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# Degree 3 Bernstein on a triangle

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In this example:
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$, $$y$$, $$x y$$, $$x^{2} y$$, $$y^{2}$$, $$x y^{2}$$, $$y^{3}$$
• $$\mathcal{L}=\{l_0,...,l_{9}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = - x^{3} - 3 x^{2} y + 3 x^{2} - 3 x y^{2} + 6 x y - 3 x - y^{3} + 3 y^{2} - 3 y + 1$$

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

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

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

$$\displaystyle \phi_{2} = y^{3}$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{3}:v\mapsto c^{e_{0}}_{1}$$
where $$v=\sum_ic^{e_{0}}_iB^{e_{0}}_i$$;
$$B^{e_{0}}_1$$ to $$B^{e_{0}}_n$$ are the degree 3 Bernstein polynomials on $$e_{0}$$;
and $$e_{0}$$ is the 0th edge.

$$\displaystyle \phi_{3} = 3 x^{2} y$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{4}:v\mapsto c^{e_{0}}_{2}$$
where $$v=\sum_ic^{e_{0}}_iB^{e_{0}}_i$$;
$$B^{e_{0}}_1$$ to $$B^{e_{0}}_n$$ are the degree 3 Bernstein polynomials on $$e_{0}$$;
and $$e_{0}$$ is the 0th edge.

$$\displaystyle \phi_{4} = 3 x y^{2}$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{5}:v\mapsto c^{e_{1}}_{1}$$
where $$v=\sum_ic^{e_{1}}_iB^{e_{1}}_i$$;
$$B^{e_{1}}_1$$ to $$B^{e_{1}}_n$$ are the degree 3 Bernstein polynomials on $$e_{1}$$;
and $$e_{1}$$ is the 1st edge.

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

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{6}:v\mapsto c^{e_{1}}_{2}$$
where $$v=\sum_ic^{e_{1}}_iB^{e_{1}}_i$$;
$$B^{e_{1}}_1$$ to $$B^{e_{1}}_n$$ are the degree 3 Bernstein polynomials on $$e_{1}$$;
and $$e_{1}$$ is the 1st edge.

$$\displaystyle \phi_{6} = 3 y^{2} \left(- x - y + 1\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{7}:v\mapsto c^{e_{2}}_{1}$$
where $$v=\sum_ic^{e_{2}}_iB^{e_{2}}_i$$;
$$B^{e_{2}}_1$$ to $$B^{e_{2}}_n$$ are the degree 3 Bernstein polynomials on $$e_{2}$$;
and $$e_{2}$$ is the 2nd edge.

$$\displaystyle \phi_{7} = 3 x \left(x^{2} + 2 x y - 2 x + y^{2} - 2 y + 1\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{8}:v\mapsto c^{e_{2}}_{2}$$
where $$v=\sum_ic^{e_{2}}_iB^{e_{2}}_i$$;
$$B^{e_{2}}_1$$ to $$B^{e_{2}}_n$$ are the degree 3 Bernstein polynomials on $$e_{2}$$;
and $$e_{2}$$ is the 2nd edge.

$$\displaystyle \phi_{8} = 3 x^{2} \left(- x - y + 1\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{9}:v\mapsto c_{5}$$
where $$v=\sum_ic_iB_i$$;
and $$B_1$$ to $$B_n$$ are the degree 3 Bernstein polynomials on the cell.

$$\displaystyle \phi_{9} = 6 x y \left(- x - y + 1\right)$$

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