an encyclopedia of finite element definitions

# Degree 1 P1 macro on a triangle

◀ Back to P1 macro definition page
In this example:
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\begin{cases} 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\1&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} x&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\x&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\x&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} y&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\y&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$, $$\begin{cases} 3 y&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 3 x - 3 y + 3&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\3 x&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$
• $$\mathcal{L}=\{l_0,...,l_{3}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

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

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

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

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

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

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{3}:\mathbf{v}\mapsto\displaystyle\int_{R}v$$
where $$R$$ is the reference element.

$$\displaystyle \phi_{3} = \begin{cases} 6 y&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\- 6 x - 6 y + 6&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\6 x&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}$$

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