# High accuracy analysis of a nonconforming discrete Stokes complex over rectangular meshes

## TL;DR

This paper develops a high-accuracy analysis of a nonconforming discrete Stokes complex on rectangular meshes, achieving superconvergence rates under certain conditions and verifying results with numerical tests.

## Contribution

It introduces a simple, low-degree nonconforming element for biharmonic problems that attains higher convergence rates on uniform rectangular meshes, with rigorous proofs and numerical validation.

## Key findings

- Achieves $O(h^2)$ convergence in discrete $H^2$-norm on uniform meshes.
- Attains $O(h^3)$ convergence in discrete $H^1$-norm for convex domains.
- Numerical tests confirm theoretical superconvergence results.

## Abstract

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.