an encyclopedia of finite element definitions

# Vector bubble enriched Lagrange

 Orders $$1\leqslant k\leqslant 2$$ Reference elements triangle Polynomial set $$\mathcal{P}_{k}^d \oplus \left(\mathcal{Z}^{(13)}_{k+2}\right)^d$$↓ Show polynomial set definitions ↓ DOFs On each vertex: point evaluations in coordinate directions On each edge: point evaluations in coordinate directions On each face: point evaluations in coordinate directions Number of DOFs triangle: $$2(k+1)^2$$ (A001105) Categories Vector-valued elements

## Implementations

 Symfem "bubble enriched vector Lagrange"↓ Show Symfem examples ↓

## Examples

triangle
order 1
triangle
order 2
• $$R$$ is the reference triangle. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y \left(- x - y + 1\right)\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{7}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle 9 x^{2} y + 9 x y^{2} - 9 x y - x - y + 1\\\displaystyle 0\end{array}\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 9 x^{2} y + 9 x y^{2} - 9 x y - x - y + 1\end{array}\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle x \left(9 x y + 9 y^{2} - 9 y + 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(9 x y + 9 y^{2} - 9 y + 1\right)\end{array}\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle y \left(9 x^{2} + 9 x y - 9 x + 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(9 x^{2} + 9 x y - 9 x + 1\right)\end{array}\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 27 x y \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{3},\tfrac{1}{3})\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 27 x y \left(- x - y + 1\right)\end{array}\right)$$

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: $$\left(\begin{array}{c}\displaystyle 1\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle y^{2}\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x^{2} y \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle x y^{2} \left(- x - y + 1\right)\\\displaystyle 0\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle 1\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle y^{2}\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y \left(- x - y + 1\right)\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x^{2} y \left(- x - y + 1\right)\end{array}\right)$$, $$\left(\begin{array}{c}\displaystyle 0\\\displaystyle x y^{2} \left(- x - y + 1\right)\end{array}\right)$$
• $$\mathcal{L}=\{l_0,...,l_{17}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{0} = \left(\begin{array}{c}\displaystyle - 16 x^{3} y - 32 x^{2} y^{2} + 24 x^{2} y + 2 x^{2} - 16 x y^{3} + 24 x y^{2} - 4 x y - 3 x + 2 y^{2} - 3 y + 1\\\displaystyle 0\end{array}\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{1} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle - 16 x^{3} y - 32 x^{2} y^{2} + 24 x^{2} y + 2 x^{2} - 16 x y^{3} + 24 x y^{2} - 4 x y - 3 x + 2 y^{2} - 3 y + 1\end{array}\right)$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{2} = \left(\begin{array}{c}\displaystyle x \left(16 x^{2} y + 16 x y^{2} - 24 x y + 2 x - 8 y^{2} + 8 y - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{3} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle x \left(16 x^{2} y + 16 x y^{2} - 24 x y + 2 x - 8 y^{2} + 8 y - 1\right)\end{array}\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{4} = \left(\begin{array}{c}\displaystyle y \left(16 x^{2} y - 8 x^{2} + 16 x y^{2} - 24 x y + 8 x + 2 y - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1)\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{5} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle y \left(16 x^{2} y - 8 x^{2} + 16 x y^{2} - 24 x y + 8 x + 2 y - 1\right)\end{array}\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{6} = \left(\begin{array}{c}\displaystyle 4 x y \left(8 x^{2} + 16 x y - 10 x + 8 y^{2} - 10 y + 3\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{7} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x y \left(8 x^{2} + 16 x y - 10 x + 8 y^{2} - 10 y + 3\right)\end{array}\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{8} = \left(\begin{array}{c}\displaystyle 4 y \left(- 8 x^{3} - 8 x^{2} y + 14 x^{2} + 6 x y - 7 x - y + 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{9} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y \left(- 8 x^{3} - 8 x^{2} y + 14 x^{2} + 6 x y - 7 x - y + 1\right)\end{array}\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0)\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{10} = \left(\begin{array}{c}\displaystyle 4 x \left(- 8 x y^{2} + 6 x y - x - 8 y^{3} + 14 y^{2} - 7 y + 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},0)\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{11} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x \left(- 8 x y^{2} + 6 x y - x - 8 y^{3} + 14 y^{2} - 7 y + 1\right)\end{array}\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{12}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{4},\tfrac{1}{4})\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{12} = \left(\begin{array}{c}\displaystyle 32 x y \left(4 x^{2} + 8 x y - 7 x + 4 y^{2} - 7 y + 3\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{13}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{4},\tfrac{1}{4})\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{13} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 32 x y \left(4 x^{2} + 8 x y - 7 x + 4 y^{2} - 7 y + 3\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{14}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{4},\tfrac{1}{2})\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{14} = \left(\begin{array}{c}\displaystyle 32 x y \left(- 4 x y + x - 4 y^{2} + 5 y - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{15}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{4},\tfrac{1}{2})\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{15} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 32 x y \left(- 4 x y + x - 4 y^{2} + 5 y - 1\right)\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{16}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{4})\cdot\left(\begin{array}{c}1\\0\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{16} = \left(\begin{array}{c}\displaystyle 32 x y \left(- 4 x^{2} - 4 x y + 5 x + y - 1\right)\\\displaystyle 0\end{array}\right)$$

This DOF is associated with face 0 of the reference element.
$$\displaystyle l_{17}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{2},\tfrac{1}{4})\cdot\left(\begin{array}{c}0\\1\end{array}\right)$$

$$\displaystyle \boldsymbol{\phi}_{17} = \left(\begin{array}{c}\displaystyle 0\\\displaystyle 32 x y \left(- 4 x^{2} - 4 x y + 5 x + y - 1\right)\end{array}\right)$$

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

## DefElement stats

 Element added 02 March 2021 Element last updated 13 June 2021