an encyclopedia of finite element definitions

# Taylor

 Alternative names discontinuous Taylor Orders $$0\leqslant k$$ Reference elements interval, triangle, tetrahedron Polynomial set $$\mathcal{P}_{k}$$↓ Show polynomial set definitions ↓ DOFs On the interior of the reference element: integral over cell, and point evaluations at midpoint of derivatives up to order $$k$$ Number of DOFs interval: $$k+1$$ (A000027)triangle: $$(k+1)(k+2)/2$$ (A000217)tetrahedron: $$(k+1)(k+2)(k+3)/6$$ (A000292) Categories Scalar-valued elements

## Implementations

 Symfem "Taylor"↓ Show Symfem examples ↓ UFL "TDG"↓ Show UFL examples ↓

## Examples

interval
order 1
interval
order 2
interval
order 3
triangle
order 1
triangle
order 2
triangle
order 3
• $$R$$ is the reference interval. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$
• $$\mathcal{L}=\{l_0,...,l_{1}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}(1)v$$
where $$R$$ is the reference element.

$$\displaystyle \phi_{0} = 1$$

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

$$\displaystyle \phi_{1} = x - \frac{1}{2}$$

This DOF is associated with edge 0 of the reference element.
• $$R$$ is the reference interval. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$
• $$\mathcal{L}=\{l_0,...,l_{2}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}(1)v$$
where $$R$$ is the reference element.

$$\displaystyle \phi_{0} = 1$$

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

$$\displaystyle \phi_{1} = x - \frac{1}{2}$$

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

$$\displaystyle \phi_{2} = \frac{x^{2}}{2} - \frac{x}{2} + \frac{1}{12}$$

This DOF is associated with edge 0 of the reference element.
• $$R$$ is the reference interval. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$x^{3}$$
• $$\mathcal{L}=\{l_0,...,l_{3}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}(1)v$$
where $$R$$ is the reference element.

$$\displaystyle \phi_{0} = 1$$

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

$$\displaystyle \phi_{1} = x - \frac{1}{2}$$

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

$$\displaystyle \phi_{2} = \frac{x^{2}}{2} - \frac{x}{2} + \frac{1}{12}$$

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

$$\displaystyle \phi_{3} = \frac{x^{3}}{6} - \frac{x^{2}}{4} + \frac{x}{8} - \frac{1}{48}$$

This DOF is associated with edge 0 of the reference element.
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$y$$
• $$\mathcal{L}=\{l_0,...,l_{2}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}(1)v$$
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.
$$\displaystyle l_{2}:v\mapsto\frac{\partial}{\partial x}v(\tfrac{1}{3},\tfrac{1}{3})$$

$$\displaystyle \phi_{2} = x - \frac{1}{3}$$

This DOF is associated with face 0 of the reference element.
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$x$$, $$x^{2}$$, $$y$$, $$x y$$, $$y^{2}$$
• $$\mathcal{L}=\{l_0,...,l_{5}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}(1)v$$
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.
$$\displaystyle l_{3}:v\mapsto\frac{\partial}{\partial x}v(\tfrac{1}{3},\tfrac{1}{3})$$

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

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

$$\displaystyle \phi_{4} = x y - \frac{x}{3} - \frac{y}{3} + \frac{5}{36}$$

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

$$\displaystyle \phi_{5} = \frac{x^{2}}{2} - \frac{x}{3} + \frac{1}{36}$$

This DOF is associated with face 0 of the reference element.
• $$R$$ 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:
$$\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}(1)v$$
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.

## DefElement stats

 Element added 01 March 2021 Element last updated 05 June 2021