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Degree 0 Gopalakrishnan–Lederer–Schöberl 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: \(\left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 1\\\displaystyle 0&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 1&\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{cc}\displaystyle 0&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{3}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{0}}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle \frac{\sqrt{2}}{2}\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2}}{2}\\\displaystyle - \frac{\sqrt{2}}{2}\end{array}\right)\)
where \(e_{0}\) is the 0th edge.
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle \frac{1}{2}&\displaystyle 0\\\displaystyle 0&\displaystyle - \frac{1}{2}\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge.
\(\displaystyle \mathbf{\Phi}_{0} = \left(\begin{array}{cc}\displaystyle \frac{1}{2}&\displaystyle 0\\\displaystyle 0&\displaystyle - \frac{1}{2}\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{1}}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle -1\\\displaystyle 0\end{array}\right)\)
where \(e_{1}\) is the 1st edge.
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle - \frac{1}{2}&\displaystyle 0\\\displaystyle -1&\displaystyle \frac{1}{2}\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge.
\(\displaystyle \mathbf{\Phi}_{1} = \left(\begin{array}{cc}\displaystyle - \frac{1}{2}&\displaystyle 0\\\displaystyle -1&\displaystyle \frac{1}{2}\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{V}\mapsto\displaystyle\int_{e_{2}}\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)^{\text{t}}\boldsymbol{V}\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\)
where \(e_{2}\) is the 2nd edge.
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle - \frac{1}{2}&\displaystyle 1\\\displaystyle 0&\displaystyle \frac{1}{2}\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge.
\(\displaystyle \mathbf{\Phi}_{2} = \left(\begin{array}{cc}\displaystyle - \frac{1}{2}&\displaystyle 1\\\displaystyle 0&\displaystyle \frac{1}{2}\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{V}\mapsto\displaystyle\int_{R}\operatorname{tr}(\boldsymbol{V})\)
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \mathbf{\Phi}_{3} = \left(\begin{array}{cc}\displaystyle 1&\displaystyle 0\\\displaystyle 0&\displaystyle 1\end{array}\right)\)
This DOF is associated with face 0 of the reference element.