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Degree 1 Lagrange 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: \(1\), \(z\), \(y\), \(y z\), \(x\), \(x z\), \(x y\), \(x y z\)
- \(\mathcal{L}=\{l_0,...,l_{7}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0,0)\)
\(\displaystyle \phi_{0} = - x y z + x y + x z - x + y z - y - z + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = - x y z + x y + x z - x + y z - y - z + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0,0)\)
\(\displaystyle \phi_{1} = x \left(y z - y - z + 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{1} = x \left(y z - y - z + 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1,0)\)
\(\displaystyle \phi_{2} = y \left(x z - x - z + 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{2} = y \left(x z - x - z + 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(1,1,0)\)
\(\displaystyle \phi_{3} = x y \left(1 - z\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \phi_{3} = x y \left(1 - z\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(0,0,1)\)
\(\displaystyle \phi_{4} = z \left(x y - x - y + 1\right)\)
This DOF is associated with vertex 4 of the reference element.
\(\displaystyle \phi_{4} = z \left(x y - x - y + 1\right)\)
This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(1,0,1)\)
\(\displaystyle \phi_{5} = x z \left(1 - y\right)\)
This DOF is associated with vertex 5 of the reference element.
\(\displaystyle \phi_{5} = x z \left(1 - y\right)\)
This DOF is associated with vertex 5 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,1,1)\)
\(\displaystyle \phi_{6} = y z \left(1 - x\right)\)
This DOF is associated with vertex 6 of the reference element.
\(\displaystyle \phi_{6} = y z \left(1 - x\right)\)
This DOF is associated with vertex 6 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(1,1,1)\)
\(\displaystyle \phi_{7} = x y z\)
This DOF is associated with vertex 7 of the reference element.
\(\displaystyle \phi_{7} = x y z\)
This DOF is associated with vertex 7 of the reference element.