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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:
\(\displaystyle l_{0}:v\mapsto v(0,0,0)\)
\(\displaystyle \phi_{0} = - x y z + x y + x z - x + y z - y - z + 1\)
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.
\(\displaystyle l_{8}:\mathbf{v}\mapsto\displaystyle\int_{e_{0}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v\)
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \phi_{8} = \sqrt{3} x \left(x y z - x y - x z + x - y z + y + z - 1\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \phi_{8} = \sqrt{3} x \left(x y z - x y - x z + x - y z + y + z - 1\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{9}:\mathbf{v}\mapsto\displaystyle\int_{e_{1}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v\)
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \phi_{9} = \sqrt{3} y \left(x y z - x y - x z + x - y z + y + z - 1\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \phi_{9} = \sqrt{3} y \left(x y z - x y - x z + x - y z + y + z - 1\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{10}:\mathbf{v}\mapsto\displaystyle\int_{e_{2}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v\)
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \phi_{10} = \sqrt{3} z \left(x y z - x y - x z + x - y z + y + z - 1\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \phi_{10} = \sqrt{3} z \left(x y z - x y - x z + x - y z + y + z - 1\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{11}:\mathbf{v}\mapsto\displaystyle\int_{e_{3}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v\)
where \(e_{3}\) is the 3rd edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{3}\).
\(\displaystyle \phi_{11} = \sqrt{3} x y \left(- y z + y + z - 1\right)\)
This DOF is associated with edge 3 of the reference element.
where \(e_{3}\) is the 3rd edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{3}\).
\(\displaystyle \phi_{11} = \sqrt{3} x y \left(- y z + y + z - 1\right)\)
This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{12}:\mathbf{v}\mapsto\displaystyle\int_{e_{4}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v\)
where \(e_{4}\) is the 4th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{4}\).
\(\displaystyle \phi_{12} = \sqrt{3} x z \left(- y z + y + z - 1\right)\)
This DOF is associated with edge 4 of the reference element.
where \(e_{4}\) is the 4th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{4}\).
\(\displaystyle \phi_{12} = \sqrt{3} x z \left(- y z + y + z - 1\right)\)
This DOF is associated with edge 4 of the reference element.
\(\displaystyle l_{13}:\mathbf{v}\mapsto\displaystyle\int_{e_{5}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v\)
where \(e_{5}\) is the 5th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{5}\).
\(\displaystyle \phi_{13} = \sqrt{3} x y \left(- x z + x + z - 1\right)\)
This DOF is associated with edge 5 of the reference element.
where \(e_{5}\) is the 5th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{5}\).
\(\displaystyle \phi_{13} = \sqrt{3} x y \left(- x z + x + z - 1\right)\)
This DOF is associated with edge 5 of the reference element.
\(\displaystyle l_{14}:\mathbf{v}\mapsto\displaystyle\int_{e_{6}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v\)
where \(e_{6}\) is the 6th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{6}\).
\(\displaystyle \phi_{14} = \sqrt{3} y z \left(- x z + x + z - 1\right)\)
This DOF is associated with edge 6 of the reference element.
where \(e_{6}\) is the 6th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{6}\).
\(\displaystyle \phi_{14} = \sqrt{3} y z \left(- x z + x + z - 1\right)\)
This DOF is associated with edge 6 of the reference element.
\(\displaystyle l_{15}:\mathbf{v}\mapsto\displaystyle\int_{e_{7}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v\)
where \(e_{7}\) is the 7th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{7}\).
\(\displaystyle \phi_{15} = \sqrt{3} x y z \left(z - 1\right)\)
This DOF is associated with edge 7 of the reference element.
where \(e_{7}\) is the 7th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{7}\).
\(\displaystyle \phi_{15} = \sqrt{3} x y z \left(z - 1\right)\)
This DOF is associated with edge 7 of the reference element.
\(\displaystyle l_{16}:\mathbf{v}\mapsto\displaystyle\int_{e_{8}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v\)
where \(e_{8}\) is the 8th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{8}\).
\(\displaystyle \phi_{16} = \sqrt{3} x z \left(- x y + x + y - 1\right)\)
This DOF is associated with edge 8 of the reference element.
where \(e_{8}\) is the 8th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{8}\).
\(\displaystyle \phi_{16} = \sqrt{3} x z \left(- x y + x + y - 1\right)\)
This DOF is associated with edge 8 of the reference element.
\(\displaystyle l_{17}:\mathbf{v}\mapsto\displaystyle\int_{e_{9}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v\)
where \(e_{9}\) is the 9th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{9}\).
\(\displaystyle \phi_{17} = \sqrt{3} y z \left(- x y + x + y - 1\right)\)
This DOF is associated with edge 9 of the reference element.
where \(e_{9}\) is the 9th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{9}\).
\(\displaystyle \phi_{17} = \sqrt{3} y z \left(- x y + x + y - 1\right)\)
This DOF is associated with edge 9 of the reference element.
\(\displaystyle l_{18}:\mathbf{v}\mapsto\displaystyle\int_{e_{10}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v\)
where \(e_{10}\) is the 10th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{10}\).
\(\displaystyle \phi_{18} = \sqrt{3} x y z \left(y - 1\right)\)
This DOF is associated with edge 10 of the reference element.
where \(e_{10}\) is the 10th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{10}\).
\(\displaystyle \phi_{18} = \sqrt{3} x y z \left(y - 1\right)\)
This DOF is associated with edge 10 of the reference element.
