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Degree 2 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: \(1\), \(z\), \(z^{2}\), \(y\), \(y z\), \(y z^{2}\), \(y^{2}\), \(y^{2} z\), \(y^{2} z^{2}\), \(x\), \(x z\), \(x z^{2}\), \(x y\), \(x y z\), \(x y z^{2}\), \(x y^{2}\), \(x y^{2} z\), \(x y^{2} z^{2}\), \(x^{2}\), \(x^{2} z\), \(x^{2} z^{2}\), \(x^{2} y\), \(x^{2} y z\), \(x^{2} y z^{2}\), \(x^{2} y^{2}\), \(x^{2} y^{2} z\), \(x^{2} y^{2} z^{2}\)
- \(\mathcal{L}=\{l_0,...,l_{26}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0,0)\)
\(\displaystyle \phi_{0} = 8 x^{2} y^{2} z^{2} - 12 x^{2} y^{2} z + 4 x^{2} y^{2} - 12 x^{2} y z^{2} + 18 x^{2} y z - 6 x^{2} y + 4 x^{2} z^{2} - 6 x^{2} z + 2 x^{2} - 12 x y^{2} z^{2} + 18 x y^{2} z - 6 x y^{2} + 18 x y z^{2} - 27 x y z + 9 x y - 6 x z^{2} + 9 x z - 3 x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = 8 x^{2} y^{2} z^{2} - 12 x^{2} y^{2} z + 4 x^{2} y^{2} - 12 x^{2} y z^{2} + 18 x^{2} y z - 6 x^{2} y + 4 x^{2} z^{2} - 6 x^{2} z + 2 x^{2} - 12 x y^{2} z^{2} + 18 x y^{2} z - 6 x y^{2} + 18 x y z^{2} - 27 x y z + 9 x y - 6 x z^{2} + 9 x z - 3 x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1,0,0)\)
\(\displaystyle \phi_{1} = x \left(8 x y^{2} z^{2} - 12 x y^{2} z + 4 x y^{2} - 12 x y z^{2} + 18 x y z - 6 x y + 4 x z^{2} - 6 x z + 2 x - 4 y^{2} z^{2} + 6 y^{2} z - 2 y^{2} + 6 y z^{2} - 9 y z + 3 y - 2 z^{2} + 3 z - 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{1} = x \left(8 x y^{2} z^{2} - 12 x y^{2} z + 4 x y^{2} - 12 x y z^{2} + 18 x y z - 6 x y + 4 x z^{2} - 6 x z + 2 x - 4 y^{2} z^{2} + 6 y^{2} z - 2 y^{2} + 6 y z^{2} - 9 y z + 3 y - 2 z^{2} + 3 z - 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(0,1,0)\)
\(\displaystyle \phi_{2} = y \left(8 x^{2} y z^{2} - 12 x^{2} y z + 4 x^{2} y - 4 x^{2} z^{2} + 6 x^{2} z - 2 x^{2} - 12 x y z^{2} + 18 x y z - 6 x y + 6 x z^{2} - 9 x z + 3 x + 4 y z^{2} - 6 y z + 2 y - 2 z^{2} + 3 z - 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{2} = y \left(8 x^{2} y z^{2} - 12 x^{2} y z + 4 x^{2} y - 4 x^{2} z^{2} + 6 x^{2} z - 2 x^{2} - 12 x y z^{2} + 18 x y z - 6 x y + 6 x z^{2} - 9 x z + 3 x + 4 y z^{2} - 6 y z + 2 y - 2 z^{2} + 3 z - 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(1,1,0)\)
\(\displaystyle \phi_{3} = x y \left(8 x y z^{2} - 12 x y z + 4 x y - 4 x z^{2} + 6 x z - 2 x - 4 y z^{2} + 6 y z - 2 y + 2 z^{2} - 3 z + 1\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \phi_{3} = x y \left(8 x y z^{2} - 12 x y z + 4 x y - 4 x z^{2} + 6 x z - 2 x - 4 y z^{2} + 6 y z - 2 y + 2 z^{2} - 3 z + 1\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(0,0,1)\)
\(\displaystyle \phi_{4} = z \left(8 x^{2} y^{2} z - 4 x^{2} y^{2} - 12 x^{2} y z + 6 x^{2} y + 4 x^{2} z - 2 x^{2} - 12 x y^{2} z + 6 x y^{2} + 18 x y z - 9 x y - 6 x z + 3 x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)\)
This DOF is associated with vertex 4 of the reference element.
