an encyclopedia of finite element definitions

# Degree 2 direct serendipity on a quadrilateral

◀ Back to direct serendipity definition page In this example: $$\displaystyle \phi_{0} = x y - x - y + 1$$

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.