\(\displaystyle l_{19}:\mathbf{v}\mapsto\displaystyle\int_{e_{11}}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v\)
where \(e_{11}\) is the 11th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{11}\).
\(\displaystyle \phi_{19} = \sqrt{3} x y z \left(x - 1\right)\)
This DOF is associated with edge 11 of the reference element.
where \(e_{11}\) is the 11th edge;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(e_{11}\).
\(\displaystyle \phi_{19} = \sqrt{3} x y z \left(x - 1\right)\)
This DOF is associated with edge 11 of the reference element.
\(\displaystyle l_{20}:\mathbf{v}\mapsto\displaystyle\int_{f_{0}}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v\)
where \(f_{0}\) is the 0th face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \phi_{20} = 3 x y \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with face 0 of the reference element.
where \(f_{0}\) is the 0th face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{0}\).
\(\displaystyle \phi_{20} = 3 x y \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{21}:\mathbf{v}\mapsto\displaystyle\int_{f_{1}}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v\)
where \(f_{1}\) is the 1st face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \phi_{21} = 3 x z \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with face 1 of the reference element.
where \(f_{1}\) is the 1st face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{1}\).
\(\displaystyle \phi_{21} = 3 x z \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with face 1 of the reference element.
\(\displaystyle l_{22}:\mathbf{v}\mapsto\displaystyle\int_{f_{2}}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v\)
where \(f_{2}\) is the 2nd face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \phi_{22} = 3 y z \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with face 2 of the reference element.
where \(f_{2}\) is the 2nd face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{2}\).
\(\displaystyle \phi_{22} = 3 y z \left(- x y z + x y + x z - x + y z - y - z + 1\right)\)
This DOF is associated with face 2 of the reference element.
\(\displaystyle l_{23}:\mathbf{v}\mapsto\displaystyle\int_{f_{3}}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v\)
where \(f_{3}\) is the 3rd face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \phi_{23} = 3 x y z \left(y z - y - z + 1\right)\)
This DOF is associated with face 3 of the reference element.
where \(f_{3}\) is the 3rd face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{3}\).
\(\displaystyle \phi_{23} = 3 x y z \left(y z - y - z + 1\right)\)
This DOF is associated with face 3 of the reference element.
\(\displaystyle l_{24}:\mathbf{v}\mapsto\displaystyle\int_{f_{4}}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v\)
where \(f_{4}\) is the 4th face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{4}\).
\(\displaystyle \phi_{24} = 3 x y z \left(x z - x - z + 1\right)\)
This DOF is associated with face 4 of the reference element.
where \(f_{4}\) is the 4th face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{4}\).
\(\displaystyle \phi_{24} = 3 x y z \left(x z - x - z + 1\right)\)
This DOF is associated with face 4 of the reference element.
\(\displaystyle l_{25}:\mathbf{v}\mapsto\displaystyle\int_{f_{5}}(10800 s_{0}^{2} s_{1}^{2} - 10800 s_{0}^{2} s_{1} + 1800 s_{0}^{2} - 10800 s_{0} s_{1}^{2} + 10800 s_{0} s_{1} - 1800 s_{0} + 1800 s_{1}^{2} - 1800 s_{1} + 300)v\)
where \(f_{5}\) is the 5th face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{5}\).
\(\displaystyle \phi_{25} = 3 x y z \left(x y - x - y + 1\right)\)
This DOF is associated with face 5 of the reference element.
where \(f_{5}\) is the 5th face;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(f_{5}\).
\(\displaystyle \phi_{25} = 3 x y z \left(x y - x - y + 1\right)\)
This DOF is associated with face 5 of the reference element.
\(\displaystyle l_{26}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{3} \cdot \left(648000 s_{0}^{2} s_{1}^{2} s_{2}^{2} - 648000 s_{0}^{2} s_{1}^{2} s_{2} + 108000 s_{0}^{2} s_{1}^{2} - 648000 s_{0}^{2} s_{1} s_{2}^{2} + 648000 s_{0}^{2} s_{1} s_{2} - 108000 s_{0}^{2} s_{1} + 108000 s_{0}^{2} s_{2}^{2} - 108000 s_{0}^{2} s_{2} + 18000 s_{0}^{2} - 648000 s_{0} s_{1}^{2} s_{2}^{2} + 648000 s_{0} s_{1}^{2} s_{2} - 108000 s_{0} s_{1}^{2} + 648000 s_{0} s_{1} s_{2}^{2} - 648000 s_{0} s_{1} s_{2} + 108000 s_{0} s_{1} - 108000 s_{0} s_{2}^{2} + 108000 s_{0} s_{2} - 18000 s_{0} + 108000 s_{1}^{2} s_{2}^{2} - 108000 s_{1}^{2} s_{2} + 18000 s_{1}^{2} - 108000 s_{1} s_{2}^{2} + 108000 s_{1} s_{2} - 18000 s_{1} + 18000 s_{2}^{2} - 18000 s_{2} + 3000\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \phi_{26} = 3 \sqrt{3} x y z \left(x y z - x y - x z + x - y z + y + z - 1\right)\)
This DOF is associated with volume 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1},s_{2}\) is a parametrisation of \(R\).
\(\displaystyle \phi_{26} = 3 \sqrt{3} x y z \left(x y z - x y - x z + x - y z + y + z - 1\right)\)
This DOF is associated with volume 0 of the reference element.