an encyclopedia of finite element definitions

# Degree 3 Bernstein on a interval

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

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

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

$$\displaystyle \phi_{1} = x^{3}$$

This DOF is associated with vertex 1 of the reference element. $$\displaystyle l_{2}:v\mapsto c_{1}$$
where $$v=\sum_ic_iB_i$$;
and $$B_1$$ to $$B_n$$ are the degree 3 Bernstein polynomials on the cell.

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

This DOF is associated with edge 0 of the reference element. $$\displaystyle l_{3}:v\mapsto c_{2}$$
where $$v=\sum_ic_iB_i$$;
and $$B_1$$ to $$B_n$$ are the degree 3 Bernstein polynomials on the cell.

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

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