Degree 1 direct serendipity on a quadrilateral

• \(R\) 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.