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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 808514, 26 pages
http://dx.doi.org/10.1155/2012/808514
Research Article

Adaptive Finite Element Method for Optimal Control Problem Governed by Linear Quasiparabolic Integrodifferential Equations

Wanfang Shen1,2Β and Hua Su2

1School of Mathematics, Shandong University, Jinan 250100, China
2School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China

Received 21 June 2012; Accepted 10 July 2012

Academic Editor: XinguangΒ Zhang

Copyright Β© 2012 Wanfang Shen and Hua Su. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The mathematical formulation for a quadratic optimal control problem governed by a linear quasiparabolic integrodifferential equation is studied. The control constrains are given in an integral sense: π‘ˆπ‘Žπ‘‘βˆ«={π‘’βˆˆπ‘‹;Ξ©π‘ˆπ‘’β©Ύ0,π‘‘βˆˆ[0,𝑇]}. Then the a posteriori error estimates in 𝐿∞(0,𝑇;𝐻1(Ξ©))-norm and 𝐿2(0,𝑇;𝐿2(Ξ©))-norm for both the state and the control approximation are given.

1. Introduction

Integrodifferential equations of quasiparabolic and their control of this nature appear in applications such as biology mechanics, nuclear reaction dynamics, heat conduction in materials with memory, and viscoelasticity. All these models express a conservation of a certain quantity in any moment for any subdomain and the historical accumulation feature in the physical models. This in many applications is the most desirable feature of the approximation method when it comes to numerical solution of the corresponding initial boundary value problem. The existence and uniqueness of the solution of the quasiparabolic Integrodifferential equations has been studied in [1]. Finite element methods for quasiparabolic Integrodifferential equations problems with a smooth kernel have been discussed in Cui [2]. Although there is so much work for the finite element approximation of this problem, to our knowledge, there has been a lack of a posteriori error estimates for finite element approximation of any quasiparabolic Integrodifferential optimal control problem.

The finite element approximation of optimal control problems has been an important topic in engineering design works. There have been extensive theoretical and numerical studies for various optimal control problems, see, for instance, [3–11], although it is impossible to give even a very brief review here. And research on finite element approximation of parabolic optimal control problems can be found in, for example, [12, 13].

Among many finite element methods, the adaptive finite element method based on a posteriori error estimates has become a central theme in scientific and engineering computations for its high efficiency. In order to obtain a numerical solution of acceptable accuracy, it is essential for the adaptive finite element method to use a posteriori error estimate indicators to guide the mesh refinement procedure. We only need refine the area where the error indicators are larger, so that a higher density of nodes are distributed over the area where the solution is difficult to approximate. In this sense, adaptive finite element approximation relies very much on the error indicators used, which are often based on a posteriori error estimates of the solutions.

The purpose of this paper is to derive the a posteriori error estimates for the semidiscrete finite element approximation of a quadratic optimal control problem governed by a linear quasiparabolic Integrodifferential equation, which paves a way to derive the a posteriori error estimates for the full discrete finite element approximation for this control problem and thus to develop its adaptive finite element schemes. We extend the existing techniques and results in [14–16] to the optimal control problem governed by the Integrodifferential equation of quasiparabolic type.

The outline of the paper is as follows. In Section 2, we first briefly introduce the optimal control problem and give the optimality conditions, then construct the finite element approximation schemes for the optimal control problem. In Section 3, we give the a posteriori error bounds in 𝐿∞(0,𝑇;𝐻1(Ξ©))-norm for the control problem. And the a posteriori error bounds in 𝐿2(0,𝑇;𝐿2(Ξ©))-norm for the control problem are derived in Section 4.

2. Optimal Control Problem and Its Finite Element Approximation

Let Ξ© and Ξ©π‘ˆ be bounded convex polygon domains in 𝑅𝑑 with Lipschitz boundary πœ•Ξ© and πœ•Ξ©π‘ˆ. In this paper, we adopt the standard notation π‘Šπ‘š,π‘ž(Ξ©) for Sobolev spaces on Ξ© with norm β€–β‹…β€–π‘š,π‘ž,Ξ©, and seminorm  |β‹…|π‘š,π‘ž,Ξ©. We set π‘Š0π‘š,π‘ž(Ξ©)={π‘€βˆˆπ‘Šπ‘š,π‘ž(Ξ©)βˆΆπ‘€|πœ•Ξ©=0}. We denote π‘Šπ‘š,2(Ξ©)(π‘Š0π‘š,2(Ξ©)) by π»π‘š(Ξ©)(π»π‘š0(Ξ©)), with normβ€‰β€‰β€–β‹…β€–π‘š,Ξ©, and seminorm  |β‹…|π‘š,Ξ©.

We denote by 𝐿𝑠(0,𝑇;π‘Šπ‘š,π‘ž(Ξ©)) the Banach space of all 𝐿𝑠 integrable functions from (0,𝑇) into π‘Šπ‘š,π‘ž(Ξ©) with norm ‖𝑣‖𝐿𝑠(0,𝑇;π‘Šπ‘š,π‘ž(Ξ©))∫=(𝑇0β€–π‘£β€–π‘ π‘Šπ‘š,π‘ž(Ξ©)𝑑𝑑)1/𝑠 for π‘ βˆˆ[1,∞) and the standard modification for 𝑠=∞. Similarly, one can define the spaces 𝐻1(0,𝑇;π‘Šπ‘š,π‘ž(Ξ©)) and πΆπ‘˜(0,𝑇;π‘Šπ‘š,π‘ž(Ξ©)). The details can be found in [17]. In addition, 𝑐 or 𝐢 denotes a general positive constant independent of the mesh size β„Ž.

In the following, we will give semi-discrete finite element approximation schemes for the optimal control problem governed by a linear quasiparabolic Integrodifferential equation.

2.1. Model Problem and Its Weak Formulation

We will take the state space π‘Š=𝐿2(0,𝑇;𝑉) with 𝑉=𝐻10(Ξ©) and the control space 𝑋=𝐿2(0,𝑇;π‘ˆ) with π‘ˆ=𝐿2(Ξ©π‘ˆ). Let the observation space π‘Œ=𝐿2(0,𝑇;𝐻) with 𝐻=𝐿2(Ξ©) and π‘ˆπ‘Žπ‘‘βŠ†π‘‹ a convex subset.

We are interested in the following optimal control problem: minπ‘’βˆˆπ‘ˆπ‘Žπ‘‘βŠ‚π‘‹1𝐽(𝑒,𝑦(𝑒))=2ξ‚»ξ€œπ‘‡0β€–β€–π‘¦βˆ’π‘§π‘‘β€–β€–20,Ξ©ξ€œπ‘‘π‘‘+𝑇0‖𝑒‖20,Ξ©π‘ˆξ‚Ό,𝑑𝑑(2.1) subject to π‘¦π‘‘ξ‚΅βˆ’divπ΄βˆ‡π‘¦π‘‘ξ€œ+π·βˆ‡π‘¦+𝑑0ξ‚Ά],[],𝐢(𝑑,𝜏)βˆ‡π‘¦(π‘₯,𝜏)π‘‘πœ=𝑓+𝐡𝑒,inΩ×(0,𝑇𝑦=0,onπœ•Ξ©Γ—0,𝑇𝑦|𝑑=0=𝑦0,inΞ©,(2.2) where 𝑒 is control, 𝑦 is state, 𝑧𝑑 is the observation, π‘ˆπ‘Žπ‘‘ is a closed convex subset, 𝑓(π‘₯,𝑑)∈𝐿2(0,𝑇;𝐿2(Ξ©)), and 𝑧𝑑 and 𝑦0∈𝐻1(Ξ©) are some suitable functions to be specified later. 𝐡 is a linear bounded operator from 𝐿2(Ξ©π‘ˆ) to 𝐿2(Ξ©) independent of 𝑑. And ξ€·π‘Žπ΄=𝐴(π‘₯)=𝑖,𝑗(β‹…)π‘›Γ—π‘›βˆˆξ‚€πΆβˆžξ‚€Ξ©ξ‚ξ‚π‘›Γ—π‘›ξ€·π‘‘,𝐷=𝐷(π‘₯)=𝑖,𝑗(β‹…)π‘›Γ—π‘›βˆˆξ‚€πΆβˆžξ‚€Ξ©ξ‚ξ‚π‘›Γ—π‘›,(2.3) such that there is a constant 𝑐>0 satisfying that for any vector π‘‹βˆˆπ‘…π‘› as follows: 𝑋𝑑𝐴𝑋β‰₯𝑐‖𝑋‖2𝑅𝑛,𝑋𝑑𝐷𝑋β‰₯𝑐‖𝑋‖2𝑅𝑛,(2.4)𝐢=𝐢(π‘₯,𝑑,𝜏)=(𝑐𝑖,𝑗(π‘₯,𝑑,𝜏))π‘›Γ—π‘›βˆˆ(𝐢∞(0,𝑇;𝐿2(Ξ©))𝑛×𝑛).

Let 𝑓1,𝑓2ξ€Έ=ξ€œΞ©π‘“1𝑓2𝑓,βˆ€1,𝑓2ξ€Έβˆˆπ»Γ—π»,(𝑒,𝑣)π‘ˆ=ξ€œΞ©π‘ˆπ‘’π‘£,βˆ€(𝑒,𝑣)βˆˆπ‘ˆΓ—π‘ˆ,π‘Ž(𝑧,πœ”)=(π΄βˆ‡π‘§,βˆ‡πœ”),𝑑(𝑧,πœ”)=(π·βˆ‡π‘§,βˆ‡πœ”),𝑐(𝑑,𝜏;𝑧,πœ”)=(𝐢(𝑑,𝜏)βˆ‡π‘§,βˆ‡πœ”),βˆ€π‘§,π‘€βˆˆπ‘‰Γ—π‘‰.(2.5) In the case that 𝑓1βˆˆπ‘‰ and 𝑓2βˆˆπ‘‰βˆ—, the dual pair (𝑓1,𝑓2) is understood as βŸ¨π‘“1,𝑓2βŸ©π‘‰Γ—π‘‰βˆ—.

Assume that there are constants 𝑐and 𝐢, such that for all 𝑑 and 𝜏 in [0,𝑇] as follows: (π‘Ž)π‘Ž(𝑧,𝑧)⩾𝑐‖𝑧‖21,Ξ©,||||(b)π‘Ž(𝑧,𝑀)⩽𝐢‖𝑧‖1,Ω‖𝑀‖1,Ξ©,||||𝑑(𝑧,𝑀)⩽𝐢‖𝑧‖1,Ω‖𝑀‖1,Ξ©,(||||c)𝑐(𝑑,𝜏;𝑧,𝑀)⩽𝐢‖𝑧‖1,Ω‖𝑀‖1,Ξ©.(2.6) for any 𝑧 and 𝑀 in 𝑉.

Then the weak form of the state equation reads as 𝑦𝑑𝑦,𝑀+π‘Žπ‘‘ξ€Έξ€œ,𝑀+𝑑(𝑦,𝑀)+𝑑0],𝑐(𝑑,𝜏;𝑦(𝜏),𝑀)π‘‘πœ=(𝑓+𝐡𝑒,𝑀)βˆ€π‘€βˆˆπ‘‰,π‘‘βˆˆ(0,𝑇𝑦|𝑑=0=𝑦0.(2.7) It is well known (see, e.g., [1]) that the above weak formulation has at least one solution inβ€‰β€‰π‘¦βˆˆπ‘Š(0,𝑇)={π‘€βˆˆπΏβˆž(0,𝑇;𝐻1(Ξ©)),π‘€ξ…žπ‘‘βˆˆπΏ2(0,𝑇;𝐻1(Ξ©))}.

