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Acta Mathematica Scientia 2003年第3期23卷 386-398页

作者：陈蔚 Academy of Mathematics System science CASBeijing 100080China School of EconomicsShandong UniversityJinan 250100China

The transient behavior of a semiconductor device consists of a Poisson equation for the electric potential and of two nonlinear parabolic equations for the electron density and hole density. The electric potential equation is discretized by a mixed finite element method. The electron and hole density equations are treated by implicit-explicit multistep finite element methods. The schemes are very efficient. The optimal order error estimates both in time and space are derived.

关键词： Semiconductor device strongly A(0)-stable multistep methods finite element methods mixed finite element methods

A Posteriori Error Estimates of Triangular mixed finite element methods for Semilinear Optimal Control Problems

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Advances in Applied Mathematics and Mechanics 2009年第2期1卷 242-256页

作者： Zuliang Lu Yanping Chen Hunan Key Laboratory for Computation and Simulation in Science and Engineering Department of MathematicsXiangtan UniversityXiangtan 411105HunanP.R.China School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongP.R.China

In this paper,we present an a posteriori error estimates of semilinear quadratic constrained optimal control problems using triangular mixed finite element methods.The state and co-state are approximated by the order k≤1 RaviartThomas mixed finite element spaces and the control is approximated by piecewise constant element.We derive a posteriori error estimates for the coupled state and control approximations.A numerical example is presented in confirmation of the theory.

关键词： Semilinear optimal control problems mixed finite element methods a posteriori error estimates

Error Estimates and Superconvergence of mixed finite element methods for Optimal Control Problems with Low Regularity

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Advances in Applied Mathematics and Mechanics 2012年第6期4卷 751-768页

作者： Yanping Chen Tianliang Hou Weishan Zheng School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongChina Hunan Key Laboratory for Computation and Simulation in Science and Engineering Department of MathematicsXiangtan UniversityXiangtan 411105HunanChina

In this paper,we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems.The state and co-state are approximated by the lowest order Raviart-Thomas mixed fi-nite element spaces and the control variable is approximated by piecewise constant functions.We derive L^(2) and L^(∞)-error estimates for the control variable.Moreover,using a recovery operator,we also derive some superconvergence results for the control variable.Finally,a numerical example is given to demonstrate the theoretical results.

关键词： Elliptic equations optimal control problems superconvergence error estimates mixed finite element methods

Superconvergence of Rectangular mixed finite element methods for Constrained Optimal Control Problem

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Advances in Applied Mathematics and Mechanics 2010年第1期2卷 56-75页

作者： Yanping Chen Li Dai Zuliang Lu School of Mathematical Sciences South China Normal UniversityGuangzhouGuangdongChina Mathematics Teaching and Research Group Xiguan Foreign Language SchoolGuangzhouGuangdongChina Hunan Key Laboratory for Computation and Simulation in Science and Engineering Department of MathematicsXiangtan UniversityXiangtanHunanChina

We investigate the superconvergence properties of the constrained quadratic elliptic optimal control problem which is solved by using rectangular mixed finite element methods.We use the lowest order Raviart-Thomas mixed finite element spaces to approximate the state and co-state variables and use piecewise constant functions to approximate the control variable.We obtain the superconvergence of O(h^(1+s))(0

关键词： Constrained optimal control problem linear elliptic equation mixed finite element methods rectangular partition superconvergence properties

Comparison of finite Difference and mixed finite element methods for Perfectly Matched Layer Models

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Communications in Computational Physics 2007年第4期2卷 806-826页

作者： V.A.Bokil M.W.Buksas Department of Mathematics Oregon State UniversityCorvallisOR 97331USA. CCS-4 Mail Stop D409Los Alamos National LaboratoryNM 87544USA

We consider the anisotropic uniaxial formulation of the perfectly matched layer(UPML)model for Maxwell’s equations in the time domain.We present and analyze a mixed finite element method for the discretization of the UPML in the time domain to simulate wave propagation on unbounded domains in two dimensions.On rectangles the spatial discretization uses bilinear finite elements for the electric field and the lowest order Raviart-Thomas divergence conforming elements for the magnetic field.We use a centered finite difference method for the time discretization.We compare the finite element technique presented to the finite difference time domain method(FDTD)via a numerical reflection coefficient analysis.We derive the numerical reflection coefficient for the case of a semi-infinite PML layer to show consistency between the numerical and continuous models,and in the case of a finite PML to study the effects of terminating the absorbing layer.Finally,we demonstrate the effectiveness of the mixed finite element scheme for the UPML by a numerical example and provide comparisons with the split field PML discretized by the FDTD method.In conclusion,we observe that the mixed finite element scheme for the UPML model has absorbing properties that are comparable to the FDTD method.

关键词： Perfectly matched layers mixed finite element methods FDTD Maxwell’s equations.

