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Degree 2 Fortin–Soulie on a triangle
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- \(R\) is the reference triangle. The following numbering of the sub-entities of the reference cell is used:
- \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(y\), \(x y\), \(y^{2}\)
- \(\mathcal{L}=\{l_0,...,l_{5}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(\tfrac{1}{2},0)\)
\(\displaystyle \phi_{0} = - \frac{12 x^{2}}{7} - \frac{12 x y}{7} + \frac{12 x}{7} + \frac{24 y^{2}}{7} - \frac{24 y}{7} + \frac{4}{7}\)
This DOF is associated with edge 0 of the reference cell.
\(\displaystyle \phi_{0} = - \frac{12 x^{2}}{7} - \frac{12 x y}{7} + \frac{12 x}{7} + \frac{24 y^{2}}{7} - \frac{24 y}{7} + \frac{4}{7}\)
This DOF is associated with edge 0 of the reference cell.
\(\displaystyle l_{1}:v\mapsto v(0,\tfrac{1}{2} - \tfrac{\sqrt{3}}{6})\)
\(\displaystyle \phi_{1} = \frac{6 \sqrt{3} x^{2}}{7} + \frac{12 x^{2}}{7} - \frac{9 x y}{7} + \frac{27 \sqrt{3} x y}{7} - \frac{13 \sqrt{3} x}{7} - \frac{12 x}{7} - \frac{3 y^{2}}{7} + \frac{9 \sqrt{3} y^{2}}{7} - \frac{16 \sqrt{3} y}{7} + \frac{3 y}{7} + \frac{3}{7} + \frac{5 \sqrt{3}}{7}\)
This DOF is associated with edge 1 of the reference cell.
\(\displaystyle \phi_{1} = \frac{6 \sqrt{3} x^{2}}{7} + \frac{12 x^{2}}{7} - \frac{9 x y}{7} + \frac{27 \sqrt{3} x y}{7} - \frac{13 \sqrt{3} x}{7} - \frac{12 x}{7} - \frac{3 y^{2}}{7} + \frac{9 \sqrt{3} y^{2}}{7} - \frac{16 \sqrt{3} y}{7} + \frac{3 y}{7} + \frac{3}{7} + \frac{5 \sqrt{3}}{7}\)
This DOF is associated with edge 1 of the reference cell.
\(\displaystyle l_{2}:v\mapsto v(0,\tfrac{\sqrt{3}}{6} + \tfrac{1}{2})\)
\(\displaystyle \phi_{2} = - \frac{6 \sqrt{3} x^{2}}{7} + \frac{12 x^{2}}{7} - \frac{27 \sqrt{3} x y}{7} - \frac{9 x y}{7} - \frac{12 x}{7} + \frac{13 \sqrt{3} x}{7} - \frac{9 \sqrt{3} y^{2}}{7} - \frac{3 y^{2}}{7} + \frac{3 y}{7} + \frac{16 \sqrt{3} y}{7} - \frac{5 \sqrt{3}}{7} + \frac{3}{7}\)
This DOF is associated with edge 1 of the reference cell.
\(\displaystyle \phi_{2} = - \frac{6 \sqrt{3} x^{2}}{7} + \frac{12 x^{2}}{7} - \frac{27 \sqrt{3} x y}{7} - \frac{9 x y}{7} - \frac{12 x}{7} + \frac{13 \sqrt{3} x}{7} - \frac{9 \sqrt{3} y^{2}}{7} - \frac{3 y^{2}}{7} + \frac{3 y}{7} + \frac{16 \sqrt{3} y}{7} - \frac{5 \sqrt{3}}{7} + \frac{3}{7}\)
This DOF is associated with edge 1 of the reference cell.
\(\displaystyle l_{3}:v\mapsto v(\tfrac{\sqrt{3}}{6} + \tfrac{1}{2},\tfrac{1}{2} - \tfrac{\sqrt{3}}{6})\)
\(\displaystyle \phi_{3} = \frac{6 \sqrt{3} x^{2}}{7} + \frac{12 x^{2}}{7} - \frac{15 \sqrt{3} x y}{7} + \frac{33 x y}{7} - \frac{12 x}{7} + \frac{\sqrt{3} x}{7} - \frac{12 \sqrt{3} y^{2}}{7} + \frac{18 y^{2}}{7} - \frac{18 y}{7} + \frac{12 \sqrt{3} y}{7} - \frac{2 \sqrt{3}}{7} + \frac{3}{7}\)
This DOF is associated with edge 2 of the reference cell.
\(\displaystyle \phi_{3} = \frac{6 \sqrt{3} x^{2}}{7} + \frac{12 x^{2}}{7} - \frac{15 \sqrt{3} x y}{7} + \frac{33 x y}{7} - \frac{12 x}{7} + \frac{\sqrt{3} x}{7} - \frac{12 \sqrt{3} y^{2}}{7} + \frac{18 y^{2}}{7} - \frac{18 y}{7} + \frac{12 \sqrt{3} y}{7} - \frac{2 \sqrt{3}}{7} + \frac{3}{7}\)
This DOF is associated with edge 2 of the reference cell.
\(\displaystyle l_{4}:v\mapsto v(\tfrac{1}{2} - \tfrac{\sqrt{3}}{6},\tfrac{\sqrt{3}}{6} + \tfrac{1}{2})\)
\(\displaystyle \phi_{4} = - \frac{6 \sqrt{3} x^{2}}{7} + \frac{12 x^{2}}{7} + \frac{15 \sqrt{3} x y}{7} + \frac{33 x y}{7} - \frac{12 x}{7} - \frac{\sqrt{3} x}{7} + \frac{18 y^{2}}{7} + \frac{12 \sqrt{3} y^{2}}{7} - \frac{12 \sqrt{3} y}{7} - \frac{18 y}{7} + \frac{3}{7} + \frac{2 \sqrt{3}}{7}\)
This DOF is associated with edge 2 of the reference cell.
\(\displaystyle \phi_{4} = - \frac{6 \sqrt{3} x^{2}}{7} + \frac{12 x^{2}}{7} + \frac{15 \sqrt{3} x y}{7} + \frac{33 x y}{7} - \frac{12 x}{7} - \frac{\sqrt{3} x}{7} + \frac{18 y^{2}}{7} + \frac{12 \sqrt{3} y^{2}}{7} - \frac{12 \sqrt{3} y}{7} - \frac{18 y}{7} + \frac{3}{7} + \frac{2 \sqrt{3}}{7}\)
This DOF is associated with edge 2 of the reference cell.
\(\displaystyle l_{5}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3})\)
\(\displaystyle \phi_{5} = - \frac{36 x^{2}}{7} - \frac{36 x y}{7} + \frac{36 x}{7} - \frac{54 y^{2}}{7} + \frac{54 y}{7} - \frac{9}{7}\)
This DOF is associated with face 0 of the reference cell.
\(\displaystyle \phi_{5} = - \frac{36 x^{2}}{7} - \frac{36 x y}{7} + \frac{36 x}{7} - \frac{54 y^{2}}{7} + \frac{54 y}{7} - \frac{9}{7}\)
This DOF is associated with face 0 of the reference cell.