  This work is devoted to the high accuracy analysis of a discrete Stokes
complex over rectangular meshes with a simple structure. The 0-form in the
complex is a non $C^0$ nonconforming element space for biharmonic problems.
This plate element contains only 12 degrees of freedom (DoFs) over a
rectangular cell with a zeroth order weak continuity for the normal derivative,
therefore only the lowest convergence order can be obtained by a standard
consistency error analysis. Nevertheless, we prove that, if the rectangular
mesh is uniform, an $O(h^2)$ convergence rate in discrete $H^2$-norm will be
achieved. Moreover, based on a duality argument, it has an $O(h^3)$ convergence
order in discrete $H^1$-norm if the solution region is convex. The 1-form and
2-form constitute a divergence-free pair for incompressible flow. We also show
its higher accuracy than that derived from a usual error estimate under uniform
partitions, which explains the phenomenon observed in our previous work.
Numerical tests verify our theoretical results.

Numerical Analysis