an encyclopedia of finite element definitions

Vector Q

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

Alternative namesvector Lagrange
Orders\(0\leqslant k\)
Reference elementsquadrilateral, hexahedron
Polynomial set\(\mathcal{Q}_{k}^d\)
↓ Show polynomial set definitions ↓
DOFsOn each vertex: point evaluations in coordinate directions
On each edge: point evaluations in coordinate directions
On each face: point evaluations in coordinate directions
On each volume: point evaluations in coordinate directions
Number of DOFsinterval: \(k+1\) (A000027)
quadrilateral: \(2(k+1)^2\) (A001105)
hexahedron: \(3(k+1)^3\) (A117642)
CategoriesVector-valued elements

Implementations

Symfem"vector Q"
↓ Show Symfem examples ↓
UFL"Q"
↓ Show UFL examples ↓

Examples

quadrilateral
order 1
quadrilateral
order 2
hexahedron
order 1
hexahedron
order 2
  • \(R\) is the reference quadrilateral. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{7}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

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

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

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

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

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

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

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

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

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

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

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

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

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

This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

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

This DOF is associated with vertex 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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y^{2}\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{17}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

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

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

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

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

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

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

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

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

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

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(4 x^{2} y - 2 x^{2} - 6 x y + 3 x + 2 y - 1\right)\end{array}\right)\)

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

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

This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

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

This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

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

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

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

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 4 y \left(- 2 x^{2} y + 2 x^{2} + 3 x y - 3 x - y + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(- 2 x^{2} y + 2 x^{2} + 3 x y - 3 x - y + 1\right)\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

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

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y \left(- 2 x y + 2 x + y - 1\right)\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 4 x y \left(- 2 x y + x + 2 y - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y \left(- 2 x y + x + 2 y - 1\right)\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 16 x y \left(x y - x - y + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 x y \left(x y - x - y + 1\right)\end{array}\right)\)

This DOF is associated with face 0 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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{23}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

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

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

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

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

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

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

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

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

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

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

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

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

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

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

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

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

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

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

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

This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

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

This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

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

This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

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

This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(x y - x - y + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(x y - x - y + 1\right)\end{array}\right)\)

This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle x z \left(1 - y\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 5 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x z \left(1 - y\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 5 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z \left(1 - y\right)\end{array}\right)\)

This DOF is associated with vertex 5 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle y z \left(1 - x\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 6 of the reference element.
\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y z \left(1 - x\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 6 of the reference element.
\(\displaystyle l_{20}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z \left(1 - x\right)\end{array}\right)\)

This DOF is associated with vertex 6 of the reference element.
\(\displaystyle l_{21}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle x y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 7 of the reference element.
\(\displaystyle l_{22}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 7 of the reference element.
\(\displaystyle l_{23}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

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

This DOF is associated with vertex 7 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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2} z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2} z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2} z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2} z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2} z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y^{2} z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{2} z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2} z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y^{2} z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y^{2} z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2} z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y^{2} z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y z^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y^{2} z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y^{2} z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y^{2} z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2} y^{2} z^{2}\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y^{2} z^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x^{2} y^{2} z^{2}\end{array}\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{80}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 8 x^{2} y^{2} z^{2} - 12 x^{2} y^{2} z + 4 x^{2} y^{2} - 12 x^{2} y z^{2} + 18 x^{2} y z - 6 x^{2} y + 4 x^{2} z^{2} - 6 x^{2} z + 2 x^{2} - 12 x y^{2} z^{2} + 18 x y^{2} z - 6 x y^{2} + 18 x y z^{2} - 27 x y z + 9 x y - 6 x z^{2} + 9 x z - 3 x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 8 x^{2} y^{2} z^{2} - 12 x^{2} y^{2} z + 4 x^{2} y^{2} - 12 x^{2} y z^{2} + 18 x^{2} y z - 6 x^{2} y + 4 x^{2} z^{2} - 6 x^{2} z + 2 x^{2} - 12 x y^{2} z^{2} + 18 x y^{2} z - 6 x y^{2} + 18 x y z^{2} - 27 x y z + 9 x y - 6 x z^{2} + 9 x z - 3 x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 8 x^{2} y^{2} z^{2} - 12 x^{2} y^{2} z + 4 x^{2} y^{2} - 12 x^{2} y z^{2} + 18 x^{2} y z - 6 x^{2} y + 4 x^{2} z^{2} - 6 x^{2} z + 2 x^{2} - 12 x y^{2} z^{2} + 18 x y^{2} z - 6 x y^{2} + 18 x y z^{2} - 27 x y z + 9 x y - 6 x z^{2} + 9 x z - 3 x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1\end{array}\right)\)

