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# Degree 1 P1-iso-P2 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: $$\begin{cases} - 2 x - 2 y + 1&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 0), (0, 1/2)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (1/2, 1/2), (1/2, 0)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Triangle}(((0, 1/2), (1/2, 1/2), (1/2, 0)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 0), (0, 1/2)))\\2 x - 1&\text{in }\operatorname{Triangle}(((1, 0), (1/2, 1/2), (1/2, 0)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Triangle}(((0, 1/2), (1/2, 1/2), (1/2, 0)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 0), (0, 1/2)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (1/2, 1/2), (1/2, 0)))\\2 y - 1&\text{in }\operatorname{Triangle}(((0, 1), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Triangle}(((0, 1/2), (1/2, 1/2), (1/2, 0)))\end{cases}$$, $$\begin{cases} 2 x&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 0), (0, 1/2)))\\- 2 x - 2 y + 2&\text{in }\operatorname{Triangle}(((1, 0), (1/2, 1/2), (1/2, 0)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 1/2), (1/2, 1/2)))\\1 - 2 y&\text{in }\operatorname{Triangle}(((0, 1/2), (1/2, 1/2), (1/2, 0)))\end{cases}$$, $$\begin{cases} 2 y&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 0), (0, 1/2)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (1/2, 1/2), (1/2, 0)))\\- 2 x - 2 y + 2&\text{in }\operatorname{Triangle}(((0, 1), (0, 1/2), (1/2, 1/2)))\\1 - 2 x&\text{in }\operatorname{Triangle}(((0, 1/2), (1/2, 1/2), (1/2, 0)))\end{cases}$$, $$\begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 0), (0, 1/2)))\\2 y&\text{in }\operatorname{Triangle}(((1, 0), (1/2, 1/2), (1/2, 0)))\\2 x&\text{in }\operatorname{Triangle}(((0, 1), (0, 1/2), (1/2, 1/2)))\\2 x + 2 y - 1&\text{in }\operatorname{Triangle}(((0, 1/2), (1/2, 1/2), (1/2, 0)))\end{cases}$$
• $$\mathcal{L}=\{l_0,...,l_{5}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = \begin{cases} - 2 x - 2 y + 1&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 0), (0, 1/2)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (1/2, 1/2), (1/2, 0)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Triangle}(((0, 1/2), (1/2, 1/2), (1/2, 0)))\end{cases}$$

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

$$\displaystyle \phi_{1} = \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 0), (0, 1/2)))\\2 x - 1&\text{in }\operatorname{Triangle}(((1, 0), (1/2, 1/2), (1/2, 0)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Triangle}(((0, 1/2), (1/2, 1/2), (1/2, 0)))\end{cases}$$

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

$$\displaystyle \phi_{2} = \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 0), (0, 1/2)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (1/2, 1/2), (1/2, 0)))\\2 y - 1&\text{in }\operatorname{Triangle}(((0, 1), (0, 1/2), (1/2, 1/2)))\\0&\text{in }\operatorname{Triangle}(((0, 1/2), (1/2, 1/2), (1/2, 0)))\end{cases}$$

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

$$\displaystyle \phi_{3} = \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 0), (0, 1/2)))\\2 y&\text{in }\operatorname{Triangle}(((1, 0), (1/2, 1/2), (1/2, 0)))\\2 x&\text{in }\operatorname{Triangle}(((0, 1), (0, 1/2), (1/2, 1/2)))\\2 x + 2 y - 1&\text{in }\operatorname{Triangle}(((0, 1/2), (1/2, 1/2), (1/2, 0)))\end{cases}$$

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

$$\displaystyle \phi_{4} = \begin{cases} 2 y&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 0), (0, 1/2)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (1/2, 1/2), (1/2, 0)))\\- 2 x - 2 y + 2&\text{in }\operatorname{Triangle}(((0, 1), (0, 1/2), (1/2, 1/2)))\\1 - 2 x&\text{in }\operatorname{Triangle}(((0, 1/2), (1/2, 1/2), (1/2, 0)))\end{cases}$$

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

$$\displaystyle \phi_{5} = \begin{cases} 2 x&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 0), (0, 1/2)))\\- 2 x - 2 y + 2&\text{in }\operatorname{Triangle}(((1, 0), (1/2, 1/2), (1/2, 0)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 1/2), (1/2, 1/2)))\\1 - 2 y&\text{in }\operatorname{Triangle}(((0, 1/2), (1/2, 1/2), (1/2, 0)))\end{cases}$$

This DOF is associated with edge 2 of the reference element.