an encyclopedia of finite element definitions

# Dual polynomial

 Abbreviated names dual Orders $$0\leqslant k\leqslant 1$$ Reference elements dual polygon Number of DOFs dual polygon(n): $$\begin{cases}1&k=0\\n&k=1\end{cases}$$ Notes These elements are defined on the barycentric dual grid. These elements are defined as a linear combination of Lagrange basis functions on the fine grid. Categories Scalar-valued elements

## Implementations

 Bempp "DUAL"↓ Show Bempp examples ↓ Symfem "dual"↓ Show Symfem examples ↓

## Examples

dual polygon(6)
order 0
dual polygon(4)
order 1
dual polygon(5)
order 1
dual polygon(6)
order 1
• $$R$$ is the reference dual polygon. The following numbering of the subentities of the reference is used:
• Basis functions:
$$\displaystyle \phi_{0} = \begin{cases} 1&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (3/4, sqrt(3)/4)))\\1&\text{in }\operatorname{Triangle}(((0, 0), (3/4, sqrt(3)/4), (1/2, sqrt(3)/2)))\\1&\text{in }\operatorname{Triangle}(((0, 0), (1/2, sqrt(3)/2), (0, sqrt(3)/2)))\\1&\text{in }\operatorname{Triangle}(((0, 0), (0, sqrt(3)/2), (-1/2, sqrt(3)/2)))\\1&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, sqrt(3)/2), (-3/4, sqrt(3)/4)))\\1&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, sqrt(3)/4), (-1, 0)))\\1&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-3/4, -sqrt(3)/4)))\\1&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, -sqrt(3)/4), (-1/2, -sqrt(3)/2)))\\1&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -sqrt(3)/2), (0, -sqrt(3)/2)))\\1&\text{in }\operatorname{Triangle}(((0, 0), (0, -sqrt(3)/2), (1/2, -sqrt(3)/2)))\\1&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -sqrt(3)/2), (3/4, -sqrt(3)/4)))\\1&\text{in }\operatorname{Triangle}(((0, 0), (3/4, -sqrt(3)/4), (1, 0)))\end{cases}$$
• $$R$$ is the reference dual polygon. The following numbering of the subentities of the reference is used:
• Basis functions:
$$\displaystyle \phi_{0} = \begin{cases} \tfrac{x}{4} + \tfrac{5 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/2, 1/2)))\\\tfrac{5 x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 1/2), (0, 1)))\\\tfrac{3 x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (-1/2, 1/2)))\\\tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, 1/2), (-1, 0)))\\\tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-1/2, -1/2)))\\\tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -1/2), (0, -1)))\\- \tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, -1), (1/2, -1/2)))\\\tfrac{x}{4} + \tfrac{3 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -1/2), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{1} = \begin{cases} - \tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/2, 1/2)))\\- \tfrac{3 x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 1/2), (0, 1)))\\- \tfrac{5 x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (-1/2, 1/2)))\\- \tfrac{x}{4} + \tfrac{5 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, 1/2), (-1, 0)))\\- \tfrac{x}{4} + \tfrac{3 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-1/2, -1/2)))\\\tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -1/2), (0, -1)))\\- \tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, -1), (1/2, -1/2)))\\- \tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -1/2), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{2} = \begin{cases} - \tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/2, 1/2)))\\- \tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 1/2), (0, 1)))\\\tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (-1/2, 1/2)))\\- \tfrac{x}{4} - \tfrac{3 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, 1/2), (-1, 0)))\\- \tfrac{x}{4} - \tfrac{5 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-1/2, -1/2)))\\- \tfrac{5 x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -1/2), (0, -1)))\\- \tfrac{3 x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, -1), (1/2, -1/2)))\\- \tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -1/2), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{3} = \begin{cases} \tfrac{x}{4} - \tfrac{3 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/2, 1/2)))\\- \tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 1/2), (0, 1)))\\\tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (-1/2, 1/2)))\\\tfrac{x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, 1/2), (-1, 0)))\\\tfrac{x}{4} + \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-1/2, -1/2)))\\\tfrac{3 x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -1/2), (0, -1)))\\\tfrac{5 x}{4} - \tfrac{y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (0, -1), (1/2, -1/2)))\\\tfrac{x}{4} - \tfrac{5 y}{4} + \tfrac{1}{4}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -1/2), (1, 0)))\end{cases}$$
