an encyclopedia of finite element definitions

# Direct serendipity

 Orders $$1\leqslant k$$ Reference elements quadrilateral Categories Scalar-valued elements

## Implementations

 Symfem "direct serendipity"↓ Show Symfem examples ↓ UFL "Sdirect"↓ Show UFL examples ↓

## Examples

order 1
order 2
order 3
• $$R$$ is the reference quadrilateral. The following numbering of the subentities of the reference is used:
• Basis functions:
$$\displaystyle \phi_{0} = x y - x - y + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle \phi_{1} = x \left(1 - y\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle \phi_{2} = y \left(1 - x\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle \phi_{3} = x y$$

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:
• Basis functions:
$$\displaystyle \phi_{0} = x y - x - y + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle \phi_{1} = x \left(1 - y\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle \phi_{2} = y \left(1 - x\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle \phi_{3} = x y$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle \phi_{4} = y \left(1 - y\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle \phi_{5} = \frac{4 x y \left(y - 1\right)}{x + 1}$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle \phi_{6} = x \left(1 - x\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle \phi_{7} = \frac{4 x y \left(x - 1\right)}{y + 1}$$

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:
• Basis functions:
$$\displaystyle \phi_{0} = x y - x - y + 1$$

This DOF is associated with vertex 0 of the reference element.
$$\displaystyle \phi_{1} = x \left(1 - y\right)$$

This DOF is associated with vertex 1 of the reference element.
$$\displaystyle \phi_{2} = y \left(1 - x\right)$$

This DOF is associated with vertex 2 of the reference element.
$$\displaystyle \phi_{3} = x y$$

This DOF is associated with vertex 3 of the reference element.
$$\displaystyle \phi_{4} = y \left(1 - y\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle \phi_{5} = y \left(- y^{2} - y + 2\right)$$

This DOF is associated with edge 1 of the reference element.
$$\displaystyle \phi_{6} = y \left(x y - x - 2 y + 2\right)$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle \phi_{7} = \frac{4 x y \left(y^{2} + y - 2\right)}{x + 1}$$

This DOF is associated with edge 2 of the reference element.
$$\displaystyle \phi_{8} = x \left(1 - x\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle \phi_{9} = x \left(- x^{2} - x + 2\right)$$

This DOF is associated with edge 0 of the reference element.
$$\displaystyle \phi_{10} = x \left(x y - 2 x - y + 2\right)$$

This DOF is associated with edge 3 of the reference element.
$$\displaystyle \phi_{11} = \frac{4 x y \left(x^{2} + x - 2\right)}{y + 1}$$

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

## References

• Arbogast, T. and Tao, Z. Direct serendipity and mixed finite elements on convex quadrilaterals, arΧiv: 1809.02192, 2018. [BibTeX]

## DefElement stats

 Element added 27 May 2021 Element last updated 13 June 2021