Degree 4 Gauss–Lobatto–Legendre on a interval

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

• \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(x^{3}\), \(x^{4}\)
• \(\mathcal{L}=\{l_0,...,l_{4}\}\)
• Functionals and basis functions:

\(\displaystyle l_{0}:v\mapsto v(0)\)

\(\displaystyle \phi_{0} = \frac{32 x^{4}}{3} - \frac{80 x^{3}}{3} + \frac{70 x^{2}}{3} - \frac{25 x}{3} + 1\)

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

\(\displaystyle l_{1}:v\mapsto v(1)\)

\(\displaystyle \phi_{1} = \frac{x \left(32 x^{3} - 48 x^{2} + 22 x - 3\right)}{3}\)

This DOF is associated with vertex 1 of the reference element.

\(\displaystyle l_{2}:v\mapsto v(\tfrac{1}{4})\)

\(\displaystyle \phi_{2} = \frac{16 x \left(- 8 x^{3} + 18 x^{2} - 13 x + 3\right)}{3}\)

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

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

\(\displaystyle \phi_{3} = 4 x \left(16 x^{3} - 32 x^{2} + 19 x - 3\right)\)

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

\(\displaystyle l_{4}:v\mapsto v(\tfrac{3}{4})\)

\(\displaystyle \phi_{4} = \frac{16 x \left(- 8 x^{3} + 14 x^{2} - 7 x + 1\right)}{3}\)

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