• $$R$$ 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 \sqrt{10} y + 55 \sqrt{2} y + 2 \sqrt{\sqrt{5} + 5} \cdot \left(3 x + 2\right)}{20 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{13 x}{10} - \tfrac{9 \sqrt{10} y \sqrt{\sqrt{5} + 5}}{40} + \tfrac{19 \sqrt{2} y \sqrt{\sqrt{5} + 5}}{40} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\tfrac{6 x + \tfrac{22 \sqrt{5} x}{5} - \tfrac{11 y \sqrt{50 - 10 \sqrt{5}}}{10} - \tfrac{2 y \sqrt{10 \sqrt{5} + 50}}{5} + y \sqrt{2 \sqrt{5} + 10} + \tfrac{5 y \sqrt{10 - 2 \sqrt{5}}}{2} + \tfrac{8 \sqrt{5}}{5}}{-10 - \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} + \tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} - \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\- \tfrac{x}{5} + \tfrac{\sqrt{5} x}{5} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\- \tfrac{x}{5} + \tfrac{\sqrt{5} x}{5} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\tfrac{\tfrac{8 \sqrt{5} x}{5} + \tfrac{2 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{2 \sqrt{5} + 10} + \tfrac{3 y \sqrt{10 \sqrt{5} + 50}}{5} + 4 + \tfrac{12 \sqrt{5}}{5}}{\sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\tfrac{- \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} + \tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} + \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\- \tfrac{x}{5} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{20} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{3 \sqrt{10} y + 25 \sqrt{2} y + 2 \sqrt{\sqrt{5} + 5} \cdot \left(3 x + 2\right)}{20 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{1} = \begin{cases} \tfrac{- 5 \sqrt{2} y + \sqrt{10} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\- \tfrac{7 x}{10} - \tfrac{\sqrt{2} y \sqrt{\sqrt{5} + 5}}{40} + \tfrac{3 \sqrt{10} y \sqrt{\sqrt{5} + 5}}{40} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\tfrac{- \tfrac{58 \sqrt{5} x}{5} + 6 x - \tfrac{11 y \sqrt{10 - 2 \sqrt{5}}}{2} - y \sqrt{2 \sqrt{5} + 10} + \tfrac{8 y \sqrt{10 \sqrt{5} + 50}}{5} + \tfrac{29 y \sqrt{50 - 10 \sqrt{5}}}{10} + \tfrac{8 \sqrt{5}}{5}}{-10 - \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \tfrac{6 x \sqrt{\sqrt{5} + 5}}{5} + \tfrac{26 x \sqrt{5 - \sqrt{5}}}{5} + 2 \sqrt{2} y + \tfrac{16 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\\tfrac{4 x}{5 + 5 \sqrt{5}} + \tfrac{\sqrt{2} y}{\sqrt{5 - \sqrt{5}}} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\- \tfrac{x}{5} + \tfrac{\sqrt{5} x}{5} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\tfrac{\tfrac{8 \sqrt{5} x}{5} + \tfrac{2 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{2 \sqrt{5} + 10} + \tfrac{3 y \sqrt{10 \sqrt{5} + 50}}{5} + 4 + \tfrac{12 \sqrt{5}}{5}}{\sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\tfrac{- \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} + \tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} + \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\- \tfrac{x}{5} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{20} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \sqrt{10} y + 5 \sqrt{2} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{2} = \begin{cases} \tfrac{- 5 \sqrt{2} y + \sqrt{10} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\- \tfrac{x}{5} - \tfrac{y \sqrt{2 \sqrt{5} + 10}}{4} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{\sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{50} + \tfrac{\sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\tfrac{- 4 x + \tfrac{12 \sqrt{5} x}{5} - \tfrac{2 y \sqrt{10 \sqrt{5} + 50}}{5} - \tfrac{3 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{10 - 2 \sqrt{5}} + \tfrac{8 \sqrt{5}}{5}}{-10 - \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \tfrac{14 x \sqrt{5 - \sqrt{5}}}{5} - \tfrac{6 x \sqrt{\sqrt{5} + 5}}{5} - 2 \sqrt{2} y - \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\- \tfrac{16 x}{5 + 5 \sqrt{5}} - \tfrac{\sqrt{2} y}{\sqrt{5 - \sqrt{5}}} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\\tfrac{- 16 x \sqrt{5 - \sqrt{5}} + 5 \sqrt{2} y \left(1 + \sqrt{5}\right) + \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\tfrac{- 20 x - \tfrac{32 \sqrt{5} x}{5} + \tfrac{2 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{10 - 2 \sqrt{5}} + \tfrac{11 y \sqrt{10 \sqrt{5} + 50}}{10} + \tfrac{5 y \sqrt{2 \sqrt{5} + 10}}{2} + 4 + \tfrac{12 \sqrt{5}}{5}}{\sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\tfrac{- \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} + \tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} + \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\- \tfrac{x}{5} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{3 y \sqrt{2 \sqrt{5} + 10}}{20} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \sqrt{10} y + 5 \sqrt{2} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{3} = \begin{cases} \tfrac{- 5 \sqrt{2} y + \sqrt{10} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\- \tfrac{x}{5} - \tfrac{y \sqrt{2 \sqrt{5} + 10}}{4} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{\sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{50} + \tfrac{\sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\tfrac{- 4 x + \tfrac{12 \sqrt{5} x}{5} - \tfrac{2 y \sqrt{10 \sqrt{5} + 50}}{5} - \tfrac{3 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{10 - 2 \sqrt{5}} + \tfrac{8 \sqrt{5}}{5}}{-10 - \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} + \tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} - \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\- \tfrac{x}{5} + \tfrac{\sqrt{5} x}{5} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\\tfrac{4 x \sqrt{5 - \sqrt{5}} - 5 \sqrt{2} y \left(1 + \sqrt{5}\right) + \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}{5 \cdot \left(1 + \sqrt{5}\right) \sqrt{5 - \sqrt{5}}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\tfrac{\tfrac{8 \sqrt{5} x}{5} + 20 x - \tfrac{29 y \sqrt{10 \sqrt{5} + 50}}{10} - \tfrac{11 y \sqrt{2 \sqrt{5} + 10}}{2} - \tfrac{8 y \sqrt{50 - 10 \sqrt{5}}}{5} - y \sqrt{10 - 2 \sqrt{5}} + 4 + \tfrac{12 \sqrt{5}}{5}}{\sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\tfrac{- \tfrac{26 x \sqrt{\sqrt{5} + 5}}{5} + \tfrac{6 x \sqrt{5 - \sqrt{5}}}{5} - \tfrac{16 \sqrt{10} y}{5} + 2 \sqrt{2} y - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\- \tfrac{7 x}{10} - \tfrac{3 y \sqrt{10 \sqrt{5} + 50}}{40} + \tfrac{y \sqrt{2 \sqrt{5} + 10}}{40} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \sqrt{10} y + 5 \sqrt{2} y + 2 \cdot \left(1 - x\right) \sqrt{\sqrt{5} + 5}}{10 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{4} = \begin{cases} \tfrac{- 25 \sqrt{2} y - 3 \sqrt{10} y + 2 \sqrt{\sqrt{5} + 5} \cdot \left(3 x + 2\right)}{20 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2)))\\- \tfrac{x}{5} - \tfrac{y \sqrt{2 \sqrt{5} + 10}}{4} - \tfrac{y \sqrt{10 \sqrt{5} + 50}}{20} + \tfrac{\sqrt{5} y \sqrt{10 \sqrt{5} + 50}}{50} + \tfrac{\sqrt{5} y \sqrt{2 \sqrt{5} + 10}}{10} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, sqrt(sqrt(5)/8 + 5/8)/2), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8))))\\\tfrac{- 4 x + \tfrac{12 \sqrt{5} x}{5} - \tfrac{2 y \sqrt{10 \sqrt{5} + 50}}{5} - \tfrac{3 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{10 - 2 \sqrt{5}} + \tfrac{8 \sqrt{5}}{5}}{-10 - \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, sqrt(sqrt(5)/8 + 5/8)), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- \tfrac{4 x \sqrt{5 - \sqrt{5}}}{5} + \tfrac{4 x \sqrt{\sqrt{5} + 5}}{5} - \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, sqrt(5/8 - sqrt(5)/8)/2 + sqrt(sqrt(5)/8 + 5/8)/2), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8))))\\- \tfrac{x}{5} + \tfrac{\sqrt{5} x}{5} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, sqrt(5/8 - sqrt(5)/8)), (-sqrt(5)/4 - 1/4, 0)))\\- \tfrac{x}{5} + \tfrac{\sqrt{5} x}{5} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8))))\\\tfrac{\tfrac{8 \sqrt{5} x}{5} + \tfrac{2 y \sqrt{50 - 10 \sqrt{5}}}{5} + y \sqrt{2 \sqrt{5} + 10} + \tfrac{3 y \sqrt{10 \sqrt{5} + 50}}{5} + 4 + \tfrac{12 \sqrt{5}}{5}}{\sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} \sqrt{\sqrt{5} + 5} + 10 + 10 \sqrt{5}}&\text{in }\operatorname{Triangle}(((0, 0), (-sqrt(5)/4 - 1/4, -sqrt(5/8 - sqrt(5)/8)), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2)))\\\tfrac{\tfrac{6 x \sqrt{5 - \sqrt{5}}}{5} + \tfrac{14 x \sqrt{\sqrt{5} + 5}}{5} - 2 \sqrt{2} y + \tfrac{4 \sqrt{10} y}{5} - \tfrac{\sqrt{5 - \sqrt{5}}}{5} + \tfrac{\sqrt{\sqrt{5} + 5}}{5} + \tfrac{\sqrt{25 - 5 \sqrt{5}}}{5} + \tfrac{\sqrt{5 \sqrt{5} + 25}}{5}}{- \sqrt{5 - \sqrt{5}} + \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4, -sqrt(sqrt(5)/8 + 5/8)/2 - sqrt(5/8 - sqrt(5)/8)/2), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8))))\\\tfrac{13 x}{10} - \tfrac{19 y \sqrt{2 \sqrt{5} + 10}}{40} + \tfrac{9 y \sqrt{10 \sqrt{5} + 50}}{40} + \tfrac{1}{5}&\text{in }\operatorname{Triangle}(((0, 0), (-1/4 + sqrt(5)/4, -sqrt(sqrt(5)/8 + 5/8)), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2)))\\\tfrac{- 55 \sqrt{2} y + 3 \sqrt{10} y + 2 \sqrt{\sqrt{5} + 5} \cdot \left(3 x + 2\right)}{20 \sqrt{\sqrt{5} + 5}}&\text{in }\operatorname{Triangle}(((0, 0), (sqrt(5)/8 + 3/8, -sqrt(sqrt(5)/8 + 5/8)/2), (1, 0)))\end{cases}$$
• $$R$$ is the reference dual polygon. The following numbering of the subentities of the reference is used:
• Basis functions:
$$\displaystyle \phi_{0} = \begin{cases} \tfrac{x}{3} + \tfrac{7 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (3/4, sqrt(3)/4)))\\\tfrac{4 x}{3} - \tfrac{2 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, sqrt(3)/4), (1/2, sqrt(3)/2)))\\x - \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, sqrt(3)/2), (0, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, sqrt(3)/2), (-1/2, sqrt(3)/2)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, sqrt(3)/2), (-3/4, sqrt(3)/4)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, sqrt(3)/4), (-1, 0)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-3/4, -sqrt(3)/4)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, -sqrt(3)/4), (-1/2, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -sqrt(3)/2), (0, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, -sqrt(3)/2), (1/2, -sqrt(3)/2)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -sqrt(3)/2), (3/4, -sqrt(3)/4)))\\\tfrac{x}{3} + \tfrac{5 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, -sqrt(3)/4), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{1} = \begin{cases} - \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (3/4, sqrt(3)/4)))\\- \tfrac{2 x}{3} + \tfrac{4 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, sqrt(3)/4), (1/2, sqrt(3)/2)))\\- x + \tfrac{5 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, sqrt(3)/2), (0, sqrt(3)/2)))\\x + \tfrac{5 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, sqrt(3)/2), (-1/2, sqrt(3)/2)))\\\tfrac{2 x}{3} + \tfrac{4 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, sqrt(3)/2), (-3/4, sqrt(3)/4)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, sqrt(3)/4), (-1, 0)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-3/4, -sqrt(3)/4)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, -sqrt(3)/4), (-1/2, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -sqrt(3)/2), (0, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, -sqrt(3)/2), (1/2, -sqrt(3)/2)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -sqrt(3)/2), (3/4, -sqrt(3)/4)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, -sqrt(3)/4), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{2} = \begin{cases} - \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (3/4, sqrt(3)/4)))\\- \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, sqrt(3)/4), (1/2, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, sqrt(3)/2), (0, sqrt(3)/2)))\\- x - \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, sqrt(3)/2), (-1/2, sqrt(3)/2)))\\- \tfrac{4 x}{3} - \tfrac{2 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, sqrt(3)/2), (-3/4, sqrt(3)/4)))\\- \tfrac{x}{3} + \tfrac{7 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, sqrt(3)/4), (-1, 