an encyclopedia of finite element definitions

# Degree 4 bubble on a triangle

◀ Back to bubble 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: $$x y \left(- x - y + 1\right)$$, $$x^{2} y \left(- x - y + 1\right)$$, $$x y^{2} \left(- x - y + 1\right)$$
• $$\mathcal{L}=\{l_0,...,l_{2}\}$$
• Functionals and basis functions: $$\displaystyle l_{0}:v\mapsto v(\tfrac{1}{4},\tfrac{1}{4})$$

$$\displaystyle \phi_{0} = 32 x y \left(4 x^{2} + 8 x y - 7 x + 4 y^{2} - 7 y + 3\right)$$

This DOF is associated with face 0 of the reference element. $$\displaystyle l_{1}:v\mapsto v(\tfrac{1}{4},\tfrac{1}{2})$$

$$\displaystyle \phi_{1} = 32 x y \left(- 4 x y + x - 4 y^{2} + 5 y - 1\right)$$

This DOF is associated with face 0 of the reference element. $$\displaystyle l_{2}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{4})$$

$$\displaystyle \phi_{2} = 32 x y \left(- 4 x^{2} - 4 x y + 5 x + y - 1\right)$$

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