an encyclopedia of finite element definitions

Transition

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Orders\(1\leqslant k\)
Reference elementstriangle, tetrahedron
DOFsOn each vertex: point evaluations
On each edge: point evaluations
On each face: point evaluations
On each volume: point evaluations
NotesThis element is used to bridge the gap between Lagrange elements of different orders
CategoriesScalar-valued elements

Implementations

Symfem string"transition"
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Examples

triangle
order 1
edge_orders=[2, 1, 1]
triangle
order 1
edge_orders=[3, 2, 1]
triangle
order 3
edge_orders=[1, 1, 1]
  • \(R\) is the reference triangle. The following numbering of the subentities of the reference is used:
  • \(\mathcal{V}\) is spanned by: \(1\), \(x\), \(y\), \(x y\)
  • \(\mathcal{L}=\{l_0,...,l_{3}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

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

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

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

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

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

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

\(\displaystyle \phi_{3} = 4 x y\)

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: \(1\), \(x\), \(y\), \(x y \left(1 - y\right)\), \(x y^{2}\), \(y \left(- x - y + 1\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{5}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

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

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

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

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

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

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

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

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

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

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

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

This DOF is associated with edge 1 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: \(1\), \(x\), \(y\), \(x y \left(- x - y + 1\right)\)
  • \(\mathcal{L}=\{l_0,...,l_{3}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

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

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

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

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

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

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

\(\displaystyle \phi_{3} = 27 x y \left(- x - y + 1\right)\)

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