This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle x \left(8 x y^{2} z^{2} - 12 x y^{2} z + 4 x y^{2} - 12 x y z^{2} + 18 x y z - 6 x y + 4 x z^{2} - 6 x z + 2 x - 4 y^{2} z^{2} + 6 y^{2} z - 2 y^{2} + 6 y z^{2} - 9 y z + 3 y - 2 z^{2} + 3 z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(8 x y^{2} z^{2} - 12 x y^{2} z + 4 x y^{2} - 12 x y z^{2} + 18 x y z - 6 x y + 4 x z^{2} - 6 x z + 2 x - 4 y^{2} z^{2} + 6 y^{2} z - 2 y^{2} + 6 y z^{2} - 9 y z + 3 y - 2 z^{2} + 3 z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x \left(8 x y^{2} z^{2} - 12 x y^{2} z + 4 x y^{2} - 12 x y z^{2} + 18 x y z - 6 x y + 4 x z^{2} - 6 x z + 2 x - 4 y^{2} z^{2} + 6 y^{2} z - 2 y^{2} + 6 y z^{2} - 9 y z + 3 y - 2 z^{2} + 3 z - 1\right)\end{array}\right)\)

This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle y \left(8 x^{2} y z^{2} - 12 x^{2} y z + 4 x^{2} y - 4 x^{2} z^{2} + 6 x^{2} z - 2 x^{2} - 12 x y z^{2} + 18 x y z - 6 x y + 6 x z^{2} - 9 x z + 3 x + 4 y z^{2} - 6 y z + 2 y - 2 z^{2} + 3 z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(8 x^{2} y z^{2} - 12 x^{2} y z + 4 x^{2} y - 4 x^{2} z^{2} + 6 x^{2} z - 2 x^{2} - 12 x y z^{2} + 18 x y z - 6 x y + 6 x z^{2} - 9 x z + 3 x + 4 y z^{2} - 6 y z + 2 y - 2 z^{2} + 3 z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y \left(8 x^{2} y z^{2} - 12 x^{2} y z + 4 x^{2} y - 4 x^{2} z^{2} + 6 x^{2} z - 2 x^{2} - 12 x y z^{2} + 18 x y z - 6 x y + 6 x z^{2} - 9 x z + 3 x + 4 y z^{2} - 6 y z + 2 y - 2 z^{2} + 3 z - 1\right)\end{array}\right)\)

This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle x y \left(8 x y z^{2} - 12 x y z + 4 x y - 4 x z^{2} + 6 x z - 2 x - 4 y z^{2} + 6 y z - 2 y + 2 z^{2} - 3 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x y \left(8 x y z^{2} - 12 x y z + 4 x y - 4 x z^{2} + 6 x z - 2 x - 4 y z^{2} + 6 y z - 2 y + 2 z^{2} - 3 z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y \left(8 x y z^{2} - 12 x y z + 4 x y - 4 x z^{2} + 6 x z - 2 x - 4 y z^{2} + 6 y z - 2 y + 2 z^{2} - 3 z + 1\right)\end{array}\right)\)

This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle z \left(8 x^{2} y^{2} z - 4 x^{2} y^{2} - 12 x^{2} y z + 6 x^{2} y + 4 x^{2} z - 2 x^{2} - 12 x y^{2} z + 6 x y^{2} + 18 x y z - 9 x y - 6 x z + 3 x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(8 x^{2} y^{2} z - 4 x^{2} y^{2} - 12 x^{2} y z + 6 x^{2} y + 4 x^{2} z - 2 x^{2} - 12 x y^{2} z + 6 x y^{2} + 18 x y z - 9 x y - 6 x z + 3 x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(8 x^{2} y^{2} z - 4 x^{2} y^{2} - 12 x^{2} y z + 6 x^{2} y + 4 x^{2} z - 2 x^{2} - 12 x y^{2} z + 6 x y^{2} + 18 x y z - 9 x y - 6 x z + 3 x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)\end{array}\right)\)

This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle x z \left(8 x y^{2} z - 4 x y^{2} - 12 x y z + 6 x y + 4 x z - 2 x - 4 y^{2} z + 2 y^{2} + 6 y z - 3 y - 2 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 5 of the reference element.
\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

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

This DOF is associated with vertex 5 of the reference element.
\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z \left(8 x y^{2} z - 4 x y^{2} - 12 x y z + 6 x y + 4 x z - 2 x - 4 y^{2} z + 2 y^{2} + 6 y z - 3 y - 2 z + 1\right)\end{array}\right)\)

