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# Degree 3 Kong–Mulder–Veldhuizen 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: $$1$$, $$x$$, $$x^{2}$$, $$y$$, $$x y$$, $$y^{2}$$, $$x y \left(- x - y + 1\right)$$
• $$\mathcal{L}=\{l_0,...,l_{6}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto \frac{1}{40} v(0,0)$$

$$\displaystyle \phi_{0} = - 120 x^{2} y + 80 x^{2} - 120 x y^{2} + 280 x y - 120 x + 80 y^{2} - 120 y + 40$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:v\mapsto \frac{1}{40} v(1,0)$$

$$\displaystyle \phi_{1} = 40 x \left(- 3 x y + 2 x - 3 y^{2} + 3 y - 1\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{2}:v\mapsto \frac{1}{40} v(0,1)$$

$$\displaystyle \phi_{2} = 40 y \left(- 3 x^{2} - 3 x y + 3 x + 2 y - 1\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{3}:v\mapsto \frac{1}{15} v(\tfrac{1}{2},\tfrac{1}{2})$$

$$\displaystyle \phi_{3} = 60 x y \left(3 x + 3 y - 2\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{4}:v\mapsto \frac{1}{15} v(0,\tfrac{1}{2})$$

$$\displaystyle \phi_{4} = 60 y \left(3 x^{2} + 3 x y - 4 x - y + 1\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{5}:v\mapsto \frac{1}{15} v(\tfrac{1}{2},0)$$

$$\displaystyle \phi_{5} = 60 x \left(3 x y - x + 3 y^{2} - 4 y + 1\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{6}:v\mapsto \frac{9}{40} v(\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{6} = 120 x y \left(- x - y + 1\right)$$

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