Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 217493 | 21 pages | https://doi.org/10.1155/2011/217493

Adaptive Mixed Finite Element Methods for Parabolic Optimal Control Problems

Academic Editor: Kue-Hong Chen
Received12 May 2011
Accepted30 Jun 2011
Published28 Aug 2011

Abstract

We will investigate the adaptive mixed finite element methods for parabolic optimal control problems. The state and the costate are approximated by the lowest-order Raviart-Thomas mixed finite element spaces, and the control is approximated by piecewise constant elements. We derive a posteriori error estimates of the mixed finite element solutions for optimal control problems. Such a posteriori error estimates can be used to construct more efficient and reliable adaptive mixed finite element method for the optimal control problems. Next we introduce an adaptive algorithm to guide the mesh refinement. A numerical example is given to demonstrate our theoretical results.

1. Introduction

Optimal control problems are very important models in science and engineering numerical simulation. Finite element method of optimal control problems plays an important role in numerical methods for these problems. Let us mention two early papers devoted to linear optimal control problems by Falk [1] and Geveci [2]. Knowles was concerned with standard finite element approximation of parabolic time optimal control problems in [3]. In [4] Gunzburger and Hou investigated the finite element approximation of a class of constrained nonlinear optimal control problems. For quadratic optimal control problem governed by linear parabolic equation, Liu and Yan derived a posteriori error estimates for both the state and the control approximation in [5]. Systematic introductions of the finite element method for optimal control problems can be found in [6–10].

Adaptive finite element approximation was the most important means of boosting the accuracy and efficiency of finite element discretization. The literature in this aspect was huge, see, for example, [11, 12]. Adaptive finite element method was widely used in engineering numerical simulation. There has been extensive studies on adaptive finite element approximation for optimal control problems. In [13], the authors have introduced some basic concept of adaptive finite element discretization for optimal control of partial differential equations. A posteriori error estimators for distributed elliptic optimal control problems were contained in Li et al. [14]. Recently an adaptive finite element method for the estimation of distributed parameter in elliptic equation was discussed by Feng et al. [15]. Note that all the above works aimed at standard finite element method.

In many control problems, the objective functional contains the gradient of the state variables. Thus, the accuracy of the gradient is important in numerical discretization of the coupled state equations. When the objective functional contains the gradient of the state variable, mixed finite element methods should be used for discretization of the state equation with which both the scalar variable and its flux variable can be approximated in the same accuracy. In [16–20] we have done some primary works on a priori error estimates and superconvergence for linear optimal control problems by mixed finite element methods. We considered a posteriori error estimates of mixed finite element methods for quadratic and general optimal control problems in [21–23].

In [24], the authors discussed the mixed finite element approximation for general optimal control problems governed by parabolic equation. And then, they derived a posteriori error estimates of mixed finite element solution. In this paper, we study the adaptive mixed finite element methods for the parabolic optimal control problems. We construct the mixed finite element discretization for the original problems and derive a useful posteriori error indicators. Furthermore, we provide an adaptive algorithm to guide the multimesh refinement. Finally, a numerical experiment shows that this algorithm works very well with the adaptive multimesh discretization.

The plan of this paper is as follows. In the next section, we construct the mixed finite element discretization for the parabolic optimal control problems. Then, we derive a posteriori error estimates for the mixed finite element solutions in Section 3. Next, we introduce an adaptive algorithm to guide the mesh refinement in Section 4. Finally, a numerical example is given to demonstrate our theoretical results in Section 5.

2. Mixed Methods of Optimal Control Problems

In this section, we investigate the mixed finite element approximation for parabolic optimal control problems. We adopt the standard notation π‘Šπ‘š,𝑝(Ξ©) for Sobolev spaces on Ξ© with a norm β€–β‹…β€–π‘š,𝑝 given by β€–π‘£β€–π‘π‘š,p=βˆ‘|𝛼|β‰€π‘šβ€–π·π›Όπ‘£β€–π‘πΏπ‘(Ξ©), a seminorm |β‹…|π‘š,𝑝 given by |𝑣|π‘π‘š,𝑝=βˆ‘|𝛼|=π‘šβ€–π·π›Όπ‘£β€–π‘πΏπ‘(Ξ©). We set π‘Š0π‘š,𝑝(Ξ©)={π‘£βˆˆπ‘Šπ‘š,𝑝(Ξ©)βˆΆπ‘£|πœ•Ξ©=0}. For 𝑝=2, we denote π»π‘š(Ξ©)=π‘Šπ‘š,2(Ξ©),π»π‘š0(Ξ©)=π‘Š0π‘š,2(Ξ©), and β€–β‹…β€–π‘š=β€–β‹…β€–π‘š,2,β€–β‹…β€–=β€–β‹…β€–0,2.

We denote by 𝐿𝑠(0,𝑇;π‘Šπ‘š,𝑝(Ξ©)) the Banach space of all 𝐿𝑠 integrable functions from 𝐽 into π‘Šπ‘š,𝑝(Ξ©) with norm ‖𝑣‖𝐿𝑠(𝐽;π‘Šπ‘š,𝑝(Ξ©))∫=(𝑇0β€–π‘£β€–π‘ π‘Šπ‘š,𝑝(Ξ©)𝑑𝑑)1/𝑠forπ‘ βˆˆ[1,∞) and the standard modification for 𝑠=∞. The details can be found in [25].

The parabolic optimal control problems that we are interested in are as follows:minπ‘’βˆˆπΎβŠ‚π‘ˆξ‚»ξ€œπ‘‡0𝑔1(𝐩)+𝑔2ξ€Έξ‚Ό,𝑦(𝑦)+𝑗(𝑒)𝑑𝑑𝑑(π‘₯,𝑑)+div𝐩(π‘₯,𝑑)=𝑓+𝑒(π‘₯,𝑑),π‘₯∈Ω,𝐩(π‘₯,𝑑)=βˆ’π΄(π‘₯)βˆ‡π‘¦(π‘₯,𝑑),π‘₯∈Ω,𝑦(π‘₯,𝑑)=0,π‘₯βˆˆπœ•Ξ©,π‘‘βˆˆπ½,𝑦(π‘₯,0)=𝑦0(π‘₯),π‘₯∈Ω,(2.1) where the bounded open set Ξ©βŠ‚β„2 is a convex polygon with the boundary πœ•Ξ©. Let 𝐾 be a closed convex set in π‘ˆ=𝐿2(𝐽;𝐿2(Ξ©)), π‘“βˆˆπΏ2(𝐽;𝐿2(Ξ©)), 𝐽=[0,𝑇], 𝑦0(π‘₯)∈𝐻10(Ξ©). Furthermore, we assume that the coefficient matrix 𝐴(π‘₯)=(π‘Žπ‘–,𝑗(π‘₯))2Γ—2∈𝐿∞(Ξ©;ℝ2Γ—2) is a symmetric 2Γ—2-matrix and there is a constant 𝑐>0 satisfying for any vector π—βˆˆβ„2, 𝐗𝑑𝐴𝐗β‰₯𝑐‖𝐗‖2ℝ2. π‘—ξ…ž is positive, π‘”ξ…ž1, π‘”ξ…ž2, and π‘—ξ…ž are locally Lipschitz on 𝐿2(Ξ©)2, π‘Š, π‘ˆ, and that there is a 𝑐>0 such that (π‘—ξ…ž(𝑒)βˆ’π‘—ξ…ž(̃𝑒),π‘’βˆ’Μƒπ‘’)β‰₯π‘β€–π‘’βˆ’Μƒπ‘’β€–0, forall𝑒,Μƒπ‘’βˆˆπ‘ˆ.

Now we will describe the mixed finite element discretization of parabolic optimal control problems (2.1). Let 𝐿𝐕=𝐻(div;Ξ©)=𝐯∈2ξ€Έ(Ξ©)2,div𝐯∈𝐿2(Ξ©),π‘Š=𝐿2(Ξ©).(2.2) The Hilbert space 𝐕 is equipped with the following norm: ‖𝐯‖div=‖𝐯‖𝐻(div;Ξ©)=‖𝐯‖20,Ξ©+β€–div𝐯‖20,Ξ©ξ€Έ1/2.(2.3)

We recast (2.1) as the following weak form: find (𝐩,𝑦,𝑒)βˆˆπ•Γ—π‘ŠΓ—πΎ such thatminπ‘’βˆˆπΎβŠ‚π‘ˆξ‚»ξ€œπ‘‡0𝑔1(𝐩)+𝑔2ξ€Έξ‚Ό,𝐴(𝑦)+𝑗(𝑒)𝑑𝑑(2.4)βˆ’1𝑦𝐩,π―βˆ’(𝑦,div𝐯)=0,βˆ€π―βˆˆπ•,(2.5)𝑑,𝑀+(div𝐩,𝑀)=(𝑓+𝑒,𝑀),βˆ€π‘€βˆˆπ‘Š,(2.6)𝑦(π‘₯,0)=𝑦0(π‘₯),βˆ€π‘₯∈Ω.(2.7)

Similar to [26], the optimal control problems (2.4)–(2.7) have a unique solution (𝐩,𝑦,𝑒), and a triplet (𝐩,𝑦,𝑒) is the solution of (2.4)–(2.7) if and only if there is a costate (πͺ,𝑧)βˆˆπ•Γ—π‘Š such that (𝐩,𝑦,πͺ,𝑧,𝑒) satisfies the following optimality conditions:ξ€·π΄βˆ’1𝑦𝐩,π―βˆ’(𝑦,div𝐯)=0,βˆ€π―βˆˆπ•,(2.8)𝑑,𝑀+(div𝐩,𝑀)=(𝑓+𝑒,𝑀),βˆ€π‘€βˆˆπ‘Š,(2.9)𝑦(π‘₯,0)=𝑦0(𝐴π‘₯),βˆ€π‘₯∈Ω,(2.10)βˆ’1𝑔πͺ,π―βˆ’(𝑧,div𝐯)=βˆ’ξ…ž1ξ€Έβˆ’ξ€·π‘§(𝐩),𝐯,βˆ€π―βˆˆπ•,(2.11)𝑑𝑔,𝑀+(divπͺ,𝑀)=ξ…ž2ξ€Έξ€œ(𝑦),𝑀,βˆ€π‘€βˆˆπ‘Š,(2.12)𝑧(π‘₯,𝑇)=0,βˆ€π‘₯∈Ω,(2.13)𝑇0ξ€·π‘—ξ…ž(𝑒)+𝑧,Μƒπ‘’βˆ’π‘’π‘ˆπ‘‘π‘‘β‰₯0,βˆ€Μƒπ‘’βˆˆπΎ,(2.14) where (β‹…,β‹…)π‘ˆ is the inner product of π‘ˆ. In the rest of the paper, we will simply write the product as (β‹…,β‹…) whenever no confusion should be caused.