This DOF is associated with vertex 5 of the reference element.
\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle y z \left(8 x^{2} y z - 4 x^{2} y - 4 x^{2} z + 2 x^{2} - 12 x y z + 6 x y + 6 x z - 3 x + 4 y z - 2 y - 2 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 6 of the reference element.
\(\displaystyle l_{19}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y z \left(8 x^{2} y z - 4 x^{2} y - 4 x^{2} z + 2 x^{2} - 12 x y z + 6 x y + 6 x z - 3 x + 4 y z - 2 y - 2 z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 6 of the reference element.
\(\displaystyle l_{20}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z \left(8 x^{2} y z - 4 x^{2} y - 4 x^{2} z + 2 x^{2} - 12 x y z + 6 x y + 6 x z - 3 x + 4 y z - 2 y - 2 z + 1\right)\end{array}\right)\)

This DOF is associated with vertex 6 of the reference element.
\(\displaystyle l_{21}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle x y z \left(8 x y z - 4 x y - 4 x z + 2 x - 4 y z + 2 y + 2 z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 7 of the reference element.
\(\displaystyle l_{22}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z \left(8 x y z - 4 x y - 4 x z + 2 x - 4 y z + 2 y + 2 z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with vertex 7 of the reference element.
\(\displaystyle l_{23}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z \left(8 x y z - 4 x y - 4 x z + 2 x - 4 y z + 2 y + 2 z - 1\right)\end{array}\right)\)

This DOF is associated with vertex 7 of the reference element.
\(\displaystyle l_{24}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{24} = \left(\begin{array}{c}\displaystyle 4 x \left(- 4 x y^{2} z^{2} + 6 x y^{2} z - 2 x y^{2} + 6 x y z^{2} - 9 x y z + 3 x y - 2 x z^{2} + 3 x z - x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{25}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{25} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x \left(- 4 x y^{2} z^{2} + 6 x y^{2} z - 2 x y^{2} + 6 x y z^{2} - 9 x y z + 3 x y - 2 x z^{2} + 3 x z - x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{26}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{26} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x \left(- 4 x y^{2} z^{2} + 6 x y^{2} z - 2 x y^{2} + 6 x y z^{2} - 9 x y z + 3 x y - 2 x z^{2} + 3 x z - x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1\right)\end{array}\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{27}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{27} = \left(\begin{array}{c}\displaystyle 4 y \left(- 4 x^{2} y z^{2} + 6 x^{2} y z - 2 x^{2} y + 4 x^{2} z^{2} - 6 x^{2} z + 2 x^{2} + 6 x y z^{2} - 9 x y z + 3 x y - 6 x z^{2} + 9 x z - 3 x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{28}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{28} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(- 4 x^{2} y z^{2} + 6 x^{2} y z - 2 x^{2} y + 4 x^{2} z^{2} - 6 x^{2} z + 2 x^{2} + 6 x y z^{2} - 9 x y z + 3 x y - 6 x z^{2} + 9 x z - 3 x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{29}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{29} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 y \left(- 4 x^{2} y z^{2} + 6 x^{2} y z - 2 x^{2} y + 4 x^{2} z^{2} - 6 x^{2} z + 2 x^{2} + 6 x y z^{2} - 9 x y z + 3 x y - 6 x z^{2} + 9 x z - 3 x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\end{array}\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{30}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{30} = \left(\begin{array}{c}\displaystyle 4 z \left(- 4 x^{2} y^{2} z + 4 x^{2} y^{2} + 6 x^{2} y z - 6 x^{2} y - 2 x^{2} z + 2 x^{2} + 6 x y^{2} z - 6 x y^{2} - 9 x y z + 9 x y + 3 x z - 3 x - 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{31}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{31} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 z \left(- 4 x^{2} y^{2} z + 4 x^{2} y^{2} + 6 x^{2} y z - 6 x^{2} y - 2 x^{2} z + 2 x^{2} + 6 x y^{2} z - 6 x y^{2} - 9 x y z + 9 x y + 3 x z - 3 x - 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{32}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{32} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z \left(- 4 x^{2} y^{2} z + 4 x^{2} y^{2} + 6 x^{2} y z - 6 x^{2} y - 2 x^{2} z + 2 x^{2} + 6 x y^{2} z - 6 x y^{2} - 9 x y z + 9 x y + 3 x z - 3 x - 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\end{array}\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{33}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{33} = \left(\begin{array}{c}\displaystyle 4 x y \left(- 4 x y z^{2} + 6 x y z - 2 x y + 4 x z^{2} - 6 x z + 2 x + 2 y z^{2} - 3 y z + y - 2 z^{2} + 3 z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{34}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{34} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y \left(- 4 x y z^{2} + 6 x y z - 2 x y + 4 x z^{2} - 6 x z + 2 x + 2 y z^{2} - 3 y z + y - 2 z^{2} + 3 z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{35}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{35} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x y \left(- 4 x y z^{2} + 6 x y z - 2 x y + 4 x z^{2} - 6 x z + 2 x + 2 y z^{2} - 3 y z + y - 2 z^{2} + 3 z - 1\right)\end{array}\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{36}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{36} = \left(\begin{array}{c}\displaystyle 4 x z \left(- 4 x y^{2} z + 4 x y^{2} + 6 x y z - 6 x y - 2 x z + 2 x + 2 y^{2} z - 2 y^{2} - 3 y z + 3 y + z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{37}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{37} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x z \left(- 4 x y^{2} z + 4 x y^{2} + 6 x y z - 6 x y - 2 x z + 2 x + 2 y^{2} z - 2 y^{2} - 3 y z + 3 y + z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{38}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{38} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x z \left(- 4 x y^{2} z + 4 x y^{2} + 6 x y z - 6 x y - 2 x z + 2 x + 2 y^{2} z - 2 y^{2} - 3 y z + 3 y + z - 1\right)\end{array}\right)\)

