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Degree 3 Lagrange 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:
\(\displaystyle l_{0}:v\mapsto v(0)\)
\(\displaystyle \phi_{0} = - \frac{9 x^{3}}{2} + 9 x^{2} - \frac{11 x}{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = - \frac{9 x^{3}}{2} + 9 x^{2} - \frac{11 x}{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(1)\)
\(\displaystyle \phi_{1} = \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{1} = \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(\tfrac{1}{3})\)
\(\displaystyle \phi_{2} = \frac{9 x \left(3 x^{2} - 5 x + 2\right)}{2}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \phi_{2} = \frac{9 x \left(3 x^{2} - 5 x + 2\right)}{2}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(\tfrac{2}{3})\)
\(\displaystyle \phi_{3} = \frac{9 x \left(- 3 x^{2} + 4 x - 1\right)}{2}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \phi_{3} = \frac{9 x \left(- 3 x^{2} + 4 x - 1\right)}{2}\)
This DOF is associated with edge 0 of the reference element.