Let π’―β„Ž be regular triangulation of Ξ©. They are assumed to satisfy the angle condition which means that there is a positive constant 𝐢 such that, for all πœβˆˆπ’―β„Ž, πΆβˆ’1β„Ž2πœβ‰€|𝜏|β‰€πΆβ„Ž2𝜏, where |𝜏| is the area of 𝜏, β„Žπœ is the diameter of 𝜏 and β„Ž=maxβ„Žπœ. In addition 𝐢 or 𝑐 denotes a general positive constant independent of β„Ž.

Let π•β„ŽΓ—π‘Šβ„ŽβŠ‚π•Γ—π‘Š denote the Raviart-Thomas space [27] of the lowest order associated with the triangulation π’―β„Ž of Ξ©. π‘ƒπ‘˜ denotes the space of polynomials of total degree at most π‘˜. Let 𝐕(𝜏)={π―βˆˆπ‘ƒ20(𝜏)+π‘₯⋅𝑃0(𝜏)}, π‘Š(𝜏)=𝑃0(𝜏). We define π•β„Žξ€½π―βˆΆ=β„Žβˆˆπ•βˆΆβˆ€πœβˆˆπ’―β„Ž,π―β„Ž|πœξ€Ύ,π‘Šβˆˆπ•(𝜏)β„Žξ€½π‘€βˆΆ=β„Žβˆˆπ‘ŠβˆΆβˆ€πœβˆˆπ’―β„Ž,π‘€β„Ž|πœξ€Ύ,πΎβˆˆπ‘Š(𝜏)β„Žξ€½βˆΆ=Μƒπ‘’β„ŽβˆˆπΎβˆΆβˆ€πœβˆˆπ’―β„Ž,Μƒπ‘’β„Ž|πœξ€Ύ.=constant(2.15)

The mixed finite element discretization of (2.4)–(2.7) is as follows: compute (π©β„Ž,π‘¦β„Ž,π‘’β„Ž)βˆˆπ•β„ŽΓ—π‘Šβ„ŽΓ—πΎβ„Ž such thatminπ‘’β„ŽβˆˆπΎβ„Žξ‚»ξ€œπ‘‡0𝑔1ξ€·π©β„Žξ€Έ+𝑔2ξ€·π‘¦β„Žξ€Έξ€·π‘’+π‘—β„Žξ‚Ό,ξ€·π΄ξ€Έξ€Έπ‘‘π‘‘βˆ’1π©β„Ž,π―β„Žξ€Έβˆ’ξ€·π‘¦β„Ž,divπ―β„Žξ€Έ=0,βˆ€π―β„Žβˆˆπ•β„Ž,ξ€·π‘¦β„Žπ‘‘,π‘€β„Žξ€Έ+ξ€·divπ©β„Ž,π‘€β„Žξ€Έ=𝑓+π‘’β„Ž,π‘€β„Žξ€Έ,βˆ€π‘€β„Žβˆˆπ‘Šβ„Ž,π‘¦β„Ž(π‘₯,0)=π‘¦β„Ž0(π‘₯),βˆ€π‘₯∈Ω,(2.16) where π‘¦β„Ž0(π‘₯)βˆˆπ‘Šβ„Ž is an approximation of 𝑦0. The optimal control problem (2.16) again has a unique solution (π©β„Ž,π‘¦β„Ž,π‘’β„Ž), and a triplet (π©β„Ž,π‘¦β„Ž,π‘’β„Ž) is the solution of (2.16) if and only if there is a costate (πͺβ„Ž,π‘§β„Ž)βˆˆπ•β„ŽΓ—π‘Šβ„Ž such that (π©β„Ž,π‘¦β„Ž,πͺβ„Ž,π‘§β„Ž,π‘’β„Ž) satisfies the following optimality conditions:ξ€·π΄βˆ’1π©β„Ž,π―β„Žξ€Έβˆ’ξ€·π‘¦β„Ž,divπ―β„Žξ€Έ=0,βˆ€π―β„Žβˆˆπ•β„Ž,ξ€·π‘¦β„Žπ‘‘,π‘€β„Žξ€Έ+ξ€·divπ©β„Ž,π‘€β„Žξ€Έ=𝑓+π‘’β„Ž,π‘€β„Žξ€Έ,βˆ€π‘€β„Žβˆˆπ‘Šβ„Ž,π‘¦β„Ž(π‘₯,0)=𝑦0(𝐴π‘₯),βˆ€π‘₯∈Ω,βˆ’1πͺβ„Ž,π―β„Žξ€Έβˆ’ξ€·π‘§β„Ž,divπ―β„Žξ€Έξ€·π‘”=βˆ’ξ…ž1ξ€·π©β„Žξ€Έ,π―β„Žξ€Έ,βˆ€π―β„Žβˆˆπ•β„Ž,βˆ’ξ€·π‘§β„Žπ‘‘,π‘€β„Žξ€Έ+ξ€·divπͺβ„Ž,π‘€β„Žξ€Έ=ξ€·π‘”ξ…ž2ξ€·π‘¦β„Žξ€Έ,π‘€β„Žξ€Έ,βˆ€π‘€β„Žβˆˆπ‘Šβ„Ž,π‘§β„Žξ€·π‘—(π‘₯,𝑇)=0,βˆ€π‘₯∈Ω,ξ…žξ€·π‘’β„Žξ€Έ+π‘§β„Ž,Μƒπ‘’β„Žβˆ’π‘’β„Žξ€Έβ‰₯0,βˆ€Μƒπ‘’β„ŽβˆˆπΎβ„Ž.(2.17)

We now consider the fully discrete approximation for the semidiscrete problem. Let Δ𝑑>0, 𝑁=𝑇/Ξ”π‘‘βˆˆβ„€, and 𝑑𝑖=𝑖Δ𝑑, π‘–βˆˆβ„€. Also, let πœ“π‘–=πœ“π‘–ξ€·(π‘₯)=πœ“π‘₯,𝑑𝑖,π‘‘π‘‘πœ“π‘–=πœ“π‘–βˆ’πœ“π‘–βˆ’1.Δ𝑑(2.18) For 𝑖=1,2,…,𝑁, construct the finite element spaces π•π‘–β„Žβˆˆπ•, π‘Šπ‘–β„Žβˆˆπ‘Š (similar as π•β„Ž). Similarly, construct the finite element spaces πΎπ‘–β„ŽβˆˆπΎβ„Ž with the mesh π’―π‘–β„Ž. Let β„Žπ‘–πœ denote the maximum diameter of the element πœπ‘– in π’―π‘–β„Ž. Define mesh functions 𝜏(β‹…) and mesh size functions β„Žπœ(β‹…) such that 𝜏(𝑑)|π‘‘βˆˆ(π‘‘π‘–βˆ’1,𝑑𝑖]=πœπ‘–, β„Žπœ(𝑑)|π‘‘βˆˆ(π‘‘π‘–βˆ’1,𝑑𝑖]=β„Žπœπ‘–. For ease of exposition, we will denote 𝜏(𝑑) and β„Žπœ(𝑑) by 𝜏 and β„Žπœ, respectively.

