an encyclopedia of finite element definitions

Taylor

Click here to read what the information on this page means.

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

Implementations

Symfem string"Taylor"
↓ Show Symfem examples ↓
UFL string"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:
\(\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}(1)v\)

\(\displaystyle \phi_{0} = 1\)

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:
\(\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}(1)v\)

\(\displaystyle \phi_{0} = 1\)

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:
\(\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}(1)v\)

\(\displaystyle \phi_{0} = 1\)

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:
\(\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}(1)v\)

\(\displaystyle \phi_{0} = 2\)

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

\(\displaystyle \phi_{1} = y - \frac{1}{3}\)

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

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

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:
\(\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}(1)v\)

\(\displaystyle \phi_{0} = 2\)

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

\(\displaystyle \phi_{1} = y - \frac{1}{3}\)

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

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

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

\(\displaystyle \phi_{3} = x - \frac{1}{3}\)

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

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

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

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

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:
\(\displaystyle l_{0}:v\mapsto\displaystyle\int_{R}(1)v\)

\(\displaystyle \phi_{0} = 2\)

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

\(\displaystyle \phi_{1} = y - \frac{1}{3}\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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