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Degree 1 vector Q on a hexahedron
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- \(R\) is the reference hexahedron. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{23}\}\)
- 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.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - x y z + x y + x z - x + y z - y - z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - x y z + x y + x z - x + y z - y - z + 1\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - x y z + x y + x z - x + y z - y - z + 1\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle x \left(y z - y - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(y z - y - z + 1\right)\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x \left(y z - y - z + 1\right)\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle y \left(x z - x - z + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(x z - x - z + 1\right)\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y \left(x z - x - z + 1\right)\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle x y \left(1 - z\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x y \left(1 - z\right)\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y \left(1 - z\right)\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle z \left(x y - x - y + 1\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle z \left(x y - x - y + 1\right)\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle z \left(x y - x - y + 1\right)\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle x z \left(1 - y\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x z \left(1 - y\right)\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x z \left(1 - y\right)\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{18} = \left(\begin{array}{c}\displaystyle y z \left(1 - x\right)\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{19} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y z \left(1 - x\right)\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{20} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y z \left(1 - x\right)\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{21} = \left(\begin{array}{c}\displaystyle x y z\\\displaystyle 0\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{22} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x y z\\\displaystyle 0\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{23} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x y z\end{array}\right)\)
This DOF is associated with vertex 7 of the reference element.