A class of nonconforming quadrilateral finite elements for incompressible flow

This paper focuses on the low-order nonconforming rectangular and quadrilateral finite elements approximation of incompressible flow. Beyond the previous research works, we propose a general strategy to construct the basis functions. Under several specific constraints, the optimal error estimates are obtained, i.e., the first order accuracy of the velocities in H1-norm and the pressure in L2-norm, as well as the second order accuracy of the velocities in L2-norm. Besides, we clarify the differences between rectangular and quadrilateral finite element approximation. In addition, we give several examples to verify the validity of our error estimates.