\(\displaystyle \phi_{4} = z \left(8 x^{2} y^{2} z - 4 x^{2} y^{2} - 12 x^{2} y z + 6 x^{2} y + 4 x^{2} z - 2 x^{2} - 12 x y^{2} z + 6 x y^{2} + 18 x y z - 9 x y - 6 x z + 3 x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)\)
This DOF is associated with vertex 4 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(1,0,1)\)
\(\displaystyle \phi_{5} = x z \left(8 x y^{2} z - 4 x y^{2} - 12 x y z + 6 x y + 4 x z - 2 x - 4 y^{2} z + 2 y^{2} + 6 y z - 3 y - 2 z + 1\right)\)
This DOF is associated with vertex 5 of the reference element.
\(\displaystyle \phi_{5} = x z \left(8 x y^{2} z - 4 x y^{2} - 12 x y z + 6 x y + 4 x z - 2 x - 4 y^{2} z + 2 y^{2} + 6 y z - 3 y - 2 z + 1\right)\)
This DOF is associated with vertex 5 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,1,1)\)
\(\displaystyle \phi_{6} = y z \left(8 x^{2} y z - 4 x^{2} y - 4 x^{2} z + 2 x^{2} - 12 x y z + 6 x y + 6 x z - 3 x + 4 y z - 2 y - 2 z + 1\right)\)
This DOF is associated with vertex 6 of the reference element.
\(\displaystyle \phi_{6} = y z \left(8 x^{2} y z - 4 x^{2} y - 4 x^{2} z + 2 x^{2} - 12 x y z + 6 x y + 6 x z - 3 x + 4 y z - 2 y - 2 z + 1\right)\)
This DOF is associated with vertex 6 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(1,1,1)\)
\(\displaystyle \phi_{7} = x y z \left(8 x y z - 4 x y - 4 x z + 2 x - 4 y z + 2 y + 2 z - 1\right)\)
This DOF is associated with vertex 7 of the reference element.
\(\displaystyle \phi_{7} = x y z \left(8 x y z - 4 x y - 4 x z + 2 x - 4 y z + 2 y + 2 z - 1\right)\)
This DOF is associated with vertex 7 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(\tfrac{1}{2},0,0)\)
\(\displaystyle \phi_{8} = 4 x \left(- 4 x y^{2} z^{2} + 6 x y^{2} z - 2 x y^{2} + 6 x y z^{2} - 9 x y z + 3 x y - 2 x z^{2} + 3 x z - x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \phi_{8} = 4 x \left(- 4 x y^{2} z^{2} + 6 x y^{2} z - 2 x y^{2} + 6 x y z^{2} - 9 x y z + 3 x y - 2 x z^{2} + 3 x z - x + 4 y^{2} z^{2} - 6 y^{2} z + 2 y^{2} - 6 y z^{2} + 9 y z - 3 y + 2 z^{2} - 3 z + 1\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{9}:v\mapsto v(0,\tfrac{1}{2},0)\)
\(\displaystyle \phi_{9} = 4 y \left(- 4 x^{2} y z^{2} + 6 x^{2} y z - 2 x^{2} y + 4 x^{2} z^{2} - 6 x^{2} z + 2 x^{2} + 6 x y z^{2} - 9 x y z + 3 x y - 6 x z^{2} + 9 x z - 3 x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \phi_{9} = 4 y \left(- 4 x^{2} y z^{2} + 6 x^{2} y z - 2 x^{2} y + 4 x^{2} z^{2} - 6 x^{2} z + 2 x^{2} + 6 x y z^{2} - 9 x y z + 3 x y - 6 x z^{2} + 9 x z - 3 x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{10}:v\mapsto v(0,0,\tfrac{1}{2})\)
\(\displaystyle \phi_{10} = 4 z \left(- 4 x^{2} y^{2} z + 4 x^{2} y^{2} + 6 x^{2} y z - 6 x^{2} y - 2 x^{2} z + 2 x^{2} + 6 x y^{2} z - 6 x y^{2} - 9 x y z + 9 x y + 3 x z - 3 x - 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \phi_{10} = 4 z \left(- 4 x^{2} y^{2} z + 4 x^{2} y^{2} + 6 x^{2} y z - 6 x^{2} y - 2 x^{2} z + 2 x^{2} + 6 x y^{2} z - 6 x y^{2} - 9 x y z + 9 x y + 3 x z - 3 x - 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{11}:v\mapsto v(1,\tfrac{1}{2},0)\)
