High accuracy analysis of a nonconforming discrete Stokes complex over rectangular meshes
Provides rigorous high-order convergence analysis for a low-degree-of-freedom nonconforming element, benefiting computational mechanics and finite element method researchers.
This paper proves that a nonconforming plate element with only 12 degrees of freedom achieves O(h^2) convergence in discrete H^2-norm and O(h^3) in discrete H^1-norm on uniform rectangular meshes, surpassing standard consistency error estimates. Numerical tests confirm the theoretical results.
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.