an encyclopedia of finite element definitions

# Degree 3 Lagrange on a triangle

◀ Back to Lagrange 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: $$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = - \frac{9 x^{3}}{2} - \frac{27 x^{2} y}{2} + 9 x^{2} - \frac{27 x y^{2}}{2} + 18 x y - \frac{11 x}{2} - \frac{9 y^{3}}{2} + 9 y^{2} - \frac{11 y}{2} + 1$$

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

$$\displaystyle \phi_{1} = \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}$$

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

$$\displaystyle \phi_{2} = \frac{y \left(9 y^{2} - 9 y + 2\right)}{2}$$

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

$$\displaystyle \phi_{3} = \frac{9 x y \left(3 x - 1\right)}{2}$$

This DOF is associated with edge 0 of the reference element. $$\displaystyle l_{4}:v\mapsto v(\tfrac{1}{3},\tfrac{2}{3})$$

$$\displaystyle \phi_{4} = \frac{9 x y \left(3 y - 1\right)}{2}$$

This DOF is associated with edge 0 of the reference element. $$\displaystyle l_{5}:v\mapsto v(0,\tfrac{1}{3})$$

$$\displaystyle \phi_{5} = \frac{9 y \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}$$

This DOF is associated with edge 1 of the reference element. $$\displaystyle l_{6}:v\mapsto v(0,\tfrac{2}{3})$$

$$\displaystyle \phi_{6} = \frac{9 y \left(- 3 x y + x - 3 y^{2} + 4 y - 1\right)}{2}$$

This DOF is associated with edge 1 of the reference element. $$\displaystyle l_{7}:v\mapsto v(\tfrac{1}{3},0)$$

$$\displaystyle \phi_{7} = \frac{9 x \left(3 x^{2} + 6 x y - 5 x + 3 y^{2} - 5 y + 2\right)}{2}$$

This DOF is associated with edge 2 of the reference element. $$\displaystyle l_{8}:v\mapsto v(\tfrac{2}{3},0)$$

$$\displaystyle \phi_{8} = \frac{9 x \left(- 3 x^{2} - 3 x y + 4 x + y - 1\right)}{2}$$

This DOF is associated with edge 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.