1.

双线形元的各向异性后验误差估计





尹丽 职桂珍《数学季刊》,2007年第22卷第4期


The main aim of this paper is to give an anisotropic posteriori error estimator. We firstly study the convergence of bilinear finite element for the second order problem under anisotropic meshes.By using some novel approaches and techniques,the optimal error estimates and some superconvergence results are obtained without the regularity assumption and quasiuniform assumption requirements on the meshes.Then,based on these results, we give an anisotropic posteriori error estimate for the second problem.

2.

CONVERGENCE ANALYSIS FOR A NONCONFORMING MEMBRANE ELEMENT ON ANISOTROPIC MESHES 被引次数：14





Dongyang Shi Shaochun Chen Ichiro Hagiwara《计算数学(英文版)》,2005年第23卷第4期


Regular assumption of finite element meshes is a basic condition of most analysis offinite element approximations both for conventional conforming elements and nonconforming elements.The aim of this paper is to present a novel approach of dealing with theapproximation of a fourdegree nonconforming finite element for the second order ellipticproblems on the anisotropic meshes.The optimal error estimates of energy norm and L~2norm without the regular assumption or quasiuniform assumption are obtained based onsome new special features of this element discovered herein.Numerical results are givento demonstrate validity of our theoretical analysis.

3.

CONVERGENCE OF A MIXED FINITE ELEMENT FOR THE STOKES PROBLEM ON ANISOTROPIC MESHES





Qingshan Li ;Huixia Sun ;Shaochun Chen《计算数学(英文版)》,2008年第5期


The main aim of this paper is to study the convergence properties of a low order mixed finite element for the Stokes problem under anisotropic meshes. We discuss the anisotropic convergence and superconvergence independent of the aspect ratio. Without the shape regularity assumption and inverse assumption on the meshes, the optimal error estimates and natural superconvergence at central points are obtained. The global superconvergence for the gradient of the velocity and the pressure is derived with the aid of a suitable postprocessing method. Furthermore, we develop a simple method to obtain the superclose properties which improves the results of the previous works .

4.

A lockingfree anisotropic nonconforming rectangular finite element approximation for the planar elasticity problem





SHI Dongyang~1 WANG Caixia~2 1 Dept.of Math. Zhengzhou Univ. Zhengzhou 450052 China. 2 Faculty of Math.and Inform.Sci. North China Univ.of Water Conservancy and Electric Power Zhengzhou 450011 China.《高校应用数学学报(英文版)》,2008年第23卷第1期


This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2norm are obtained. The restrictions of regularity assumption and quasiuniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.

5.

ON THE ANISOTROPIC ACCURACY ANALYSIS OF ACMES NONCONFORMING FINITE ELEMENT





Dongyang Shi Shipeng Mao Shaochun Chen《计算数学(英文版)》,2005年第23卷第6期


The main aim of this paper is to study the superconvergence accuracy analysis of the famous ACM＇s nonconforming finite element for biharmonic equation under anisotropic meshes. By using some novel approaches and techniques, the optimal anisotropic interpolation error and consistency error estimates are obtained. The global error is of order O（h^2）. Lastly, some numerical tests are presented to verify the theoretical analysis.

6.

Nonconforming rotated Q_1 element on nontensor product anisotropic meshes





MAO Shipeng & SHI Zhongci Institute of Computational Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences PO Box 2719 Beijing 100080 China《中国科学A辑(英文版)》,2006年第49卷第10期


In this paper, we consider the nonconforming rotated Q1 element for the second order elliptic problem on the nontensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.

7.

Nonconforming <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>Galerkin mixed FEM for Sobolev equations on anisotropic meshes





Dongyang Shi Haihong Wang《应用数学学报(英文版)》,2009年第25卷第2期


A nonconforming H^1Calerkin mixed finite element method is analyzed for Sobolev equations on anisotropic meshes. The error estimates are obtained without using RitzVolterra projection.

8.

