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Acta Mathematicae Applicatae Sinica 2020年第2期36卷 471-481页

作者： Hong-bo GUAN Dong-yang SHI College of Mathematics and Information Sciences Zhengzhou University of Light IndustryZhengzhou 450002China School of Mathematics and Statistics Zhengzhou UniversityZhengzhou 450001China

In this paper,nonconforming finite element methods(FEMs)are proposed for the constrained optimal control problems(OCPs)governed by the nonsmooth elliptic equations,in which the popular EQr1 ot element is employed to approximate the state and adjoint state,and the piecewise constant element is used to approximate the control.Firstly,the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary problem.Secondly,the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error estimate.Lastly,the methods are extended to some other well-known nonconforming elements.

关键词： nonconforming finite element supercloseness and superconvergence optimal control problems nonsmooth elliptic equations goal-oriented error estimate

nonconforming rotated Q_1 element on non-tensor product anisotropic meshes

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Science China Mathematics 2006年第10期49卷 1363-1375页

作者： MAO Shipeng SHI Zhongci Institute of Computational Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences PO Box 2719 Beijing 100080 China Institute of Computational Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences PO Box 2719 Beijing 100080 China

In this paper, we consider the nonconforming rotated Q1 element for the second order elliptic problem on the non-tensor 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.

关键词： anisotropic, interpolation error, consistency error, nonconforming rotated Q1 element.

nonconforming FEMs for the p-Laplace Problem

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Advances in Applied Mathematics and Mechanics 2018年第6期10卷 1365-1383页

作者： D.J.Liu A.Q.Li Z.R.Chen Department of Mathematics Shanghai UniversityShanghai 200444China

The p-Laplace problems in topology optimization eventually lead to a degenerate convex minimization problem E(v):=∫_(Ω)W(■v)dx-∫_(Ω)fvdx for v∈W_(0)^(1,p)(Ω)with unique minimizer u and stress s:=DW(ru).This paper proposes the discrete Raviart-Thomas mixed finite element method(dRT-MFEM)and establishes its equivalence with the Crouzeix-Raviart nonconforming finite element method(CR-NCFEM).The sharper quasi-norm a priori and a posteriori error estimates of this two methods are presented.Numerical experiments are provided to verify the analysis.

关键词： Adaptive finite element methods nonconforming p-Laplace problem dual energy

Axisymmetric nonconforming element method and its convergence analysis

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Science China Mathematics 1997年第4期40卷 422-431页

作者：陈万吉高岩 Department of Engineering Mechanics Dalian University of Technology Dalian 116023 China

By virtue of the weighted Sobolev space theory, three convergence tests of the axisymmetric non-conforming element method are established. They consist of the generalized patch test, the F-E-M test and a test which could be used conveniently, called the strong patch test (SPT). In the light of SPT, a class of axisymmetric nonconforming elements is established.

关键词： finite element nonconforming element axisymmetric convergence.

SUPERCONVERGENCE ANALYSIS FOR TIME-FRACTIONAL DIFFUSION EQUATIONS WITH nonconforming MIXED FINITE ELEMENT METHOD

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Journal of Computational Mathematics 2019年第4期37卷 488-505页

作者： Houchao Zhang Dongyang Shi School of Mathematics and Statistics Pingdingshan University Pingdingshan 467000 China School of Mathematics and Statistics Zhengzhou University Zhengzhou 450001 China

In this paper, a fully discrete scheme based on the LI approximation in temporal direction for the fractional derivative of order in (0,1) and nonconforming mixed finite element method (MFEM) in spatial direction is established. First, we prove a novel result of the consistency error estimate with order O(h^2)of EQ1^rot element (see Lemma 2.3). Then, by using the proved character of EQ1^rot element, we present the superconvergent estimates for the original variable u in the broken H^1-norm and the flux →p =△u in the (L^2)^2-norm under a weaker regularity of the exact solution. Finally, numerical results are provided to confirm the theoretical analysis.

关键词： nonconforming MFEM LI method Time-fractional diffusion equations Superconvergence

A Cubic H3-nonconforming Finite Element

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Communications on Applied Mathematics and Computation 2019年第1期1卷 81-100页

作者： Jun Hu Shangyou Zhang LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871.China Department of Mathematical Sciences University of DelawareNewarkDE 19716USA

The lowest degree of polynomial for a finite element to solve a 2^th-order elliptic equation is k.The Morley element is such a finite element,of polynomial degree 2,for solving a fourth-order biharmonic equation.We design a cubic H3-nonconforming macro-element on two-dimensional triangular grids,solving a sixth-order tri-harmonic equation.We also write down explicitly the 12 basis functions on each macro-element.A convergence theory is established and verified by numerical tests.

关键词： nonconforming macro-element Minimum element Tri-harmonic equation

Residual-based a posteriori error estimates of nonconforming finite element method for elliptic problems with Dirac delta source terms

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Science China Mathematics 2008年第8期51卷 1440-1460页

作者： DU ShaoHong XIE XiaoPing School of Mathematics Sichuan UniversityChengdu 610064China School of Science Chongqing Jiaotong UniversityChongqiong 400047China Yangtze Center of Mathematics Sichuan UniversityChengdu 610064China

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.

关键词： Crouzeix-Raviart element nonconforming FEM a posteriori error estimator longest edge bisection

nonconforming <i>H</i><sup>1</sup>-Galerkin Mixed Finite Element Method for Pseudo-Hyperbolic Equations

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American Journal of Computational Mathematics 2012年第4期2卷 269-273页

作者： Yadong Zhang Yuqi Niu Dongwei Shi Department of Mathematics Henan Institute of Science and Technology Xinxiang China School of Mathematics and Statistics Xuchang University Xuchang China

Based on H1-Galerkin mixed finite element method with nonconforming quasi-Wilson element, a numerical approximate scheme is established for pseudo-hyperbolic equations under arbitrary quadrilateral meshes. The corresponding optimal order error estimate is derived by the interpolation technique instead of the generalized elliptic projection which is necessary for classical error estimates of finite element analysis.

关键词： Pseudo-Hyperbolic Equation nonconforming H1-Galerkin Mixed Finite Element Error Estimate

nonconforming Mixed Finite Element Method for Nonlinear Hyperbolic Equations

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Applied Mathematics 2012年第3期3卷 231-234页

作者： Haihong Wang Cheng Guo Department of Mathematical and Information Scientific Henan University of Economics and Low Zhengzhou China Department of Mathematics Zhengzhou Normal University Zhengzhou China

A nonconforming mixed finite element method for nonlinear hyperbolic equations is discussed. Existence and uniqueness of the solution to the discrete problem are proved. Priori estimates of optimal order are derived f... 详细信息

关键词： nonconforming Mixed Finite Element Hyperbolic Equations Semi-Discrete Scheme Error Estimates

CONVERGENCE OF THE nonconforming WILSON'S BRICK FOR ARBITRARY HEXAHEDRAL MESHES

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Numerical Mathematics A Journal of Chinese Universities(English Series) 1997年第2期6卷 184-199页

作者：刘发旺 Conor J. Fitzsimons John J.H.Miller Numerical Analysis Group Department of MathematicsTrinity CollegeDublinIreland.

The nonconforming Wilson’s brick classically is restricted to regular hexahedral meshes. Lesaint and Zlamal[6] relaxed this constraint for the two-dimensional analonue of this element In this paper we extend their results to three dimensions and prove that and where u is the exact solution, u_h is the approximate solution and is the usual norm for the Sobolev space H^1(?).

关键词： Convergence nonconforming Wilson’s brick hexahedral meshes.