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Degree 1 dual polynomial on a dual polygon
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- \(R\) is the reference dual polygon. The following numbering of the subentities of the reference is used:
- Basis functions:
\(\displaystyle \phi_{0} = \begin{cases}
\tfrac{3 x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/2, 1/2)))\\\tfrac{3 x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 1/2), (0, 1)))\\\tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (-1/2, 1/2)))\\\tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, 1/2), (-1, 0)))\\\tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-1/2, -1/2)))\\\tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -1/2), (0, -1)))\\\tfrac{3 x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, -1), (1/2, -1/2)))\\\tfrac{3 x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -1/2), (1, 0)))\end{cases}\)
\(\displaystyle \phi_{1} = \begin{cases}
- \tfrac{x}{4} + \tfrac{3 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/2, 1/2)))\\- \tfrac{x}{4} + \tfrac{3 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 1/2), (0, 1)))\\\tfrac{x}{4} + \tfrac{3 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (-1/2, 1/2)))\\\tfrac{x}{4} + \tfrac{3 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, 1/2), (-1, 0)))\\\tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-1/2, -1/2)))\\\tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -1/2), (0, -1)))\\- \tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, -1), (1/2, -1/2)))\\- \tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -1/2), (1, 0)))\end{cases}\)
\(\displaystyle \phi_{2} = \begin{cases}
- \tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/2, 1/2)))\\- \tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 1/2), (0, 1)))\\- \tfrac{3 x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (-1/2, 1/2)))\\- \tfrac{3 x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, 1/2), (-1, 0)))\\- \tfrac{3 x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-1/2, -1/2)))\\- \tfrac{3 x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -1/2), (0, -1)))\\- \tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, -1), (1/2, -1/2)))\\- \tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -1/2), (1, 0)))\end{cases}\)
\(\displaystyle \phi_{3} = \begin{cases}
- \tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/2, 1/2)))\\- \tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 1/2), (0, 1)))\\\tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (-1/2, 1/2)))\\\tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, 1/2), (-1, 0)))\\\tfrac{x}{4} - \tfrac{3 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-1/2, -1/2)))\\\tfrac{x}{4} - \tfrac{3 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -1/2), (0, -1)))\\- \tfrac{x}{4} - \tfrac{3 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, -1), (1/2, -1/2)))\\- \tfrac{x}{4} - \tfrac{3 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -1/2), (1, 0)))\end{cases}\)