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Degree 3 Taylor 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:
\(\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}v\)
where \(R\) is the reference element.
\(\displaystyle \phi_{0} = 1\)
This DOF is associated with edge 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \phi_{0} = 1\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v'(\tfrac{1}{2})\)
\(\displaystyle \phi_{1} = x - \frac{1}{2}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \phi_{1} = x - \frac{1}{2}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:v\mapsto v'(\tfrac{1}{2})\)
\(\displaystyle \phi_{2} = \frac{x^{2}}{2} - \frac{x}{2} + \frac{1}{12}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \phi_{2} = \frac{x^{2}}{2} - \frac{x}{2} + \frac{1}{12}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{3}:v\mapsto v'(\tfrac{1}{2})\)
\(\displaystyle \phi_{3} = \frac{x^{3}}{6} - \frac{x^{2}}{4} + \frac{x}{8} - \frac{1}{48}\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \phi_{3} = \frac{x^{3}}{6} - \frac{x^{2}}{4} + \frac{x}{8} - \frac{1}{48}\)
This DOF is associated with edge 0 of the reference element.