Therefore, the weak form of the control problem (2.1) and (2.2) reads as (OCP) minπ‘’βˆˆπ‘ˆπ‘Žπ‘‘ξ€·π‘¦π½(𝑒,𝑦(𝑒)),𝑑𝑦,𝑀+π‘Žπ‘‘ξ€Έξ€œ,𝑀+𝑑(𝑦,𝑀)+𝑑0],𝑐(𝑑,𝜏;𝑦(𝜏),𝑀)π‘‘πœ=(𝑓+𝐡𝑒,𝑀)βˆ€π‘€βˆˆπ‘‰,π‘‘βˆˆ(0,𝑇𝑦|𝑑=0=𝑦0.(2.8) In the following, we first give the existence and uniqueness of the solution of the system (2.8).

Theorem 2.1. Assume that the condition (2.6) (a)–(c) holds. There exists the unique solution (𝑒,𝑦) for the minimization problem (2.8) such that π‘’βˆˆπΏ2(0,𝑇;𝐿2(Ξ©π‘ˆ)), π‘¦βˆˆπΏβˆž(0,𝑇;𝐻1(Ξ©)), and π‘¦π‘‘βˆˆπΏ2(0,𝑇;𝐻1(Ξ©)).

Proof. Let {(𝑒𝑛,𝑦𝑛)}βˆžπ‘›=1 be a minimization sequence for the system (2.8), then the sequence {𝑒𝑛}βˆžπ‘›=1 is bounded in 𝐿2(0,𝑇;𝐿2(Ξ©π‘ˆ)). Thus there is a subsequence of {𝑒𝑛}βˆžπ‘›=1 (still denote by {𝑒𝑛}βˆžπ‘›=1 ) such that 𝑒𝑛 converges to π‘’βˆ— weakly in 𝐿2(0,𝑇;𝐿2(Ξ©π‘ˆ)). For the subsequence {𝑒𝑛}βˆžπ‘›=1, we have 𝑦𝑛𝑑𝑦,𝑀+π‘Žπ‘›π‘‘ξ€Έ,𝑀+𝑑(π‘¦π‘›ξ€œ,𝑀)+𝑑0𝑐(𝑑,𝜏;𝑦𝑛(𝜏),𝑀(𝑑))π‘‘πœ=(𝑓+𝐡𝑒𝑛].,𝑀)βˆ€π‘€βˆˆπ‘‰,π‘‘βˆˆ(0,𝑇(2.9) By setting 𝑀=𝑦𝑛 and integrating from 0 to 𝑑 in (2.9), we give ‖𝑦𝑛(𝑑)β€–21,Ξ©+ξ€œπ‘‘0‖𝑦𝑛‖21,Ξ©ξ‚»β€–β€–π‘¦π‘‘πœβ©½πΆ0β€–β€–1,Ξ©ξ€œ+𝐢𝑑0‖𝑓‖2βˆ’1,Ξ©+‖𝑒𝑛‖20,Ξ©π‘ˆξ‚ξ€œπ‘‘π‘‘+𝑑0ξ€œπœ0‖𝑦(𝑠)β€–21,Ξ©ξ‚Ό.π‘‘π‘ π‘‘πœ(2.10) Applying Gronwall's inequality to (2.10) yields ‖𝑦𝑛‖2πΏβˆžξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)+‖𝑦𝑛‖2𝐿2ξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)‖‖𝑦⩽𝐢0β€–β€–21,Ξ©+ξ€œπ‘‡0‖𝑓‖2βˆ’1,Ξ©+‖𝑒𝑛‖20,Ξ©π‘ˆξ‚ξ‚Ό.(2.11) So {𝑒𝑛}βˆžπ‘›=1 is a bounded set in 𝐿2(0,𝑇;𝐿2(Ξ©π‘ˆ)) and {𝑦𝑛}βˆžπ‘›=1 is a bounded set in 𝐿∞(0,𝑇;𝐻1(Ξ©)). Thus π‘’π‘›βŸΆπ‘’weaklyin𝐿2ξ€·0,𝑇;𝐿2ξ€·Ξ©π‘ˆ,π‘¦ξ€Έξ€Έπ‘›βŸΆπ‘¦weaklyinπΏβˆžξ€·0,𝑇;𝐻1ξ€Έ,𝑦(Ξ©)𝑛(𝑇)βŸΆπ‘¦(𝑇)weaklyin𝐻1(Ξ©).(2.12) Let π‘Š={𝑀;π‘€βˆˆπΏβˆž(0,𝑇;𝐻1(Ξ©)),π‘€ξ…žπ‘‘βˆˆπΏ2(0,𝑇;𝐻1(Ξ©))}.
By integrating time from 0 to 𝑇 in (2.9) and taking limit as π‘›β†’βˆž, we obtain ξ€œ(𝑦(𝑇),𝑀(𝑇))+π‘Ž(𝑦(𝑇),𝑀(𝑇))βˆ’π‘‡0𝑦,π‘€ξ…žπ‘‘ξ€Έξ€·+π‘Žπ‘¦,π‘€ξ…žπ‘‘ξ€Έξ€»+ξ€œ+𝑑(𝑦,𝑀)𝑇0ξ€œπ‘‘0=𝑦𝑐(𝑑,𝜏;𝑦(𝜏),𝑀(𝑑))π‘‘πœπ‘‘π‘‘0𝑦,𝑀(0)+π‘Ž0ξ€Έ+ξ€œ,𝑀(0)𝑇0(𝑓+𝐡𝑒,𝑀),βˆ€π‘€βˆˆπ‘Š.(2.13) Then, 𝑦𝑑𝑦,𝑀+π‘Žπ‘‘ξ€Έξ€œ,𝑀+𝑑(𝑦,𝑀)+𝑑0].𝑐(𝑑,𝜏;𝑦(𝜏),𝑀)π‘‘πœ=(𝑓+𝐡𝑒,𝑀),βˆ€π‘€βˆˆπ‘‰,π‘‘βˆˆ(0,𝑇(2.14) Furthermore, we have ξ€œπ‘‡0𝑦𝑑,𝑦𝑑𝑦+π‘Žπ‘‘,𝑦𝑑+𝑑𝑦,𝑦𝑑+ξ€œπ‘‘0𝑐𝑑,𝜏;𝑦(𝜏),π‘¦π‘‘ξ€Έξ‚Ήξ€œ(𝑑)π‘‘πœπ‘‘π‘‘=𝑇0𝑓+𝐡𝑒,𝑦𝑑.(2.15) Then, we get ξ€œπ‘‡0‖‖𝑦𝑑‖‖21,Ξ©ξ€œβ©½πΆπ‘‡0‖𝑓‖2βˆ’1,Ξ©+‖𝑒‖20,Ξ©π‘ˆ+‖𝑦‖21,Ξ©+ξ€œπ‘‘0‖𝑦‖21,Ξ©ξ‚Ή.π‘‘πœ(2.16) This means π‘¦π‘‘βˆˆπΏ2(0,𝑇;𝐻1(Ξ©)). So (𝑒,𝑦) is one solution of (2.8).
Since βˆ«π‘‡0β€–π‘¦βˆ’π‘§π‘‘β€–20,Ξ© is a convex function on space 𝐿2(0,𝑇;𝐿2(Ξ©)) and ∫(𝛼/2)𝑇0‖𝑒‖20,Ξ©π‘ˆ is a strictly convex function on π‘ˆ, hence 𝐽(𝑒,𝑦(𝑒)) is a strictly convex function on π‘ˆ, so the minimization problem (2.8) has one unique solution.

2.2. Optimality Conditions and Their Finite Element Approximation

By the theory of optimal control problem (see [18]), we can similarly deduce the following optimality conditions of the problem (2.8).

Theorem 2.2. A pair (𝑦,𝑒)∈𝐿2(0,𝑇;𝐻10(Ξ©))×𝐿2(0,𝑇;𝐿2(Ξ©π‘ˆ)) is the solution of the optimal control problem (2.8), if and only if there exists a costate π‘βˆˆπΏ2(0,𝑇;𝐻10(Ξ©)) such that the triple (𝑦,𝑝,𝑒) satisfies the following optimality conditions: 𝑦𝑑𝑦,𝑀+π‘Žπ‘‘ξ€Έξ€œ,𝑀+𝑑(𝑦,𝑀)+𝑑0],𝑐(𝑑,𝜏;𝑦(𝜏),𝑀(𝑑))π‘‘πœ=(𝑓+𝐡𝑒,𝑀)βˆ€π‘€βˆˆπ‘‰,π‘‘βˆˆ(0,𝑇𝑦|𝑑=0=𝑦0;βˆ’ξ€·(2.17)π‘ž,π‘π‘‘ξ€Έξ€·βˆ’π‘Žπ‘ž,π‘π‘‘ξ€Έξ€œ+𝑑(π‘ž,𝑝)+𝑇𝑑𝑐(𝜏,𝑑;π‘ž(𝑑),𝑝(𝜏))π‘‘πœ=π‘¦βˆ’π‘§π‘‘ξ€Έ[,π‘žβˆ€π‘žβˆˆπ‘‰,π‘‘βˆˆ0,𝑇),𝑝|𝑑=π‘‡ξ€œ=0;(2.18)𝑇0𝑒+π΅βˆ—ξ€Έπ‘,π‘£βˆ’π‘’π‘ˆπ‘‘π‘‘β©Ύ0,βˆ€π‘£βˆˆπ‘ˆπ‘Žπ‘‘,(2.19) where π΅βˆ— is the adjoint operator of 𝐡.

Let us consider the semi-discrete finite element approximation of the control problem (2.8). Here, we only consider triangular and conforming elements.

Let Ξ©β„Ž be a polygonal approximation to Ξ© with boundaryβ€‰β€‰πœ•Ξ©β„Ž. Let π‘‡β„Ž be a partitioning of Ξ©β„Ž into disjoint regular 𝑛-simplices 𝜏, so that Ξ©β„Ž=β‹ƒπœβˆˆπ‘‡β„Žπœ. Each element has at most one face on πœ•Ξ©β„Ž, and 𝜏 and πœβ€² have either only one common vertex or a whole edge or face if 𝜏 and πœβ€²βˆˆπ‘‡β„Ž. We further require that π‘ƒπ‘–βˆˆπœ•Ξ©β„Žβ‡’π‘ƒπ‘–βˆˆπœ•Ξ©β€‰β€‰where 𝑃𝑖(𝑖=1,…,𝐽) is the vertex set associated with the triangulation π‘‡β„Ž. As usual, β„Ž denotes the diameter of the triangulation π‘‡β„Ž. For simplicity, we assume that Ξ© is a convex polygon so that Ξ©=Ξ©β„Ž.

Associated with π‘‡β„Ž is a finite-dimensional subspace π‘†β„Ž of 𝐢(Ξ©β„Ž), such that πœ’|𝜏 are polynomials of order π‘š(π‘šβ‰₯1) for all πœ’βˆˆπ‘†β„Ž and πœβˆˆπ‘‡β„Ž. Let π‘‰β„Ž={π‘£β„Žβˆˆπ‘†β„ŽβˆΆπ‘£β„Ž(𝑃𝑖)=0(𝑖=1,…,𝐽)},π‘Šβ„Ž=𝐿2(0,𝑇;π‘‰β„Ž). Note that we do not impose a continuity requirement. It is easy to see that π‘‰β„ŽβŠ‚π‘‰,π‘Šβ„ŽβŠ‚π‘Š.