EQ 1 rot nonconforming finite element approximation to Signorini problem





SHI DongYang XU Chao《中国科学 数学(英文版)》,2013年第56卷第6期


In this paper, we apply EQ rot 1 nonconforming finite element to approximate Signorini problem. If the exact solution u∈H5/2(Ω), the error estimate of order O(h) about the broken energy norm is obtained for quadrilateral meshes satisfying regularity assumption and bisection condition. Furthermore, the superconvergence results of order O(h3/2) are derived for rectangular meshes. Numerical results are presented to confirm the considered theory.

9.

Superconvergence of a Nonconforming Finite Element Approximation to Viscoelasticity Type Equations on Anisotropic Meshes 被引次数：3





Dongyang Shi Yucheng Peng Shaochun Chen《高等学校计算数学学报(英文版)》,2006年第15卷第4期


The main aim of this paper is to study the approximation to viscoelasticity type equations with a CrouzeixRaviart type nonconforming finite element on the anisotropic meshes. The superclose property of the exact solution and the optimal error estimate of its derivative with respect to time are derived by using some novel techniques. Moreover, employing a postprocessing technique, the global superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself is studied.

10.

CONVERGENCE AND SUPERCONVERGENCE ANALYSIS OF LAGRANGE RECTANGULAR ELEMENTS WITH ANY ORDER ON ARBITRARY RECTANGULAR MESHES





Mingxia Li Xiaofei Guan Shipeng Mao《计算数学(英文版)》,2014年第2期


This paper is to study the convergence and superconvergence of rectangular finite elements under anisotropic meshes. By using of the orthogonal expansion method, an anisotropic Lagrange interpolation is presented. The family of Lagrange rectangular elements with all the possible shape function spaces are considered, which cover the Intermediate families, Tensorproduct families and Serendipity families. It is shown that the anisotropic interpolation error estimates hold for any order Sobolev norm. We extend the convergence and superconvergence result of rectangular finite elements to arbitrary rectangular meshes in a unified way.

11.

Superconvergence analysis of lower order anisotropic finite element





朱国庆 石东洋 陈绍春《应用数学和力学(英文版)》,2007年第28卷第8期


The convergence analysis of the lower order nonconforming element pro posed by Park and Sheen is applied to the secondorder elliptic problem under anisotropic meshes.The corresponding error estimation is obtained.Moreover,by using the interpo lation postprocessing technique,a global superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself is derived.Numerical results are also given to verify the theoretical analysis.

12.

Anisotropic rectangular nonconforming finite element analysis for Sobolev equations





石东洋 王海红 郭城《应用数学和力学(英文版)》,2008年第29卷第9期


An anisotropic rectangular nonconforming finite element method for solving the Sobolev equations is discussed under semidiscrete and full discrete schemes. The corresponding optimal convergence error estimates and superclose property are derived, which are the same as the traditional conforming finite elements. Furthermore, the global superconvergence is obtained using a postprocessing technique. The numerical results show the validity of the theoretical analysis.

13.

A NONCONFORMING QUADRILATERAL FINITE ELEMENT APPROXIMATION TO NONLINEAR SCHRDINGER EQUATION 被引次数：1





Dongyang SHI Xin LIAO Lele WANG《数学物理学报(B辑英文版)》,2017年第37卷第3期


In this article, a nonconforming quadrilateral element(named modified quasiWilson element) is applied to solve the nonlinear schr¨odinger equation(NLSE). On the basis of a special character of this element, that is, its consistency error is of order O(h~3) for broken H1norm on arbitrary quadrilateral meshes, which is two order higher than its interpolation error, the optimal order error estimate and superclose property are obtained. Moreover,the global superconvergence result is deduced with the help of interpolation postprocessing technique. Finally, some numerical results are provided to verify the theoretical analysis.

14.

ON THE ANISOTROPIC ACCURACY ANALYSIS OF ACM'S NONCONFORMING FINITE ELEMENT 被引次数：6





Dongyang Shi Shipeng Mao Shaochun Chen Department of Mathematics Zhengzhou University Zhengzhou 450052 China《计算数学(英文版)》,2005年第6期


The main aim of this paper is to study the superconvergence accuracy analysis of thefamous ACM's nonconforming finite element for biharmonic equation under anisotropicmeshes. By using some novel approaches and techniques, the optimal anisotropic interpolation error and consistency error estimates are obtained. The global error is of orderO(h~2). Lastly, some numerical tests are presented to verify the theoretical analysis.

