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Degree 2 dPc on a quadrilateral
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- \(R\) is the reference quadrilateral. The following numbering of the subentities of the reference 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:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)
\(\displaystyle \phi_{0} = 2 x^{2} + 4 x y - 3 x + 2 y^{2} - 3 y + 1\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{0} = 2 x^{2} + 4 x y - 3 x + 2 y^{2} - 3 y + 1\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto v(\tfrac{1}{2},0)\)
\(\displaystyle \phi_{1} = 4 x \left(- x - y + 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{1} = 4 x \left(- x - y + 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:v\mapsto v(1,0)\)
\(\displaystyle \phi_{2} = x \left(2 x - 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{2} = x \left(2 x - 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(0,\tfrac{1}{2})\)
\(\displaystyle \phi_{3} = 4 y \left(- x - y + 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{3} = 4 y \left(- x - y + 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2})\)
\(\displaystyle \phi_{4} = 4 x y\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{4} = 4 x y\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(0,1)\)
\(\displaystyle \phi_{5} = y \left(2 y - 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{5} = y \left(2 y - 1\right)\)
This DOF is associated with face 0 of the reference element.