an encyclopedia of finite element definitions

# Serendipity

 Abbreviated names S Orders $$1\leqslant k$$ Reference elements interval, quadrilateral, hexahedron Polynomial set $$\mathcal{P}_{k} \oplus \mathcal{X}_{k}$$↓ Show polynomial set definitions ↓ DOFs On 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 DOFs interval: $$k+1$$ (A000027)quadrilateral: $$\begin{cases}4&k=1\\k(k+3)/2+3&k>1\end{cases}$$ (A340266)hexahedron: $$\begin{cases}12k-4&k=1,2,3\\3k^2-3k+14&k=4,5\\k(k-1)(k+1)/6+k^2+5k+4&k>6\end{cases}$$ Categories Scalar-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
order 1
order 2
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

• Arnold, D. N. and Awanou, G. The serendipity family of finite elements, Foundations of Computational Mathematics 11(3), 337–344, 2011. [DOI: 10.1007/s10208-011-9087-3] [BibTeX]