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Gauss–Lobatto–Legendre

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Abbreviated namesGLL
Orders\(1\leqslant k\)
Reference elementsinterval, quadrilateral, hexahedron
Polynomial set\(\mathcal{Q}_{k}\)
↓ Show polynomial set definitions ↓
DOFsOn each vertex: point evaluations
On each edge: point evaluations at GLL points
On each face: point evaluations at GLL points
On each volume: point evaluations at GLL points
Number of DOFsinterval: \(k+1\) (A000027)
quadrilateral: \((k+1)^2\) (A000290)
hexahedron: \((k+1)^3\) (A000578)
CategoriesScalar-valued elements

Implementations

Symfem"Lagrange", variant="lobatto"
↓ Show Symfem examples ↓
Basixbasix.ElementFamily.P, ..., basix.LagrangeVariant.gll_warped
↓ Show Basix examples ↓

Examples

interval
order 1
interval
order 2
interval
order 3
interval
order 4
quadrilateral
order 1
quadrilateral
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 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:
\(\displaystyle l_{0}:v\mapsto v(0)\)

\(\displaystyle \phi_{0} = - \frac{9 x^{3}}{2} + 9 x^{2} - \frac{11 x}{2} + 1\)

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

\(\displaystyle \phi_{1} = \frac{x \left(9 x^{2} - 9 x + 2\right)}{2}\)

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

\(\displaystyle \phi_{2} = \frac{9 x \left(3 x^{2} - 5 x + 2\right)}{2}\)

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

\(\displaystyle \phi_{3} = \frac{9 x \left(- 3 x^{2} + 4 x - 1\right)}{2}\)

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: \(1\), \(x\), \(x^{2}\), \(x^{3}\), \(x^{4}\)
  • \(\mathcal{L}=\{l_0,...,l_{4}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0)\)

\(\displaystyle \phi_{0} = \frac{32 x^{4}}{3} - \frac{80 x^{3}}{3} + \frac{70 x^{2}}{3} - \frac{25 x}{3} + 1\)

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

\(\displaystyle \phi_{1} = \frac{x \left(32 x^{3} - 48 x^{2} + 22 x - 3\right)}{3}\)

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

\(\displaystyle \phi_{2} = \frac{16 x \left(- 8 x^{3} + 18 x^{2} - 13 x + 3\right)}{3}\)

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

\(\displaystyle \phi_{3} = 4 x \left(16 x^{3} - 32 x^{2} + 19 x - 3\right)\)

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

\(\displaystyle \phi_{4} = \frac{16 x \left(- 8 x^{3} + 14 x^{2} - 7 x + 1\right)}{3}\)

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 added10 February 2021
Element last updated28 September 2021