\(\displaystyle \phi_{11} = 4 x y \left(- 4 x y z^{2} + 6 x y z - 2 x y + 4 x z^{2} - 6 x z + 2 x + 2 y z^{2} - 3 y z + y - 2 z^{2} + 3 z - 1\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle \phi_{11} = 4 x y \left(- 4 x y z^{2} + 6 x y z - 2 x y + 4 x z^{2} - 6 x z + 2 x + 2 y z^{2} - 3 y z + y - 2 z^{2} + 3 z - 1\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{12}:v\mapsto v(1,0,\tfrac{1}{2})\)
\(\displaystyle \phi_{12} = 4 x z \left(- 4 x y^{2} z + 4 x y^{2} + 6 x y z - 6 x y - 2 x z + 2 x + 2 y^{2} z - 2 y^{2} - 3 y z + 3 y + z - 1\right)\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle \phi_{12} = 4 x z \left(- 4 x y^{2} z + 4 x y^{2} + 6 x y z - 6 x y - 2 x z + 2 x + 2 y^{2} z - 2 y^{2} - 3 y z + 3 y + z - 1\right)\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{13}:v\mapsto v(\tfrac{1}{2},1,0)\)
\(\displaystyle \phi_{13} = 4 x y \left(- 4 x y z^{2} + 6 x y z - 2 x y + 2 x z^{2} - 3 x z + x + 4 y z^{2} - 6 y z + 2 y - 2 z^{2} + 3 z - 1\right)\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle \phi_{13} = 4 x y \left(- 4 x y z^{2} + 6 x y z - 2 x y + 2 x z^{2} - 3 x z + x + 4 y z^{2} - 6 y z + 2 y - 2 z^{2} + 3 z - 1\right)\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{14}:v\mapsto v(0,1,\tfrac{1}{2})\)
\(\displaystyle \phi_{14} = 4 y z \left(- 4 x^{2} y z + 4 x^{2} y + 2 x^{2} z - 2 x^{2} + 6 x y z - 6 x y - 3 x z + 3 x - 2 y z + 2 y + z - 1\right)\)
This DOF is associated with edge 6 of the reference element.
\(\displaystyle \phi_{14} = 4 y z \left(- 4 x^{2} y z + 4 x^{2} y + 2 x^{2} z - 2 x^{2} + 6 x y z - 6 x y - 3 x z + 3 x - 2 y z + 2 y + z - 1\right)\)
This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{15}:v\mapsto v(1,1,\tfrac{1}{2})\)
\(\displaystyle \phi_{15} = 4 x y z \left(- 4 x y z + 4 x y + 2 x z - 2 x + 2 y z - 2 y - z + 1\right)\)
This DOF is associated with edge 7 of the reference element.
\(\displaystyle \phi_{15} = 4 x y z \left(- 4 x y z + 4 x y + 2 x z - 2 x + 2 y z - 2 y - z + 1\right)\)
This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{16}:v\mapsto v(\tfrac{1}{2},0,1)\)
\(\displaystyle \phi_{16} = 4 x z \left(- 4 x y^{2} z + 2 x y^{2} + 6 x y z - 3 x y - 2 x z + x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)\)
This DOF is associated with edge 8 of the reference element.
\(\displaystyle \phi_{16} = 4 x z \left(- 4 x y^{2} z + 2 x y^{2} + 6 x y z - 3 x y - 2 x z + x + 4 y^{2} z - 2 y^{2} - 6 y z + 3 y + 2 z - 1\right)\)
This DOF is associated with edge 8 of the reference element.
\(\displaystyle l_{17}:v\mapsto v(0,\tfrac{1}{2},1)\)
\(\displaystyle \phi_{17} = 4 y z \left(- 4 x^{2} y z + 2 x^{2} y + 4 x^{2} z - 2 x^{2} + 6 x y z - 3 x y - 6 x z + 3 x - 2 y z + y + 2 z - 1\right)\)
This DOF is associated with edge 9 of the reference element.
