an encyclopedia of finite element definitions

# Degree 3 Taylor on a triangle

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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:
$$\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}v$$
where $$R$$ is the reference element.

$$\displaystyle \phi_{0} = 2$$

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

$$\displaystyle \phi_{1} = y - \frac{1}{3}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{2}:v\mapsto\frac{\partial^{2}}{\partial y^{2}}v(\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{2} = \frac{y^{2}}{2} - \frac{y}{3} + \frac{1}{36}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{3}:v\mapsto\frac{\partial^{3}}{\partial y^{3}}v(\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{3} = \frac{y^{3}}{6} - \frac{y^{2}}{6} + \frac{y}{18} - \frac{1}{135}$$

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

$$\displaystyle \phi_{4} = x - \frac{1}{3}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{5}:v\mapsto\frac{\partial^{2}}{\partial x\partial y}v(\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{5} = x y - \frac{x}{3} - \frac{y}{3} + \frac{5}{36}$$

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

$$\displaystyle \phi_{6} = \frac{x y^{2}}{2} - \frac{x y}{3} + \frac{x}{18} - \frac{y^{2}}{6} + \frac{y}{9} - \frac{1}{60}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{7}:v\mapsto\frac{\partial^{2}}{\partial x^{2}}v(\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{7} = \frac{x^{2}}{2} - \frac{x}{3} + \frac{1}{36}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{8}:v\mapsto\frac{\partial^{3}}{\partial x^{2}\partial y}v(\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{8} = \frac{x^{2} y}{2} - \frac{x^{2}}{6} - \frac{x y}{3} + \frac{x}{9} + \frac{y}{18} - \frac{1}{60}$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{9}:v\mapsto\frac{\partial^{3}}{\partial x^{3}}v(\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{9} = \frac{x^{3}}{6} - \frac{x^{2}}{6} + \frac{x}{18} - \frac{1}{135}$$

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