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Serendipity

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Abbreviated namesS
Orders\(1\leqslant k\)
Reference elementsinterval, quadrilateral, hexahedron
Polynomial set\(\mathcal{P}_{k} \oplus \mathcal{X}_{k}\)
↓ Show polynomial set definitions ↓
DOFsOn each vertex: point evaluations
On each edge: integral moments with an order \(k-2\) dPc space
On each face: integral moments with an order \(k-4\) dPc space
On each volume: integral moments with an order \(k-6\) dPc space
Number of DOFsinterval: \(k+1\) (A000027)
quadrilateral: \(\begin{cases}4&k=1\\k(k+3)/2+3&k>1\end{cases}\) (A340266)
hexahedron: \(\begin{cases}12k-4&k=1,2,3\\3k^2-3k+14&k=4,5\\k(k-1)(k+1)/6+k^2+5k+4&k>6\end{cases}\)
CategoriesScalar-valued elements

Implementations

Symfem string"serendipity"
↓ Show Symfem examples ↓
Basix string"Serendipity"
↓ Show Basix examples ↓
UFL string"S"
↓ Show UFL examples ↓

Examples

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

\(\displaystyle \phi_{0} = 3 x^{2} - 4 x + 1\)

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

\(\displaystyle \phi_{1} = x \left(3 x - 2\right)\)

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

\(\displaystyle \phi_{2} = 6 x \left(1 - x\right)\)

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:
\(\displaystyle l_{0}:v\mapsto v(0)\)

\(\displaystyle \phi_{0} = - 10 x^{3} + 18 x^{2} - 9 x + 1\)

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

\(\displaystyle \phi_{1} = x \left(10 x^{2} - 12 x + 3\right)\)

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

\(\displaystyle \phi_{2} = 12 x \left(5 x^{2} - 8 x + 3\right)\)

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

\(\displaystyle \phi_{3} = 12 x \left(- 5 x^{2} + 7 x - 2\right)\)

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

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

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: \(1\), \(x\), \(x^{2}\), \(y\), \(x y\), \(y^{2}\), \(x y^{2}\), \(x^{2} y\)
  • \(\mathcal{L}=\{l_0,...,l_{7}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

\(\displaystyle \phi_{0} = - 3 x^{2} y + 3 x^{2} - 3 x y^{2} + 7 x y - 4 x + 3 y^{2} - 4 y + 1\)

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

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

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

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

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

\(\displaystyle \phi_{3} = x y \left(3 x + 3 y - 5\right)\)

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

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

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

\(\displaystyle \phi_{5} = 6 y \left(x y - x - y + 1\right)\)

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

\(\displaystyle \phi_{6} = 6 x y \left(1 - y\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{7}:v\mapsto\displaystyle\int_{e_{3}}(1)v\)

\(\displaystyle \phi_{7} = 6 x y \left(1 - x\right)\)

This DOF is associated with edge 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: \(1\), \(x\), \(x^{2}\), \(x^{3}\), \(y\), \(x y\), \(x^{2} y\), \(y^{2}\), \(x y^{2}\), \(y^{3}\), \(x y^{3}\), \(x^{3} y\)
  • \(\mathcal{L}=\{l_0,...,l_{11}\}\)
  • Functionals and basis functions:
\(\displaystyle l_{0}:v\mapsto v(0,0)\)

\(\displaystyle \phi_{0} = 10 x^{3} y - 10 x^{3} - 18 x^{2} y + 18 x^{2} + 10 x y^{3} - 18 x y^{2} + 17 x y - 9 x - 10 y^{3} + 18 y^{2} - 9 y + 1\)

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

\(\displaystyle \phi_{1} = x \left(- 10 x^{2} y + 10 x^{2} + 12 x y - 12 x - 10 y^{3} + 18 y^{2} - 11 y + 3\right)\)

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

\(\displaystyle \phi_{2} = y \left(- 10 x^{3} + 18 x^{2} - 10 x y^{2} + 12 x y - 11 x + 10 y^{2} - 12 y + 3\right)\)

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

\(\displaystyle \phi_{3} = x y \left(10 x^{2} - 12 x + 10 y^{2} - 12 y + 5\right)\)

This DOF is associated with vertex 3 of the reference element.
\(\displaystyle l_{4}:v\mapsto\displaystyle\int_{e_{0}}(1 - t_{0})v\)

\(\displaystyle \phi_{4} = 12 x \left(- 5 x^{2} y + 5 x^{2} + 8 x y - 8 x - 3 y + 3\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{5}:v\mapsto\displaystyle\int_{e_{0}}(t_{0})v\)

\(\displaystyle \phi_{5} = 12 x \left(5 x^{2} y - 5 x^{2} - 7 x y + 7 x + 2 y - 2\right)\)

This DOF is associated with edge 0 of the reference element.
\(\displaystyle l_{6}:v\mapsto\displaystyle\int_{e_{1}}(1 - t_{0})v\)

\(\displaystyle \phi_{6} = 12 y \left(- 5 x y^{2} + 8 x y - 3 x + 5 y^{2} - 8 y + 3\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{7}:v\mapsto\displaystyle\int_{e_{1}}(t_{0})v\)

\(\displaystyle \phi_{7} = 12 y \left(5 x y^{2} - 7 x y + 2 x - 5 y^{2} + 7 y - 2\right)\)

This DOF is associated with edge 1 of the reference element.
\(\displaystyle l_{8}:v\mapsto\displaystyle\int_{e_{2}}(1 - t_{0})v\)

\(\displaystyle \phi_{8} = 12 x y \left(5 y^{2} - 8 y + 3\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{9}:v\mapsto\displaystyle\int_{e_{2}}(t_{0})v\)

\(\displaystyle \phi_{9} = 12 x y \left(- 5 y^{2} + 7 y - 2\right)\)

This DOF is associated with edge 2 of the reference element.
\(\displaystyle l_{10}:v\mapsto\displaystyle\int_{e_{3}}(1 - t_{0})v\)

\(\displaystyle \phi_{10} = 12 x y \left(5 x^{2} - 8 x + 3\right)\)

This DOF is associated with edge 3 of the reference element.
\(\displaystyle l_{11}:v\mapsto\displaystyle\int_{e_{3}}(t_{0})v\)

\(\displaystyle \phi_{11} = 12 x y \left(- 5 x^{2} + 7 x - 2\right)\)

This DOF is associated with edge 3 of the reference element.

References