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Degree 2 Brezzi–Douglas–Marini 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: \(\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)\), \(\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)\)
- \(\mathcal{L}=\{l_0,...,l_{11}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle x \left(- 4 x - 4 y + 3\right)\\\displaystyle y \left(- 4 x - 4 y + 3\right)\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle x \left(- 4 x - 4 y + 3\right)\\\displaystyle y \left(- 4 x - 4 y + 3\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(\sqrt{3} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle \sqrt{3} x \left(3 x + 2 y - 2\right)\\\displaystyle \sqrt{3} y \left(- 2 x - 3 y + 2\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle \sqrt{3} x \left(3 x + 2 y - 2\right)\\\displaystyle \sqrt{3} y \left(- 2 x - 3 y + 2\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{0}}\boldsymbol{v}\cdot(\sqrt{5} \left(6 s_{0}^{2} - 6 s_{0} + 1\right))\hat{\boldsymbol{n}}_{0}\)
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \sqrt{5} x \left(- x + 2 y\right)\\\displaystyle \sqrt{5} y \left(2 x - y\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
where \(e_{0}\) is the 0th edge;
\(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{0}\).
\(\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle \sqrt{5} x \left(- x + 2 y\right)\\\displaystyle \sqrt{5} y \left(2 x - y\right)\end{array}\right)\)
This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - 4 x^{2} + 5 x - 1\\\displaystyle y \left(1 - 4 x\right)\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle - 4 x^{2} + 5 x - 1\\\displaystyle y \left(1 - 4 x\right)\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(\sqrt{3} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \sqrt{3} \left(3 x^{2} + 6 x y - 4 x - 2 y + 1\right)\\\displaystyle \sqrt{3} y \left(- 2 x + y\right)\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle \sqrt{3} \left(3 x^{2} + 6 x y - 4 x - 2 y + 1\right)\\\displaystyle \sqrt{3} y \left(- 2 x + y\right)\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{1}}\boldsymbol{v}\cdot(\sqrt{5} \left(6 s_{0}^{2} - 6 s_{0} + 1\right))\hat{\boldsymbol{n}}_{1}\)
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \sqrt{5} \left(- x^{2} - 6 x y + 2 x - 6 y^{2} + 6 y - 1\right)\\\displaystyle \sqrt{5} y \left(2 x + 3 y - 2\right)\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
where \(e_{1}\) is the 1st edge;
\(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{1}\).
\(\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle \sqrt{5} \left(- x^{2} - 6 x y + 2 x - 6 y^{2} + 6 y - 1\right)\\\displaystyle \sqrt{5} y \left(2 x + 3 y - 2\right)\end{array}\right)\)
This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle x \left(4 y - 1\right)\\\displaystyle 4 y^{2} - 5 y + 1\end{array}\right)\)
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.
\(\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle x \left(4 y - 1\right)\\\displaystyle 4 y^{2} - 5 y + 1\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(\sqrt{3} \left(2 s_{0} - 1\right))\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \sqrt{3} x \left(- x + 2 y\right)\\\displaystyle \sqrt{3} \left(- 6 x y + 2 x - 3 y^{2} + 4 y - 1\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle \sqrt{3} x \left(- x + 2 y\right)\\\displaystyle \sqrt{3} \left(- 6 x y + 2 x - 3 y^{2} + 4 y - 1\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\displaystyle\int_{e_{2}}\boldsymbol{v}\cdot(\sqrt{5} \left(6 s_{0}^{2} - 6 s_{0} + 1\right))\hat{\boldsymbol{n}}_{2}\)
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \sqrt{5} x \left(- 3 x - 2 y + 2\right)\\\displaystyle \sqrt{5} \left(6 x^{2} + 6 x y - 6 x + y^{2} - 2 y + 1\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
where \(e_{2}\) is the 2nd edge;
\(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2;
and \(s_{0},s_{1}\) is a parametrisation of \(e_{2}\).
\(\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle \sqrt{5} x \left(- 3 x - 2 y + 2\right)\\\displaystyle \sqrt{5} \left(6 x^{2} + 6 x y - 6 x + y^{2} - 2 y + 1\right)\end{array}\right)\)
This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}- s_{1}\\s_{0}\end{array}\right)\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 12 x \left(- x - 4 y + 1\right)\\\displaystyle 12 y \left(4 x + y - 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 12 x \left(- x - 4 y + 1\right)\\\displaystyle 12 y \left(4 x + y - 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}s_{1}\\1 - s_{0}\end{array}\right)\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 12 x \left(x + 2 y - 1\right)\\\displaystyle 12 y \left(- 4 x - 3 y + 3\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 12 x \left(x + 2 y - 1\right)\\\displaystyle 12 y \left(- 4 x - 3 y + 3\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\displaystyle\int_{R}\boldsymbol{v}\cdot\left(\begin{array}{c}1 - s_{1}\\s_{0}\end{array}\right)\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 12 x \left(- 3 x - 4 y + 3\right)\\\displaystyle 12 y \left(2 x + y - 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 12 x \left(- 3 x - 4 y + 3\right)\\\displaystyle 12 y \left(2 x + y - 1\right)\end{array}\right)\)
This DOF is associated with face 0 of the reference element.