This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{39}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},1,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{39} = \left(\begin{array}{c}\displaystyle 4 x y \left(- 4 x y z^{2} + 6 x y z - 2 x y + 2 x z^{2} - 3 x z + x + 4 y z^{2} - 6 y z + 2 y - 2 z^{2} + 3 z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{40}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},1,0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{40} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y \left(- 4 x y z^{2} + 6 x y z - 2 x y + 2 x z^{2} - 3 x z + x + 4 y z^{2} - 6 y z + 2 y - 2 z^{2} + 3 z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{41}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{41} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x y \left(- 4 x y z^{2} + 6 x y z - 2 x y + 2 x z^{2} - 3 x z + x + 4 y z^{2} - 6 y z + 2 y - 2 z^{2} + 3 z - 1\right)\end{array}\right)\)

This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{42}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{42} = \left(\begin{array}{c}\displaystyle 4 y z \left(- 4 x^{2} y z + 4 x^{2} y + 2 x^{2} z - 2 x^{2} + 6 x y z - 6 x y - 3 x z + 3 x - 2 y z + 2 y + z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{43}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{43} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y z \left(- 4 x^{2} y z + 4 x^{2} y + 2 x^{2} z - 2 x^{2} + 6 x y z - 6 x y - 3 x z + 3 x - 2 y z + 2 y + z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{44}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{44} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 y z \left(- 4 x^{2} y z + 4 x^{2} y + 2 x^{2} z - 2 x^{2} + 6 x y z - 6 x y - 3 x z + 3 x - 2 y z + 2 y + z - 1\right)\end{array}\right)\)

This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{45}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{45} = \left(\begin{array}{c}\displaystyle 4 x y z \left(- 4 x y z + 4 x y + 2 x z - 2 x + 2 y z - 2 y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{46}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{46} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y z \left(- 4 x y z + 4 x y + 2 x z - 2 x + 2 y z - 2 y - z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{47}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,1,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{47} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x y z \left(- 4 x y z + 4 x y + 2 x z - 2 x + 2 y z - 2 y - z + 1\right)\end{array}\right)\)

This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{48}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{48} = \left(\begin{array}{c}\displaystyle 4 x z \left(- 4 x y^{2} z + 2 x y^{2} + 6 x y z - 3 x y - 2 x z + x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 8 of the reference element.
\(\displaystyle l_{49}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{49} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x z \left(- 4 x y^{2} z + 2 x y^{2} + 6 x y z - 3 x y - 2 x z + x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 8 of the reference element.
\(\displaystyle l_{50}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{50} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x z \left(- 4 x y^{2} z + 2 x y^{2} + 6 x y z - 3 x y - 2 x z + x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)\end{array}\right)\)

This DOF is associated with edge 8 of the reference element.
\(\displaystyle l_{51}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{51} = \left(\begin{array}{c}\displaystyle 4 y z \left(- 4 x^{2} y z + 2 x^{2} y + 4 x^{2} z - 2 x^{2} + 6 x y z - 3 x y - 6 x z + 3 x - 2 y z + y + 2 z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 9 of the reference element.
\(\displaystyle l_{52}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{52} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y z \left(- 4 x^{2} y z + 2 x^{2} y + 4 x^{2} z - 2 x^{2} + 6 x y z - 3 x y - 6 x z + 3 x - 2 y z + y + 2 z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 9 of the reference element.
\(\displaystyle l_{53}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{53} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 y z \left(- 4 x^{2} y z + 2 x^{2} y + 4 x^{2} z - 2 x^{2} + 6 x y z - 3 x y - 6 x z + 3 x - 2 y z + y + 2 z - 1\right)\end{array}\right)\)