Let π‘‡β„Žπ‘ˆ be a partitioning of Ξ©β„Žπ‘ˆ into disjoint regular 𝑛-simplices πœπ‘ˆ, so that Ξ©β„Žπ‘ˆ=β‹ƒπœπ‘ˆβˆˆπ‘‡β„Žπ‘ˆπœπ‘ˆ. πœπ‘ˆ and πœξ…žπ‘ˆ have either only one common vertex or a whole edge or face if πœπ‘ˆ and πœξ…žπ‘ˆβˆˆπ‘‡β„Žπ‘ˆ. We further require that π‘ƒπ‘–βˆˆπœ•Ξ©β„Žπ‘ˆβ‡’π‘ƒπ‘–βˆˆπœ•Ξ©π‘ˆ, where 𝑃𝑖(𝑖=1,…,𝐽) is the vertex set associated with the triangulation π‘‡β„Žπ‘ˆ. For simplicity, we again assume that Ξ©π‘ˆ is a convex polygon so that Ξ©π‘ˆ=Ξ©β„Žπ‘ˆ.

Associated with π‘‡β„Žπ‘ˆ is another finite-dimensional subspace π‘ˆβ„Ž of 𝐿2(Ξ©β„Žπ‘ˆ), such that πœ’|πœπ‘ˆ are polynomials of order π‘š(π‘šβ©Ύ0) for all πœ’βˆˆπ‘ˆβ„Ž and πœπ‘ˆβˆˆπ‘‡β„Žπ‘ˆ. Here there is no requirement for the continuity. Let π‘‹β„Ž=𝐿2(0,𝑇;π‘ˆβ„Ž). It is easy to see that π‘‹β„ŽβŠ‚π‘‹. Let β„Žπœ(β„Žπœπ‘ˆ) denote the maximum diameter of the element 𝜏(πœπ‘ˆ) in π‘‡β„Ž(π‘‡β„Žπ‘ˆ).

Due to the limited regularity of the optimal control 𝑒 in general, there will be no advantage in considering higher-order finite element spaces than the piecewise constant space for the control. We therefore only consider the piecewise constant finite element space for the approximation of the control, though higher-order finite element spaces will be used to approximate the state and the co-state. Let 𝑃0(Ξ©) denote all the 0-order polynomial over Ξ©. Therefore, we always take π‘‹β„Ž={π‘’βˆˆπ‘‹βˆΆπ‘’(π‘₯,𝑑)|π‘₯βˆˆπœπ‘ˆβˆˆπ‘ƒ0(πœπ‘ˆ),forallπ‘‘βˆˆ[0,𝑇]}. π‘ˆβ„Žπ‘Žπ‘‘ is a closed convex set in π‘‹β„Ž. For ease of exposition, in this paper, we assume that π‘ˆβ„Žπ‘Žπ‘‘βŠ‚π‘ˆπ‘Žπ‘‘βˆ©π‘‹β„Ž.

Then a possible semi-discrete finite element approximation of (OCP) is as follows (OCP)β„ŽβˆΆminπ‘’β„Žβˆˆπ‘ˆβ„Žπ‘Žπ‘‘π½ξ€·π‘’β„Ž,π‘¦β„Žξ€Έ=12ξ‚»ξ€œπ‘‡0β€–β€–π‘¦β„Žβˆ’π‘§π‘‘β€–β€–20,Ξ©+ξ€œπ‘‡0β€–β€–π‘’β„Žβ€–β€–20,Ξ©π‘ˆξ‚Ό,(2.20) such that ξ‚΅πœ•π‘¦β„Žπœ•π‘‘,π‘€β„Žξ‚Άξ‚΅+π‘Žπœ•π‘¦β„Žπœ•π‘‘,π‘€β„Žξ‚Άξ€·π‘¦+π‘‘β„Ž,π‘€β„Žξ€Έ+ξ€œπ‘‘0𝑐𝑑,𝜏;π‘¦β„Ž(𝜏),π‘€β„Ž(𝑑)π‘‘πœ=𝑓+π΅π‘’β„Ž,π‘€β„Žξ€Έ,βˆ€π‘€β„Žβˆˆπ‘‰β„Ž],𝑦,π‘‘βˆˆ(0,π‘‡β„Ž||𝑑=0=𝑦0β„Ž,(2.21) where π‘¦β„Žβˆˆπ‘Šβ„Ž and 𝑦0β„Žβˆˆπ‘‰β„Ž is the approximation of 𝑦0.

In the same way of proving Theorem 2.1, we can easily prove that the problem (2.20)-(2.21) has a unique solution (π‘¦β„Ž,π‘’β„Ž)βˆˆπ‘Šβ„ŽΓ—π‘ˆβ„Žπ‘Žπ‘‘.

It is well known (see [18]) that a pair (π‘¦β„Ž,π‘’β„Ž)βˆˆπ‘Šβ„ŽΓ—π‘ˆβ„Žπ‘Žπ‘‘ is a solution of (2.20)-(2.21), if and only if there exists a co-state π‘β„Žβˆˆπ‘Šβ„Ž such that the triple (π‘¦β„Ž,π‘β„Ž,π‘’β„Ž) satisfies the following optimality conditions: ξ‚΅πœ•π‘¦β„Žπœ•π‘‘,π‘€β„Žξ‚Άξ‚΅+π‘Žπœ•π‘¦β„Žπœ•π‘‘,π‘€β„Žξ‚Άξ€·π‘¦+π‘‘β„Ž,π‘€β„Žξ€Έ+ξ€œπ‘‘0𝑐𝑑,𝜏;π‘¦β„Ž(𝜏),π‘€β„Ž(𝑑)π‘‘πœ=𝑓+π΅π‘’β„Ž,π‘€β„Žξ€Έ,βˆ€π‘€β„Žβˆˆπ‘‰β„Ž,π‘¦β„Ž|𝑑=0=𝑦0β„Ž,βˆ’ξ‚΅π‘ž(2.22)β„Ž,πœ•π‘β„Žξ‚Άξ‚΅π‘žπœ•π‘‘βˆ’π‘Žβ„Ž,πœ•π‘β„Žξ‚Άξ€·π‘žπœ•π‘‘+π‘‘β„Ž,π‘β„Žξ€Έ+ξ€œπ‘‡π‘‘π‘ξ€·πœ,𝑑;π‘žβ„Ž,π‘β„Žξ€Έξ€·π‘¦(𝜏)π‘‘πœ=β„Žβˆ’π‘§π‘‘,π‘žβ„Žξ€Έ,βˆ€π‘žβ„Žβˆˆπ‘‰β„Ž,π‘β„Ž|𝑑=π‘‡ξ€œ=0,(2.23)𝑇0ξ€·π‘’β„Ž+π΅βˆ—π‘β„Ž,π‘£β„Žβˆ’π‘’β„Žξ€Έπ‘ˆπ‘‘π‘‘β©Ύ0,βˆ€π‘£β„Žβˆˆπ‘ˆβ„Žπ‘Žπ‘‘.(2.24)

The optimality conditions in (2.22)–(2.24) are the semi-discrete approximation to the problem (2.17)–(2.19).

Introduce the local averaging operator πœ‹β„Ž given by ξ€·πœ‹β„Žπ‘€ξ€Έ|πœπ‘ˆβˆ«βˆΆ=πœπ‘ˆπ‘€βˆ«πœπ‘ˆ1,βˆ€πœπ‘ˆβˆˆπ‘‡β„Žπ‘ˆ.(2.25) Then, we have βˆ«Ξ©π‘ˆβˆ«π‘€=Ξ©π‘ˆπœ‹β„Žπ‘€ for any π‘€βˆˆπΏ2(0,𝑇;𝐿2(Ξ©π‘ˆ)),π‘‘βˆˆ[0,𝑇] and (2.24) is equivalent to ξ€œπ‘‡0ξ€·π‘’β„Ž+πœ‹β„Žξ€·π΅βˆ—π‘β„Žξ€Έ,π‘£β„Žβˆ’π‘’β„Žξ€Έπ‘ˆπ‘‘π‘‘β©Ύ0,βˆ€π‘£β„Žβˆˆπ‘ˆβ„Žπ‘Žπ‘‘.(2.26) In the following, we derive the a posteriori error estimates for semi-discrete finite element approximation (2.22)–(2.24), allowing different meshes to be used for the state and the control.

The following lemmas are important in deriving the a posteriori error estimates of residual type.

Lemma 2.3 (see [19]). Let ξπœ‹β„Ž be the standard Lagrange interpolation operator. For π‘š=0 or 1, π‘ž>𝑛/2 and π‘£βˆˆπ‘Š2,π‘ž(Ξ©) as ||π‘£βˆ’ξπœ‹β„Žπ‘£||π‘š,π‘ž,Ξ©β©½πΆβ„Ž2βˆ’π‘š|𝑣|2,π‘ž,Ξ©.(2.27)

Lemma 2.4 (see [20]). Let πœ‹β„Ž be the average interpolation operator defined in (2.25). For π‘š=0 or 1, 1β©½π‘žβ©½βˆž and forallπ‘£βˆˆπ‘Š1,π‘ž(Ξ©β„Ž) as ||π‘£βˆ’πœ‹β„Žπ‘£||π‘š,π‘ž,πœβ©½ξ“β‹‚πœξ…žπœβ‰ βˆ…πΆβ„Žπœ1βˆ’π‘š|𝑣|1,π‘ž,πœβ€².(2.28)

Lemma 2.5 (see [21]). For all π‘£βˆˆπ‘Š1,π‘ž(Ξ©),1β©½π‘ž<∞ as ‖𝑣‖0,π‘ž,πœ•πœξ‚€β„Žβ©½πΆπœβˆ’1/π‘žβ€–π‘£β€–0,π‘ž,𝜏+β„Žπœ1βˆ’1/π‘ž|𝑣|1,π‘ž,πœξ‚.(2.29)

3. A Posteriori Error Estimates in 𝐿∞(0,𝑇;𝐻1(Ξ©))-Norm

In this paper, the control constraints are given in an integral sense as follows: π‘ˆπ‘Žπ‘‘=ξ‚»ξ€œπ‘£βˆˆπ‘‹;Ξ©π‘ˆ[]ξ‚Ό.𝑣⩾0,π‘‘βˆˆ0,𝑇(3.1) The following lemma is the first step to derive the a posteriori error estimates of residual type.