15.

RECOVERY A POSTERIORI ERROR ESTIMATES FOR GENERAL CONVEX ELLIPTIC OPTIMAL CONTROL PROBLEMS SUBJECT TO POINTWISE CONTROL CONSTRAINTS





Yanping Chen Yao Fu Huanwen Liu Yongquan Dai Huayi Wei《计算数学(英文版)》,2009年第4期


Superconvergence and recovery a posteriori error estimates of the finite element ap proximation for general convex optimal control problems are investigated in this paper. We obtain the superconvergence properties of finite element solutions, and by using the superconvergence results we get recovery a posteriori error estimates which are asymptotically exact under some regularity conditions. Some numerical examples are provided to verify the theoretical results.

16.

Superconvergence analysis of splitting positive definite nonconforming mixed finite element method for pseudohyperbolic equations





Dongyang Shi Qili Tang《应用数学学报(英文版)》,2013年第29卷第4期


In this paper, a new splitting positive definite nonconforming mixed finite element method is proposed for pseudohyperbolic equations, in which a quasiWilson quadrilateral element is used for the flux p, and the bilinear element is used for u. Superconvergence results in ｜｜·｜｜div,h norm for p and optimal error estimates in L2 norm for u are derived for both semidiscrete and fully discrete schemes under almost uniform meshes.

17.

AN ANISOTROPIC NONCONFORMING FINITE ELEMENT METHOD FOR APPROXIMATING A CLASS OF NONLINEAR SOBOLEV EQUATIONS





Dongyang Shi Haihong Wang Yuepeng Du《计算数学(英文版)》,2009年第2期


An anisotropic nonconforming finite element method is presented for a class of nonlinear Sobolev equations. The optimal error estimates and supercloseness are obtained for both semidiscrete and fullydiscrete approximate schemes, which are the same as the traditional finite element methods. In addition, the global superconvergence is derived through the postprocessing technique. Numerical experiments are included to illustrate the feasibility of the proposed method.

18.

一类非正则网格上的非协调Mortar元的高精度分析





WU Jingzhu SHI Dongyang 《数学季刊》,2005年第20卷第3期


In this paper, the nonconforming mortar finite element with a class of meshes is studied without considering the global regularity condition or quasiuniformly assumption. Meanwhile, the superclose result coincides with conventional methods is obtained by means of integral identities techniques.

19.

Nonconforming stabilized combined finite element method for ReissnerMindlin plate 被引次数：1





冯民富 杨艳 周天孝《应用数学和力学(英文版)》,2009年第30卷第2期


Based on combination of two variational principles, a nonconforming stabilized finite element method is presented for the ReissnerMindlin plates. The method is convergent when the finite element space is energycompatible. Error estimates are derived. In particular, three finite element spaces are applied in the computation. Numerical results show that the method is insensitive to the mesh distortion and has better performence than the MITC4 and DKQ methods. With properly chosen parameters, high accuracy can be obtained at coarse meshes.

20.

Superconvergence and recovery type a posteriori error estimation for hybrid stress finite element method





YanHong Bai YongKe Wu XiaoPing Xie《中国科学 数学(英文版)》,2016年第59卷第9期


Superconvergence and recovery type a posteriori error estimators are analyzed for Pian and Sumihara's 4node hybrid stress quadrilateral finite element method for linear elasticity problems. Superconvergence of order O(h~(1+min){α,1}) is established for both the displacement approximation in H~1norm and the stress approximation in L~2norm under a mesh assumption, where α 0 is a parameter characterizing the distortion of meshes from parallelograms to quadrilaterals. Recovery type approximations for the displacement gradients and the stress tensor are constructed, and a posteriori error estimators based on the recovered quantities are shown to be asymptotically exact. Numerical experiments confirm the theoretical results.
