an encyclopedia of finite element definitions

# Gauss–Legendre

 Orders $$0\leqslant k$$ Reference elements interval, quadrilateral, hexahedron Polynomial set $$\mathcal{Q}_{k}$$↓ Show polynomial set definitions ↓ DOFs On each vertex: point evaluations On each edge: point evaluations at Gauss–Legendre points On each face: point evaluations at Gauss–Legendre points On each volume: point evaluations at Gauss–Legendre points Number of DOFs interval: $$k+1$$ (A000027)quadrilateral: $$(k+1)^2$$ (A000290)hexahedron: $$(k+1)^3$$ (A000578) Categories Scalar-valued elements

## Implementations

 Symfem "Lagrange", variant="legendre"↓ Show Symfem examples ↓

## Examples

interval
order 1
interval
order 2
order 1
order 2
• $$R$$ 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:
$$\displaystyle l_{0}:v\mapsto v(0)$$

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

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

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

This DOF is associated with vertex 1 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:
$$\displaystyle l_{0}:v\mapsto v(0)$$

$$\displaystyle \phi_{0} = 2 x^{2} - 3 x + 1$$

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

$$\displaystyle \phi_{1} = x \left(2 x - 1\right)$$

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

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

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

$$\displaystyle \phi_{0} = x y - x - y + 1$$

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

$$\displaystyle \phi_{1} = x \left(1 - y\right)$$

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

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

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

$$\displaystyle \phi_{3} = x y$$

This DOF is associated with vertex 3 of the reference element.
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• $$\mathcal{V}$$ is spanned by: $$1$$, $$y$$, $$y^{2}$$, $$x$$, $$x y$$, $$x y^{2}$$, $$x^{2}$$, $$x^{2} y$$, $$x^{2} y^{2}$$
• $$\mathcal{L}=\{l_0,...,l_{8}\}$$
• Functionals and basis functions:
$$\displaystyle l_{0}:v\mapsto v(0,0)$$

$$\displaystyle \phi_{0} = 4 x^{2} y^{2} - 6 x^{2} y + 2 x^{2} - 6 x y^{2} + 9 x y - 3 x + 2 y^{2} - 3 y + 1$$

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

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

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

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

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

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

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

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

This DOF is associated with edge 0 of the reference element.
$$\displaystyle l_{5}:v\mapsto v(0,\tfrac{1}{2})$$

$$\displaystyle \phi_{5} = 4 y \left(- 2 x^{2} y + 2 x^{2} + 3 x y - 3 x - y + 1\right)$$

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

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

This DOF is associated with edge 2 of the reference element.
$$\displaystyle l_{7}:v\mapsto v(\tfrac{1}{2},1)$$

$$\displaystyle \phi_{7} = 4 x y \left(- 2 x y + x + 2 y - 1\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle l_{8}:v\mapsto v(\tfrac{1}{2},\tfrac{1}{2})$$

$$\displaystyle \phi_{8} = 16 x y \left(x y - x - y + 1\right)$$

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

## DefElement stats

 Element added 20 February 2021 Element last updated 03 July 2021