XueQin Yang

XiangYun Meng, XueQin Yang, Shuo Zhang

In this paper, we present the convergence analysis of the rectangular Morley element scheme utilised on the second order problem in arbitrary dimensions. Specifically, we prove that the convergence of the scheme is of $\mathcal{O}(h)$ order in energy norm and of $\mathcal{O}(h^2)$ order in $L^2$ norm on general $d$-rectangular grids. Moreover, when the grid is uniform, the convergence rate can be of $\mathcal{O}(h^2)$ order in energy norm, and the convergence rate in $L^2$ norm is still of $\mathcal{O}(h^2)$ order, which can not be improved. Numerical examples are presented to demonstrate our theoretical results.

Jun Hu, Zhongci Shi, Xueqin Yang

In the present paper, superconvergence of second order, after an appropriate postprocessing, is achieved for both the two and three dimensional first order rectangular Morley elements of biharmonic equations. The analysis is dependent on superconvergence of second order for the consistency error and a corrected canonical interpolation operator, which help to establish supercloseness of second order for the corrected canonical interpolation. Then the final superconvergence follows a standard postprocessing. For first order nonconforming finite element methods of both two and three dimensional fourth order elliptic problems, it is the first time that full superconvergence of second order is obtained without an extra boundary condition imposed on exact solutions. It is also the first time that superconvergence is established for nonconforming finite element methods of three dimensional fourth order elliptic problems. Numerical results are presented to demonstrate the theoretical results.

Jun Hu, Xueqin Yang, Shuo Zhang

This paper is devoted to the convergence analysis of the Adini element scheme for the fourth order problem in arbitrary dimension. We prove that, the Adini element scheme is $\mathcal{O}(h^2)$ order convergent in energy norm provided the exact solution is in $H^4$, and the convergence rate in $L^2$ norm can not be nontrivially higher than $\mathcal{O}(h^2)$ order. Numerical verifications are presented.

Jun Hu, Xueqin Yang

In this paper, we analyze the lower bound property of the discrete eigenvalues by the rectangular Morley elements of the biharmonic operators in both two and three dimensions. The analysis relies on an identity for the errors of eigenvalues. We explore a refined property of the canonical interpolation operators and use it to analyze the key term in this identity. In particular, we show that such a term is of higher order for two dimensions, and is negative and of second order for three dimensions, which causes a main difficulty. To overcome it, we propose a novel decomposition of the first term in the aforementioned identity. Finally, we establish a saturation condition to show that the discrete eigenvalues are smaller than the exact ones. We present some numerical results to demonstrate the theoretical results.