an encyclopedia of finite element definitions

# Degree 1 Regge on a triangle

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In this example:
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 1\\\displaystyle 1&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle x&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle x\\\displaystyle x&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle y&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle y\\\displaystyle y&\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle y\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{8}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{2}{3},\tfrac{1}{3})\left(\begin{array}{c}-1\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle \frac{1}{2} - \frac{3 x}{2}\\\displaystyle \frac{1}{2} - \frac{3 x}{2}&\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{1}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},\tfrac{2}{3})\left(\begin{array}{c}-1\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle \frac{1}{2} - \frac{3 y}{2}\\\displaystyle \frac{1}{2} - \frac{3 y}{2}&\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{2}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,\tfrac{1}{3})\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle - \frac{3 x}{2} - \frac{3 y}{2} + 1\\\displaystyle - \frac{3 x}{2} - \frac{3 y}{2} + 1&\displaystyle - 3 x - 3 y + 2\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{3}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,\tfrac{2}{3})\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle \frac{3 y}{2} - \frac{1}{2}\\\displaystyle \frac{3 y}{2} - \frac{1}{2}&\displaystyle 3 y - 1\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{4}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},0)\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle - 3 x - 3 y + 2&\displaystyle - \frac{3 x}{2} - \frac{3 y}{2} + 1\\\displaystyle - \frac{3 x}{2} - \frac{3 y}{2} + 1&\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{5}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{2}{3},0)\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle 3 x - 1&\displaystyle \frac{3 x}{2} - \frac{1}{2}\\\displaystyle \frac{3 x}{2} - \frac{1}{2}&\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{6}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},\tfrac{1}{3})\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle 3 y&\displaystyle \frac{3 y}{2}\\\displaystyle \frac{3 y}{2}&\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{7}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},\tfrac{1}{3})\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle \frac{3 x}{2}\\\displaystyle \frac{3 x}{2}&\displaystyle 3 x\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{8}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},\tfrac{1}{3})\left(\begin{array}{c}-1\\1\end{array}\right)$$

$$\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle 0&\displaystyle \frac{3 x}{2} + \frac{3 y}{2} - \frac{3}{2}\\\displaystyle \frac{3 x}{2} + \frac{3 y}{2} - \frac{3}{2}&\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.