Lemma 3.1. Let (𝑦,𝑝,𝑒) and (π‘¦β„Ž,π‘β„Ž,π‘’β„Ž) be the solutions of (2.17)–(2.19) and (2.22)–(2.24). Then, we have β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€·Ξ©π‘ˆξ€Έξ€Έβ©½πΆπœ‚21‖‖𝑝+πΆβ„Žξ€·π‘’βˆ’π‘β„Žξ€Έβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€Έ(Ξ©),(3.2) where πœ‚21=ξ€œπ‘‡0ξƒ―ξ“πœπ‘ˆξ€œπœπ‘ˆξ€·π΅βˆ—π‘β„Žβˆ’π‘ƒβ„Žξ€·π΅βˆ—π‘β„Žξ€Έξ€Έ2𝑑𝑑,(3.3)π‘ƒβ„Ž is the 𝐿2-projection from 𝐿2(Ξ©) to π‘ˆβ„Ž, and 𝑝(π‘’β„Ž) is defined by the following system: ξ‚€πœ•π‘¦ξ€·π‘’πœ•π‘‘β„Žξ€Έξ‚ξ‚€πœ•,πœ”+π‘Žπ‘¦ξ€·π‘’πœ•π‘‘β„Žξ€Έξ‚ξ€·π‘¦ξ€·π‘’,πœ”+π‘‘β„Žξ€Έξ€Έ+ξ€œ,πœ”π‘‘0𝑐𝑒𝑑,𝜏;π‘¦β„Žξ€Έ(ξ€Έ=ξ€·πœ),πœ”(𝑑)π‘‘πœπ‘“+π΅π‘’β„Žξ€Έπ‘¦ξ€·π‘’,πœ”,βˆ€πœ”βˆˆπ‘‰,β„Žξ€Έ(0)=π‘¦β„Ž0(βˆ’ξ‚€πœ•π‘₯),π‘₯∈Ω,(3.4)π‘ž,π‘ξ€·π‘’πœ•π‘‘β„Žξ€Έξ‚ξ‚€πœ•βˆ’π‘Žπ‘ž,π‘ξ€·π‘’πœ•π‘‘β„Žξ€Έξ‚ξ€·ξ€·π‘’+π‘‘π‘ž,π‘β„Ž+ξ€œξ€Έξ€Έπ‘‡π‘‘π‘ξ€·ξ€·π‘’πœ,𝑑;π‘ž(𝑑),π‘β„Žξ€Έξ€Έ=𝑦𝑒(𝜏)π‘‘πœβ„Žξ€Έβˆ’π‘§π‘‘ξ€Έ,π‘ž,βˆ€π‘žβˆˆπ‘‰.(3.5)

Proof. From (2.19), we have 𝑒,π‘’βˆ’π‘’β„Žξ€Έπ‘ˆξ€·π΅β©½βˆ’βˆ—π‘,π‘’βˆ’π‘’β„Žξ€Έπ‘ˆ.(3.6) Then, by (2.24) and (3.6), we have β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€·Ξ©π‘ˆξ€Έξ€Έ=ξ€œπ‘‡0𝑒,π‘’βˆ’π‘’β„Žξ€Έπ‘ˆβˆ’ξ€·π‘’β„Ž,π‘’βˆ’π‘’β„Žξ€Έπ‘ˆξ€»ξ€œπ‘‘π‘‘=𝑇0βˆ’ξ€·π΅βˆ—π‘+π‘’β„Ž,π‘’βˆ’π‘’β„Žξ€Έπ‘ˆξ€œπ‘‘π‘‘=βˆ’π‘‡0ξ€·π΅βˆ—π‘β„Ž+π‘’β„Ž,π‘’βˆ’π‘£β„Žξ€Έπ‘ˆξ€œπ‘‘π‘‘βˆ’π‘‡0ξ€·π΅βˆ—π‘β„Ž+π‘’β„Ž,π‘£β„Žβˆ’π‘’β„Žξ€Έπ‘ˆ+ξ€œπ‘‘π‘‘π‘‡0ξ€·π΅βˆ—π‘β„Žβˆ’π΅βˆ—π‘ξ€·π‘’β„Žξ€Έ,π‘’βˆ’π‘’β„Žξ€Έπ‘ˆξ€œπ‘‘π‘‘+𝑇0ξ€·π΅βˆ—π‘ξ€·π‘’β„Žξ€Έβˆ’π΅βˆ—π‘,π‘’βˆ’π‘’β„Žξ€Έπ‘ˆπ‘‘π‘‘β©½infπ‘£β„Žβˆˆπ‘ˆβ„Žπ‘Žπ‘‘ξ€œπ‘‡0ξ€·π΅βˆ—π‘β„Ž+π‘’β„Ž,π‘£β„Žξ€Έβˆ’π‘’π‘ˆ+ξ€œπ‘‘π‘‘π‘‡0ξ€·π΅βˆ—ξ€·π‘β„Žξ€·π‘’βˆ’π‘β„Žξ€Έξ€Έ,π‘’βˆ’π‘’β„Žξ€Έπ‘ˆξ€œπ‘‘π‘‘+𝑇0ξ€·π΅βˆ—ξ€·π‘ξ€·π‘’β„Žξ€Έξ€Έβˆ’π‘,π‘’βˆ’π‘’β„Žξ€Έπ‘ˆπ‘‘π‘‘=𝐼1+𝐼2+𝐼3.(3.7) Next, we will estimate 𝐼1,𝐼2,and𝐼3, respectively.
(1) We first estimate 𝐼1. Let π‘ƒβ„Ž be the 𝐿2-projection from 𝐿2(Ξ©) to π‘ˆβ„Ž.
We have ξ€œΞ©π‘ˆξ€·π‘ƒβ„Žξ€Έπ‘£βˆ’π‘£πœ™=0,βˆ€πœ™βˆˆπ‘‹β„Ž,π‘£βˆˆπ‘ˆπ‘Žπ‘‘].,π‘‘βˆˆ(0,𝑇(3.8) Since π‘£βˆˆπ‘ˆπ‘Žπ‘‘, so βˆ«Ξ©π‘ˆπ‘ƒβ„Žπ‘£β©Ύ0, then π‘ƒβ„Žπ‘£βˆˆπ‘ˆβ„Žπ‘Žπ‘‘. So that we can take π‘£β„Ž=π‘ƒβ„Žπ‘’ in 𝐼1.
For given π‘‘βˆˆ(0,𝑇], let π‘’β„Ž=π‘ƒβ„Žξƒ©βˆ’π΅βˆ—π‘β„Žξƒ―βˆ«+max0,Ξ©π‘ˆπ΅βˆ—π‘β„Žβˆ«Ξ©π‘ˆ1ξƒ°ξƒͺ.(3.9) We have π‘’β„Žβˆˆπ‘‹β„Ž. We will show that π‘’β„Ž is the solution of the variational inequality in (2.24) assuming π‘β„Ž is known.
Since βˆ«Ξ©π‘ˆ[π‘ƒβ„Ž(βˆ’π΅βˆ—π‘β„Žβˆ«+max{0,Ξ©π‘ˆπ΅βˆ—π‘β„Ž/βˆ«Ξ©π‘ˆ1})βˆ’(βˆ’π΅βˆ—π‘β„Žβˆ«+max{0,Ξ©π‘ˆπ΅βˆ—π‘β„Ž/βˆ«Ξ©π‘ˆ1})]=0, we have ξ€œΞ©π‘ˆπ‘’β„Žξ€œ=βˆ’Ξ©π‘ˆπ΅βˆ—π‘β„Ž+ξ€œΞ©π‘ˆξƒ―βˆ«max0,Ξ©π‘ˆπ΅βˆ—π‘β„Žβˆ«Ξ©π‘ˆ1ξƒ°=ξƒ―βˆ’βˆ«Ξ©π‘ˆπ΅βˆ—π‘β„Ž,βˆ«Ξ©π‘ˆπ΅βˆ—π‘β„Žβˆ«<0,0,Ξ©π‘ˆπ΅βˆ—π‘β„Žβ©Ύ0.(3.10) Thus, βˆ«Ξ©π‘ˆπ‘’β„Žβ©Ύ0, we have π‘’β„Žβˆˆπ‘ˆβ„Žπ‘Žπ‘‘. Note that for all π‘£β„Žβˆˆπ‘ˆβ„Žπ‘Žπ‘‘,π‘‘βˆˆ(0,𝑇], we have ξ€·π‘’β„Ž+π΅βˆ—π‘β„Ž,π‘£β„Žβˆ’π‘’β„Žξ€Έπ‘ˆ=ξ€œΞ©π‘ˆξƒ¬π‘ƒβ„Žξƒ©βˆ’π΅βˆ—π‘β„Žξƒ―βˆ«+max0,Ξ©π‘ˆπ΅βˆ—π‘β„Žβˆ«Ξ©π‘ˆ1βˆ’ξƒ©ξƒ°ξƒͺβˆ’π΅βˆ—π‘β„Žξƒ―βˆ«+max0,Ξ©π‘ˆπ΅βˆ—π‘β„Žβˆ«Ξ©π‘ˆ1ξƒ―βˆ«ξƒ°ξƒͺ+max0,Ξ©π‘ˆπ΅βˆ—π‘β„Žβˆ«Ξ©π‘ˆ1ξ€·π‘£ξƒ°ξƒ­β„Žβˆ’π‘’β„Žξ€Έ=ξ€œΞ©π‘ˆξƒ―βˆ«max0,Ξ©π‘ˆπ΅βˆ—π‘β„Žβˆ«Ξ©π‘ˆ1ξƒ°ξ€·π‘£β„Žβˆ’π‘’β„Žξ€Έ.(3.11) If βˆ«Ξ©π‘ˆπ΅βˆ—π‘β„Ž<0, then ξ€·π‘’β„Ž+π΅βˆ—π‘β„Ž,π‘£β„Žβˆ’π‘’β„Žξ€Έπ‘ˆ=ξ€œΞ©π‘ˆξ€·π‘£0β‹…β„Žβˆ’π‘’β„Žξ€Έ=0.(3.12) If βˆ«Ξ©π‘ˆπ΅βˆ—π‘β„Žβ©Ύ0, since ξ€œΞ©π‘ˆπ‘’β„Ž=ξ€œΞ©π‘ˆξƒ©βˆ’π΅βˆ—π‘β„Žξƒ―βˆ«+max0,Ξ©π‘ˆπ΅βˆ—π‘β„Žβˆ«Ξ©π‘ˆ1ξƒ°ξƒͺ=0.(3.13) we have ξ€·π‘’β„Ž+π΅βˆ—π‘β„Ž,π‘£β„Žβˆ’π‘’β„Žξ€Έπ‘ˆ=βˆ«Ξ©π‘ˆπ΅βˆ—π‘β„Žβˆ«Ξ©π‘ˆ1ξ€œΞ©π‘ˆξ€·π‘£β„Žβˆ’π‘’β„Žξ€Έ=βˆ«Ξ©π‘ˆπ΅βˆ—π‘β„Žβˆ«Ξ©π‘ˆ1β‹…ξ€œΞ©π‘ˆπ‘£β„Žβ©Ύ0.(3.14) From (3.11)–(3.14), we obtain ξ€·π‘’β„Ž+π΅βˆ—π‘β„Ž,π‘£β„Žβˆ’π‘’β„Žξ€Έπ‘ˆβ©Ύ0,βˆ€π‘£β„Žβˆˆπ‘ˆβ„Žπ‘Žπ‘‘.(3.15) So π‘’β„Ž=π‘ƒβ„Ž(βˆ’π΅βˆ—π‘β„Žβˆ«+max{0,Ξ©π‘ˆπ΅βˆ—π‘β„Ž/βˆ«Ξ©π‘ˆ1}) is the solution of the variational inequality in (2.24) assuming π‘β„Ž is known.
Then, 𝐼1β©½ξ€œπ‘‡0ξ€·π΅βˆ—π‘β„Ž+π‘’β„Ž,π‘ƒβ„Žξ€Έπ‘’βˆ’π‘’π‘ˆ=ξ€œπ‘‘π‘‘π‘‡0ξƒ―ξ“πœπ‘ˆξ€œπœπ‘ˆξƒ¬π‘ƒβ„Žξƒ©βˆ’π΅βˆ—π‘β„Žξƒ―βˆ«+max0,Ξ©π‘ˆπ΅βˆ—π‘β„Žβˆ«Ξ©π‘ˆ1ξƒ°ξƒͺ+π΅βˆ—π‘β„Žξƒ­ξ€·π‘ƒβ„Žξ€Έξƒ°π‘’βˆ’π‘’π‘‘π‘‘.(3.16) Since βˆ«πœπ‘ˆ(π‘ƒβ„Žπ‘’βˆ’π‘’)=0, we have 𝐼1β©½ξ€œπ‘‡0ξƒ―ξ“πœπ‘ˆξ€œπœπ‘ˆξ€·βˆ’π‘ƒβ„Žξ€·π΅βˆ—π‘β„Žξ€Έ+π΅βˆ—π‘β„Žπ‘ƒξ€Έξ€·β„Žξ€Έξƒ°=ξ€œπ‘’βˆ’π‘’π‘‘π‘‘π‘‡0ξƒ―ξ“πœπ‘ˆξ€œπœπ‘ˆξ€·βˆ’π‘ƒβ„Žξ€·π΅βˆ—π‘β„Žξ€Έ+π΅βˆ—π‘β„Žπ‘ƒξ€Έξ€·β„Žξ€·π‘’βˆ’π‘’β„Žξ€Έβˆ’ξ€·π‘’βˆ’π‘’β„Žξƒ°ξ€œξ€Έξ€Έπ‘‘π‘‘β©½πΆ(𝛿)𝑇0ξƒ―ξ“πœπ‘ˆξ€œπœπ‘ˆξ€·βˆ’π‘ƒβ„Žξ€·π΅βˆ—π‘β„Žξ€Έ+π΅βˆ—π‘β„Žξ€Έ2‖‖𝑑𝑑+π›Ώπ‘’βˆ’π‘’β„Žβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€·Ξ©π‘ˆξ€Έξ€Έ=πΆπœ‚21β€–β€–+π›Ώπ‘’βˆ’π‘’β„Žβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€·Ξ©π‘ˆξ€Έξ€Έ.(3.17)(2) Consider 𝐼2=ξ€œπ‘‡0ξ€·π΅βˆ—ξ€·π‘β„Žξ€·π‘’βˆ’π‘β„Žξ€Έξ€Έ,π‘’βˆ’π‘’β„Žξ€Έπ‘ˆβ€–β€–π‘π‘‘π‘‘β©½πΆβ„Žξ€·π‘’βˆ’π‘β„Žξ€Έβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€Έ(Ξ©)β€–β€–+π›Ώπ‘’βˆ’π‘’β„Žβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€·Ξ©π‘ˆξ€Έξ€Έ.(3.18)(3) By (3.4) and (2.17), we have for π‘‘βˆˆ(0,𝑇]ξ‚€πœ•ξ€·ξ€·π‘’πœ•π‘‘π‘¦βˆ’π‘¦β„Žξ‚ξ‚€πœ•ξ€Έξ€Έ,πœ”+π‘Žξ€·ξ€·π‘’πœ•π‘‘π‘¦βˆ’π‘¦β„Žξ‚ξ€·ξ€·π‘’ξ€Έξ€Έ,πœ”+π‘‘π‘¦βˆ’π‘¦β„Žξ€Έξ€Έ+ξ€œ,𝑀𝑑0𝑐𝑒𝑑,𝜏;π‘¦βˆ’π‘¦β„Ž(ξ€Έξ€·π΅ξ€·ξ€Έξ€Έπœ),πœ”(𝑑)π‘‘πœ=π‘’βˆ’π‘’β„Žξ€Έξ€Έ,πœ”,βˆ€πœ”βˆˆπ‘‰,(3.19) and from (3.5) and (2.18), we have βˆ’ξ‚€πœ•π‘ž,ξ€·ξ€·π‘’πœ•π‘‘π‘βˆ’π‘β„Žξ‚ξ‚€πœ•ξ€Έξ€Έβˆ’π‘Žπ‘ž,ξ€·ξ€·π‘’πœ•π‘‘π‘βˆ’π‘β„Žξ‚ξ€·ξ€·π‘’ξ€Έξ€Έ+π‘‘π‘ž,π‘βˆ’π‘β„Ž+ξ€œξ€Έξ€Έπ‘‡π‘‘π‘ξ€·ξ€·ξ€·π‘’πœ,𝑑;π‘ž(𝑑),π‘βˆ’π‘β„Žξ€Έξ€·ξ€·π‘’ξ€Έξ€Έ(𝜏)π‘‘πœ=π‘¦βˆ’π‘¦β„Žξ€Έξ€Έ,π‘ž,βˆ€π‘žβˆˆπ‘‰.(3.20) Then, from (3.19), (3.20), and integrating by part we have 𝐼3=ξ€œπ‘‡0ξ€·π΅βˆ—ξ€·π‘ξ€·π‘’β„Žξ€Έξ€Έβˆ’π‘,π‘’βˆ’π‘’β„Žξ€Έπ‘ˆξ€œπ‘‘π‘‘=𝑇0ξ€·π‘ξ€·π‘’β„Žξ€Έξ€·βˆ’π‘,π΅π‘’βˆ’π‘’β„Žξ€Έξ€Έπ‘ˆ=ξ€œπ‘‘π‘‘π‘‡0πœ•ξ‚ƒξ‚€ξ€·ξ€·π‘’πœ•π‘‘π‘¦βˆ’π‘¦β„Žξ€·π‘’ξ€Έξ€Έ,π‘β„Žξ€Έξ‚ξ‚€πœ•βˆ’π‘+π‘Žξ€·ξ€·π‘’πœ•π‘‘π‘¦βˆ’π‘¦β„Žξ€·π‘’ξ€Έξ€Έ,π‘β„Žξ€Έξ‚ξ€·ξ€·π‘’βˆ’π‘+π‘‘π‘¦βˆ’π‘¦β„Žξ€Έξ€·π‘’,π‘β„Žξ€Έξ€Έ+ξ€œβˆ’π‘π‘‘0𝑐𝑒𝑑,𝜏;π‘¦βˆ’π‘¦β„Žξ€·π‘ξ€·π‘’ξ€Έξ€Έ(𝜏),β„Žξ€Έξ€Έξ€Έξ‚Ή=ξ€œβˆ’π‘(𝑑)π‘‘πœπ‘‘π‘‘π‘‡0ξ‚ƒβˆ’ξ‚€ξ€·π‘’π‘¦βˆ’π‘¦β„Žξ€Έ,πœ•ξ€·π‘ξ€·π‘’πœ•π‘‘β„Žξ€Έξ€Έξ‚ξ‚€ξ€·π‘’βˆ’π‘βˆ’π‘Žπ‘¦βˆ’π‘¦β„Žξ€Έ,πœ•ξ€·π‘ξ€·π‘’πœ•π‘‘β„Žξ€Έξ€Έξ‚ξ€·ξ€·π‘’βˆ’π‘+π‘‘π‘¦βˆ’π‘¦β„Žξ€Έξ€·π‘’,π‘β„Žξ€Έξ€Έ+ξ€œβˆ’π‘π‘‡π‘‘π‘ξ€·ξ€·ξ€·π‘’πœ,𝑑;π‘¦βˆ’π‘¦β„Ž(𝑝𝑒𝑑),β„Žξ€Έξ€Έ(ξ€Έξ‚Ή=ξ€œβˆ’π‘πœ)π‘‘πœπ‘‘π‘‘π‘‡0βˆ’ξ€·ξ€·π‘’π‘¦βˆ’π‘¦β„Žξ€Έξ€·π‘’,π‘¦βˆ’π‘¦β„Žξ€Έξ€Έπ‘‘π‘‘β©½0.(3.21) Following from (3.17)–(3.21), let 𝛿 be small enough as β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€·Ξ©π‘ˆξ€Έξ€Έβ©½πΆπœ‚21‖‖𝑝+πΆβ„Žξ€·π‘’βˆ’π‘β„Žξ€Έβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€Έ(Ξ©).(3.22) This completes the proof.

