Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation

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     

Local discontinuous galerkin approximation of convection‐dominated diffusion optimal control problems with control constraints

In this article, we investigate local discontinuous Galerkin approximation of stationary convection‐dominated diffusion optimal control problems with distributed control constraints. The state variable and adjoint state variable are approximated by piecewise linear polynomials without continuity requirement, whereas the control variable is discretized by variational discretization concept. The discrete first‐order optimality condition is derived. We show that optimization and discretization are commutative for the local discontinuous Galerkin approximation. Because the solutions to convection‐dominated diffusion equations often admit interior or boundary layers, residual type a posteriori error estimate in L² norm is proved, which can be used to guide mesh refinement. Finally, numerical examples are presented to illustrate the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 339–360, 2014     

A characteristic finite element method for optimal control problems governed by convection-diffusion equations

In this paper we analyze a characteristic finite element approximation of convex optimal control problems governed by linear convection-dominated diffusion equations with pointwise inequality constraints on the control variable, where the state and co-state variables are discretized by piecewise linear continuous functions and the control variable is approximated by either piecewise constant functions or piecewise linear discontinuous functions. A priori error estimates are derived for the state, co-state and the control. Numerical examples are given to show the efficiency of the characteristic finite element method.     

Superconvergence for Optimal Control Problem Governed by Nonlinear Elliptic Equations

In this article, we investigate the superconvergence of the finite element approximation for optimal control problem governed by nonlinear elliptic equations. The state and co-state are discretized by piecewise linear functions and control is approximated by piecewise constant functions. We give the superconvergence analysis for both the control variable and the state variables. Finally, the numerical experiments show the theoretical results.     

Linear and Discontinuous Approximations for Optimal Control Problems

Abstract

An optimal control problem for 2D and 3D elliptic equations is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise linear but discontinuous functions. The state and the adjoint state are discretized by linear finite elements. The paper is focused on similarities and differences to piecewise constant and piecewise linear (continuous) approximation of the controls. Approximation of order h in the L ^∞-norm is proved in the main result.     

分布参数最优控制的边界元共轭梯度算法

研究了一类线性椭圆型分布参数最优控制问题的数值解算法.得到最优控制对应的最优性方程组,在凸性条件下,证明了最优控制的唯一存在性问题.将最优控制问题化为以控制函数和状态函数为局中人的递阶式(Stackelberg)非合作对策问题,其平衡点是最优控制的解.进一步得到求平衡点的边界元共轭梯度算法.最后,研究算法中边界元离散的误差估计,以算例验证该算法.     

A global superconvergent $L^{\infty}$-error estimate of mixed finite element methods for semilinear elliptic optimal control problems

In this paper, we discuss the superconvergence of mixed finite element methods for a semilinear elliptic control problem with an integral constraint. The state and costate are approximated by the order $k=1$ Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Approximation of the optimal control of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that this approximation has convergence order $h^{2}$ in $L^{\infty}$-norm. Finally, a numerical example is given to demonstrate the theoretical results.     

An Error Analysis of Discontinuous Finite Element Methods for the Optimal Control Problems Governed by Stokes Equation

In this article, an abstract framework for the error analysis of discontinuous finite element method is developed for the distributed and Neumann boundary control problems governed by the stationary Stokes equation with control constraints. A priori error estimates of optimal order are derived for velocity and pressure in the energy norm and the L²-norm, respectively. Moreover, a reliable and efficient a posteriori error estimator is derived. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. In particular, we consider the abstract results with suitable stable pairs of velocity and pressure spaces like as the lowest-order Crouzeix–Raviart finite element and piecewise constant spaces, piecewise linear and constant finite element spaces. The theoretical results are illustrated by the numerical experiments.     

L ∞-error estimates of triangular mixed finite element methods for optimal control problems governed by semilinear elliptic equations

We look at L ^∞-error estimates for convex quadratic optimal control problems governed by nonlinear elliptic partial differential equations. In so doing, use is made of mixed finite element methods. The state and costate are approximated by the lowest order Raviart-Thomas mixed finite element spaces, and the control, by piecewise constant functions. L ^∞-error estimates of optimal order are derived for a mixed finite element approximation of a semilinear elliptic optimal control problem. Finally, numerical tests are presented which confirm our theoretical results.     

