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Degree 3 Hermite on a interval
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- \(R\) is the reference interval. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(x^{3}\)
- \(\mathcal{L}=\{l_0,...,l_{3}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0)\)
\(\displaystyle \phi_{0} = 2 x^{3} - 3 x^{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = 2 x^{3} - 3 x^{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v'(0)\)
\(\displaystyle \phi_{1} = x \left(x^{2} - 2 x + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{1} = x \left(x^{2} - 2 x + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(1)\)
\(\displaystyle \phi_{2} = x^{2} \cdot \left(3 - 2 x\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{2} = x^{2} \cdot \left(3 - 2 x\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{3}:v\mapsto v'(1)\)
\(\displaystyle \phi_{3} = x^{2} \left(x - 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{3} = x^{2} \left(x - 1\right)\)
This DOF is associated with vertex 1 of the reference element.