an encyclopedia of finite element definitions

# Degree 3 Hermite on a triangle

◀ Back to Hermite definition page In this example:
• $$R$$ 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. $$\displaystyle l_{1}:v\mapsto\frac{\partial}{\partial x}v(0,0)$$

$$\displaystyle \phi_{1} = x \left(x^{2} + 3 x y - 2 x + 2 y^{2} - 3 y + 1\right)$$

This DOF is associated with vertex 0 of the reference element. $$\displaystyle l_{2}:v\mapsto\frac{\partial}{\partial y}v(0,0)$$

$$\displaystyle \phi_{2} = y \left(2 x^{2} + 3 x y - 3 x + y^{2} - 2 y + 1\right)$$

This DOF is associated with vertex 0 of the reference element. $$\displaystyle l_{3}:v\mapsto v(1,0)$$

$$\displaystyle \phi_{3} = x \left(- 2 x^{2} + 7 x y + 3 x + 7 y^{2} - 7 y\right)$$

This DOF is associated with vertex 1 of the reference element. $$\displaystyle l_{4}:v\mapsto\frac{\partial}{\partial x}v(1,0)$$

$$\displaystyle \phi_{4} = x \left(x^{2} - 2 x y - x - 2 y^{2} + 2 y\right)$$

This DOF is associated with vertex 1 of the reference element. $$\displaystyle l_{5}:v\mapsto\frac{\partial}{\partial y}v(1,0)$$

$$\displaystyle \phi_{5} = x y \left(2 x + y - 1\right)$$

This DOF is associated with vertex 1 of the reference element. $$\displaystyle l_{6}:v\mapsto v(0,1)$$

$$\displaystyle \phi_{6} = y \left(7 x^{2} + 7 x y - 7 x - 2 y^{2} + 3 y\right)$$

This DOF is associated with vertex 2 of the reference element. $$\displaystyle l_{7}:v\mapsto\frac{\partial}{\partial x}v(0,1)$$

$$\displaystyle \phi_{7} = x y \left(x + 2 y - 1\right)$$

This DOF is associated with vertex 2 of the reference element. $$\displaystyle l_{8}:v\mapsto\frac{\partial}{\partial y}v(0,1)$$

$$\displaystyle \phi_{8} = y \left(- 2 x^{2} - 2 x y + 2 x + y^{2} - y\right)$$

This DOF is associated with vertex 2 of the reference element. $$\displaystyle l_{9}:v\mapsto v(\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{9} = 27 x y \left(- x - y + 1\right)$$

This DOF is associated with face 0 of the reference element.