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# Degree 3 serendipity on a interval

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In this example:
• $$R$$ 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:
$$\displaystyle l_{0}:v\mapsto v(0)$$

$$\displaystyle \phi_{0} = - 10 x^{3} + 18 x^{2} - 9 x + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto v(1)$$

$$\displaystyle \phi_{1} = x \left(10 x^{2} - 12 x + 3\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{2}:v\mapsto\displaystyle\int_{R}(1 - s_{0})v$$
where $$R$$ is the reference element;
and $$s_{0}$$ is a parametrisation of $$R$$.

$$\displaystyle \phi_{2} = 12 x \left(5 x^{2} - 8 x + 3\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{3}:v\mapsto\displaystyle\int_{R}(s_{0})v$$
where $$R$$ is the reference element;
and $$s_{0}$$ is a parametrisation of $$R$$.

$$\displaystyle \phi_{3} = 12 x \left(- 5 x^{2} + 7 x - 2\right)$$

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