The following fully discrete approximation scheme is to find (π©π‘–β„Ž,π‘¦π‘–β„Ž,π‘’π‘–β„Ž)βˆˆπ•π‘–β„ŽΓ—π‘Šπ‘–β„ŽΓ—πΎπ‘–β„Ž,𝑖=1,2,…,𝑁, such thatminπ‘’π‘–β„ŽβˆˆπΎπ‘–β„Žξƒ―π‘ξ“π‘–=1ξ€œπ‘‘π‘–π‘‘π‘–βˆ’1𝑔1ξ€·π©π‘–β„Žξ€Έ+𝑔2ξ€·π‘¦π‘–β„Žξ€Έξ€·π‘’+π‘—π‘–β„Žξƒ°,𝐴(2.19)βˆ’1π©π‘–β„Ž,π―β„Žξ€Έβˆ’ξ€·π‘¦π‘–β„Ž,divπ―β„Žξ€Έ=0,βˆ€π―β„Žβˆˆπ•π‘–β„Ž,𝑑(2.20)π‘‘π‘¦π‘–β„Ž,π‘€β„Žξ€Έ+ξ€·divπ©π‘–β„Ž,π‘€β„Žξ€Έ=𝑓π‘₯,𝑑𝑖+π‘’π‘–β„Ž,π‘€β„Žξ€Έ,βˆ€π‘€β„Žβˆˆπ‘Šπ‘–β„Žπ‘¦,(2.21)0β„Ž(π‘₯,0)=π‘¦β„Ž0(π‘₯),βˆ€π‘₯∈Ω.(2.22) It follows that the optimal control problems (2.19)–(2.22) have a unique solution (π©π‘–β„Ž,π‘¦π‘–β„Ž,π‘’π‘–β„Ž),𝑖=1,2,…,𝑁, and that a triplet (π©π‘–β„Ž,π‘¦π‘–β„Ž,π‘’π‘–β„Ž)βˆˆπ•π‘–β„ŽΓ—π‘Šπ‘–β„ŽΓ—πΎπ‘–β„Ž,  𝑖=1,2,…,𝑁, is the solution of (2.19)–(2.22) if and only if there is a costate (πͺβ„Žπ‘–βˆ’1,π‘§β„Žπ‘–βˆ’1)βˆˆπ•π‘–β„ŽΓ—π‘Šπ‘–β„Ž such that (π©π‘–β„Ž,π‘¦π‘–β„Ž,πͺβ„Žπ‘–βˆ’1,π‘§β„Žπ‘–βˆ’1,π‘’π‘–β„Ž)∈(π•π‘–β„ŽΓ—π‘Šπ‘–β„Ž)2Γ—πΎπ‘–β„Ž satisfies the following optimality conditions:ξ€·π΄βˆ’1π©π‘–β„Ž,π―β„Žξ€Έβˆ’ξ€·π‘¦π‘–β„Ž,divπ―β„Žξ€Έ=0,βˆ€π―β„Žβˆˆπ•π‘–β„Ž,𝑑(2.23)π‘‘π‘¦π‘–β„Ž,π‘€β„Žξ€Έ+ξ€·divπ©π‘–β„Ž,π‘€β„Žξ€Έ=𝑓π‘₯,𝑑𝑖+π‘’π‘–β„Ž,π‘€β„Žξ€Έ,βˆ€π‘€β„Žβˆˆπ‘Šπ‘–β„Žπ‘¦,(2.24)0β„Ž(π‘₯,0)=π‘¦β„Ž0𝐴(π‘₯),βˆ€π‘₯∈Ω,(2.25)βˆ’1πͺβ„Žπ‘–βˆ’1,π―β„Žξ€Έβˆ’ξ€·π‘§β„Žπ‘–βˆ’1,divπ―β„Žξ€Έξ€·π‘”=βˆ’ξ…ž1ξ€·π©π‘–β„Žξ€Έ,π―β„Žξ€Έ,βˆ€π―β„Žβˆˆπ•π‘–β„Ž,βˆ’ξ€·π‘‘(2.26)π‘‘π‘§π‘–β„Ž,π‘€β„Žξ€Έ+ξ€·divπͺβ„Žπ‘–βˆ’1,π‘€β„Žξ€Έ=ξ€·π‘”ξ…ž2ξ€·π‘¦π‘–β„Žξ€Έ,π‘€β„Žξ€Έ,βˆ€π‘€β„Žβˆˆπ‘Šπ‘–β„Ž,𝑧(2.27)π‘β„Žξ€·π‘—(π‘₯,𝑇)=0,βˆ€π‘₯∈Ω,(2.28)ξ…žξ€·π‘’π‘–β„Žξ€Έ+π‘§β„Žπ‘–βˆ’1,Μƒπ‘’β„Žβˆ’π‘’π‘–β„Žξ€Έβ‰₯0,βˆ€Μƒπ‘’β„ŽβˆˆπΎπ‘–β„Ž.(2.29)

For 𝑖=1,2,…,𝑁, let π‘Œβ„Ž|(π‘‘π‘–βˆ’1,𝑑𝑖]=π‘‘ξ€·ξ€·π‘–ξ€Έπ‘¦βˆ’π‘‘β„Žπ‘–βˆ’1+ξ€·π‘‘βˆ’π‘‘π‘–βˆ’1ξ€Έπ‘¦π‘–β„Žξ€Έ,π‘Ξ”π‘‘β„Ž|(π‘‘π‘–βˆ’1,𝑑𝑖]=π‘‘ξ€·ξ€·π‘–ξ€Έπ‘§βˆ’π‘‘β„Žπ‘–βˆ’1+ξ€·π‘‘βˆ’π‘‘π‘–βˆ’1ξ€Έπ‘§π‘–β„Žξ€Έ,π‘ƒΞ”π‘‘β„Ž|(π‘‘π‘–βˆ’1,𝑑𝑖]=π‘‘ξ€·ξ€·π‘–ξ€Έπ©βˆ’π‘‘β„Žπ‘–βˆ’1+ξ€·π‘‘βˆ’π‘‘π‘–βˆ’1ξ€Έπ©π‘–β„Žξ€Έ,π‘„Ξ”π‘‘β„Ž|(π‘‘π‘–βˆ’1,𝑑𝑖]=𝑑𝑖πͺβˆ’π‘‘β„Žπ‘–βˆ’1+ξ€·π‘‘βˆ’π‘‘π‘–βˆ’1ξ€Έπͺπ‘–β„Žξ€Έ,π‘ˆΞ”π‘‘β„Ž|(π‘‘π‘–βˆ’1,𝑑𝑖]=π‘’π‘–β„Ž.(2.30)

For any function π‘€βˆˆπΆ(𝐽;𝐿2(Ξ©)), let 𝑀(π‘₯,𝑑)|π‘‘βˆˆ(π‘‘π‘–βˆ’1,𝑑𝑖]ξ€·=𝑀π‘₯,𝑑𝑖,𝑀(π‘₯,𝑑)|π‘‘βˆˆ(π‘‘π‘–βˆ’1,𝑑𝑖]ξ€·=𝑀π‘₯,π‘‘π‘–βˆ’1ξ€Έ.(2.31)

Then the optimality conditions (2.23)–(2.29) can be restated as.ξ‚€π΄βˆ’1ξπ‘ƒβ„Ž,π―β„Žξ‚βˆ’ξ‚€ξπ‘Œβ„Ž,divπ―β„Žξ‚=0,βˆ€π―β„Žβˆˆπ•π‘–β„Žξ€·π‘Œ,(2.32)β„Žπ‘‘,π‘€β„Žξ€Έ+𝑃divβ„Ž,π‘€β„Žξ‚=𝑓+π‘ˆβ„Ž,π‘€β„Žξ‚,βˆ€π‘€β„Žβˆˆπ‘Šπ‘–β„Žπ‘Œ,(2.33)β„Ž(π‘₯,0)=π‘¦β„Ž0𝐴(π‘₯),βˆ€π‘₯∈Ω,(2.34)βˆ’1ξ‚π‘„β„Ž,π―β„Žξ‚βˆ’ξ‚€ξ‚π‘β„Ž,divπ―β„Žξ‚ξ‚€π‘”=βˆ’ξ…ž1ξ‚€ξπ‘ƒβ„Žξ‚,π―β„Žξ‚,βˆ€π―β„Žβˆˆπ•π‘–β„Žβˆ’ξ€·π‘,(2.35)β„Žπ‘‘,π‘€β„Žξ€Έ+𝑄divβ„Ž,π‘€β„Žξ‚=ξ‚€π‘”ξ…ž2ξ‚€ξπ‘Œβ„Žξ‚,π‘€β„Žξ‚,βˆ€π‘€β„Žβˆˆπ‘Šπ‘–β„Žπ‘,(2.36)β„Žξ‚€π‘—(π‘₯,𝑇)=0,βˆ€π‘₯∈Ω,(2.37)ξ…žξ€·π‘ˆβ„Žξ€Έ+ξ‚π‘β„Ž,Μƒπ‘’β„Žβˆ’π‘ˆβ„Žξ‚β‰₯0,βˆ€Μƒπ‘’β„ŽβˆˆπΎπ‘–β„Ž.(2.38)

In the rest of the paper, we will use some intermediate variables. For any control function π‘ˆβ„ŽβˆˆπΎ, we first define the state solution (𝐩(π‘ˆβ„Ž),𝑦(π‘ˆβ„Ž),πͺ(π‘ˆβ„Ž),𝑧(π‘ˆβ„Ž)) which satisfiesξ€·π΄βˆ’1π©ξ€·π‘ˆβ„Žξ€Έξ€Έβˆ’ξ€·π‘¦ξ€·π‘ˆ,π―β„Žξ€Έξ€Έξ€·π‘¦,div𝐯=0,βˆ€π―βˆˆπ•,(2.39)π‘‘ξ€·π‘ˆβ„Žξ€Έξ€Έ+ξ€·ξ€·π‘ˆ,𝑀divπ©β„Žξ€Έξ€Έ=ξ€·,𝑀𝑓+π‘ˆβ„Žξ€Έπ‘¦ξ€·π‘ˆ,𝑀,βˆ€π‘€βˆˆπ‘Š,(2.40)β„Žξ€Έ(π‘₯,0)=𝑦0𝐴(π‘₯),βˆ€π‘₯∈Ω,(2.41)βˆ’1πͺξ€·π‘ˆβ„Žξ€Έξ€Έβˆ’ξ€·π‘§ξ€·π‘ˆ,π―β„Žξ€Έξ€Έξ€·π‘”,div𝐯=βˆ’ξ…ž1ξ€·π©ξ€·π‘ˆβ„Žξ€Έβˆ’ξ€·π‘§ξ€Έξ€Έ,𝐯,βˆ€π―βˆˆπ•,(2.42)π‘‘ξ€·π‘ˆβ„Žξ€Έξ€Έ+ξ€·ξ€·π‘ˆ,𝑀divπͺβ„Žξ€Έξ€Έ=𝑔,π‘€ξ…ž2ξ€·π‘¦ξ€·π‘ˆβ„Žξ€Έπ‘§ξ€·π‘ˆξ€Έξ€Έ,𝑀,βˆ€π‘€βˆˆπ‘Š,(2.43)β„Žξ€Έ(π‘₯,𝑇)=0,βˆ€π‘₯∈Ω.(2.44)