\(\displaystyle \phi_{17} = 4 y z \left(- 4 x^{2} y z + 2 x^{2} y + 4 x^{2} z - 2 x^{2} + 6 x y z - 3 x y - 6 x z + 3 x - 2 y z + y + 2 z - 1\right)\)
This DOF is associated with edge 9 of the reference element.
\(\displaystyle l_{18}:v\mapsto v(1,\tfrac{1}{2},1)\)
\(\displaystyle \phi_{18} = 4 x y z \left(- 4 x y z + 2 x y + 4 x z - 2 x + 2 y z - y - 2 z + 1\right)\)
This DOF is associated with edge 10 of the reference element.
\(\displaystyle \phi_{18} = 4 x y z \left(- 4 x y z + 2 x y + 4 x z - 2 x + 2 y z - y - 2 z + 1\right)\)
This DOF is associated with edge 10 of the reference element.
\(\displaystyle l_{19}:v\mapsto v(\tfrac{1}{2},1,1)\)
\(\displaystyle \phi_{19} = 4 x y z \left(- 4 x y z + 2 x y + 2 x z - x + 4 y z - 2 y - 2 z + 1\right)\)
This DOF is associated with edge 11 of the reference element.
\(\displaystyle \phi_{19} = 4 x y z \left(- 4 x y z + 2 x y + 2 x z - x + 4 y z - 2 y - 2 z + 1\right)\)
This DOF is associated with edge 11 of the reference element.
\(\displaystyle l_{20}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},0)\)
\(\displaystyle \phi_{20} = 16 x y \left(2 x y z^{2} - 3 x y z + x y - 2 x z^{2} + 3 x z - x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{20} = 16 x y \left(2 x y z^{2} - 3 x y z + x y - 2 x z^{2} + 3 x z - x - 2 y z^{2} + 3 y z - y + 2 z^{2} - 3 z + 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{21}:v\mapsto v(\tfrac{1}{2},0,\tfrac{1}{2})\)
\(\displaystyle \phi_{21} = 16 x z \left(2 x y^{2} z - 2 x y^{2} - 3 x y z + 3 x y + x z - x - 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle \phi_{21} = 16 x z \left(2 x y^{2} z - 2 x y^{2} - 3 x y z + 3 x y + x z - x - 2 y^{2} z + 2 y^{2} + 3 y z - 3 y - z + 1\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{22}:v\mapsto v(0,\tfrac{1}{2},\tfrac{1}{2})\)
\(\displaystyle \phi_{22} = 16 y z \left(2 x^{2} y z - 2 x^{2} y - 2 x^{2} z + 2 x^{2} - 3 x y z + 3 x y + 3 x z - 3 x + y z - y - z + 1\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle \phi_{22} = 16 y z \left(2 x^{2} y z - 2 x^{2} y - 2 x^{2} z + 2 x^{2} - 3 x y z + 3 x y + 3 x z - 3 x + y z - y - z + 1\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{23}:v\mapsto v(1,\tfrac{1}{2},\tfrac{1}{2})\)
\(\displaystyle \phi_{23} = 16 x y z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle \phi_{23} = 16 x y z \left(2 x y z - 2 x y - 2 x z + 2 x - y z + y + z - 1\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{24}:v\mapsto v(\tfrac{1}{2},1,\tfrac{1}{2})\)
\(\displaystyle \phi_{24} = 16 x y z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)\)
This DOF is associated with face 4 of the reference element.
\(\displaystyle \phi_{24} = 16 x y z \left(2 x y z - 2 x y - x z + x - 2 y z + 2 y + z - 1\right)\)
This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{25}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},1)\)
\(\displaystyle \phi_{25} = 16 x y z \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)\)
This DOF is associated with face 5 of the reference element.
\(\displaystyle \phi_{25} = 16 x y z \left(2 x y z - x y - 2 x z + x - 2 y z + y + 2 z - 1\right)\)
This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{26}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})\)
\(\displaystyle \phi_{26} = 64 x y z \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with volume 0 of the reference element.
\(\displaystyle \phi_{26} = 64 x y z \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with volume 0 of the reference element.