Degree 1 Buffa–Christiansen 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:

\(\displaystyle \boldsymbol{\phi}_{0} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{7 x}{4} - \tfrac{3}{8}\\\displaystyle \tfrac{7 y}{4} - \tfrac{3}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/2, 1/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{3}{8} - \tfrac{x}{4}\\\displaystyle - \tfrac{y}{4} - \tfrac{1}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 1/2), (0, 1)))\\\left(\begin{array}{c}\displaystyle \tfrac{1}{8} - \tfrac{x}{4}\\\displaystyle \tfrac{3}{8} - \tfrac{y}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (-1/2, 1/2)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x}{4} - \tfrac{1}{8}\\\displaystyle \tfrac{1}{8} - \tfrac{y}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, 1/2), (-1, 0)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x}{4} - \tfrac{1}{8}\\\displaystyle - \tfrac{y}{4} - \tfrac{1}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-1/2, -1/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{1}{8} - \tfrac{x}{4}\\\displaystyle - \tfrac{y}{4} - \tfrac{3}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -1/2), (0, -1)))\\\left(\begin{array}{c}\displaystyle \tfrac{3}{8} - \tfrac{x}{4}\\\displaystyle \tfrac{1}{8} - \tfrac{y}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, -1), (1/2, -1/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{7 x}{4} - \tfrac{3}{8}\\\displaystyle \tfrac{7 y}{4} + \tfrac{3}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -1/2), (1, 0)))\end{cases}\)

\(\displaystyle \boldsymbol{\phi}_{1} = \begin{cases} \left(\begin{array}{c}\displaystyle - \tfrac{x}{4} - \tfrac{1}{8}\\\displaystyle \tfrac{3}{8} - \tfrac{y}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/2, 1/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{7 x}{4} - \tfrac{3}{8}\\\displaystyle \tfrac{7 y}{4} - \tfrac{3}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 1/2), (0, 1)))\\\left(\begin{array}{c}\displaystyle \tfrac{7 x}{4} + \tfrac{3}{8}\\\displaystyle \tfrac{7 y}{4} - \tfrac{3}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (-1/2, 1/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{1}{8} - \tfrac{x}{4}\\\displaystyle \tfrac{3}{8} - \tfrac{y}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, 1/2), (-1, 0)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x}{4} - \tfrac{3}{8}\\\displaystyle \tfrac{1}{8} - \tfrac{y}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-1/2, -1/2)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x}{4} - \tfrac{1}{8}\\\displaystyle - \tfrac{y}{4} - \tfrac{1}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -1/2), (0, -1)))\\\left(\begin{array}{c}\displaystyle \tfrac{1}{8} - \tfrac{x}{4}\\\displaystyle - \tfrac{y}{4} - \tfrac{1}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, -1), (1/2, -1/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{3}{8} - \tfrac{x}{4}\\\displaystyle \tfrac{1}{8} - \tfrac{y}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -1/2), (1, 0)))\end{cases}\)

\(\displaystyle \boldsymbol{\phi}_{2} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{1}{8} - \tfrac{x}{4}\\\displaystyle \tfrac{1}{8} - \tfrac{y}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/2, 1/2)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x}{4} - \tfrac{1}{8}\\\displaystyle \tfrac{3}{8} - \tfrac{y}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 1/2), (0, 1)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x}{4} - \tfrac{3}{8}\\\displaystyle - \tfrac{y}{4} - \tfrac{1}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (-1/2, 1/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{7 x}{4} + \tfrac{3}{8}\\\displaystyle \tfrac{7 y}{4} - \tfrac{3}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, 1/2), (-1, 0)))\\\left(\begin{array}{c}\displaystyle \tfrac{7 x}{4} + \tfrac{3}{8}\\\displaystyle \tfrac{7 y}{4} + \tfrac{3}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-1/2, -1/2)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x}{4} - \tfrac{3}{8}\\\displaystyle \tfrac{1}{8} - \tfrac{y}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -1/2), (0, -1)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x}{4} - \tfrac{1}{8}\\\displaystyle - \tfrac{y}{4} - \tfrac{3}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, -1), (1/2, -1/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{1}{8} - \tfrac{x}{4}\\\displaystyle - \tfrac{y}{4} - \tfrac{1}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -1/2), (1, 0)))\end{cases}\)

\(\displaystyle \boldsymbol{\phi}_{3} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{3}{8} - \tfrac{x}{4}\\\displaystyle - \tfrac{y}{4} - \tfrac{1}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1, 0), (1/2, 1/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{1}{8} - \tfrac{x}{4}\\\displaystyle \tfrac{1}{8} - \tfrac{y}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1/2, 1/2), (0, 1)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x}{4} - \tfrac{1}{8}\\\displaystyle \tfrac{1}{8} - \tfrac{y}{4}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, 1), (-1/2, 1/2)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x}{4} - \tfrac{3}{8}\\\displaystyle - \tfrac{y}{4} - \tfrac{1}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, 1/2), (-1, 0)))\\\left(\begin{array}{c}\displaystyle \tfrac{1}{8} - \tfrac{x}{4}\\\displaystyle - \tfrac{y}{4} - \tfrac{3}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1, 0), (-1/2, -1/2)))\\\left(\begin{array}{c}\displaystyle \tfrac{7 x}{4} + \tfrac{3}{8}\\\displaystyle \tfrac{7 y}{4} + \tfrac{3}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (-1/2, -1/2), (0, -1)))\\\left(\begin{array}{c}\displaystyle \tfrac{7 x}{4} - \tfrac{3}{8}\\\displaystyle \tfrac{7 y}{4} + \tfrac{3}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (0, -1), (1/2, -1/2)))\\\left(\begin{array}{c}\displaystyle - \tfrac{x}{4} - \tfrac{1}{8}\\\displaystyle - \tfrac{y}{4} - \tfrac{3}{8}\end{array}\right)&\text{in }\operatorname{Triangle}(((0, 0), (1/2, -1/2), (1, 0)))\end{cases}\)