Lemma 3.2. Let (𝑦,𝑝,𝑒) and (π‘¦β„Ž,π‘β„Ž,π‘’β„Ž) be the solutions of (2.17)–(2.19), and (2.22)–(2.24) respectively. Then, there hold the a posteriori error estimates as β€–β€–π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–2πΏβˆžξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)+β€–β€–β€–πœ•ξ€·π‘¦πœ•π‘‘β„Žξ€·π‘’βˆ’π‘¦β„Žβ€–β€–β€–ξ€Έξ€Έ2𝐿2ξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)+β€–β€–π‘β„Žξ€·π‘’βˆ’π‘β„Žξ€Έβ€–β€–2πΏβˆžξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)+β€–β€–β€–πœ•ξ€·π‘πœ•π‘‘β„Žξ€·π‘’βˆ’π‘β„Žβ€–β€–β€–ξ€Έξ€Έ2𝐿2ξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)⩽𝐢6𝑖=2πœ‚2𝑖,(3.23) where πœ‚22=ξ€œπ‘‡0ξƒ―ξ“πœβ„Ž2πœξ€œπœξ‚΅πœ•π‘β„Žξ‚΅π΄πœ•π‘‘βˆ’divβˆ—βˆ‡πœ•π‘β„Žξ‚Άξ€·π·πœ•π‘‘+divβˆ—βˆ‡π‘β„Žξ€Έ+ξ€œπ‘‡π‘‘ξ€·πΆdivβˆ—(𝜏,𝑑)βˆ‡π‘β„Ž(ξ€Έπœ)π‘‘πœ+π‘¦β„Žβˆ’π‘§π‘‘ξ‚Ά2ξƒ°πœ‚π‘‘πœπ‘‘π‘‘,23=ξ€œπ‘‡0ξ“πœβ„Žπ‘™ξ€œπœ•πœξ‚Έβˆ’ξ‚΅π΄βˆ—βˆ‡πœ•π‘β„Žξ‚Άξ€·π·πœ•π‘‘β‹…π‘›+βˆ—βˆ‡π‘β„Žξ€Έξ€œβ‹…π‘›+π‘‡π‘‘ξ€·πΆβˆ—(𝜏,𝑑)βˆ‡π‘β„Žξ€Έξ‚Ή(𝜏)β‹…π‘›π‘‘πœ2πœ‚π‘‘π‘™π‘‘π‘‘,24=ξ€œπ‘‡0ξƒ―ξ“πœβ„Ž2πœξ€œπœξ‚΅πœ•π‘¦πœ•π‘‘β„Žξ‚΅βˆ’divπ΄βˆ‡πœ•π‘¦β„Žξ‚Άξ€·πœ•π‘‘βˆ’divπ·βˆ‡π‘¦β„Žξ€Έβˆ’ξ€œπ‘‘0ξ€·div𝐢(𝑑,𝜏)βˆ‡π‘¦β„Žξ€Έ(𝜏)π‘‘πœβˆ’π‘“βˆ’π΅π‘’β„Žξ‚Ά2ξƒ°πœ‚π‘‘π‘‘,25=ξ€œπ‘‡0ξ“πœβ„Žπ‘™ξ€œπœ•πœξ‚Έξ‚΅π΄βˆ‡πœ•π‘¦β„Žξ‚Άξ€·πœ•π‘‘β‹…π‘›+π΄βˆ‡π‘¦β„Žξ€Έξ€œβ‹…π‘›+𝑑0𝐢(𝑑,𝜏)βˆ‡π‘¦β„Žξ€Έξ‚Ή(𝜏)β‹…π‘›π‘‘πœ2πœ‚π‘‘π‘™π‘‘π‘‘,26=β€–β€–π‘¦β„Ž0βˆ’π‘¦0β€–β€–21,Ξ©,(3.24) where 𝑙 is a face of an element 𝜏, β„Žπ‘™ is the maximum diameter of 𝑙,  and  [βˆ‡π‘β„Žβ‹…π‘›] and [βˆ‡π‘¦β„Žβ‹…π‘›] are the normal derivative jumps over the interior face 𝑙 defined by ξ€Ίβˆ‡π‘β„Žξ€»β‹…π‘›π‘™=ξ‚€βˆ‡π‘β„Žβ„Ž|𝜏1π‘™βˆ’βˆ‡π‘β„Ž|𝜏2𝑙⋅𝑛,βˆ‡π‘¦β„Žξ€»β‹…π‘›π‘™=ξ‚€βˆ‡π‘¦β„Ž|𝜏1π‘™βˆ’βˆ‡π‘¦β„Ž|𝜏2𝑙⋅𝑛,(3.25) where 𝑛 is the unit normal vector on 𝑙=𝜏1π‘™βˆ©πœ2𝑙 outwards 𝜏1𝑙. For later convenience, one can define [βˆ‡π‘β„Žβ‹…π‘›]𝑙=0 and [βˆ‡π‘¦β„Žβ‹…π‘›]𝑙=0 when π‘™βŠ‚πœ•Ξ©.