This DOF is associated with edge 9 of the reference element.
\(\displaystyle l_{54}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,\tfrac{1}{2},1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{54} = \left(\begin{array}{c}\displaystyle 4 x y z \left(- 4 x y z + 2 x y + 4 x z - 2 x + 2 y z - y - 2 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 10 of the reference element.
\(\displaystyle l_{55}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,\tfrac{1}{2},1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{55} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y z \left(- 4 x y z + 2 x y + 4 x z - 2 x + 2 y z - y - 2 z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 10 of the reference element.
\(\displaystyle l_{56}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,\tfrac{1}{2},1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{56} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x y z \left(- 4 x y z + 2 x y + 4 x z - 2 x + 2 y z - y - 2 z + 1\right)\end{array}\right)\)

This DOF is associated with edge 10 of the reference element.
\(\displaystyle l_{57}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},1,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{57} = \left(\begin{array}{c}\displaystyle 4 x y z \left(- 4 x y z + 2 x y + 2 x z - x + 4 y z - 2 y - 2 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 11 of the reference element.
\(\displaystyle l_{58}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},1,1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{58} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y z \left(- 4 x y z + 2 x y + 2 x z - x + 4 y z - 2 y - 2 z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with edge 11 of the reference element.
\(\displaystyle l_{59}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},1,1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{59} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x y z \left(- 4 x y z + 2 x y + 2 x z - x + 4 y z - 2 y - 2 z + 1\right)\end{array}\right)\)

This DOF is associated with edge 11 of the reference element.
\(\displaystyle l_{60}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{60} = \left(\begin{array}{c}\displaystyle 16 x y \left(2 x y z^{2} - 3 x y z + x y - 2 x z^{2} + 3 x z - x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{61}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{61} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 x y \left(2 x y z^{2} - 3 x y z + x y - 2 x z^{2} + 3 x z - x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{62}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{62} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 x y \left(2 x y z^{2} - 3 x y z + x y - 2 x z^{2} + 3 x z - x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\end{array}\right)\)

This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{63}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{63} = \left(\begin{array}{c}\displaystyle 16 x z \left(2 x y^{2} z - 2 x y^{2} - 3 x y z + 3 x y + x z - x - 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{64}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{64} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 x z \left(2 x y^{2} z - 2 x y^{2} - 3 x y z + 3 x y + x z - x - 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{65}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{65} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 x z \left(2 x y^{2} z - 2 x y^{2} - 3 x y z + 3 x y + x z - x - 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\end{array}\right)\)

This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{66}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{66} = \left(\begin{array}{c}\displaystyle 16 y z \left(2 x^{2} y z - 2 x^{2} y - 2 x^{2} z + 2 x^{2} - 3 x y z + 3 x y + 3 x z - 3 x + y z - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{67}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{67} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 y z \left(2 x^{2} y z - 2 x^{2} y - 2 x^{2} z + 2 x^{2} - 3 x y z + 3 x y + 3 x z - 3 x + y z - y - z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{68}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{68} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 y z \left(2 x^{2} y z - 2 x^{2} y - 2 x^{2} z + 2 x^{2} - 3 x y z + 3 x y + 3 x z - 3 x + y z - y - z + 1\right)\end{array}\right)\)

This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{69}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{69} = \left(\begin{array}{c}\displaystyle 16 x y z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{70}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{70} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 x y z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{71}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{71} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 x y z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)\end{array}\right)\)

This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{72}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},1,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{72} = \left(\begin{array}{c}\displaystyle 16 x y z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{73}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},1,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{73} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 x y z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{74}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},1,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{74} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 x y z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)\end{array}\right)\)

This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{75}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{75} = \left(\begin{array}{c}\displaystyle 16 x y z \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{76}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},1)\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{76} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 16 x y z \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{77}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},1)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{77} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 16 x y z \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)\end{array}\right)\)

This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{78}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{78} = \left(\begin{array}{c}\displaystyle 64 x y z \left(- x y z + x y + x z - x + y z - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{79}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{79} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 64 x y z \left(- x y z + x y + x z - x + y z - y - z + 1\right)\\\displaystyle 0\end{array}\right)\)

This DOF is associated with volume 0 of the reference element.
\(\displaystyle l_{80}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{80} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 64 x y z \left(- x y z + x y + x z - x + y z - y - z + 1\right)\end{array}\right)\)

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

DefElement stats

Element added30 December 2020
Element last updated05 June 2021