3. A Posteriori Error Estimates

In this section we study a posteriori error estimates of the mixed finite element approximation for the parabolic optimal control problems. Fixed given π‘’βˆˆπΎ, let β„³1, β„³2 be the inverse operators of the state equation (2.6), such that 𝐩(𝑒)=β„³1𝑒 and 𝑦(𝑒)=β„³2𝑒 are the solutions of the state equations (2.6). Similarly, for given π‘ˆβ„ŽβˆˆπΎβ„Ž, π‘ƒβ„Ž(π‘ˆβ„Ž)=β„³1β„Žπ‘ˆβ„Ž, π‘Œβ„Ž(π‘ˆβ„Ž)=β„³2β„Žπ‘ˆβ„Ž are the solutions of the discrete state equation (2.33). Let𝐹(𝑒)=𝑔1ξ€·β„³1π‘ˆξ€Έ+𝑔2ξ€·β„³2π‘ˆξ€ΈπΉ+𝑗(𝑒),β„Žξ€·π‘ˆβ„Žξ€Έ=𝑔1ξ€·β„³1β„Žπ‘ˆβ„Žξ€Έ+𝑔2ξ€·β„³2β„Žπ‘ˆβ„Žξ€Έξ€·π‘ˆ+π‘—β„Žξ€Έ.(3.1) It can be shown thatξ€·πΉξ…žξ€Έ=𝑗(𝑒),π‘£ξ…žξ€Έ,𝐹(𝑒)+𝑧,π‘£ξ…žξ€·π‘ˆβ„Žξ€Έξ€Έ=𝑗,π‘£ξ…žξ€·π‘ˆβ„Žξ€Έξ€·π‘ˆ+π‘§β„Žξ€Έξ€Έ,𝐹,π‘£ξ…žβ„Žξ€·π‘ˆβ„Žξ€Έξ€Έ=𝑗,π‘£ξ…žξ€·π‘ˆβ„Žξ€Έ+ξ‚π‘β„Žξ‚.,𝑣(3.2)

It is clear that 𝐹 and πΉβ„Ž are well defined and continuous on 𝐾 and πΎπ‘–β„Ž. Also the functional πΉβ„Ž can be naturally extended on 𝐾. Then (2.4) and (2.19) can be represented asminπ‘’βˆˆπΎ{𝐹(𝑒)},(3.3)minπ‘ˆβ„ŽβˆˆπΎπ‘–β„Žξ€½πΉβ„Žξ€·π‘ˆβ„Ž.ξ€Έξ€Ύ(3.4)

In many application, 𝐹(β‹…) is uniform convex near the solution 𝑒. The convexity of 𝐹(β‹…) is closely related to the second order sufficient conditions of the optimal control problems, which are assumed in many studies on numerical methods of the problem. For instance, in many applications, 𝑒→𝑔1(β„³1π‘ˆ) and 𝑒→𝑔2(β„³2π‘ˆ) are convex. Then there is a constant 𝑐>0, independent of β„Ž, such thatξ€œπ‘‡0ξ€·πΉξ…ž(𝑒)βˆ’πΉξ…žξ€·π‘ˆβ„Žξ€Έ,π‘’βˆ’π‘ˆβ„Žξ€Έπ‘ˆβ€–β€–β‰₯π‘π‘’βˆ’π‘ˆβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©).(3.5)

Theorem 3.1. Let 𝑒 and π‘ˆβ„Ž be the solutions of (3.3) and (3.4), respectively. Assume that πΎπ‘–β„ŽβŠ‚πΎ. In addition, assume that (πΉξ…žβ„Ž(π‘ˆβ„Ž))|πœβˆˆπ»π‘ (𝜏),forallπœβˆˆπ’―β„Ž,(𝑠=0,1), and there is a π‘£β„ŽβˆˆπΎπ‘–β„Ž such that ||ξ€·πΉξ…žβ„Žξ€·π‘ˆβ„Žξ€Έ,π‘£β„Žξ€Έ||ξ“βˆ’π‘’β‰€πΆπœβˆˆπ’―β„Žβ„Žπœβ€–β€–πΉξ…žβ„Žξ€·π‘ˆβ„Žξ€Έβ€–β€–π»π‘ (𝜏)β€–β€–π‘’βˆ’π‘ˆβ„Žβ€–β€–π‘ πΏ2(𝜏).(3.6) Then one has β€–β€–π‘’βˆ’π‘ˆβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)β‰€πΆπœ‚21β€–β€–+𝐢𝑧(π‘ˆβ„Žξ‚π‘)βˆ’β„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©),(3.7) where πœ‚21=ξ€œπ‘‡0ξ“πœβˆˆπ’―β„Žβ„Žπœ1+π‘ β€–β€–π‘—ξ…ž(π‘ˆβ„Žξ‚π‘)+β„Žβ€–β€–π»1+𝑠1(𝜏).(3.8)

Proof. It follows from (3.3) and (3.4) that ξ€œπ‘‡0ξ€·πΉξ…žξ€Έξ€œ(𝑒),π‘’βˆ’π‘£β‰€0,βˆ€π‘£βˆˆπΎ,𝑇0ξ€·πΉξ…žβ„Žξ€·π‘ˆβ„Žξ€Έ,π‘ˆβ„Žβˆ’π‘£β„Žξ€Έβ‰€0,βˆ€π‘£β„ŽβˆˆπΎπ‘–β„ŽβŠ‚πΎ.(3.9) Then it follows from assumptions (3.5), (3.6) and Schwartz inequality that π‘β€–β€–π‘’βˆ’π‘ˆβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)β‰€ξ€œπ‘‡0ξ€·πΉξ…ž(𝑒)βˆ’πΉξ…žξ€·π‘ˆβ„Žξ€Έ,π‘’βˆ’π‘ˆβ„Žξ€Έβ‰€ξ€œπ‘‡0πΉξ€½ξ€·ξ…žβ„Žξ€·π‘ˆβ„Žξ€Έ,π‘£β„Žξ€Έ+ξ€·πΉβˆ’π‘’ξ…žβ„Žξ€·π‘ˆβ„Žξ€Έβˆ’πΉξ…žξ€·π‘ˆβ„Žξ€Έ,π‘’βˆ’π‘ˆβ„Žξ€œξ€Έξ€Ύβ‰€πΆπ‘‡0⎧βŽͺ⎨βŽͺβŽ©ξ“πœβˆˆπ’―β„Žβ„Žπœ1+π‘ β€–β€–πΉξ…žβ„Žξ€·π‘ˆβ„Žξ€Έβ€–β€–π»1+𝑠𝑠(𝜏)+β€–β€–πΉξ…žβ„Žξ€·π‘ˆβ„Žξ€Έβˆ’πΉξ…žξ€·π‘ˆβ„Žξ€Έβ€–β€–2𝐿2(Ξ©)⎫βŽͺ⎬βŽͺ⎭+𝑐2β€–β€–π‘’βˆ’π‘ˆβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©).(3.10) It is not difficult to show πΉξ…žβ„Žξ€·π‘ˆβ„Žξ€Έ=π‘—ξ…žξ€·π‘ˆβ„Žξ€Έ+ξ‚π‘β„Ž,πΉξ…žξ€·π‘ˆβ„Žξ€Έ=π‘—ξ…žξ€·π‘ˆβ„Žξ€Έξ€·π‘ˆ+π‘§β„Žξ€Έ,(3.11) where 𝑧(π‘ˆβ„Ž) is defined in (2.39)–(2.44). Thanks to (3.11), it is easy to derive β€–β€–πΉξ…žβ„Žξ€·π‘ˆβ„Žξ€Έβˆ’πΉξ…žξ€·π‘ˆβ„Žξ€Έβ€–β€–πΏ2(Ξ©)β€–β€–ξ‚π‘β‰€πΆβ„Žξ€·π‘ˆβˆ’π‘§β„Žξ€Έβ€–β€–πΏ2(Ξ©).(3.12) Then by estimates (3.10) and (3.12) we can prove the requested result (3.7).

In order to estimate the a posteriori error of the mixed finite element approximation solution, we will use the following dual equations:βˆ’πœ‘π‘‘],βˆ’div(π΄βˆ‡πœ‘)=𝐺,π‘₯∈Ω,π‘‘βˆˆ(0,π‘‡πœ‘|πœ•Ξ©πœ“=0,π‘‘βˆˆπ½,πœ‘(π‘₯,𝑇)=0,π‘₯∈Ω,(3.13)π‘‘ξ€·π΄βˆ’divβˆ—ξ€Έ],βˆ‡πœ“=𝐺,π‘₯∈Ω,π‘‘βˆˆ(0,π‘‡πœ“|πœ•Ξ©=0,π‘‘βˆˆπ½,πœ“(π‘₯,0)=0,π‘₯∈Ω.(3.14)

The following well-known stability results are presented in [28].

Lemma 3.2. Let πœ‘ and πœ“ be the solutions of (3.13), and (3.14), respectively. Then, for 𝑣=πœ‘ or 𝑣=πœ“, β€–π‘£β€–πΏβˆž(𝐽;𝐿2(Ξ©))≀𝐢‖𝐺‖𝐿2(𝐽;𝐿2(Ξ©)),β€–βˆ‡π‘£β€–πΏ2(𝐽;𝐿2(Ξ©))≀𝐢‖𝐺‖𝐿2(𝐽;𝐿2(Ξ©)),‖‖𝐷2𝑣‖‖𝐿2(𝐽;𝐿2(Ξ©))≀𝐢‖𝐺‖𝐿2(𝐽;𝐿2(Ξ©)),‖‖𝑣𝑑‖‖𝐿2(𝐽;𝐿2(Ξ©))≀𝐢‖𝐺‖𝐿2(𝐽;𝐿2(Ξ©)),(3.15) where 𝐷2𝑣=πœ•2𝑣/πœ•π‘₯π‘–πœ•π‘₯𝑗,1≀𝑖,𝑗≀2.

