an encyclopedia of finite element definitions
Degree 3 Hermite on a triangle
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- \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
- \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(x^{2}\), \(x^{3}\), \(y\), \(x y\), \(x^{2} y\), \(y^{2}\), \(x y^{2}\), \(y^{3}\)
- \(\mathcal{L}=\{l_0,...,l_{9}\}\)
- Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)
\(\displaystyle \phi_{0} = 2 x^{3} + 13 x^{2} y - 3 x^{2} + 13 x y^{2} - 13 x y + 2 y^{3} - 3 y^{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{0} = 2 x^{3} + 13 x^{2} y - 3 x^{2} + 13 x y^{2} - 13 x y + 2 y^{3} - 3 y^{2} + 1\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{1}:v\mapsto\frac{\partial}{\partial x}v(0,0)\)
\(\displaystyle \phi_{1} = x \left(x^{2} + 3 x y - 2 x + 2 y^{2} - 3 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{1} = x \left(x^{2} + 3 x y - 2 x + 2 y^{2} - 3 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{2}:v\mapsto\frac{\partial}{\partial y}v(0,0)\)
\(\displaystyle \phi_{2} = y \left(2 x^{2} + 3 x y - 3 x + y^{2} - 2 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle \phi_{2} = y \left(2 x^{2} + 3 x y - 3 x + y^{2} - 2 y + 1\right)\)
This DOF is associated with vertex 0 of the reference element.
\(\displaystyle l_{3}:v\mapsto v(1,0)\)
\(\displaystyle \phi_{3} = x \left(- 2 x^{2} + 7 x y + 3 x + 7 y^{2} - 7 y\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{3} = x \left(- 2 x^{2} + 7 x y + 3 x + 7 y^{2} - 7 y\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{4}:v\mapsto\frac{\partial}{\partial x}v(1,0)\)
\(\displaystyle \phi_{4} = x \left(x^{2} - 2 x y - x - 2 y^{2} + 2 y\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{4} = x \left(x^{2} - 2 x y - x - 2 y^{2} + 2 y\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{5}:v\mapsto\frac{\partial}{\partial y}v(1,0)\)
\(\displaystyle \phi_{5} = x y \left(2 x + y - 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle \phi_{5} = x y \left(2 x + y - 1\right)\)
This DOF is associated with vertex 1 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(0,1)\)
\(\displaystyle \phi_{6} = y \left(7 x^{2} + 7 x y - 7 x - 2 y^{2} + 3 y\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{6} = y \left(7 x^{2} + 7 x y - 7 x - 2 y^{2} + 3 y\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{7}:v\mapsto\frac{\partial}{\partial x}v(0,1)\)
\(\displaystyle \phi_{7} = x y \left(x + 2 y - 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{7} = x y \left(x + 2 y - 1\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{8}:v\mapsto\frac{\partial}{\partial y}v(0,1)\)
\(\displaystyle \phi_{8} = y \left(- 2 x^{2} - 2 x y + 2 x + y^{2} - y\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle \phi_{8} = y \left(- 2 x^{2} - 2 x y + 2 x + y^{2} - y\right)\)
This DOF is associated with vertex 2 of the reference element.
\(\displaystyle l_{9}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3})\)
\(\displaystyle \phi_{9} = 27 x y \left(- x - y + 1\right)\)
This DOF is associated with face 0 of the reference element.
\(\displaystyle \phi_{9} = 27 x y \left(- x - y + 1\right)\)
This DOF is associated with face 0 of the reference element.