Two residual-based a posteriori error estimators of the nonconforming Crouzeix-Raviart element are derived for elliptic problems with Dirac delta source terms. One estimator is shown to be reliable and efficient, which yields global upper and lower bounds for the error in piecewise W1,p- seminorm. The other one is proved to give a global upper bound of the error in Lp-norm. By taking the two estimators as refinement indicators, adaptive algorithms are suggested, which are experimentally shown to attain optimal convergence orders.

In this paper, a combined hybrid method is appplied to finite element discretization of plate bending problems. It is shown that the resultant schemes are stabilized, i.e., the convergence of the schemes is independent of inf-sup conditions and any other patch test. Based on this, two new series of plate elements are proposed.

Several methods have been developed in the literatures of computational mechanics to improve the performance of the conventional lower-order displacement finite elements which yield poor results for problems with bending and for nearly incompressible medium. This paper is devoted to a unified analysis of convergence for Pian–Sumiharas, Chen-Cheungs and Piltner-Taylor s enhanced stress/strain schemes. By virtue of the energy compatibility and the rank condition, error estimates for these typical finite elements of high prformance are obtained in a unified framework, and especially, weakly locking-free error estimates with respect to the Poisson s ratio m in energy norms are obtained uniformly for v ≤ (1-Ch) =2 as h → 0, where C is a constant independent of m and the mesh size h. Very much the same about the three methods is pointed out.

The combined hybrid nite element method for plate bending problems allows arbitrary combinations f de ection interpolation and bending moment approximations. A novel expression of the approach discloses the energy-adjustable mechanism of the hybrid variational principle to enhance accuracy and stability of displacement-based nite element models. For a given displacement approximation, appropriate choices of the bending moment mode and the combination parameter ∈(0; 1) can lead to accurate energy approximation which generally yields numerically high accuracy of the displacement and bending moment approximations. By virtue of this mechanism, improvement of Zienkiewicz' s triangular plate-element is discussed. The de ection is approximated by Zienkiewicz incomplete cubic interpolation. And three kinds of bending moments approximations are considered: a 3-parameter constant mode, a 5-parameter incomplete linear mode, and a 9-parameter linear mode. Since the parameters of the assumed bending moments modes can be eliminated at an element level, the computational cost of the combined hybrid counterparts of Zienkiewicz' s triangle are as same as that of Zienkiewicz' s triangle. Numerical experiments show that the combined hybrid versions can attain high accuracy at coarse meshes.

从克服Locking发散性到高性能格式的研究发展是过去十年有限元法研究的一个重要方面。本文认为所谓高性能有限元方法是一种特别的低阶位移工，它具有如下的理论和计算优点：（1）保持物理力学问题固有的数学物理特征；（2）因此，应用于工程数值模拟，它是“鲁棒”的、高明效的。按此立场，评论介绍了在计算结构力学和计算流体力学中发展的克服两类Locking发散性的有限方法