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In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results. 相似文献
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Thomas Apel Thomas G. Flaig Serge Nicaise 《Numerical Functional Analysis & Optimization》2013,34(2):153-176
The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In this article, we introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two-dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other and appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions. We establish a regularity result and estimate for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use linear, quadratic or cubic Lagrange elements in the other dimension cubic Hermite elements. For these elements, we prove the error bound O(h 2 + τ k ) in the energy norm and O((h 2 + τ k )(h 2 + τ)) in the L 2(Q)-norm. 相似文献
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Yue Shen & Chang Jin 《高等学校计算数学学报(英文版)》2020,13(2):400-432
In this paper, we study the finite element methods for distributed optimal control problems governed by the biharmonic operator. Motivated from reducing the regularity of solution space, we use the decoupled mixed element method which was used to approximate the solution of biharmonic equation to solve the fourth order optimal control problems. Two finite element schemes, i.e., Lagrange conforming element combined with full control discretization and the nonconforming Crouzeix-Raviart element combined with variational control discretization, are used to discretize the decoupled optimal control system. The corresponding a priori error estimates are derived under appropriate norms which are then verified by extensive numerical experiments. 相似文献
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Radouen Ghanem & Yazid Dendani 《计算数学(英文版)》2023,41(4):717-740
In this paper we deal with the convergence analysis of the finite element method for an elliptic penalized unilateral obstacle optimal control problem where the control and the obstacle coincide. Error estimates are established for both state and control variables. We apply a fixed point type iteration method to solve the discretized problem.To corroborate our error estimations and the efficiency of our algorithms, the convergence results and numerical experiments are illustrated by concrete examples. 相似文献
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In this article,the authors discuss the optimal conditions of the linear fractional programming problem and prove that a locally optional solution is a globally optional so- lution and the locally optimal solution can be attained at a basic feasible solution with constraint condition. 相似文献
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We investigate the spectral approximation of optimal control governed by Stokes equations with integral state constraint. A good choice for basis functions leads the discrete system with sparse matrices. The optimality conditions are derived, a priori and a posteriori error estimates are presented in both H1 and L2 norms. Numerical experiment indicates the high precision can be achieved with the proposed method. 相似文献
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本文讨论了四阶障碍问题的稳定化混合有限元方法.首先,引入网格依赖范数,通过加罚方法得到了与四阶障碍问题的等价的混合变分形式.随后给出了基于C~0协调有限元空间(W_h,V_h)的混合有限元逼近,例如P_k-P_k三角形有限元.在网格依赖范数下,(W_h,V_h)满足离散的inf-sup条件.最后,我们在不同的假设下,得到了一些误差估计. 相似文献
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Hongfei Fu & Hongxing Rui 《计算数学(英文版)》2015,33(2):113-127
In this paper, a constrained distributed optimal control problem governed by a first-order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient (PCG) and algebraic multi-grid (AMG) can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in $L^2(Ω)$-norm, for the original state and adjoint state in $H^1(Ω)$-norm, and for the flux state and adjoint flux state in $H$(div; $Ω$)-norm. Finally, we use one numerical example to validate the theoretical findings. 相似文献
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In this article, we consider an application of the abstract error estimate for a class of optimal control systems described by a linear partial differential equation (as stated in Numer. Funct. Anal. Optim. 2009; 30:523–547). The control is applied at the boundary and we consider both, Neumann and Dirichlet optimal control problems. Finite element methods are proposed to approximate the optimal control considering an approximation of the variational inequality resulting from the optimality conditions; this approach is known as classical one. We obtain optimal order error estimates for the control variable and numerical examples, taken from the literature, are included to illustrate the results. 相似文献
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An inverse problem of reconstructing the initial condition for a time fractional diffusion equation is investigated. On the basis of the optimal control framework,
the uniqueness and first order necessary optimality condition of the minimizer for the
objective functional are established, and a time-space spectral method is proposed to
numerically solve the resulting minimization problem. The contribution of the paper
is threefold: 1) a priori error estimate for the spectral approximation is derived; 2) a
conjugate gradient optimization algorithm is designed to efficiently solve the inverse
problem; 3) some numerical experiments are carried out to show that the proposed
method is capable to find out the optimal initial condition, and that the convergence
rate of the method is exponential if the optimal initial condition is smooth. 相似文献
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This work is concerned with the maximum principle for optimal control problem governed by magnetohydrodynamic equations, which describe the motion of a viscous incompressible conducting fluid in a magnetic field and consist of a subtle coupling of the Navier-Stokes equation of viscous incompressible fluid flow and the Maxwell equation of electromagnetic field. An integral type state constraint is considered. 相似文献
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带状态约束的抛物型变分不等式的最优控制 总被引:1,自引:1,他引:0
利用非光滑分析和半变分不等式的一些方法和结果,研究了一类带状态约束的具有非线性、不连续以及非单调多值项的抛物型变分不等式的优化控制问题以及它的逼近等,推广了一些已有的结果. 相似文献
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讨论了差分-流线扩散法(FDSD)求解线性对流占优扩散问题解的精度,利用插值后处理技术,使该格式解的空间精间达到最优. 相似文献
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Some Optimal Control Problems Governed by Elliptic Variational Inequalities with Control and State Constraint on the Boundary 总被引:1,自引:0,他引:1
This work deals with the necessary conditions of optimality for some optimal control problems governed by elliptic variational inequalities. Boundary control and state constrained problems are considered. The techniques used are based on those in Ref. 1 and a new penalty functional is defined in this paper. 相似文献
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Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation
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Huipo Liu Shuanghu Wang Hongbin Han Lan Yuan 《Numerical Methods for Partial Differential Equations》2017,33(5):1493-1512
This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493–1512, 2017 相似文献
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B. Vexler 《Numerical Functional Analysis & Optimization》2013,34(7-8):957-973
We develop a priori error analysis for the finite element Galerkin discretization of elliptic Dirichlet optimal control problems. The state equation is given by an elliptic partial differential equation and the finite dimensional control variable enters the Dirichlet boundary conditions. We prove the optimal order of convergence and present a numerical example confirming our results. 相似文献