0)))\\- \tfrac{x}{3} + \tfrac{5 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-3/4, -sqrt(3)/4)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, -sqrt(3)/4), (-1/2, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -sqrt(3)/2), (0, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, -sqrt(3)/2), (1/2, -sqrt(3)/2)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -sqrt(3)/2), (3/4, -sqrt(3)/4)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, -sqrt(3)/4), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{3} = \begin{cases} - \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (3/4, sqrt(3)/4)))\\- \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, sqrt(3)/4), (1/2, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, sqrt(3)/2), (0, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, sqrt(3)/2), (-1/2, sqrt(3)/2)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, sqrt(3)/2), (-3/4, sqrt(3)/4)))\\- \tfrac{x}{3} - \tfrac{5 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, sqrt(3)/4), (-1, 0)))\\- \tfrac{x}{3} - \tfrac{7 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-3/4, -sqrt(3)/4)))\\- \tfrac{4 x}{3} + \tfrac{2 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, -sqrt(3)/4), (-1/2, -sqrt(3)/2)))\\- x + \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -sqrt(3)/2), (0, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, -sqrt(3)/2), (1/2, -sqrt(3)/2)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -sqrt(3)/2), (3/4, -sqrt(3)/4)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, -sqrt(3)/4), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{4} = \begin{cases} - \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (3/4, sqrt(3)/4)))\\- \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, sqrt(3)/4), (1/2, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, sqrt(3)/2), (0, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, sqrt(3)/2), (-1/2, sqrt(3)/2)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, sqrt(3)/2), (-3/4, sqrt(3)/4)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, sqrt(3)/4), (-1, 0)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-3/4, -sqrt(3)/4)))\\\tfrac{2 x}{3} - \tfrac{4 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, -sqrt(3)/4), (-1/2, -sqrt(3)/2)))\\x - \tfrac{5 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -sqrt(3)/2), (0, -sqrt(3)/2)))\\- x - \tfrac{5 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, -sqrt(3)/2), (1/2, -sqrt(3)/2)))\\- \tfrac{2 x}{3} - \tfrac{4 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -sqrt(3)/2), (3/4, -sqrt(3)/4)))\\- \tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, -sqrt(3)/4), (1, 0)))\end{cases}$$
$$\displaystyle \phi_{5} = \begin{cases} \tfrac{x}{3} - \tfrac{5 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (3/4, sqrt(3)/4)))\\- \tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, sqrt(3)/4), (1/2, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, sqrt(3)/2), (0, sqrt(3)/2)))\\- \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, sqrt(3)/2), (-1/2, sqrt(3)/2)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, sqrt(3)/2), (-3/4, sqrt(3)/4)))\\\tfrac{x}{6} - \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, sqrt(3)/4), (-1, 0)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-3/4, -sqrt(3)/4)))\\\tfrac{x}{6} + \tfrac{\sqrt{3} y}{18} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-3/4, -sqrt(3)/4), (-1/2, -sqrt(3)/2)))\\\tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -sqrt(3)/2), (0, -sqrt(3)/2)))\\x + \tfrac{\sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (0, -sqrt(3)/2), (1/2, -sqrt(3)/2)))\\\tfrac{4 x}{3} + \tfrac{2 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -sqrt(3)/2), (3/4, -sqrt(3)/4)))\\\tfrac{x}{3} - \tfrac{7 \sqrt{3} y}{9} + \tfrac{1}{6}&\text{in }\operatorname{Triangle}(((0, 0), (3/4, -sqrt(3)/4), (1, 0)))\end{cases}$$

## References

• Buffa, Annalisa and Christiansen, Snorre H. A dual finite element complex on the barycentric refinement, Mathematics of Computation 76, 1743–1769, 2007. [DOI: 10.1016/j.crma.2004.12.022] [BibTeX]

## DefElement stats

 Element added 24 January 2021 Element last updated 10 February 2022