Degree 3 Lagrange on a interval

• $R$ is the reference interval. The following numbering of the subentities of the reference is used:

• $\mathcal{V}$ is spanned by: $1$, $x$, $x^2$, $x^3$
• $\mathcal{L}=\{l_0,...,l_3\}$
• Functionals and basis functions:

${\displaystyle l_0:v\mapsto v(0)}$

${\displaystyle {\varphi }_0=1-x}$

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

${\displaystyle l_1:v\mapsto v(1)}$

${\displaystyle {\varphi }_1=x}$

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

${\displaystyle l_2:\mathbf{v}\mapsto {\displaystyle {\int }_R(\sqrt{3}(60s_0^2-60s_0+10))v}}$
where $R$ is the reference element;
and $s_0$ is a parametrisation of $R$.

${\displaystyle {\varphi }_2=\sqrt{3}x(x-1)}$

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

${\displaystyle l_3:\mathbf{v}\mapsto {\displaystyle {\int }_R(\sqrt{5}(280s_0^3-420s_0^2+168s_0-14))v}}$
where $R$ is the reference element;
and $s_0$ is a parametrisation of $R$.

${\displaystyle {\varphi }_3=\sqrt{5}x(2x^2-3x+1)}$

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