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Degree 1 Gauss–Legendre 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: \(1\), \(y\), \(x\), \(x y\)
- \(\mathcal{L}=\{l_0,...,l_{3}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:\mathbf{v}\mapsto\displaystyle\int_{R}v\)
where \(R\) is the reference element.
\(\displaystyle \phi_{0} = 1\)
This DOF is associated with face 0 of the reference element.
where \(R\) is the reference element.
\(\displaystyle \phi_{0} = 1\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{1}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{3} \cdot \left(2 s_{1} - 1\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \phi_{1} = \sqrt{3} \cdot \left(2 y - 1\right)\)
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 \(R\).
\(\displaystyle \phi_{1} = \sqrt{3} \cdot \left(2 y - 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{3} \cdot \left(2 s_{0} - 1\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \phi_{2} = \sqrt{3} \cdot \left(2 x - 1\right)\)
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 \(R\).
\(\displaystyle \phi_{2} = \sqrt{3} \cdot \left(2 x - 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:\mathbf{v}\mapsto\displaystyle\int_{R}(12 s_{0} s_{1} - 6 s_{0} - 6 s_{1} + 3)v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \phi_{3} = 12 x y - 6 x - 6 y + 3\)
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 \(R\).
\(\displaystyle \phi_{3} = 12 x y - 6 x - 6 y + 3\)
This DOF is associated with face 0 of the reference element.