Proof. Let ξ«π‘…ξ€·π‘’β„Žξ€Έξ¬ξ‚€πœ•,𝑣=βˆ’π‘£,ξ€·π‘πœ•π‘‘β„Žξ€·π‘’βˆ’π‘β„Žξ‚ξ‚€πœ•ξ€Έξ€Έβˆ’π‘Žπ‘£,ξ€·π‘πœ•π‘‘β„Žξ€·π‘’βˆ’π‘β„Žξ‚ξ€·ξ€Έξ€Έ+𝑑𝑣,π‘β„Žξ€·π‘’βˆ’π‘β„Ž+ξ€œξ€Έξ€Έπ‘‡π‘‘π‘ξ€·ξ€·π‘πœ,𝑑;𝑣(𝑑),β„Žξ€·π‘’βˆ’π‘β„Žξ€Έξ€Έξ€Έ(𝜏)π‘‘πœ,(3.26) and πœ‹β„Ž the average interpolation operator defined as in (2.25) and 𝑒=π‘β„Žβˆ’π‘(π‘’β„Ž). Then, it follows from (2.23) and (3.5) that βˆ’ξ‚€π‘žβ„Ž,πœ•ξ€·π‘πœ•π‘‘β„Žξ€·π‘’βˆ’π‘β„Žξ‚ξ‚€π‘žξ€Έξ€Έβˆ’π‘Žβ„Ž,πœ•ξ€·π‘πœ•π‘‘β„Žξ€·π‘’βˆ’π‘β„Žξ‚ξ€·π‘žξ€Έξ€Έ+π‘‘β„Ž,π‘β„Žξ€·π‘’βˆ’π‘β„Ž+ξ€œξ€Έξ€Έπ‘‡π‘‘π‘ξ€·πœ,𝑑;π‘žβ„Žξ€·π‘(𝑑),β„Žξ€·π‘’βˆ’π‘β„Žξ€Έξ€·π‘¦ξ€Έξ€Έ(𝜏)π‘‘πœ=β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έ,π‘žβ„Žξ€Έ,βˆ€π‘žβ„Žβˆˆπ‘‰β„Ž.(3.27) We have ξ«π‘…ξ€·π‘’β„Žξ€Έξ¬ξ‚€,𝑣=βˆ’π‘£βˆ’πœ‹β„Žπœ•π‘£,ξ€·π‘πœ•π‘‘β„Žξ€·π‘’βˆ’π‘β„Žξ‚ξ‚€ξ€Έξ€Έβˆ’π‘Žπ‘£βˆ’πœ‹β„Žπœ•π‘£,ξ€·π‘πœ•π‘‘β„Žξ€·π‘’βˆ’π‘β„Žξ‚ξ€·ξ€Έξ€Έ+π‘‘π‘£βˆ’πœ‹β„Žπ‘£,π‘β„Žξ€·π‘’βˆ’π‘β„Ž+ξ€œξ€Έξ€Έπ‘‡π‘‘π‘ξ€·ξ€·πœ,𝑑;π‘£βˆ’πœ‹β„Žπ‘£ξ€Έξ€·π‘(𝑑),β„Žξ€·π‘’βˆ’π‘β„Žξ€Έξ‚€πœ‹ξ€Έξ€Έ(𝜏)π‘‘πœβˆ’β„Žπœ•π‘£,ξ€·π‘πœ•π‘‘β„Žξ€·π‘’βˆ’π‘β„Žξ‚ξ‚€πœ‹ξ€Έξ€Έβˆ’π‘Žβ„Žπœ•π‘£,ξ€·π‘πœ•π‘‘β„Žξ€·π‘’βˆ’π‘β„Žξ‚ξ€·πœ‹ξ€Έξ€Έ+π‘‘β„Žπ‘£,π‘β„Žξ€·π‘’βˆ’π‘β„Ž+ξ€œξ€Έξ€Έπ‘‡π‘‘π‘ξ€·πœ,𝑑;πœ‹β„Žξ€·π‘π‘£(𝑑),β„Žξ€·π‘’βˆ’π‘β„Žξ€Έξ‚΅ξ€Έξ€Έ(𝜏)π‘‘πœ=βˆ’π‘£βˆ’πœ‹β„Žπ‘£,πœ•π‘β„Žξ‚Άξ‚΅πœ•π‘‘βˆ’π‘Žπ‘£βˆ’πœ‹β„Žπ‘£,πœ•π‘β„Žξ‚Άξ€·πœ•π‘‘+π‘‘π‘£βˆ’πœ‹β„Žπ‘£,π‘β„Žξ€Έ+ξ€œπ‘‡π‘‘π‘ξ€·ξ€·πœ,𝑑;π‘£βˆ’πœ‹β„Žπ‘£ξ€Έ(𝑑),π‘β„Žξ€Έβˆ’ξ€·π‘¦ξ€·π‘’(𝜏)π‘‘πœβ„Žξ€Έβˆ’π‘§π‘‘,π‘£βˆ’πœ‹β„Žπ‘£ξ€Έ+ξ€·π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έ,πœ‹β„Žπ‘£ξ€Έ=ξ“πœξ€œπœξ‚΅βˆ’πœ•π‘β„Žξ‚΅π΄πœ•π‘‘+divβˆ—βˆ‡πœ•π‘β„Žξ‚Άξ€·π·πœ•π‘‘βˆ’divβˆ—βˆ‡π‘β„Žξ€Έβˆ’ξ€œπ‘‡π‘‘ξ€·πΆdivβˆ—(𝜏,𝑑)βˆ‡π‘β„Žξ€Έ(𝜏)π‘‘πœβˆ’π‘¦β„Ž+π‘§π‘‘ξ‚ΆΓ—ξ€·π‘£βˆ’πœ‹β„Žπ‘£ξ€Έ+ξ“πœξ€œπœ•πœξ‚Έβˆ’ξ‚΅π΄βˆ—βˆ‡πœ•π‘β„Žξ‚Άξ€·π·πœ•π‘‘β‹…π‘›+βˆ—βˆ‡π‘β„Žξ€Έξ€œβ‹…π‘›+π‘‡π‘‘ξ€·πΆβˆ—(𝜏,𝑑)βˆ‡π‘β„Žξ€Έξ‚ΉΓ—ξ€·(𝜏)β‹…π‘›π‘‘πœπ‘£βˆ’πœ‹β„Žπ‘£ξ€Έ+ξ€·π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έξ€Έβ©½ξƒ―ξ“,π‘£πœξ€œπœβ„Ž2πœξ‚΅βˆ’πœ•π‘β„Žξ‚΅π΄πœ•π‘‘+divβˆ—βˆ‡πœ•π‘β„Žξ‚Άξ€·π·πœ•π‘‘βˆ’divβˆ—βˆ‡π‘β„Žξ€Έβˆ’ξ€œπ‘‡π‘‘ξ€·πΆdivβˆ—(𝜏,𝑑)βˆ‡π‘β„Ž(ξ€Έπœ)π‘‘πœβˆ’π‘¦β„Ž+𝑧𝑑2+ξ“πœξ€œπœ•πœβ„Žπ‘™ξ‚Έβˆ’ξ‚΅π΄βˆ—βˆ‡πœ•π‘β„Žξ‚Άξ€·π·πœ•π‘‘β‹…π‘›+βˆ—βˆ‡π‘β„Žξ€Έξ€œβ‹…π‘›+π‘‡π‘‘ξ€·πΆβˆ—(𝜏,𝑑)βˆ‡π‘β„Žξ€Έξ‚Ή(𝜏)β‹…π‘›π‘‘πœ2ξƒ°1/2×‖𝑣‖1,Ξ©+ξ€·π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έξ€Έ.,𝑣(3.28) Taking 𝑣=π‘β„Žβˆ’π‘(π‘’β„Ž) in (3.28) and from (2.6), we have βˆ’12π‘‘β€–β€–π‘π‘‘π‘‘β„Žβˆ’π‘(π‘’β„Ž)β€–β€–20,Ξ©βˆ’12π‘‘π‘Žξ€·π‘π‘‘π‘‘β„Žξ€·π‘’βˆ’π‘β„Žξ€Έ,π‘β„Žξ€·π‘’βˆ’π‘β„Žβ€–β€–π‘ξ€Έξ€Έ+π‘β„Žξ€·π‘’βˆ’π‘β„Žξ€Έβ€–β€–21,Ξ©β©½ξƒ―ξ“πœξ€œπœβ„Ž2πœξ‚΅βˆ’πœ•π‘β„Žξ‚΅π΄πœ•π‘‘+divβˆ—βˆ‡πœ•π‘β„Žξ‚Άξ€·π·πœ•π‘‘βˆ’divβˆ—βˆ‡π‘β„Žξ€Έβˆ’ξ€œπ‘‡π‘‘ξ€·πΆdivβˆ—(𝜏,𝑑)βˆ‡π‘β„Ž(ξ€Έπœ)π‘‘πœβˆ’π‘¦β„Ž+𝑧𝑑2+ξ“πœξ€œπœ•πœβ„Žπ‘™ξ‚Έβˆ’ξ‚΅π΄βˆ—βˆ‡πœ•π‘β„Žξ‚Άξ€·π·πœ•π‘‘β‹…π‘›+βˆ—βˆ‡π‘β„Žξ€Έξ‚Ήβ‹…π‘›2ξƒ°1/2Γ—β€–β€–π‘β„Žξ€·π‘’βˆ’π‘β„Žξ€Έβ€–β€–1,Ξ©+ξ€·π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έ,π‘β„Žξ€·π‘’βˆ’π‘β„Žβˆ’ξ€œξ€Έξ€Έπ‘‡π‘‘π‘ξ€·ξ€·π‘πœ,𝑑;β„Žξ€·π‘’βˆ’π‘β„Žξ€·π‘ξ€Έξ€Έ(𝑑),β„Žξ€·π‘’βˆ’π‘β„Žξ€Έξ€Έξ€Έ(𝜏)π‘‘πœ.(3.29) Integrating time from 𝑑 to 𝑇 in (3.29) and by Schwartz inequality, Lemmas 2.4 and 2.5, we have 12β€–β€–π‘β„Žξ€·π‘’βˆ’π‘β„Žξ€Έβ€–β€–20,Ω‖‖𝑝+π‘β„Žξ€·π‘’βˆ’π‘β„Žξ€Έβ€–β€–21,Ξ©ξ€œ+π‘π‘‡π‘‘β€–β€–π‘β„Žξ€·π‘’βˆ’π‘β„Žξ€Έβ€–β€–21,Ξ©β©½ξ€œπ‘‘πœπ‘‡π‘‘ξ“πœβ„Ž2πœΓ—ξ€œπœξ‚΅πœ•π‘β„Žξ‚΅π΄πœ•π‘‘βˆ’divβˆ—βˆ‡πœ•π‘β„Žξ‚Άξ€·π·πœ•π‘‘+divβˆ—βˆ‡π‘β„Žξ€Έ+ξ€œπ‘‡πœξ€·πΆdivβˆ—(𝑠,𝜏)βˆ‡π‘β„Žξ€Έ(𝑠)𝑑𝑠+π‘¦β„Žβˆ’π‘§π‘‘ξ‚Ά2+ξ€œπ‘‘πœπ‘‡π‘‘ξ“πœβ„Žπ‘™ξ€œπœ•πœξ‚Έβˆ’ξ‚΅π΄βˆ—βˆ‡πœ•π‘β„Žξ‚Άξ€·π·πœ•π‘‘β‹…π‘›+βˆ—βˆ‡π‘β„Žξ€Έξ€œβ‹…π‘›+π‘‡πœξ€·πΆβˆ—(𝑠,𝜏)βˆ‡π‘β„Ž(𝑠)⋅𝑛𝑑𝑠2ξ€œπ‘‘πœ+π›Ώπ‘‡π‘‘β€–β€–π‘β„Žξ€·π‘’βˆ’π‘β„Žξ€Έβ€–β€–21,Ξ©ξ€œπ‘‘πœ+πΆπ‘‡π‘‘β€–β€–π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–20,Ξ©ξ€œπ‘‘πœ+πΆπ‘‡π‘‘ξ€œπ‘‡πœβ€–β€–ξ€·π‘β„Žξ€·π‘’βˆ’π‘β„Ž(‖‖𝑠)21,Ξ©π‘‘π‘ π‘‘πœ.(3.30) Letting 𝛿 be small enough, we have ξ€œπ‘‡π‘‘β€–β€–π‘β„Žβˆ’π‘(π‘’β„Ž)β€–β€–21,Ξ©ξ€œπ‘‘πœβ©½πΆπ‘‡π‘‘ξ“πœβ„Ž2πœξ€œπœξ‚΅πœ•π‘β„Žξ‚΅π΄πœ•π‘‘βˆ’divβˆ—βˆ‡πœ•π‘β„Žξ‚Άξ€·π·πœ•π‘‘+divβˆ—βˆ‡π‘β„Žξ€Έ+ξ€œπ‘‡πœξ€·πΆdivβˆ—(𝑠,𝜏)βˆ‡π‘β„Žξ€Έ(𝑠)𝑑𝑠+π‘¦β„Žβˆ’π‘§π‘‘ξ‚Ά2ξ€œπ‘‘πœ+πΆπ‘‡π‘‘ξ“πœβ„Žπ‘™ξ€œπœ•πœξ‚Έβˆ’ξ‚΅π΄βˆ—βˆ‡πœ•π‘β„Žξ‚Άξ€·π·πœ•π‘‘β‹…π‘›+βˆ—βˆ‡π‘β„Žξ€Έ+ξ€œβ‹…π‘›π‘‡πœξ€·πΆβˆ—(𝑠,𝜏)βˆ‡π‘β„Ž(𝑠)⋅𝑛𝑑𝑠2ξ€œπ‘‘πœ+πΆπ‘‡π‘‘β€–β€–π‘¦β„Žβˆ’π‘¦(π‘’β„Ž)β€–β€–20,Ξ©ξ€œπ‘‘πœ+πΆπ‘‡π‘‘ξ€œπ‘‡πœβ€–β€–ξ€·π‘β„Žξ€·π‘’βˆ’π‘β„Žβ€–β€–ξ€Έξ€Έ(𝑠)21,Ξ©π‘‘π‘ π‘‘πœ.(3.31) Then, from Gronwall inequality and (3.28)–(3.31) we have β€–β€–π‘β„Žβˆ’π‘(π‘’β„Ž)β€–β€–2𝐿2ξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)β©½πΆπœ‚22+πΆπœ‚23‖‖𝑦+πΆβ„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€Έ(Ξ©).(3.32) Similarly, β€–β€–π‘β„Žξ€·π‘’βˆ’π‘β„Žξ€Έβ€–β€–2𝐿∞(0,𝑇;𝐻1(Ξ©))ξ‚€πœ‚β©½πΆ22+πœ‚23+β€–β€–π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€Έ(Ξ©)ξ‚ξ€œ+𝐢𝑇0ξ€œπ‘‡π‘‘β€–β€–ξ€·π‘β„Žξ€·π‘’βˆ’π‘β„Žβ€–β€–ξ€Έξ€Έ(𝜏)21,Ξ©π‘‘πœπ‘‘π‘‘β©½πΆπœ‚22+πΆπœ‚23‖‖𝑦+πΆβ„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€Έ(Ξ©).