A second-order convergent nonconforming polynomial stokes element on quadrilateral meshes

In this paper, we construct a new mixed finite element for the Stokes problem on general convex quadrilateral partitions. The velocity is approximated by piecewise polynomial element space, and the pressure is approximated by piecewise constant. Moreover, we assert that the discrete velocity is second-order convergent in discrete \(H^{1}\) $H^1$ seminorm, and the convergence order of the pressure solution can be improved to second by a post-processing for Stokes problems. Lastly, numerical tests verify the convergence analysis.