Now, we are able to derive the main result.

Theorem 3.3. Let (π‘Œβ„Ž,π‘ƒβ„Ž,π‘β„Ž,π‘„β„Ž,π‘ˆβ„Ž) and (𝑦(π‘ˆβ„Ž),𝐩(π‘ˆβ„Ž),𝑧(π‘ˆβ„Ž),πͺ(π‘ˆβ„Ž),π‘ˆβ„Ž) be the solutions of (2.32)–(2.38) and (2.39)–(2.44). Then, β€–β€–π‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–π‘ƒβ„Žξ€·π‘ˆβˆ’π©β„Žξ€Έβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)β‰€πΆπœ‚22,(3.16) where πœ‚22=β€–β€–π‘Œβ„Žπ‘‘ξπ‘ƒ+divβ„Žβˆ’ξπ‘“βˆ’π‘ˆβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–ξ€·π‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žβ€–β€–ξ€Έξ€Έ(π‘₯,0)2𝐿2(Ξ©)+β€–β€–ξβ€–β€–π‘“βˆ’π‘“2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–β€–ξ‚€ξπ‘Œβ„Žβˆ’π‘Œβ„Žξ‚π‘‘β€–β€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–ξπ‘ƒβ„Žβˆ’π‘ƒβ„Žβ€–β€–2𝐿2(𝐽;𝐻(div;Ξ©)).(3.17)

Proof. Letting πœ‘ be the solution of (3.13) with 𝐺=π‘Œβ„Žβˆ’π‘¦(π‘ˆβ„Ž), we infer β€–β€–π‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)=ξ€œπ‘‡0ξ€·π‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έξ€Έ=ξ€œ,𝐹𝑑𝑑𝑇0ξ€·π‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έ,βˆ’πœ‘π‘‘ξ€Έ=ξ€œβˆ’div(π΄βˆ‡πœ‘)𝑑𝑑𝑇0π‘Œξ€·ξ€·ξ€·β„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έξ€Έπ‘‘ξ€Έβˆ’ξ€·π‘ƒ,πœ‘β„Žξ€·π‘ˆβˆ’π©β„Žξ€Έ+β€–β€–ξ€·π‘Œ,βˆ‡πœ‘ξ€Έξ€Έπ‘‘π‘‘β„Žξ€·π‘ˆβˆ’π‘¦β„Žβ€–β€–ξ€Έξ€Έ(π‘₯,0)2𝐿2(Ξ©)=ξ€œπ‘‡0π‘Œξ€·ξ€·β„Žπ‘‘βˆ’π‘¦π‘‘ξ€·π‘ˆβ„Žξ€Έξ€Έ+𝑃,πœ‘divβ„Žξ€·π‘ˆβˆ’π©β„Ž+β€–β€–ξ€·π‘Œξ€Έξ€Έ,πœ‘ξ€Έξ€Έπ‘‘π‘‘β„Žξ€·π‘ˆβˆ’π‘¦β„Žβ€–β€–ξ€Έξ€Έ(π‘₯,0)2𝐿2(Ξ©).(3.18) Then it follows from (2.39)-(2.40) that β€–β€–π‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)=ξ€œπ‘‡0ξ‚€ξ€·π‘Œβ„Žπ‘‘ξ€Έ+𝑃,πœ‘divβ„Žξ‚βˆ’ξ€·π‘¦,πœ‘π‘‘ξ€·π‘ˆβ„Žξ€Έξ€Έβˆ’ξ€·ξ€·π‘ˆ,πœ‘divπ©β„Žξ€Έξ€Έ+𝑃,πœ‘divβ„Žβˆ’ξπ‘ƒβ„Žξ‚+β€–β€–ξ€·π‘Œ,πœ‘ξ‚ξ‚π‘‘π‘‘β„Žξ€·π‘ˆβˆ’π‘¦β„Ž(β€–β€–ξ€Έξ€Έπ‘₯,0)2𝐿2(Ξ©)=ξ€œπ‘‡0π‘Œξ‚€ξ‚€β„Žπ‘‘ξπ‘ƒ+divβ„Žβˆ’ξπ‘“βˆ’π‘ˆβ„Žξ‚++𝑃,πœ‘π‘“βˆ’π‘“,πœ‘divβ„Žβˆ’ξπ‘ƒβ„Žξ‚+β€–β€–ξ€·π‘Œ,πœ‘ξ‚ξ‚π‘‘π‘‘β„Žξ€·π‘ˆβˆ’π‘¦β„Žβ€–β€–ξ€Έξ€Έ(π‘₯,0)2𝐿2(Ξ©)=ξ€œπ‘‡0π‘Œξ‚€ξ‚€β„Žπ‘‘ξπ‘ƒ+divβ„Žβˆ’ξπ‘“βˆ’π‘ˆβ„Žξ‚++𝑃,πœ‘π‘“βˆ’π‘“,πœ‘divβ„Žβˆ’ξπ‘ƒβ„Žξ‚+β€–β€–ξ€·π‘Œ,πœ‘ξ‚ξ‚π‘‘π‘‘β„Žξ€·π‘ˆβˆ’π‘¦β„Ž(β€–β€–ξ€Έξ€Έπ‘₯,0)2𝐿2(Ξ©)β€–β€–π‘Œβ‰€πΆ(𝛿)β„Žπ‘‘ξπ‘ƒ+divβ„Žβˆ’ξπ‘“βˆ’π‘ˆβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)‖‖‖‖+𝐢(𝛿)π‘“βˆ’π‘“2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)‖‖𝑃+𝐢(𝛿)β„Žβˆ’π‘ƒβ„Žβ€–β€–2𝐿2(𝐽;𝐻(div;Ξ©))β€–β€–ξπ‘Œ+𝐢(𝛿)β„Žβˆ’π‘Œβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–ξ€·π‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žβ€–β€–ξ€Έξ€Έ(π‘₯,0)2𝐿2(Ξ©)+πΆπ›Ώβ€–πœ‘β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©),(3.19) where and after, 𝛿 is an arbitrary positive number, 𝐢(𝛿) is the constant dependent on π›Ώβˆ’1.
Now, we are in the position of estimating the error β€–π‘ƒβ„Žβˆ’π©(π‘ˆβ„Ž)β€–2𝐿2(𝐽;𝐿2(Ξ©)). First, we derive from (2.32)-(2.33) and (2.39)-(2.40) the following useful error equations: ξ‚€π΄βˆ’1ξ‚€ξπ‘ƒβ„Žξ€·π‘ˆβˆ’π©β„Žξ€Έξ‚,π―β„Žξ‚βˆ’ξ‚€ξπ‘Œβ„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έ,divπ―β„Žξ‚ξπ‘Œ=0,(3.20)ξ‚€ξ‚€β„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έξ‚π‘‘,π‘€β„Žξ‚+𝑃divβ„Žξ€·π‘ˆβˆ’π©β„Žξ€Έξ‚,π‘€β„Žξ‚=ξ‚€ξπ‘“βˆ’π‘“,π‘€β„Žξ‚βˆ’π‘Œξ‚€ξ‚€β„Žβˆ’ξπ‘Œβ„Žξ‚π‘‘,π‘€β„Žξ‚,(3.21) where π―β„Žβˆˆπ•β„Ž, π‘€β„Žβˆˆπ‘Šβ„Ž. Choose π‘£β„Ž=ξπ‘ƒβ„Žβˆ’π©(π‘ˆβ„Ž) and π‘€β„Ž=ξπ‘Œβ„Žβˆ’π‘¦(π‘ˆβ„Ž) as the test functions and add the two relations of (3.20)-(3.21) such that ξ‚€π΄βˆ’1ξ‚€ξπ‘ƒβ„Žξ€·π‘ˆβˆ’π©β„Žξ€Έξ‚,ξπ‘ƒβ„Žξ€·π‘ˆβˆ’π©β„Žξ€Έξ‚+ξπ‘Œξ‚€ξ‚€β„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έξ‚π‘‘,ξπ‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έξ‚=ξ‚€ξξπ‘Œπ‘“βˆ’π‘“,β„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έξ‚βˆ’π‘Œξ‚€ξ‚€β„Žβˆ’ξπ‘Œβ„Žξ‚π‘‘,ξπ‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έξ‚.(3.22) Then, using πœ–-Cauchy inequality, we can find an estimate as follows: π‘β€–β€–ξπ‘ƒβ„Žξ€·π‘ˆβˆ’π©β„Žξ€Έβ€–β€–2𝐿2(Ξ©)+ξπ‘Œξ‚€ξ‚€β„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έξ‚π‘‘,ξπ‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έξ‚β€–β€–ξπ‘Œβ‰€πΆβ„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2(Ξ©)‖‖‖‖+πΆπ‘“βˆ’π‘“2𝐿2(Ξ©)β€–β€–β€–ξ‚€ξπ‘Œ+πΆβ„Žβˆ’π‘Œβ„Žξ‚π‘‘β€–β€–β€–2𝐿2(Ξ©).(3.23) Note that ξπ‘Œξ‚€ξ‚€β„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έξ‚π‘‘,ξπ‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έξ‚=12πœ•β€–β€–ξπ‘Œπœ•π‘‘β„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2(Ξ©),(3.24) then, using the assumption on 𝐴, we verify that π‘β€–β€–ξπ‘ƒβ„Žβˆ’π©(π‘ˆβ„Ž)β€–β€–2𝐿2(Ξ©)+12πœ•β€–β€–ξπ‘Œπœ•π‘‘β„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2(Ξ©)β€–β€–ξπ‘Œβ‰€πΆβ„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2(Ξ©)‖‖‖‖+πΆπ‘“βˆ’π‘“2𝐿2(Ξ©)β€–β€–β€–ξ‚€ξπ‘Œ+πΆβ„Žβˆ’π‘Œβ„Žξ‚π‘‘β€–β€–β€–2𝐿2(Ξ©).(3.25) Integrating (3.25) in time and since ξπ‘Œβ„Ž(0)βˆ’π‘¦(π‘ˆβ„Ž)(0)=0, applying Gronwall's lemma, we can easily obtain the following error estimate: β€–β€–ξπ‘ƒβ„Žξ€·π‘ˆβˆ’π©β„Žξ€Έβ€–β€–πΏ2(𝐽;𝐿2(Ξ©))+β€–β€–ξπ‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έβ€–β€–πΏβˆž(𝐽;𝐿2(Ξ©))β€–β€–ξβ€–β€–β‰€πΆπ‘“βˆ’π‘“πΏ2(𝐽;𝐿2(Ξ©))β€–β€–β€–ξ‚€ξπ‘Œ+πΆβ„Žβˆ’π‘Œβ„Žξ‚π‘‘β€–β€–β€–πΏ2(𝐽;𝐿2(Ξ©)).(3.26) Using the triangle inequality and (3.26), we deduce that β€–β€–π‘ƒβ„Žξ€·π‘ˆβˆ’π©β„Žξ€Έβ€–β€–πΏ2(𝐽;𝐿2(Ξ©))ξ‚΅β€–β€–ξβ€–β€–β‰€πΆπ‘“βˆ’π‘“πΏ2(𝐽;𝐿2(Ξ©))+β€–β€–β€–ξ‚€ξπ‘Œβ„Žβˆ’π‘Œβ„Žξ‚π‘‘β€–β€–β€–πΏ2(𝐽;𝐿2(Ξ©))+β€–β€–ξπ‘ƒβ„Žβˆ’π‘ƒβ„Žβ€–β€–πΏ2(𝐽;𝐿2(Ξ©))ξ‚Ά.(3.27) Then letting 𝛿 be small enough, it follows from (3.18)–(3.27) that β€–β€–π‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–π‘ƒβ„Žξ€·π‘ˆβˆ’π©β„Žξ€Έβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)β‰€πΆπœ‚22.(3.28) This proves (3.16).

