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Degree 1 P1 macro on a triangle
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- \(R\) 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.
\(\displaystyle \phi_{0} = \begin{cases} - x - 2 y + 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- 2 x - y + 1&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
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.
\(\displaystyle \phi_{1} = \begin{cases} x - y&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\2 x + y - 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\0&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
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.
\(\displaystyle \phi_{2} = \begin{cases} 0&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/3, 1/3)))\\x + 2 y - 1&\text{in }\operatorname{Triangle}(((1, 0), (0, 1), (1/3, 1/3)))\\- x + y&\text{in }\operatorname{Triangle}(((0, 1), (0, 0), (1/3, 1/3)))\end{cases}\)
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.
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.