(3.33) In the same way of getting (3.32),by setting 𝑣=(πœ•/πœ•π‘‘)(π‘β„Žβˆ’π‘(π‘’β„Ž)) in (3.28), we have β€–β€–β€–πœ•ξ€·π‘πœ•π‘‘β„Žξ€·π‘’βˆ’π‘β„Žβ€–β€–β€–ξ€Έξ€Έ2𝐿2ξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)β©½πΆπœ‚22+πΆπœ‚23‖‖𝑦+πΆβ„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€Έ(Ξ©).(3.34) Similarly analysis for β€–π‘¦β„Žβˆ’π‘¦(π‘’β„Ž)β€–πΏβˆž(0,𝑇;𝐻1(Ξ©)), we let ξ«π‘„ξ€·π‘’β„Žξ€Έξ¬=ξ‚΅πœ•,π‘£πœ•π‘‘ξ€·π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ‚Άξ‚΅πœ•ξ€Έξ€Έ,𝑣+π‘Žπœ•π‘‘ξ€·π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ‚Άξ€·π‘¦ξ€Έξ€Έ,𝑣+π‘‘β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έξ€Έ+ξ€œ,𝑣𝑑0𝑐𝑦𝑑,𝜏;β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έξ€Έξ€Έ(𝜏),𝑣(𝑑)π‘‘πœ.(3.35) From (2.22) and (3.4), we obtain ξ‚€πœ”β„Ž,πœ•ξ€·π‘¦πœ•π‘‘β„Žξ€·π‘’βˆ’π‘¦β„Žξ‚ξ‚€πœ•ξ€Έξ€Έ+π‘Žξ€·π‘¦πœ•π‘‘β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έξ€Έ,πœ”β„Žξ‚ξ€·π‘¦+π‘‘β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έ,πœ”β„Žξ€Έ+ξ€œπ‘‘0𝑐𝑦𝑑,𝜏;β„Žξ€·π‘’βˆ’π‘¦β„Ž(ξ€Έξ€Έπœ),πœ”β„Ž(𝑑)π‘‘πœ=0,βˆ€πœ”β„Žβˆˆπ‘‰β„Ž.(3.36) We have ξ«π‘„ξ€·π‘’β„Žξ€Έξ¬=ξ‚€πœ•,π‘£ξ€·π‘¦πœ•π‘‘β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έξ€Έ,π‘£βˆ’πœ‹β„Žπ‘£ξ‚ξ‚€πœ•+π‘Žξ€·π‘¦πœ•π‘‘β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έξ€Έ,π‘£βˆ’πœ‹β„Žπ‘£ξ‚ξ€·π‘¦+π‘‘β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έ,π‘£βˆ’πœ‹β„Žπ‘£ξ€Έ+ξ€œπ‘‘0𝑐𝑦𝑑,𝜏;β„Žξ€·π‘’βˆ’π‘¦β„Žξ€·ξ€Έξ€Έ(𝜏),π‘£βˆ’πœ‹β„Žπ‘£ξ€Έξ€Έ=ξ‚΅(𝑑)π‘‘πœπœ•π‘¦β„Žπœ•π‘‘,π‘£βˆ’πœ‹β„Žπ‘£ξ‚Άξ‚΅+π‘Žπœ•π‘¦β„Žπœ•π‘‘,π‘£βˆ’πœ‹β„Žπ‘£ξ‚Άξ€·π‘¦+π‘‘β„Ž,π‘£βˆ’πœ‹β„Žπ‘£ξ€Έ+ξ€œπ‘‘0𝑐𝑑,𝜏;π‘¦β„Žξ€·(𝜏),π‘£βˆ’πœ‹β„Žπ‘£ξ€Έξ€Έξ€·(𝑑)π‘‘πœβˆ’π‘“+π΅π‘’β„Ž,π‘£βˆ’πœ‹β„Žπ‘£ξ€Έ=ξ“πœξ€œπœξ‚΅πœ•π‘¦β„Žξ‚΅πœ•π‘‘βˆ’divπ΄βˆ‡πœ•π‘¦β„Žξ‚Άξ€·πœ•π‘‘βˆ’divπ·βˆ‡π‘¦β„Žξ€Έβˆ’ξ€œπ‘‘0ξ€·div𝐢(𝑑,𝜏)βˆ‡π‘¦β„Žξ€Έπ‘‘πœβˆ’π‘“βˆ’π΅π‘’β„Žξ‚Άξ€·π‘£βˆ’πœ‹β„Žπ‘£ξ€Έ+ξ“πœξ€œπœ•πœξ‚Έξ‚΅π΄βˆ‡πœ•π‘¦β„Žξ‚Άξ€·πœ•π‘‘β‹…π‘›+π·βˆ‡π‘¦β„Žξ€Έξ€œβ‹…π‘›+𝑑0𝐢(𝑑,𝜏)βˆ‡π‘¦β„Žξ€Έξ‚Ήξ€·β‹…π‘›π‘‘πœπ‘£βˆ’πœ‹β„Žπ‘£ξ€Έβ©½ξƒ―ξ“πœξ€œπœβ„Ž2πœξ‚΅πœ•π‘¦πœ•π‘‘β„Žξ‚΅βˆ’divπ΄βˆ‡πœ•π‘¦β„Žξ‚Άξ€·πœ•π‘‘βˆ’divπ·βˆ‡π‘¦β„Žξ€Έβˆ’ξ€œπ‘‘0ξ€·div𝐢(𝑑,𝜏)βˆ‡π‘¦β„Žξ€Έπ‘‘πœβˆ’π‘“βˆ’π΅π‘’β„Žξ‚Ά2+ξ“πœξ€œπœ•πœβ„Žπ‘™ξ‚Έξ‚΅π΄βˆ‡πœ•π‘¦β„Žξ‚Άξ€·πœ•π‘‘β‹…π‘›+π·βˆ‡π‘¦β„Žξ€Έξ€œβ‹…π‘›+𝑑0𝐢(𝑑,𝜏)βˆ‡π‘¦β„Žξ€Έξ‚Ήβ‹…π‘›π‘‘πœ2ξƒ°1/2‖𝑣‖1,Ξ©.(3.37) By setting 𝑣=π‘¦β„Žβˆ’π‘¦(π‘’β„Ž) and Swartz inequality, we have 12π‘‘β€–β€–π‘¦π‘‘π‘‘β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–20,Ξ©+12π‘‘π‘Žξ€·π‘¦π‘‘π‘‘β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έ,π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žβ€–β€–π‘¦ξ€Έξ€Έ+π‘β„Žβˆ’π‘¦(π‘’β„Ž)β€–β€–21,Ξ©β©½ξ“πœξ€œπœβ„Ž2πœξ‚΅πœ•π‘¦β„Žξ‚΅πœ•π‘‘βˆ’divπ΄βˆ‡πœ•π‘¦β„Žξ‚Άξ€·πœ•π‘‘βˆ’divπ·βˆ‡π‘¦β„Žξ€Έβˆ’ξ€œπ‘‘0ξ€·div𝐢(𝑑,𝜏)βˆ‡π‘¦β„Žξ€Έπ‘‘πœβˆ’π‘“βˆ’π΅π‘’β„Žξ‚Ά2+ξ“πœξ€œπœ•πœβ„Žπ‘™ξ‚Έξ‚΅π΄βˆ‡πœ•π‘¦β„Žξ‚Άξ€·πœ•π‘‘β‹…π‘›+π·βˆ‡π‘¦β„Žξ€Έξ€œβ‹…π‘›+𝑑0𝐢(𝑑,𝜏)βˆ‡π‘¦β„Žξ€Έξ‚Ήβ‹…π‘›π‘‘πœ2‖‖𝑦+π›Ώβ„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–21,Ξ©βˆ’ξ€œπ‘‘0𝑐𝑦𝑑,𝜏;β„Žξ€·π‘’βˆ’π‘¦β„Žξ€·π‘¦ξ€Έξ€Έ(𝜏),β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έξ€Έξ€Έ(𝑑)π‘‘πœ.(3.38) Integrating time from 0 to 𝑑 in (3.38), we obtain β€–β€–π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–21,Ξ©ξ€œ+𝑐𝑑0β€–β€–π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–21,Ξ©ξƒ―ξ€œπ‘‘πœβ©½πΆπ‘‘0ξ“πœβ„Ž2πœξ€œπœξ‚΅πœ•π‘¦β„Žξ‚΅πœ•π‘‘βˆ’divπ΄βˆ‡πœ•π‘¦β„Žξ‚Άξ€·πœ•π‘‘βˆ’divπ·βˆ‡π‘¦β„Žξ€Έβˆ’ξ€œπœ0ξ€·div𝐢(𝜏,𝑠)βˆ‡π‘¦β„Žξ€Έπ‘‘π‘ βˆ’π‘“βˆ’π΅π‘’β„Žξ‚Ά2+ξ€œπ‘‘πœπ‘‘0ξ“πœβ„Žπ‘™ξ€œπœ•πœξ‚Έξ‚΅π΄βˆ‡πœ•π‘¦β„Žξ‚Άξ€·πœ•π‘‘β‹…π‘›+π·βˆ‡π‘¦β„Žξ€Έξ€œβ‹…π‘›+𝜏0𝐢(𝜏,𝑠)βˆ‡π‘¦β„Žξ€Έξ‚Ήβ‹…π‘›π‘‘π‘ 2ξƒ°ξ€œπ‘‘π‘‘+𝛿𝑑0β€–β€–π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–21,Ξ©ξ€œπ‘‘πœ+𝐢𝑑0ξ€œπœ0β€–β€–π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–21,Ξ©β€–β€–π‘¦π‘‘π‘ π‘‘πœ+𝐢0βˆ’π‘¦β„Ž0β€–β€–21,Ξ©.(3.39) Since 𝛿 is small enough, then from (3.39) and Gronwall inequality, we have ξ€œπ‘‘0β€–β€–π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–21,Ξ©ξ€œπ‘‘π‘‘β©½πΆπ‘‘0ξ“πœβ„Ž2πœξ€œπœξ‚΅πœ•π‘¦β„Žξ‚΅πœ•π‘‘βˆ’divπ΄βˆ‡πœ•π‘¦β„Žξ‚Άξ€·πœ•π‘‘βˆ’divπ·βˆ‡π‘¦β„Žξ€Έβˆ’ξ€œπœ0ξ€·div𝐢(𝜏,𝑠)βˆ‡π‘¦β„Žξ€Έπ‘‘π‘ βˆ’π‘“βˆ’π΅π‘’β„Žξ‚Ά2ξ€œπ‘‘πœ+𝐢𝑑0ξ“πœβ„Žπ‘™ξ€œπœ•πœξ‚Έξ‚΅π΄βˆ‡πœ•π‘¦β„Žξ‚Άξ€·πœ•π‘‘β‹…π‘›+π·βˆ‡π‘¦β„Žξ€Έξ€œβ‹…π‘›+𝜏0𝐢(𝜏,𝑠)βˆ‡π‘¦β„Žξ€Έξ‚Ήβ‹…π‘›π‘‘π‘ 2‖‖𝑦𝑑𝑑+𝐢0βˆ’π‘¦β„Ž0β€–β€–21,Ξ©.(3.40) Then, β€–β€–π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2ξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)ξ€·πœ‚β©½πΆ24+πœ‚25+πœ‚26ξ€Έ,β€–β€–π‘¦β„Žξ€·π‘’βˆ’π‘¦β„Žξ€Έβ€–β€–2πΏβˆžξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)ξ€·πœ‚β©½πΆ24+πœ‚25+πœ‚26ξ€Έ.(3.41) In the same way of getting (3.34), we can similarly obtain β€–β€–β€–πœ•ξ€·π‘¦πœ•π‘‘β„Žξ€·π‘’βˆ’π‘¦β„Žβ€–β€–β€–ξ€Έξ€Έ2𝐿2ξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)ξ€·πœ‚β©½πΆ24+πœ‚25+πœ‚26ξ€Έ.(3.42) Then the desired results (3.23) follow from (3.32)–(3.34) and (3.41)-(3.42).

