Xinchen Zhou

Xinchen Zhou, Zhaoliang Meng, Xin Fan et al.

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.

Xinchen Zhou, Zhaoliang Meng, Xin Fan et al.

This work introduces two 11-node triangular prism elements for 3D elliptic problems. The degrees of freedom (DoFs) of both elements are at the vertices and face centroids of a prism cell. The first element is $H^1$-nonconforming and works for second order problems, which achieves a second order convergence rate in discrete $H^1$-norm. The other is $H^2$-nonconforming and solves fourth order problems, with a first order convergence rate in discrete $H^2$-norm. Numerical examples verify our theoretical results.

Xinchen Zhou, Zhaoliang Meng, Xin Fan et al.

This work proposes two nodal type nonconforming finite elements over convex quadrilaterals, which are parts of a finite element exact sequence. Both elements are of 12 degrees of freedom (DoFs) with polynomial shape function spaces selected. The first one is designed for fourth order elliptic singular perturbation problems, and the other works for Brinkman problems. Numerical examples are also provided.