an encyclopedia of finite element definitions

# Degree 3 Hermite on a interval

◀ Back to Hermite 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: $$\displaystyle l_{0}:v\mapsto v(0)$$

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

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

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

This DOF is associated with vertex 0 of the reference element. $$\displaystyle l_{2}:v\mapsto v(1)$$

$$\displaystyle \phi_{2} = x^{2} \cdot \left(3 - 2 x\right)$$

This DOF is associated with vertex 1 of the reference element. $$\displaystyle l_{3}:v\mapsto v'(1)$$

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

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