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nonconforming Finite Element Methods for the Constrained Optimal Control Problems Governed by Nonsmooth Elliptic Equations

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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 ***,the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary ***,the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error ***,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 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 *** experiments are provided to verify the analysis.

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

SUPERCONVERGENCE ANALYSIS OF A nonconforming TRIANGULAR ELEMENT ON ANISOTROPIC MESHES

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Journal of Systems Science & Complexity 2007年第4期20卷 536-544页

作者： Dongyang SHI Hui LIANG Caixia WANG Department of Mathematics Zhengzhou University Zhengzhou 450052 China Department of Mathematics Harbin Institute of Technology Harbin 150001 China

The class of anisotropic meshes we conceived abandons the regular assumption. Some distinct properties of Carey＇s element are used to deal with the superconvergence for a class of two- dimensional second-order elliptic boundary value problems on anisotropic meshes. The optimal results are obtained and numerical examples are given to confirm our theoretical analysis.

关键词： Anisotropic meshes Carey's element nonconforming superclose superconvergence.

A Posterior Error Analysis for the nonconforming Discretization of Stokes Eigenvalue Problem

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Acta Mathematica Sinica,English Series 2014年第6期30卷 949-967页

作者： Shang Hui JIA Fu Sheng LUO He Hu XIE School of Statistic and Mathematics Central University of Finance and Economics The Third Institute of Oceanography SOA LSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences

In this paper, we present a posteriori error estimator for the nonconforming finite element approximation, including using Crouzeix–Raviart element and extended Crouzeix–Raviart element, of the Stokes eigenvalue problem. With the technique of Helmholtz decomposition, we first give out a posteriori error estimator and prove it as the global upper bound and local lower bound of the approximation error. Then, by deleting a jump term in the indicator, another simpler but equivalent indicator is obtained. Some numerical experiments are provided to verify our analysis.

关键词： A posteriori error estimate adaptive finite element method nonconforming Stokes eigen-value problem

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

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.

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 *** Morley element is such a finite element,of polynomial degree 2,for solving a fourth-order biharmonic *** design a cubic H3-nonconforming macro-element on two-dimensional triangular grids,solving a sixth-order tri-harmonic *** 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

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

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