Similarly, letting πœ“ be the solution of (3.14) with 𝐺=π‘β„Žβˆ’π‘§(π‘ˆβ„Ž), with the aid of (2.26)-(2.27), (2.42)-(2.43), we can conclude the following.

Theorem 3.4. Let (π‘Œβ„Ž,π‘ƒβ„Ž,π‘β„Ž,π‘„β„Ž,π‘ˆβ„Ž) and (𝑦(π‘ˆβ„Ž),𝐩(π‘ˆβ„Ž),𝑧(π‘ˆβ„Ž),πͺ(π‘ˆβ„Ž),π‘ˆβ„Ž) be the solutions of (2.32)–(2.38) and (2.39)–(2.44). Then, β€–β€–π‘β„Žξ€·π‘ˆβˆ’π‘§β„Žξ€Έβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–π‘„β„Žξ€·π‘ˆβˆ’πͺβ„Žξ€Έβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)ξ€·πœ‚β‰€πΆ22+πœ‚23ξ€Έ,(3.29) where πœ‚23=β€–β€–π‘”ξ…ž2ξ‚€ξπ‘Œβ„Žξ‚+π‘β„Žπ‘‘ξ‚€ξ‚π‘„βˆ’divβ„Žξ‚β€–β€–2𝐿2(𝐽;𝐿2(Ξ©))+β€–β€–ξ‚π‘β„Žβˆ’π‘β„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–β€–ξ‚€ξ‚π‘β„Žβˆ’π‘β„Žξ‚π‘‘β€–β€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–π‘Œβ„Žβˆ’ξπ‘Œβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–ξ‚π‘„β„Žβˆ’π‘„β„Žβ€–β€–2𝐿2(𝐽;𝐻(div;Ξ©)).(3.30)

Let (𝐩,𝑦,πͺ,𝑧,𝑒) and (π‘ƒβ„Ž,π‘Œβ„Ž,π‘„β„Ž,π‘β„Ž,π‘ˆβ„Ž) be the solutions of (2.8)–(2.14) and (2.32)–(2.38), respectively. We decompose the errors as follows: π©βˆ’π‘ƒβ„Žξ€·π‘ˆ=π©βˆ’π©β„Žξ€Έξ€·π‘ˆ+π©β„Žξ€Έβˆ’π‘ƒβ„ŽβˆΆ=πœ–1+πœ€1,π‘¦βˆ’π‘Œβ„Žξ€·π‘ˆ=π‘¦βˆ’π‘¦β„Žξ€Έξ€·π‘ˆ+π‘¦β„Žξ€Έβˆ’π‘Œβ„ŽβˆΆ=π‘Ÿ1+𝑒1,πͺβˆ’π‘„β„Žξ€·π‘ˆ=πͺβˆ’πͺβ„Žξ€Έξ€·π‘ˆ+πͺβ„Žξ€Έβˆ’π‘„β„ŽβˆΆ=πœ–2+πœ€2,π‘§βˆ’π‘β„Žξ€·π‘ˆ=π‘§βˆ’π‘§β„Žξ€Έξ€·π‘ˆ+π‘§β„Žξ€Έβˆ’π‘β„ŽβˆΆ=π‘Ÿ2+𝑒2.(3.31)

From (2.8)–(2.13) and (2.39)–(2.44), we derive the error equations:ξ€·π΄βˆ’1πœ–1ξ€Έβˆ’ξ€·π‘Ÿ,𝐯1ξ€Έξ€·π‘Ÿ,div𝐯=0,βˆ€π―βˆˆπ•,(3.32)1𝑑+ξ€·,𝑀divπœ–1ξ€Έ=ξ€·,π‘€π‘’βˆ’π‘ˆβ„Žξ€Έξ€·π΄,𝑀,βˆ€π‘€βˆˆπ‘Š,(3.33)βˆ’1πœ–2ξ€Έβˆ’ξ€·π‘Ÿ,𝐯2𝑔,div𝐯=βˆ’ξ…ž1(𝐩)βˆ’π‘”ξ…ž1ξ€·π©ξ€·π‘ˆβ„Žπ―ξ€Έξ€·π‘Ÿξ€Έξ€Έ,βˆ€π―βˆˆπ•,(3.34)2𝑑+ξ€·,𝑀divπœ–2ξ€Έ=𝑔,π‘€ξ…ž2(𝑦)βˆ’π‘”ξ…ž2ξ€·π‘¦ξ€·π‘ˆβ„Žξ€Έξ€Έξ€Έ,𝑀,βˆ€π‘€βˆˆπ‘Š.(3.35)

Theorem 3.5. There is a constant 𝐢>0, independent of β„Ž, such that β€–β€–πœ–1‖‖𝐿2(𝐽;𝐿2(Ξ©))+β€–β€–π‘Ÿ1‖‖𝐿2(𝐽;𝐿2(Ξ©))β€–β€–β‰€πΆπ‘’βˆ’π‘ˆβ„Žβ€–β€–πΏ2(𝐽;𝐿2(Ξ©)),β€–β€–πœ–(3.36)2‖‖𝐿2(𝐽;𝐿2(Ξ©))+β€–β€–π‘Ÿ2‖‖𝐿2(𝐽;𝐿2(Ξ©))β€–β€–β‰€πΆπ‘’βˆ’π‘ˆβ„Žβ€–β€–πΏ2(𝐽;𝐿2(Ξ©)).(3.37)

Proof. Part I
Choose 𝐯=πœ–1 and 𝑀=π‘Ÿ1 as the test functions and add the two relations of (3.32)-(3.33), then we have ξ€·π΄βˆ’1πœ–1,πœ–1ξ€Έ+ξ€·π‘Ÿ1𝑑,π‘Ÿ1ξ€Έ=ξ€·π‘’βˆ’π‘ˆβ„Ž,π‘Ÿ1ξ€Έ.(3.38) Then, using the Cauchy inequality, we can find an estimate as follows: ξ€·π΄βˆ’1πœ–1,πœ–1ξ€Έ+ξ€·π‘Ÿ1𝑑,π‘Ÿ1ξ€Έξ‚€β€–β€–π‘Ÿβ‰€πΆ1β€–β€–2𝐿2(Ξ©)+β€–β€–π‘’βˆ’π‘ˆβ„Žβ€–β€–2𝐿2(Ξ©).(3.39) Note that ξ€·π‘Ÿ1𝑑,π‘Ÿ1ξ€Έ=12πœ•β€–β€–π‘Ÿπœ•π‘‘1β€–β€–2𝐿2(Ξ©),(3.40) then, using the assumption on 𝐴, we can obtain that β€–β€–πœ–1β€–β€–2𝐿2(Ξ©)+12πœ•β€–β€–π‘Ÿπœ•π‘‘1β€–β€–2𝐿2(Ξ©)ξ‚€β€–β€–π‘Ÿβ‰€πΆ1β€–β€–2𝐿2(Ξ©)+β€–β€–π‘’βˆ’π‘ˆβ„Žβ€–β€–2𝐿2(Ξ©).(3.41) Integrating (3.41) in time and since π‘Ÿ1(0)=0, applying the Gronwall's lemma, we can easily obtain the following error estimate β€–β€–πœ–1β€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–π‘Ÿ1β€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)β€–β€–β‰€πΆπ‘’βˆ’π‘ˆβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©).(3.42) This implies (3.36).

