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# Degree 3 Lagrange on a interval

◀ Back to Lagrange definition page 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: $$\displaystyle l_{0}:v\mapsto v(0)$$

$$\displaystyle \phi_{0} = 1 - x$$

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

$$\displaystyle \phi_{1} = x$$

This DOF is associated with vertex 1 of the reference element. $$\displaystyle l_{2}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{3} \cdot \left(60 s_{0}^{2} - 60 s_{0} + 10\right))v$$
where $$R$$ is the reference element;
and $$s_{0}$$ is a parametrisation of $$R$$.

$$\displaystyle \phi_{2} = \sqrt{3} x \left(x - 1\right)$$

This DOF is associated with edge 0 of the reference element. $$\displaystyle l_{3}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{5} \cdot \left(280 s_{0}^{3} - 420 s_{0}^{2} + 168 s_{0} - 14\right))v$$
where $$R$$ is the reference element;
and $$s_{0}$$ is a parametrisation of $$R$$.

$$\displaystyle \phi_{3} = \sqrt{5} x \left(2 x^{2} - 3 x + 1\right)$$

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