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Degree 1 Regge on a triangle
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- \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(\left(\begin{array}{cc}\displaystyle1&\displaystyle0\\\displaystyle0&\displaystyle0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle0&\displaystyle1\\\displaystyle1&\displaystyle0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle0&\displaystyle0\\\displaystyle0&\displaystyle1\end{array}\right)\), \(\left(\begin{array}{cc}\displaystylex&\displaystyle0\\\displaystyle0&\displaystyle0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle0&\displaystylex\\\displaystylex&\displaystyle0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle0&\displaystyle0\\\displaystyle0&\displaystylex\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyley&\displaystyle0\\\displaystyle0&\displaystyle0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle0&\displaystyley\\\displaystyley&\displaystyle0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle0&\displaystyle0\\\displaystyle0&\displaystyley\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{8}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{2}{3},\tfrac{1}{3})\left(\begin{array}{c}-1\\1\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle0&\displaystyle\frac{1}{2} - \frac{3 x}{2}\\\displaystyle\frac{1}{2} - \frac{3 x}{2}&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle0&\displaystyle\frac{1}{2} - \frac{3 x}{2}\\\displaystyle\frac{1}{2} - \frac{3 x}{2}&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},\tfrac{2}{3})\left(\begin{array}{c}-1\\1\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle0&\displaystyle\frac{1}{2} - \frac{3 y}{2}\\\displaystyle\frac{1}{2} - \frac{3 y}{2}&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle0&\displaystyle\frac{1}{2} - \frac{3 y}{2}\\\displaystyle\frac{1}{2} - \frac{3 y}{2}&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,\tfrac{1}{3})\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle0&\displaystyle- \frac{3 x}{2} - \frac{3 y}{2} + 1\\\displaystyle- \frac{3 x}{2} - \frac{3 y}{2} + 1&\displaystyle- 3 x - 3 y + 2\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle0&\displaystyle- \frac{3 x}{2} - \frac{3 y}{2} + 1\\\displaystyle- \frac{3 x}{2} - \frac{3 y}{2} + 1&\displaystyle- 3 x - 3 y + 2\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{3}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(0,\tfrac{2}{3})\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle0&\displaystyle\frac{3 y}{2} - \frac{1}{2}\\\displaystyle\frac{3 y}{2} - \frac{1}{2}&\displaystyle3 y - 1\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle0&\displaystyle\frac{3 y}{2} - \frac{1}{2}\\\displaystyle\frac{3 y}{2} - \frac{1}{2}&\displaystyle3 y - 1\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},0)\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle- 3 x - 3 y + 2&\displaystyle- \frac{3 x}{2} - \frac{3 y}{2} + 1\\\displaystyle- \frac{3 x}{2} - \frac{3 y}{2} + 1&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \mathbf{\Phi}_{4} = \left(\begin{array}{cc}\displaystyle- 3 x - 3 y + 2&\displaystyle- \frac{3 x}{2} - \frac{3 y}{2} + 1\\\displaystyle- \frac{3 x}{2} - \frac{3 y}{2} + 1&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{5}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{2}{3},0)\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle3 x - 1&\displaystyle\frac{3 x}{2} - \frac{1}{2}\\\displaystyle\frac{3 x}{2} - \frac{1}{2}&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle \mathbf{\Phi}_{5} = \left(\begin{array}{cc}\displaystyle3 x - 1&\displaystyle\frac{3 x}{2} - \frac{1}{2}\\\displaystyle\frac{3 x}{2} - \frac{1}{2}&\displaystyle0\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{6}:\mathbf{V}\mapsto\left(\begin{array}{c}1\\0\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},\tfrac{1}{3})\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle3 y&\displaystyle\frac{3 y}{2}\\\displaystyle\frac{3 y}{2}&\displaystyle0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \mathbf{\Phi}_{6} = \left(\begin{array}{cc}\displaystyle3 y&\displaystyle\frac{3 y}{2}\\\displaystyle\frac{3 y}{2}&\displaystyle0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{7}:\mathbf{V}\mapsto\left(\begin{array}{c}0\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},\tfrac{1}{3})\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle0&\displaystyle\frac{3 x}{2}\\\displaystyle\frac{3 x}{2}&\displaystyle3 x\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \mathbf{\Phi}_{7} = \left(\begin{array}{cc}\displaystyle0&\displaystyle\frac{3 x}{2}\\\displaystyle\frac{3 x}{2}&\displaystyle3 x\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{8}:\mathbf{V}\mapsto\left(\begin{array}{c}-1\\1\end{array}\right)^{\text{t}}\mathbf{V}(\tfrac{1}{3},\tfrac{1}{3})\left(\begin{array}{c}-1\\1\end{array}\right)\)
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle0&\displaystyle\frac{3 x}{2} + \frac{3 y}{2} - \frac{3}{2}\\\displaystyle\frac{3 x}{2} + \frac{3 y}{2} - \frac{3}{2}&\displaystyle0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \mathbf{\Phi}_{8} = \left(\begin{array}{cc}\displaystyle0&\displaystyle\frac{3 x}{2} + \frac{3 y}{2} - \frac{3}{2}\\\displaystyle\frac{3 x}{2} + \frac{3 y}{2} - \frac{3}{2}&\displaystyle0\end{array}\right)\)
This DOF is associated with face 0 of the reference element.