From Lemmas 3.1 and 3.2, we have the following results.

Theorem 3.3. Let(𝑦,𝑝,𝑒) and (π‘¦β„Ž,π‘β„Ž,π‘’β„Ž) be the solutions of(2.17)–(2.19) and (2.22)–(2.24) respectively. Then, there hold the a posteriori error estimates as β€–β€–π‘’βˆ’π‘’β„Žβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€·Ξ©π‘ˆξ€Έξ€Έ+β€–β€–π‘¦βˆ’π‘¦β„Žβ€–β€–2πΏβˆžξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)+β€–β€–β€–πœ•ξ€·πœ•π‘‘π‘¦βˆ’π‘¦β„Žξ€Έβ€–β€–β€–2𝐿2ξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)+β€–β€–π‘βˆ’π‘β„Žβ€–β€–2πΏβˆžξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)+β€–β€–β€–πœ•ξ€·πœ•π‘‘π‘βˆ’π‘β„Žξ€Έβ€–β€–β€–2𝐿2ξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)⩽𝐢6𝑖=1πœ‚2𝑖,(3.43) where πœ‚21 is defined in Lemma 3.1.

Proof. First,  from (3.27) and (3.36), and [2], we have the following stability results: β€–β€–ξ€·π‘’π‘¦βˆ’π‘¦β„Žξ€Έβ€–β€–2πΏβˆžξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)+β€–β€–β€–πœ•ξ€·ξ€·π‘’πœ•π‘‘π‘¦βˆ’π‘¦β„Žβ€–β€–β€–ξ€Έξ€Έ2𝐿2ξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)β€–β€–β©½πΆπ‘’βˆ’π‘’β„Žβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€·Ξ©π‘ˆξ€Έξ€Έ,β€–β€–ξ€·π‘’π‘βˆ’π‘β„Žξ€Έβ€–β€–2πΏβˆžξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)+β€–β€–β€–πœ•ξ€·ξ€·π‘’πœ•π‘‘π‘βˆ’π‘β„Žβ€–β€–β€–ξ€Έξ€Έ2𝐿2ξ€·0,𝑇;𝐻1ξ€Έ(Ξ©)β€–β€–ξ€·π‘’β©½πΆπ‘¦βˆ’π‘¦β„Žξ€Έβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€Έ(Ξ©)β€–β€–β©½πΆπ‘’βˆ’π‘’β„Žβ€–β€–2𝐿2ξ€·0,𝑇;𝐿2ξ€·Ξ©π‘ˆξ€Έξ€Έ.(3.44) Then, the desired results (3.43) follows from triangle inequality, (3.44) and Lemmas 3.1 and 3.2.
This completes the proof.

4. A Posteriori Error Estimates in 𝐿2(0,𝑇;𝐿2(Ξ©))-Norm

In the following, we will derive the a posteriori error estimates in 𝐿2(0,𝑇;𝐿2(Ξ©))-norm.

We will introduce the following dual equations.

For given 𝐹∈𝐿2(0,𝑇;𝐿2(Ξ©)), we have πœ•πœ™ξ‚΅πœ•π‘‘βˆ’divπ΄βˆ‡πœ•πœ™ξ‚Άξ€œπœ•π‘‘βˆ’div(π·βˆ‡πœ™)βˆ’π‘‘0],πœ™div(𝐢(𝑑,𝜏)βˆ‡πœ™(𝜏))π‘‘πœ=𝐹,(π‘₯,𝑑)βˆˆΞ©Γ—(0,𝑇],πœ•Ξ©=0,π‘‘βˆˆ(0,π‘‡πœ™(π‘₯,0)=0,π‘₯∈Ω,(4.1) and its dual equation βˆ’πœ•πœ“ξ‚΅π΄πœ•π‘‘+divβˆ—βˆ‡πœ•πœ“ξ‚Άξ€·π·πœ•π‘‘βˆ’divβˆ—ξ€Έβˆ’ξ€œβˆ‡πœ“π‘‡π‘‘ξ€·πΆdivβˆ—ξ€Έ],πœ“(𝜏,𝑑)βˆ‡πœ“(𝜏)π‘‘πœ=𝐹,(π‘₯,𝑑)βˆˆΞ©Γ—(0,𝑇],πœ•Ξ©=0,π‘‘βˆˆ(0,π‘‡πœ“(π‘₯,𝑇)=0,π‘₯∈Ω.(4.2) From [1, 2], we have the following stability results.

Lemma 4.1. Assume that Ξ© is a convex domain. Let πœ™ and πœ“ be the solution of (4.1) and (4.2), respectively. Then, β€–πœ™β€–πΏβˆž(0,𝑇;𝐿2(Ξ©))⩽𝐢‖𝐹‖𝐿2(0,𝑇;𝐿2(Ξ©)),β€–βˆ‡πœ™β€–πΏ2(0,𝑇;𝐿2(Ξ©))⩽𝐢‖𝐹‖𝐿2(0,𝑇;𝐿2(Ξ©)),‖‖𝐷2πœ™β€–β€–πΏ2(0,𝑇;𝐿2(Ξ©))⩽𝐢‖𝐹‖𝐿2(0,𝑇;𝐿2(Ξ©)),β€–β€–β€–πœ•πœ™β€–β€–β€–πœ•π‘‘πΏ2(0,𝑇;𝐿2(Ξ©))⩽𝐢‖𝐹‖