Higher Order Triangular mixed finite element methods for Semilinear Quadratic Optimal Control Problems

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Numerical Mathematics(Theory,methods and Applications) 2011年第2期4卷 180-196页

作者： Kang Deng Yanping Chen Zuliang Lu School of Mathematical Sciences Hunan University of Science and TechnologyXiangtan 411201P.R.China School of Mathematical Sciences South China Normal UniversityGuangzhou 510631P.R.China College of Mathematics and Computer Sciences Chongqing Three Gorges UniversityChongqing 404000P.R.China College of Civil Engineering and Mechanics Xiangtan UniversityXiangtan 411105P.R.China

In this paper,we investigate a priori error estimates for the quadratic optimal control problems governed by semilinear elliptic partial differential equations using higher order triangular mixed finite element methods.The state and the co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k≥0).A priori error estimates for the mixed finite element approximation of semilinear control problems are obtained.Finally,we present some numerical examples which confirm our theoretical results.

关键词： a priori error estimates semilinear optimal control problems higher order triangular elements mixed finite element methods

UNCONDITIONAL CONVERGENCE AND ERROR ESTIMATES OF A FULLY DISCRETE finite element METHOD FOR THE MICROPOLAR NAVIER-STOKES EQUATIONS

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Journal of Computational Mathematics 2024年第1期42卷 71-110页

作者： Shipeng Mao Jiaao Sun Wendong Xue NCMIS LSECInstitute of Computational Mathematics and Scientific/Enginnering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

In this paper,we consider the initial-boundary value problem(IBVP)for the micropolar Naviers-Stokes equations(MNSE)and analyze a first order fully discrete mixed finite element scheme.We first establish some regularity results for the solution of MNSE,which seem to be not available in the literature.Next,we study a semi-implicit time-discrete scheme for the MNSE and prove L2-H1 error estimates for the time discrete solution.Furthermore,certain regularity results for the time discrete solution are establishes rigorously.Based on these regularity results,we prove the unconditional L2-H1 error estimates for the finite element solution of MNSE.Finally,some numerical examples are carried out to demonstrate both accuracy and efficiency of the fully discrete finite element scheme.

关键词： Micropolar fluids Regularity estimates Euler semi-implicit scheme mixed finite element methods Unconditional convergence

Error Estimates of Variational Discretization and mixed finite element methods for Quasilinear Optimal Control Problems

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Error Estimates of Variational Discretization and Mixed Fini...

ON DISCRETE ENERGY DISSIPATION OF MAXWELL’S EQUATIONS IN A COLE-COLE DISPERSIVE MEDIUM

第24届中国控制与决策会议

作者： Zuliang Lu 1,2 , Xiao Huang 3 1. School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404000, China 2. College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105, China3. College of Electronics and Information Engineering, Chongqing Three Gorges University, Chongqing 404100, China

In this paper we study a variational discretization and mixed finite element approximation of optimal control problems governed by quasilinear elliptic equations. The state and co-state are approximated by the lowest order Raviart- Thomas mixed finite element spaces and the control is not discreted. We derive a priori error estimates both for the state variables and the control variable. Finally, a numerical example is given to demonstrate the theoretical results.

关键词： Error estimates quasilinear optimal control problems variational discretization mixed finite element methods

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Journal of Computational Mathematics 2023年第5期41卷 980-1002页

作者： Baoli Yin Yang Liu Hong Li Zhimin Zhang School of Mathematical Sciences Inner Mongolia UniversityHohhot 010021China Beijing Computational Science Research Center Beijing 100193China Department of Mathematics Wayne State UniversityDetroitMI 48202USA

A simple criterion is studied for the first time for identifying the discrete energy dissipation of the Crank-Nicolson scheme for Maxwell’s equations in a Cole-Cole dispersive medium.Several numerical formulas that approximate the time fractional derivatives are investigated based on this criterion,including the L1 formula,the fractional BDF-2,and the shifted fractional trapezoidal rule(SFTR).Detailed error analysis is provided within the framework of time domain mixed finite element methods for smooth solutions.The convergence results and discrete energy dissipation law are confirmed by numerical tests.For nonsmooth solutions,the method SFTR can still maintain the optimal convergence order at a positive time on uniform meshes.Authors believe this is the first appearance that a second-order time-stepping method can restore the optimal convergence rate for Maxwell’s equations in a Cole-Cole dispersive medium regardless of the initial singularity of the solution.

关键词： Discrete energy dissipation Crank-Nicolson scheme Maxwell’s equations Shifted fractional trapezoidal rule mixed finite element methods

A POSTERIORI ERROR ESTIMATES FOR finite element APPROXIMATIONS OF THE CAHN-HILLIARD EQUATION AND THE HELE-SHAW FLOW

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Journal of Computational Mathematics 2008年第6期26卷 767-796页

作者： Xiaobing Feng Haijun Wu Department of Mathematics The University of Tennessee Knoxville TN 37996 USA Department of Mathematics Nanjing University Nanjing 210093 China

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut ＋ △（ε△Au-ε^-1f（u）） = 0. It is shown that the a posteriori error bounds depends on ε^-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct at2 adaptive algorithm for computing the solution of the Cahn- Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.

关键词： Cahn-Hilliard equation Hele-Shaw flow Phase transition Conforming elements mixed finite element methods A posteriori error estimates Adaptivity.