Error estimates of expanded mixed methods for optimal control problems governed by hyperbolic integro‐differential equations

In this article, we investigate the L^∞(L²) ‐error estimates of the semidiscrete expanded mixed finite element methods for quadratic optimal control problems governed by hyperbolic integrodifferential equations. The state and the costate are discretized by the order k Raviart‐Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of order k(k ≥ 0). We derive error estimates for both the state and the control approximation. Numerical experiments are presented to test the theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Numerical analysis for the approximation of optimal control problems with pointwise observations

In this paper, we study the numerical methods for optimal control problems governed by elliptic PDEs with pointwise observations of the state. The first order optimality conditions as well as regularities of the solutions are derived. The optimal control and adjoint state have low regularities due to the pointwise observations. For the finite dimensional approximation, we use the standard conforming piecewise linear finite elements to approximate the state and adjoint state variables, whereas variational discretization is applied to the discretization of the control. A priori and a posteriori error estimates for the optimal control, the state and adjoint state are obtained. Numerical experiments are also provided to confirm our theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability analysis of parabolic partial differential equations with piecewise continuous arguments

This article deals with the analytic and numerical stability of numerical methods for a parabolic partial differential equation with piecewise continuous arguments of alternately retarded and advanced type. First, application of the theory of separation of variables in matrix form and the Fourier method, the necessary and sufficient condition under which the analytic solution is asymptotically stable is derived. Then, the θ‐methods are applied to solve the corresponding initial value problem, the sufficient conditions for the asymptotic stability of numerical methods are obtained. Finally, several numerical examples are presented to support the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 531–545, 2017     

Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems

We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.     

半线性抛物最优控制问题全离散插值系数有限元方法的收敛性分析

本文考虑全离散插值系数有限元方法求解半线性抛物最优控制问题,其中控制变量用分片常数函数逼近,状态变量和对偶状态变量用分片线性函数逼近.对于方程中的半线性项,先用插值系数技巧处理,再用牛顿迭代法求解.通过引入一些辅助变量和投影算子,并利用有限元空间的逼近性质,得到半线性抛物最优控制问题插值系数有限元方法的收敛性结果;数值算例结果验证了理论结果的正确性.     

Superconvergence and $L^{\infty}$-Error Estimates of RT1 Mixed Methods for Semilinear Elliptic Control Problems with an Integral Constraint

In this paper, we investigate the superconvergence property and the $L^{\infty}$-error estimates of mixed finite element methods for a semilinear elliptic control problem with an integral constraint. The 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 functions. We derive some superconvergence results for the control variable and the state variables when the control is approximated by piecewise constant functions. Moreover, we derive $L^{\infty}$-error estimates for both the control variable and the state variables when the control is discretized by piecewise linear functions. Finally, some numerical examples are given to demonstrate the theoretical results.     

Approximation of relaxed nonlinear parabolic optimal control problems

We consider a relaxed optimal control problem for systems defined by nonlinear parabolic partial differential equations with distributed control. The problem is completely discretized by using a finite-element approximation scheme with piecewise linear states and piecewise constant controls. Existence of optimal controls and necessary conditions for optimality are derived for both the continuous and the discrete problem. We then prove that accumulation points of sequences of discrete optimal [resp. extremal] controls are optimal [resp. extremal] for the continuous problem.     

Error estimates of mixed finite element methods for quadratic optimal control problems

In this paper, we investigate the error estimates for the solutions of optimal control problems by mixed finite element methods. The state and costate are approximated by Raviart-Thomas mixed finite element spaces of order k and the control is approximated by piecewise polynomials of order k. Under the special constraint set, we will show that the control variable can be smooth in the whole domain. We derive error estimates of optimal order both for the state variables and the control variable.     

Two-grid methods for –P1 mixed finite element approximation of general elliptic optimal control problems with low regularity

In this paper, we present a two-grid mixed finite element scheme for distributed optimal control governed by general elliptic equations. –P₁ mixed finite elements are used for the discretization of the state and co-state variables, whereas piecewise constant function is used to approximate the control variable. We first use a new approach to obtain the superclose property between the centroid interpolation and the numerical solution of the optimal control u with order h² under the low regularity. Based on the superclose property, we derive the optimal a priori error estimates. Then, using a postprocessing projection operator, we get a second-order superconvergent result for the control u. Next, we construct a two-grid mixed finite element scheme and analyze a priori error estimates. In the two-grid scheme, the solution of the elliptic optimal control problem on a fine grid is reduced to the solution of the elliptic optimal control problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. Finally, a numerical example is presented to verify the theoretical results.     

Piecewise optimal distributed Controls for 2D Boussinesq equations

In this article, we study the dynamics of a piecewise (in time) distributed optimal control problem for the Boussinesq equations which model velocity tracking over time coupled to thermal dynamics. We also study the dynamics of semidiscrete approximation of this problem. We prove that the rates of velocity tracking coupled to thermal dynamics are exponential and that the difference between the solution of the semi‐discrete piecewise optimal control problem and the desired states in L² and H¹ norms decay to zero exponentially as n→∞. Copyright © 2000 John Wiley & Sons, Ltd.     

A superconvergent $L^{\infty}$-error estimate of RT1 mixed methods for elliptic control problems with an integral constraint

In this paper, we investigate the superconvergence property of mixed finite element methods for a linear elliptic control problem with an integral constraint. The 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 functions. A superconvergent approximation of the control variable $u$ will be constructed by a projection of the discrete adjoint state. It is proved that this approximation have convergence order $h^{2}$ in $L^{\infty}$-norm. Finally, a numerical example is given to demonstrate the theoretical results.