an encyclopedia of finite element definitions
Degree 3 Bogner–Fox–Schmitt 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: \(1\), \(y\), \(y^{2}\), \(y^{3}\), \(x\), \(x y\), \(x y^{2}\), \(x y^{3}\), \(x^{2}\), \(x^{2} y\), \(x^{2} y^{2}\), \(x^{2} y^{3}\), \(x^{3}\), \(x^{3} y\), \(x^{3} y^{2}\), \(x^{3} y^{3}\)
- \(\mathcal{L}=\{l_0,...,l_{15}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)
\(\displaystyle \phi_{0} = 4 x^{3} y^{3} - 6 x^{3} y^{2} + 2 x^{3} - 6 x^{2} y^{3} + 9 x^{2} y^{2} - 3 x^{2} + 2 y^{3} - 3 y^{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = 4 x^{3} y^{3} - 6 x^{3} y^{2} + 2 x^{3} - 6 x^{2} y^{3} + 9 x^{2} y^{2} - 3 x^{2} + 2 y^{3} - 3 y^{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto\frac{\partial}{\partial x}v(0,0)\)
\(\displaystyle \phi_{1} = x \left(2 x^{2} y^{3} - 3 x^{2} y^{2} + x^{2} - 4 x y^{3} + 6 x y^{2} - 2 x + 2 y^{3} - 3 y^{2} + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{1} = x \left(2 x^{2} y^{3} - 3 x^{2} y^{2} + x^{2} - 4 x y^{3} + 6 x y^{2} - 2 x + 2 y^{3} - 3 y^{2} + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:v\mapsto\frac{\partial}{\partial y}v(0,0)\)
\(\displaystyle \phi_{2} = y \left(2 x^{3} y^{2} - 4 x^{3} y + 2 x^{3} - 3 x^{2} y^{2} + 6 x^{2} y - 3 x^{2} + y^{2} - 2 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{2} = y \left(2 x^{3} y^{2} - 4 x^{3} y + 2 x^{3} - 3 x^{2} y^{2} + 6 x^{2} y - 3 x^{2} + y^{2} - 2 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{3}:v\mapsto\frac{\partial^{2}}{\partial x\partial y}v(0,0)\)
\(\displaystyle \phi_{3} = x y \left(x^{2} y^{2} - 2 x^{2} y + x^{2} - 2 x y^{2} + 4 x y - 2 x + y^{2} - 2 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{3} = x y \left(x^{2} y^{2} - 2 x^{2} y + x^{2} - 2 x y^{2} + 4 x y - 2 x + y^{2} - 2 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(1,0)\)
\(\displaystyle \phi_{4} = x^{2} \left(- 4 x y^{3} + 6 x y^{2} - 2 x + 6 y^{3} - 9 y^{2} + 3\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{4} = x^{2} \left(- 4 x y^{3} + 6 x y^{2} - 2 x + 6 y^{3} - 9 y^{2} + 3\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{5}:v\mapsto\frac{\partial}{\partial x}v(1,0)\)
\(\displaystyle \phi_{5} = x^{2} \cdot \left(2 x y^{3} - 3 x y^{2} + x - 2 y^{3} + 3 y^{2} - 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{5} = x^{2} \cdot \left(2 x y^{3} - 3 x y^{2} + x - 2 y^{3} + 3 y^{2} - 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{6}:v\mapsto\frac{\partial}{\partial y}v(1,0)\)
\(\displaystyle \phi_{6} = x^{2} y \left(- 2 x y^{2} + 4 x y - 2 x + 3 y^{2} - 6 y + 3\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{6} = x^{2} y \left(- 2 x y^{2} + 4 x y - 2 x + 3 y^{2} - 6 y + 3\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{7}:v\mapsto\frac{\partial^{2}}{\partial x\partial y}v(1,0)\)
\(\displaystyle \phi_{7} = x^{2} y \left(x y^{2} - 2 x y + x - y^{2} + 2 y - 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{7} = x^{2} y \left(x y^{2} - 2 x y + x - y^{2} + 2 y - 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(0,1)\)
\(\displaystyle \phi_{8} = y^{2} \left(- 4 x^{3} y + 6 x^{3} + 6 x^{2} y - 9 x^{2} - 2 y + 3\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{8} = y^{2} \left(- 4 x^{3} y + 6 x^{3} + 6 x^{2} y - 9 x^{2} - 2 y + 3\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{9}:v\mapsto\frac{\partial}{\partial x}v(0,1)\)
\(\displaystyle \phi_{9} = x y^{2} \left(- 2 x^{2} y + 3 x^{2} + 4 x y - 6 x - 2 y + 3\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{9} = x y^{2} \left(- 2 x^{2} y + 3 x^{2} + 4 x y - 6 x - 2 y + 3\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{10}:v\mapsto\frac{\partial}{\partial y}v(0,1)\)
\(\displaystyle \phi_{10} = y^{2} \cdot \left(2 x^{3} y - 2 x^{3} - 3 x^{2} y + 3 x^{2} + y - 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{10} = y^{2} \cdot \left(2 x^{3} y - 2 x^{3} - 3 x^{2} y + 3 x^{2} + y - 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{11}:v\mapsto\frac{\partial^{2}}{\partial x\partial y}v(0,1)\)
\(\displaystyle \phi_{11} = x y^{2} \left(x^{2} y - x^{2} - 2 x y + 2 x + y - 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{11} = x y^{2} \left(x^{2} y - x^{2} - 2 x y + 2 x + y - 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{12}:v\mapsto v(1,1)\)
\(\displaystyle \phi_{12} = x^{2} y^{2} \cdot \left(4 x y - 6 x - 6 y + 9\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \phi_{12} = x^{2} y^{2} \cdot \left(4 x y - 6 x - 6 y + 9\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{13}:v\mapsto\frac{\partial}{\partial x}v(1,1)\)
\(\displaystyle \phi_{13} = x^{2} y^{2} \left(- 2 x y + 3 x + 2 y - 3\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \phi_{13} = x^{2} y^{2} \left(- 2 x y + 3 x + 2 y - 3\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{14}:v\mapsto\frac{\partial}{\partial y}v(1,1)\)
\(\displaystyle \phi_{14} = x^{2} y^{2} \left(- 2 x y + 2 x + 3 y - 3\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \phi_{14} = x^{2} y^{2} \left(- 2 x y + 2 x + 3 y - 3\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{15}:v\mapsto\frac{\partial^{2}}{\partial x\partial y}v(1,1)\)
\(\displaystyle \phi_{15} = x^{2} y^{2} \left(x y - x - y + 1\right)\)
This DOF is associated with vertex 3 of the reference element.
\(\displaystyle \phi_{15} = x^{2} y^{2} \left(x y - x - y + 1\right)\)
This DOF is associated with vertex 3 of the reference element.