2 x + 6 y z - 4 y - 3 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.

\(\displaystyle l_{33}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{4}}\boldsymbol{v}\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)
where \(f_{4}\) is the 4th face.

\(\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 8 y z \left(6 x z - 4 x + 3 y z - 2 y - 3 z + 2\right)\\\displaystyle 8 x z \left(- 6 x z + 4 x - 3 y z + 2 y + 6 z - 4\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.

\(\displaystyle l_{34}:\mathbf{V}\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 48 y z \left(x z - x + 2 y z - 2 y - 2 z + 2\right)\\\displaystyle 48 x z \left(- x z + x - 2 y z + 2 y + z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.

\(\displaystyle l_{35}:\mathbf{V}\mapsto\displaystyle\int_{R}(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right))v\)
where \(R\) is the reference element.

\(\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle 48 y z \left(- 2 x z + 2 x - y z + y + z - 1\right)\\\displaystyle 48 x z \left(2 x z - 2 x + y z - y - 2 z + 2\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 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]