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Degree 1 Bernardi–Raugel 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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle - \frac{\sqrt{2} x y}{2}\\\displaystyle - \frac{\sqrt{2} x y}{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y \left(x + y - 1\right)\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(- x - y + 1\right)\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{8}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 3 x y - x + 3 y^{2} - 4 y + 1\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 3 x y - x + 3 y^{2} - 4 y + 1\\\displaystyle 0\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 x^{2} + 3 x y - 4 x - y + 1\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 3 x^{2} + 3 x y - 4 x - y + 1\end{array}\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{x \left(2 - 3 y\right)}{2}\\\displaystyle - \frac{3 x y}{2}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \frac{x \left(2 - 3 y\right)}{2}\\\displaystyle - \frac{3 x y}{2}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - \frac{3 x y}{2}\\\displaystyle \frac{x \left(6 x + 3 y - 4\right)}{2}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - \frac{3 x y}{2}\\\displaystyle \frac{x \left(6 x + 3 y - 4\right)}{2}\end{array}\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{y \left(3 x + 6 y - 4\right)}{2}\\\displaystyle - \frac{3 x y}{2}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \frac{y \left(3 x + 6 y - 4\right)}{2}\\\displaystyle - \frac{3 x y}{2}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)\)
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - \frac{3 x y}{2}\\\displaystyle \frac{y \left(2 - 3 x\right)}{2}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle - \frac{3 x y}{2}\\\displaystyle \frac{y \left(2 - 3 x\right)}{2}\end{array}\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle - 3 x y\\\displaystyle - 3 x y\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle - 3 x y\\\displaystyle - 3 x y\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 6 y \left(x + y - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 6 y \left(x + y - 1\right)\\\displaystyle 0\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x \left(- x - y + 1\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 6 x \left(- x - y + 1\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.