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Radau

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

Implementations

Symfem"Lagrange", variant="radau"
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Examples

interval
order 1
interval
order 2
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 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:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

\(\displaystyle \phi_{0} = \frac{4 \sqrt{6} x^{2} y^{2}}{3} + \frac{14 x^{2} y^{2}}{3} - \frac{20 x^{2} y}{3} - \frac{5 \sqrt{6} x^{2} y}{3} + \frac{\sqrt{6} x^{2}}{3} + 2 x^{2} - \frac{20 x y^{2}}{3} - \frac{5 \sqrt{6} x y^{2}}{3} + 2 \sqrt{6} x y + \frac{29 x y}{3} - 3 x - \frac{\sqrt{6} x}{3} + \frac{\sqrt{6} y^{2}}{3} + 2 y^{2} - 3 y - \frac{\sqrt{6} y}{3} + 1\)

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

\(\displaystyle \phi_{1} = - \frac{2 \sqrt{6} x^{2} y^{2}}{3} + 6 x^{2} y^{2} - 10 x^{2} y + \frac{5 \sqrt{6} x^{2} y}{3} - \sqrt{6} x^{2} + 4 x^{2} - 4 x y^{2} + \sqrt{6} x y^{2} - 2 \sqrt{6} x y + 7 x y - 3 x + \sqrt{6} x\)

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

\(\displaystyle \phi_{2} = - \frac{2 \sqrt{6} x^{2} y^{2}}{3} + 6 x^{2} y^{2} - 4 x^{2} y + \sqrt{6} x^{2} y - 10 x y^{2} + \frac{5 \sqrt{6} x y^{2}}{3} - 2 \sqrt{6} x y + 7 x y - \sqrt{6} y^{2} + 4 y^{2} - 3 y + \sqrt{6} y\)

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

\(\displaystyle \phi_{3} = - 8 \sqrt{6} x^{2} y^{2} + 22 x^{2} y^{2} - 18 x^{2} y + 7 \sqrt{6} x^{2} y - 18 x y^{2} + 7 \sqrt{6} x y^{2} - 6 \sqrt{6} x y + 15 x y\)

This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{4}:v\mapsto v(\tfrac{3}{5} - \tfrac{\sqrt{6}}{10},0)\)

\(\displaystyle \phi_{4} = - \frac{32 x^{2} y^{2}}{3} - \frac{2 \sqrt{6} x^{2} y^{2}}{3} + \frac{50 x^{2} y}{3} - 6 x^{2} + \frac{2 \sqrt{6} x^{2}}{3} + \frac{2 \sqrt{6} x y^{2}}{3} + \frac{32 x y^{2}}{3} - \frac{50 x y}{3} - \frac{2 \sqrt{6} x}{3} + 6 x\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{5}:v\mapsto v(0,\tfrac{3}{5} - \tfrac{\sqrt{6}}{10})\)

\(\displaystyle \phi_{5} = - \frac{32 x^{2} y^{2}}{3} - \frac{2 \sqrt{6} x^{2} y^{2}}{3} + \frac{2 \sqrt{6} x^{2} y}{3} + \frac{32 x^{2} y}{3} + \frac{50 x y^{2}}{3} - \frac{50 x y}{3} - 6 y^{2} + \frac{2 \sqrt{6} y^{2}}{3} - \frac{2 \sqrt{6} y}{3} + 6 y\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{6}:v\mapsto v(1,\tfrac{3}{5} - \tfrac{\sqrt{6}}{10})\)

\(\displaystyle \phi_{6} = - 28 x^{2} y^{2} + \frac{26 \sqrt{6} x^{2} y^{2}}{3} - \frac{26 \sqrt{6} x^{2} y}{3} + 28 x^{2} y - 8 \sqrt{6} x y^{2} + 22 x y^{2} - 22 x y + 8 \sqrt{6} x y\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{7}:v\mapsto v(\tfrac{3}{5} - \tfrac{\sqrt{6}}{10},1)\)

\(\displaystyle \phi_{7} = - 28 x^{2} y^{2} + \frac{26 \sqrt{6} x^{2} y^{2}}{3} - 8 \sqrt{6} x^{2} y + 22 x^{2} y - \frac{26 \sqrt{6} x y^{2}}{3} + 28 x y^{2} - 22 x y + 8 \sqrt{6} x y\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{8}:v\mapsto v(\tfrac{3}{5} - \tfrac{\sqrt{6}}{10},\tfrac{3}{5} - \tfrac{\sqrt{6}}{10})\)

\(\displaystyle \phi_{8} = - 8 \sqrt{6} x^{2} y^{2} + \frac{116 x^{2} y^{2}}{3} - \frac{116 x^{2} y}{3} + 8 \sqrt{6} x^{2} y - \frac{116 x y^{2}}{3} + 8 \sqrt{6} x y^{2} - 8 \sqrt{6} x y + \frac{116 x y}{3}\)

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

DefElement stats

Element added20 February 2021
Element last updated03 July 2021