Degree 3 bubble on a interval

• \(R\) is the reference interval. The following numbering of the subentities of the reference is used:

• \(\mathcal{V}\) is spanned by: \(x \left(x^{2} - 2 x + 1\right)\), \(x^{2} \left(1 - x\right)\)
• \(\mathcal{L}=\{l_0,...,l_{1}\}\)
• Functionals and basis functions:

\(\displaystyle l_{0}:v\mapsto v(\tfrac{1}{3})\)

\(\displaystyle \phi_{0} = \frac{9 x \left(3 x^{2} - 5 x + 2\right)}{2}\)

This DOF is associated with edge 0 of the reference element.

\(\displaystyle l_{1}:v\mapsto v(\tfrac{2}{3})\)

\(\displaystyle \phi_{1} = \frac{9 x \left(- 3 x^{2} + 4 x - 1\right)}{2}\)

This DOF is associated with edge 0 of the reference element.