an encyclopedia of finite element definitions

# Degree 3 direct serendipity on a quadrilateral

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In this example:
• $$R$$ 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.