Degree 1 dual polynomial on a dual polygon

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In this example:

\(R\) is the reference dual polygon. The following numbering of the subentities of the reference is used:

Basis functions:

\(\displaystyle \phi_{0} = \begin{cases} \tfrac{5 x}{6} - \tfrac{7 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (3/4, sqrt(3)/4)))\\\tfrac{5 x}{6} - \tfrac{7 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, sqrt(3)/4), (1/2, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, sqrt(3)/2), (0, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, sqrt(3)/2), (-1/2, sqrt(3)/2)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, sqrt(3)/2), (-3/4, sqrt(3)/4)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, sqrt(3)/4), (-1, 0)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-3/4, -sqrt(3)/4)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, -sqrt(3)/4), (-1/2, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -sqrt(3)/2), (0, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, -sqrt(3)/2), (1/2, -sqrt(3)/2)))\\\tfrac{5 x}{6} + \tfrac{7 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -sqrt(3)/2), (3/4, -sqrt(3)/4)))\\\tfrac{5 x}{6} + \tfrac{7 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, -sqrt(3)/4), (1, 0)))\end{cases}\)

\(\displaystyle \phi_{1} = \begin{cases} - \tfrac{x}{6} + \tfrac{11 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (3/4, sqrt(3)/4)))\\- \tfrac{x}{6} + \tfrac{11 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, sqrt(3)/4), (1/2, sqrt(3)/2)))\\x + \tfrac{2 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, sqrt(3)/2), (0, sqrt(3)/2)))\\x + \tfrac{2 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, sqrt(3)/2), (-1/2, sqrt(3)/2)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, sqrt(3)/2), (-3/4, sqrt(3)/4)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, sqrt(3)/4), (-1, 0)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-3/4, -sqrt(3)/4)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, -sqrt(3)/4), (-1/2, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -sqrt(3)/2), (0, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, -sqrt(3)/2), (1/2, -sqrt(3)/2)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -sqrt(3)/2), (3/4, -sqrt(3)/4)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, -sqrt(3)/4), (1, 0)))\end{cases}\)

\(\displaystyle \phi_{2} = \begin{cases} - \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (3/4, sqrt(3)/4)))\\- \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, sqrt(3)/4), (1/2, sqrt(3)/2)))\\- x + \tfrac{2 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, sqrt(3)/2), (0, sqrt(3)/2)))\\- x + \tfrac{2 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, sqrt(3)/2), (-1/2, sqrt(3)/2)))\\\tfrac{x}{6} + \tfrac{11 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, sqrt(3)/2), (-3/4, sqrt(3)/4)))\\\tfrac{x}{6} + \tfrac{11 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, sqrt(3)/4), (-1, 0)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-3/4, -sqrt(3)/4)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, -sqrt(3)/4), (-1/2, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -sqrt(3)/2), (0, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, -sqrt(3)/2), (1/2, -sqrt(3)/2)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -sqrt(3)/2), (3/4, -sqrt(3)/4)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, -sqrt(3)/4), (1, 0)))\end{cases}\)

\(\displaystyle \phi_{3} = \begin{cases} - \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (3/4, sqrt(3)/4)))\\- \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, sqrt(3)/4), (1/2, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, sqrt(3)/2), (0, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, sqrt(3)/2), (-1/2, sqrt(3)/2)))\\- \tfrac{5 x}{6} - \tfrac{7 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, sqrt(3)/2), (-3/4, sqrt(3)/4)))\\- \tfrac{5 x}{6} - \tfrac{7 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, sqrt(3)/4), (-1, 0)))\\- \tfrac{5 x}{6} + \tfrac{7 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-3/4, -sqrt(3)/4)))\\- \tfrac{5 x}{6} + \tfrac{7 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, -sqrt(3)/4), (-1/2, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -sqrt(3)/2), (0, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, -sqrt(3)/2), (1/2, -sqrt(3)/2)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -sqrt(3)/2), (3/4, -sqrt(3)/4)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, -sqrt(3)/4), (1, 0)))\end{cases}\)

\(\displaystyle \phi_{4} = \begin{cases} - \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (3/4, sqrt(3)/4)))\\- \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, sqrt(3)/4), (1/2, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, sqrt(3)/2), (0, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, sqrt(3)/2), (-1/2, sqrt(3)/2)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, sqrt(3)/2), (-3/4, sqrt(3)/4)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, sqrt(3)/4), (-1, 0)))\\\tfrac{x}{6} - \tfrac{11 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-3/4, -sqrt(3)/4)))\\\tfrac{x}{6} - \tfrac{11 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, -sqrt(3)/4), (-1/2, -sqrt(3)/2)))\\- x - \tfrac{2 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -sqrt(3)/2), (0, -sqrt(3)/2)))\\- x - \tfrac{2 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, -sqrt(3)/2), (1/2, -sqrt(3)/2)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -sqrt(3)/2), (3/4, -sqrt(3)/4)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, -sqrt(3)/4), (1, 0)))\end{cases}\)

\(\displaystyle \phi_{5} = \begin{cases} - \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (3/4, sqrt(3)/4)))\\- \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, sqrt(3)/4), (1/2, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, sqrt(3)/2), (0, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, sqrt(3)/2), (-1/2, sqrt(3)/2)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, sqrt(3)/2), (-3/4, sqrt(3)/4)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, sqrt(3)/4), (-1, 0)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-3/4, -sqrt(3)/4)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, -sqrt(3)/4), (-1/2, -sqrt(3)/2)))\\x - \tfrac{2 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -sqrt(3)/2), (0, -sqrt(3)/2)))\\x - \tfrac{2 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, -sqrt(3)/2), (1/2, -sqrt(3)/2)))\\- \tfrac{x}{6} - \tfrac{11 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -sqrt(3)/2), (3/4, -sqrt(3)/4)))\\- \tfrac{x}{6} - \tfrac{11 \sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, -sqrt(3)/4), (1, 0)))\end{cases}\)