Part II
Similarly, choose 𝑣=πœ–2 and 𝑀=π‘Ÿ2 as the test functions and add the two relations of (3.34)-(3.35), then we can obtain that ξ€·π΄βˆ’1πœ–2,πœ–2ξ€Έ+ξ€·π‘Ÿ2𝑑,π‘Ÿ2ξ€Έ=ξ€·π‘”ξ…ž2(𝑦)βˆ’π‘”ξ…ž2ξ€·π‘¦ξ€·π‘ˆβ„Žξ€Έξ€Έ,π‘Ÿ2ξ€Έβˆ’ξ€·π‘”ξ…ž1(𝐩)βˆ’π‘”ξ…ž1ξ€·π©ξ€·π‘ˆβ„Žξ€Έξ€Έ,πœ–2ξ€Έ.(3.43) Then, using the Cauchy inequality, we can find an estimate as follows: ξ€·π΄βˆ’1πœ–2,πœ–2ξ€Έ+ξ€·π‘Ÿ2𝑑,π‘Ÿ2ξ€Έξ‚€β€–β€–π‘Ÿβ‰€πΆ1β€–β€–2𝐿2(Ξ©)+β€–β€–π‘Ÿ2β€–β€–2𝐿2(Ξ©)+β€–β€–πœ–1β€–β€–2𝐿2(Ξ©)+𝑐2β€–β€–πœ–2β€–β€–2𝐿2(Ξ©).(3.44) Note that ξ€·π‘Ÿ2𝑑,π‘Ÿ2ξ€Έ=12πœ•β€–β€–π‘Ÿπœ•π‘‘2β€–β€–2𝐿2(Ξ©),(3.45) then, using the assumption on 𝐴, we verify that β€–β€–πœ–2β€–β€–2𝐿2(Ξ©)+12πœ•β€–β€–π‘Ÿπœ•π‘‘2β€–β€–2𝐿2(Ξ©)ξ‚€β€–β€–π‘Ÿβ‰€πΆ1β€–β€–2𝐿2(Ξ©)+β€–β€–π‘Ÿ2β€–β€–2𝐿2(Ξ©)+β€–β€–πœ–1β€–β€–2𝐿2(Ξ©).(3.46) Integrating (3.46) in time and since π‘Ÿ2(𝑇)=0, applying the Gronwall's lemma, we can easily obtain the following error estimate β€–β€–πœ–2β€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–π‘Ÿ2β€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)β€–β€–β‰€πΆπ‘’βˆ’π‘ˆβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©).(3.47) Then (3.37) follows from (3.47) and the previous statements immediately.

Collecting Theorems 3.1–3.5, we can derive the following results.

Theorem 3.6. Let (𝐩,𝑦,πͺ,𝑧,𝑒) and (π‘ƒβ„Ž,π‘Œβ„Ž,π‘„β„Ž,π‘β„Ž,π‘ˆβ„Ž) be the solutions of (2.8)–(2.14) and (2.32)–(2.38), respectively. In addition, assume that (π‘—ξ…ž(π‘ˆβ„Žξ‚π‘)+β„Ž)|πœβˆˆπ»π‘ (𝜏),forallπœβˆˆπ’―β„Ž,(𝑠=0,1), and that there is a π‘£β„ŽβˆˆπΎβ„Ž such that |||ξ‚€π‘—ξ…žξ€·π‘ˆβ„Žξ€Έ+ξ‚π‘β„Ž,π‘£β„Žξ‚|||ξ“βˆ’π‘’β‰€πΆπœβˆˆπ’―β„Žβ„Žπœβ€–β€–π‘—ξ…žξ€·π‘ˆβ„Žξ€Έ+ξ‚π‘β„Žβ€–β€–π»π‘ (𝜏)β€–β€–π‘’βˆ’π‘ˆβ„Žβ€–β€–π‘ πΏ2(𝜏).(3.48) Then one has that, forallπ‘‘βˆˆ(0,𝑇], β€–β€–π‘’βˆ’π‘ˆβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–π‘¦βˆ’π‘Œβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–π©βˆ’π‘ƒβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–π‘§βˆ’π‘β„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)+β€–β€–πͺβˆ’π‘„β„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(Ξ©)≀𝐢3𝑖=1πœ‚2𝑖,(3.49) where πœ‚1, πœ‚2, and πœ‚3 are defined in Theorems 3.1–3.4.

4. An Adaptive Algorithm

In the section, we introduce an adaptive algorithm to guide the mesh refine process. A posteriori error estimates which have been derived in Section 3 are used as an error indicator to guide the mesh refinement in adaptive finite element method.

Now, we discuss the adaptive mesh refinement strategy. The general idea is to refine the mesh such that the error indicator like πœ‚ is equally distributed over the computational mesh. Assume that an a posteriori error estimator πœ‚ has the form of πœ‚2=βˆ‘πœπ‘–πœ‚2πœπ‘–, where πœπ‘– is the finite elements. At each iteration, an average quantity of all πœ‚2πœπ‘– is calculated, and each πœ‚2πœπ‘– is then compared with this quantity. The element πœπ‘– is to be refined if πœ‚2πœπ‘– is larger than this quantity. As πœ‚2πœπ‘– represents the total approximation error over πœπ‘–, this strategy makes sure that higher density of nodes is distributed over the area where the error is higher.

Based on this principle, we define an adaptive algorithm of the optimal control problems (2.1) as follows.

Starting from initial triangulations π’―β„Ž0 of Ξ©, we construct a sequence of refined triangulation π’―β„Žπ‘— as follows. Given π’―β„Žπ‘—, we compute the solutions (π‘ƒβ„Ž,π‘Œβ„Ž,π‘„β„Ž,π‘β„Ž,π‘ˆβ„Ž) of the system (2.32)–(2.38) and their error estimator πœ‚2𝜏=ξ€œπ‘‡0ξ“πœβˆˆπ’―β„Žβ„Žπœ1+π‘ β€–β€–π‘—ξ…žξ€·π‘ˆβ„Žξ€Έ+ξ‚π‘β„Žβ€–β€–π»1+𝑠1(𝜏)+β€–β€–π‘Œβ„Žπ‘‘ξπ‘ƒ+divβ„Žβˆ’ξπ‘“βˆ’π‘ˆβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(𝜏)+β€–β€–ξ€·π‘Œβ„Žξ€·π‘ˆβˆ’π‘¦β„Žβ€–β€–ξ€Έξ€Έ(π‘₯,0)2𝐿2(𝜏)+β€–β€–ξβ€–β€–π‘“βˆ’π‘“2𝐿2𝐽;𝐿2ξ€Έ(𝜏)+β€–β€–β€–ξ‚€ξπ‘Œβ„Žβˆ’π‘Œβ„Žξ‚π‘‘β€–β€–β€–2𝐿2𝐽;𝐿2ξ€Έ(𝜏)+β€–β€–ξπ‘ƒβ„Žβˆ’π‘ƒβ„Žβ€–β€–2𝐿2(𝐽;𝐻(div;𝜏))+β€–β€–π‘”ξ…ž2ξ‚€ξπ‘Œβ„Žξ‚+π‘β„Žπ‘‘ξ‚€ξ‚π‘„βˆ’divβ„Žξ‚β€–β€–2𝐿2𝐽;𝐿2ξ€Έ(𝜏)+β€–β€–ξ‚π‘β„Žβˆ’π‘β„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(𝜏)+β€–β€–β€–ξ‚€ξ‚π‘β„Žβˆ’π‘β„Žξ‚π‘‘β€–β€–β€–2𝐿2𝐽;𝐿2ξ€Έ(𝜏)+β€–β€–π‘Œβ„Žβˆ’ξπ‘Œβ„Žβ€–β€–2𝐿2𝐽;𝐿2ξ€Έ(𝜏)+β€–β€–ξ‚π‘„β„Žβˆ’π‘„β„Žβ€–β€–2𝐿2(𝐽;𝐻(div;𝜏)),𝐸𝑗=ξ“πœβˆˆπ’―β„Žπœ‚2𝜏.(4.1)

Then we adopt the following mesh refinement strategy. All the triangles πœβˆˆπ’―β„Žπ‘— satisfying πœ‚2𝜏β‰₯(𝛼𝐸𝑗/𝑛) are divided into four new triangles in π’―β„Žπ‘—+1 by joining the midpoints of the edges, where 𝑛 is the numbers of the elements of π’―β„Žπ‘—, 𝛼 is a given constant. In order to maintain the new triangulation π’―β„Žπ‘—+1 to be regular and conformal, some additional triangles need to be divided into two or four new triangles depending on whether they have one or more neighbor which have refined. Then we obtain the new mesh π’―β„Žπ‘—+1. The above procedure will continue until 𝐸𝑗≀tol, where tol is a given tolerance error.

