an encyclopedia of finite element definitions
Degree 2 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: \(1\), \(y\), \(y^{2}\), \(x\), \(x y\), \(x y^{2}\), \(x^{2}\), \(x^{2} y\), \(x^{2} y^{2}\)
- \(\mathcal{L}=\{l_0,...,l_{8}\}\)
- 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.
\(\displaystyle l_{1}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{3} \left(2 s_{1} - 1\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \phi_{1} = \sqrt{3} \left(2 y - 1\right)\)
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 \(R\).
\(\displaystyle \phi_{1} = \sqrt{3} \left(2 y - 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{2}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{5} \left(6 s_{1}^{2} - 6 s_{1} + 1\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \phi_{2} = \sqrt{5} \left(6 y^{2} - 6 y + 1\right)\)
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 \(R\).
\(\displaystyle \phi_{2} = \sqrt{5} \left(6 y^{2} - 6 y + 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{3}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{3} \left(2 s_{0} - 1\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \phi_{3} = \sqrt{3} \left(2 x - 1\right)\)
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 \(R\).
\(\displaystyle \phi_{3} = \sqrt{3} \left(2 x - 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{4}:\mathbf{v}\mapsto\displaystyle\int_{R}(12 s_{0} s_{1} - 6 s_{0} - 6 s_{1} + 3)v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \phi_{4} = 12 x y - 6 x - 6 y + 3\)
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 \(R\).
\(\displaystyle \phi_{4} = 12 x y - 6 x - 6 y + 3\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{5}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{15} \left(12 s_{0} s_{1}^{2} - 12 s_{0} s_{1} + 2 s_{0} - 6 s_{1}^{2} + 6 s_{1} - 1\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \phi_{5} = \sqrt{15} \left(12 x y^{2} - 12 x y + 2 x - 6 y^{2} + 6 y - 1\right)\)
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 \(R\).
\(\displaystyle \phi_{5} = \sqrt{15} \left(12 x y^{2} - 12 x y + 2 x - 6 y^{2} + 6 y - 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{6}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{5} \left(6 s_{0}^{2} - 6 s_{0} + 1\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \phi_{6} = \sqrt{5} \left(6 x^{2} - 6 x + 1\right)\)
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 \(R\).
\(\displaystyle \phi_{6} = \sqrt{5} \left(6 x^{2} - 6 x + 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{7}:\mathbf{v}\mapsto\displaystyle\int_{R}(\sqrt{15} \left(12 s_{0}^{2} s_{1} - 6 s_{0}^{2} - 12 s_{0} s_{1} + 6 s_{0} + 2 s_{1} - 1\right))v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \phi_{7} = \sqrt{15} \left(12 x^{2} y - 6 x^{2} - 12 x y + 6 x + 2 y - 1\right)\)
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 \(R\).
\(\displaystyle \phi_{7} = \sqrt{15} \left(12 x^{2} y - 6 x^{2} - 12 x y + 6 x + 2 y - 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle l_{8}:\mathbf{v}\mapsto\displaystyle\int_{R}(180 s_{0}^{2} s_{1}^{2} - 180 s_{0}^{2} s_{1} + 30 s_{0}^{2} - 180 s_{0} s_{1}^{2} + 180 s_{0} s_{1} - 30 s_{0} + 30 s_{1}^{2} - 30 s_{1} + 5)v\)
where \(R\) is the reference element;
and \(s_{0},s_{1}\) is a parametrisation of \(R\).
\(\displaystyle \phi_{8} = 180 x^{2} y^{2} - 180 x^{2} y + 30 x^{2} - 180 x y^{2} + 180 x y - 30 x + 30 y^{2} - 30 y + 5\)
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 \(R\).
\(\displaystyle \phi_{8} = 180 x^{2} y^{2} - 180 x^{2} y + 30 x^{2} - 180 x y^{2} + 180 x y - 30 x + 30 y^{2} - 30 y + 5\)
This DOF is associated with face 0 of the reference element.