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Numerical Mathematics(Theory,Methods and Applications) 2013年第3期6卷 479-498页

作者： tianliang hou School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongChina

In this paper, we investigate the superconvergence property and the L∞-errorestimates of mixed finite element methods for a semilinear elliptic control problem. Thestate and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant *** derive some superconvergence results for the control variable. Moreover, we derive L^(∞)-error estimates both for the control variable and the state variables. Finally, anumerical example is given to demonstrate the theoretical results.

关键词： Semilinear elliptic equations distributed optimal control problems superconvergence L^(∞)-error estimates mixed finite element methods.

MIXED DISCONTINUOUS GALERKIN TIME-STEPPING METHOD FOR LINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS

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Journal of Computational Mathematics 2015年第2期33卷 158-178页

作者： tianliang hou Yanping Chen School of Mathematical Sciences South China Normal University Guangzhou 510631 China School of Mathematics and Statistics Beihua University Jilin 132013 China

In this paper, we discuss the mixed discontinuous Galerkin （DG） finite element ap- proximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time dis- cretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We de- rive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori L2（O, T; L2（Ω）） error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.

关键词： A priori error estimates A posteriori error estimates Mixed finite element,Discontinuous Galerkin method Parabolic control problems.

VARIATIONAL DISCRETIZATION FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY PARABOLIC EQUATIONS

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Journal of Systems Science & Complexity 2013年第6期26卷 902-924页

作者： CHEN Yanping hou tianliang YI Nianyu School of Mathematical Sciences South China Normal University School of Mathematics and Computational Science Xiangtan University

This paper considers the variational discretization for the constrained optimal control problem governed by linear parabolic *** state and co-state are approximated by RaviartThomas mixed finite element spaces,and the authors do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control.A priori error estimates are derived for the state,the co-state,and the *** numerical examples are presented to confirm the theoretical investigations.

关键词： A priori error estimates mixed finite element methods optimal control problems,parabolic equations variational discretization.

Error Estimates and Superconvergence of RT0Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems

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Numerical Mathematics(Theory,Methods and Applications) 2013年第4期6卷 637-656页

作者： Yanping Chen tianliang hou School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongChina Research Institute of Hong Kong Baptist University in Shenzhen Shenzhen 518057GuangdongChina

In this paper,we will investigate the error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on *** state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element and the control variable is approximated by piecewise constant *** derive some superconvergence properties for the control variable and the state ***,we derive L∞-and H−1-error estimates both for the control variable and the state ***,a numerical example is given to demonstrate the theoretical results.

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

Superconvergence and L^(∞)-Error Estimates of RT1Mixed Methods for Semilinear Elliptic Control Problems with an Integral Constraint

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Numerical Mathematics(Theory,Methods and Applications) 2012年第3期5卷 423-446页

作者： Yanping Chen tianliang hou 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 superconvergence property and the L∞-error estimates of mixed finite element methods for a semilinear elliptic control problem with an integral *** state and co-state are approximated by the order one Raviart-Thomas mixed finite element space and the control variable is approximated by piecewise constant functions or piecewise linear *** derive some superconvergence results for the control variable and the state variables when the control is approximated by piecewise constant ***,we derive L∞-error estimates for both the control variable and the state variables when the control is discretized by piecewise linear ***,some numerical examples are given to demonstrate the theoretical results.

关键词： Semilinear elliptic equations optimal control problems superconvergence L∞-error estimates mixed finite element methods postprocessing

Error Estimates ofMixedMethods forOptimal Control Problems Governed by General Elliptic Equations

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Advances in Applied Mathematics and Mechanics 2016年第6期8卷 1050-1071页

作者： tianliang hou Li Li School of Mathematics and Statistics Beihua UniversityJilin 132013China Key Laboratory for Nonlinear Science and System Structure School of Mathematics and StatisticsChongqing Three Gorges UniversityWanzhou 404100China

In this paper,we investigate the error estimates of mixed finite element methods for optimal control problems governed by general elliptic *** state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant *** derive L2 and H−1-error estimates both for the control variable and the state ***,a numerical example is given to demonstrate the theoretical results.

关键词： General elliptic equations optimal control problems superconvergence error estimates,mixed finite element methods

Superconvergence of RT1 mixed finite element approximations for elliptic control problems

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Science China Mathematics 2013年第2期56卷 267-281页

作者： hou tianliang CHEN YanPing Department of Mathematics Xiangtan University School of Mathematical Sciences South China Normal University

In this paper,we investigate the superconvergence property of the numerical solution to a quadratic elliptic control problem by using mixed finite element *** state and co-state are approximated by the order k=1 Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant *** prove the superconvergence error estimate of h3/2 in L2-norm between the approximated solution and the average L2 projection of the ***,by the postprocessing technique,a quadratic superconvergence result of the control is derived.

关键词： elliptic equations optimal control problems superconvergence mixed finite element methods postprocessing

Fully Discrete H^(1) -Galerkin Mixed Finite Element Methods for Parabolic Optimal Control Problems

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Numerical Mathematics(Theory,Methods and Applications) 2019年第1期12卷 134-153页

作者： tianliang hou Chunmei Liu Hongbo Chen School of Mathematics and Statistics Northeast Normal UniversityChangchun 130024China School of Mathematics and Statistics Beihua UniversityJilin 132013China Institute for Computational Mathematics College of ScienceHunan University of Science and EngineeringYongzhou 425199HunanChina

In this paper,we investigate a priori and a posteriori error estimates of fully discrete H^(1)-Galerkin mixed finite element methods for parabolic optimal control *** state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element,and the control vari-able is approximated by piecewise constant *** time discretization of the state and co-state are based on finite difference ***,we derive a priori error estimates for the control variable,the state variables and the adjoint state ***,by use of energy approach,we derive a posteriori error estimates for optimal control problems,assuming that only the underlying mesh is static.A numerical example is presented to verify the theoretical results on a priori error estimates.

关键词： Parabolic equations optimal control problems a priori error estimates a posteriori error estimates H^(1)-Galerkin mixed finite element methods

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 *** 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 *** derive L^(2) and L^(∞)-error estimates for the control ***,using a recovery operator,we also derive some superconvergence results for the control ***,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

A Posteriori Error Estimates of Mixed Methods for Quadratic Optimal Control Problems Governed by Parabolic Equations

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Numerical Mathematics(Theory,Methods and Applications) 2011年第4期4卷 439-458页

作者： tianliang hou Yanping Chen Yunqing Huang Hunan Key Laboratory for Computation and Simulation in Science and Engineering Department of MathematicsXiangtan UniversityXiangtan 411105HunanChina School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongChina

In this paper,we discuss the a posteriori error estimates of the mixed finite element method for quadratic optimal control problems governed by linear parabolic *** state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant *** derive a posteriori error estimates for both the state and the control *** estimates,which are apparently not available in the literature,are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.

关键词： A posteriori error estimates quadratic optimal control problems parabolic equations mixed finite element methods