5. Numerical Example

The purpose of this section is to illustrate our theoretical results by numerical example. Our numerical example is the following optimal control problem:minπ‘’βˆˆπΎβŠ‚π‘ˆξ‚»12ξ€œπ‘‡0ξ‚€β€–β€–π©βˆ’π©π‘‘β€–β€–2+β€–β€–π‘¦βˆ’π‘¦π‘‘β€–β€–2+β€–β€–π‘’βˆ’π‘’0β€–β€–2,𝑦𝑑𝑑(5.1)𝑑+div𝐩=𝑓+𝑒,𝐩=βˆ’βˆ‡π‘¦,π‘₯∈Ω,𝑦|πœ•Ξ©=0,𝑦(π‘₯,0)=0,(5.2)βˆ’π‘§π‘‘+divπͺ=π‘¦βˆ’π‘¦π‘‘ξ€·,πͺ=βˆ’βˆ‡π‘§+π©βˆ’π©π‘‘ξ€Έ,π‘₯∈Ω,𝑧|πœ•Ξ©=0,𝑧(π‘₯,𝑇)=0.(5.3)

In our example, we choose the domain Ξ©=[0,1]Γ—[0,1]. Let Ξ© be partitioned into π’―β„Ž as described Section 2. For the constrained optimization problem,minπ‘’βˆˆπΎπΉ(𝑒),(5.4) where 𝐹(𝑒) is a convex functional on π‘ˆ and 𝐾={π‘’βˆˆπΏ2(Ξ©)βˆΆπ‘’β‰₯0π‘Ž.𝑒.inΩ×𝐽}, the iterative scheme reads (𝑛=0,1,2,…)𝑏𝑒𝑛+(1/2)𝑒,𝑣=𝑏𝑛,π‘£βˆ’πœŒπ‘›ξ€·πΉξ…žξ€·π‘’π‘›ξ€Έξ€Έπ‘’,𝑣,βˆ€π‘£βˆˆπ‘ˆ,𝑛+1=𝑃𝑏𝐾𝑒𝑛+(1/2)ξ€Έ,(5.5) where 𝑏(β‹…,β‹…) is a symmetric and positive definite bilinear form such that there exist constants 𝑐0 and 𝑐1 satisfying||||𝑏(𝑒,𝑣)≀𝑐1β€–π‘’β€–π‘ˆβ€–π‘£β€–π‘ˆ,βˆ€π‘’,π‘£βˆˆπ‘ˆ,𝑏(𝑒,𝑒)β‰₯𝑐0‖𝑒‖2π‘ˆ,(5.6) and the projection operator π‘ƒπ‘πΎπ‘ˆβ†’πΎ is defined. For given π‘€βˆˆπ‘ˆ find π‘ƒπ‘πΎπ‘€βˆˆπΎ such thatπ‘ξ€·π‘ƒπ‘πΎπ‘€βˆ’π‘€,π‘ƒπ‘πΎξ€Έπ‘€βˆ’π‘€=minπ‘’βˆˆπΎπ‘(π‘’βˆ’π‘€,π‘’βˆ’π‘€).(5.7) The bilinear form 𝑏(β‹…,β‹…) provides suitable preconditioning for the projection algorithm. An application of (5.5) to the discretized parabolic optimal control problem yields the following algorithm: 𝑏𝑒𝑛+(1/2),π‘£β„Žξ€Έξ€·π‘’=𝑏𝑛,π‘£β„Žξ€Έβˆ’πœŒπ‘›ξ€œπ‘‡0𝑒𝑛+𝑧𝑛,π‘£β„Žξ€Έ,βˆ€π‘£β„Žβˆˆπ‘ˆβ„Ž,ξ€œπ‘‡0𝐩𝑛,π―β„Žξ€Έβˆ’ξ€·π‘¦π‘›,divπ―β„Žξ€Έξ€Έ=0,βˆ€π―β„Žβˆˆπ•β„Ž,ξ€œπ‘‡0𝑦𝑛𝑑,π‘€β„Žξ€Έ+ξ€·div𝐩𝑛,π‘€β„Ž+𝑦𝑛=ξ€œ(0),𝑀(0)𝑇0𝑓+𝑒𝑛,π‘€β„Žξ€Έ,βˆ€π‘€β„Žβˆˆπ‘Šβ„Ž,ξ€œπ‘‡0πͺ𝑛,π―β„Žξ€Έβˆ’ξ€·π‘§π‘›,divπ―β„Žξ€œξ€Έξ€Έ=βˆ’π‘‡0ξ€·π©π‘›βˆ’π©π‘‘,π‘£β„Žξ€Έ,βˆ€π―β„Žβˆˆπ•β„Ž,ξ€œπ‘‡0ξ€·βˆ’ξ€·π‘§π‘›π‘‘,π‘€β„Žξ€Έ+ξ€·divπͺ𝑛,π‘€β„Ž+𝑧𝑛(𝑇),π‘€β„Ž(ξ€Έ=ξ€œπ‘‡)𝑇0ξ€·π‘¦π‘›βˆ’π‘¦π‘‘,π‘€β„Žξ€Έ,βˆ€π‘€β„Žβˆˆπ‘Šβ„Ž,𝑒𝑛+1=𝑃𝑏𝐾𝑒𝑛+(1/2)ξ€Έ,𝑒𝑛+(1/2),π‘’π‘›βˆˆπ‘ˆβ„Ž.(5.8) The main computational effort is to solve the four state and costate equations and to compute the projection 𝑃𝑏𝐾𝑒𝑛+(1/2). In this paper we use a fast algebraic multigrid solver to solve the state and costate equations. Then it is clear that the key to saving computing time is how to compute 𝑃𝑏𝐾𝑒𝑛+(1/2) efficiently. If one uses the 𝐢0 finite elements to approximate to the control, then one has to solve a global variational inequality, via, for example, semismooth Newton method. The computational load is not trivial. For the piecewise constant elements, πΎβ„Ž={π‘’β„ŽβˆΆπ‘’β„Žβ‰₯0} and 𝑏(𝑒,𝑣)=(𝑒,𝑣)π‘ˆ, then 𝑃𝑏𝐾𝑒𝑛+(1/2)|πœξ€·ξ€·π‘’=max0,avg𝑛+(1/2)ξ€Έ|πœξ€Έ,(5.9) where avg(𝑒𝑛+(1/2))|𝜏 is the average of 𝑒𝑛+(1/2) over 𝜏.

In solving our discretized optimal control problem, we use the preconditioned projection gradient method with 𝑏(𝑒,𝑣)=(𝑒,𝑣)π‘ˆ and a fixed step size 𝜌=0.8. We now briefly describe the solution algorithm to be used for solving the numerical examples in this section.

5.1. Algorithm
(1)Solve the discretized optimization problem with the projection gradient method on the current meshes, and calculate the error estimators πœ‚π‘–.(2)Adjust the meshes using the estimators, and update the solution on new meshes, as described.

Now, we present a numerical example to illustrate our theoretical results.

Example 5.1. We choose the state function by 𝑦π‘₯1,π‘₯2ξ€Έ=sinπœ‹π‘₯1sinπœ‹π‘₯2sinπœ‹π‘‘(5.10) and the function 𝑓(π‘₯1,π‘₯2)=𝑦𝑑+divπ©βˆ’π‘’ with 𝐩π‘₯1,π‘₯2ξ€Έξ€·=βˆ’πœ‹cosπœ‹π‘₯1sinπœ‹π‘₯2sinπœ‹π‘‘,πœ‹sinπœ‹π‘₯1cosπœ‹π‘₯2ξ€Έ,πͺξ€·π‘₯sinπœ‹π‘‘1,π‘₯2ξ€Έ=𝐩𝑑π‘₯1,π‘₯2ξ€Έξ€·π‘₯=𝐩1,π‘₯2ξ€Έ.(5.11)

The costate function can be chosen as𝑧π‘₯1,π‘₯2ξ€Έ=sinπœ‹π‘₯1sinπœ‹π‘₯2sinπœ‹π‘‘.(5.12) It follows from (5.2)-(5.3) that𝑦𝑑π‘₯1,π‘₯2ξ€Έ=𝑦+π‘§π‘‘βˆ’divπͺ.(5.13)

We assume thatξƒ―πœ†=0.5,π‘₯1+π‘₯2>1.0,0.0,π‘₯1+π‘₯2𝑒≀1.0,0ξ€·π‘₯1,π‘₯2ξ€Έ=1βˆ’sinπœ‹π‘₯12βˆ’sinπœ‹π‘₯22+πœ†.(5.14) Thus, the control function is given by𝑒π‘₯1,π‘₯2𝑒=max0ξ€Έβˆ’π‘§,0.(5.15)

In this example, the optimal control has a strong discontinuity, introduced by 𝑒0. The exact solution for the control 𝑒 is plotted in Figure 1. The control function 𝑒 is discretized by piecewise constant functions, whereas the state (𝑦,𝐩) and the costate (𝑧,πͺ) were approximation by the lowest-order Raviart-Thomas mixed finite elements. In Table 1, numerical results of 𝑒, 𝑦, and 𝑧 on uniform and adaptive meshes are presented. It can be founded that the adaptive meshes generated using our error indicators can save substantial computational work, in comparison with the uniform meshes. At the same time, for the discontinuous control variable 𝑒, the accuracy has been improved obviously from the uniform meshes to the adaptive meshes in Table 1.


On uniform mesh On adaptive mesh
𝑒 𝑦 𝑧 𝑒 𝑦 𝑧

Nodes 8097 8097 8097 1089 1393 1393
Sides 239682396823968 2825 3819 3819
Elements 15872 15872 15872 63482423 2423
Dofs 15872 15872 158726348 2423 2423
Total 𝐿 2 error 6 . 9 1 5 e βˆ’ 0 3 4 . 0 6 5 e βˆ’ 3 4 . 0 1 8 𝑒 βˆ’ 3 6 . 5 2 7 e βˆ’ 0 3 4 . 3 4 6 e βˆ’ 3 4 . 3 2 3 e βˆ’ 3

In Figure 2, the adaptive mesh for 𝑒 at 𝑑=0.25 is shown. In the computing, we use πœ‚1βˆ’πœ‚3 as the error indicators in the adaptive finite element method. It can be founded that the mesh adapts well to be neighborhood of the discontinuity, and a higher density of node points is indeed distributed along them.

Acknowledgments

The authors express their thanks to the referees for their helpful suggestions, which led to improvement of the presentation. This work was supported by National Nature Science Foundation under Grant 10971074 and Hunan Provincial Innovation Foundation For Postgraduate CX2009B119.

References

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