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

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: 𝑈_𝑎𝑑={𝑢∈𝑋;∫_Ω𝑈𝑢⩾0,𝑡∈[0,𝑇]}. Then the a posteriori error estimates in 𝐿^∞(0,𝑇;𝐻¹(Ω))-norm and 𝐿²(0,𝑇;𝐿²(Ω))-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(Ω))-norm for the control problem. And the a posteriori error bounds in 𝐿²(0,𝑇;𝐿²(Ω))-norm for the control problem are derived in Section 4.

2. Optimal Control Problem and Its Finite Element Approximation

Let Ω and Ω_𝑈 be bounded convex polygon domains in 𝑅^𝑑 with Lipschitz boundary 𝜕Ω and 𝜕Ω_𝑈. In this paper, we adopt the standard notation 𝑊^𝑚,𝑞(Ω) for Sobolev spaces on Ω with norm ‖⋅‖_𝑚,𝑞,Ω, and seminorm  |⋅|_𝑚,𝑞,Ω. We set 𝑊^0𝑚,𝑞(Ω)={𝑤∈𝑊^𝑚,𝑞(Ω)∶𝑤|_𝜕Ω=0}. We denote 𝑊^𝑚,2(Ω)(𝑊^0𝑚,2(Ω)) by 𝐻^𝑚(Ω)(𝐻^𝑚0(Ω)), with norm  ‖⋅‖_𝑚,Ω, and seminorm  |⋅|_𝑚,Ω.

We denote by 𝐿^𝑠(0,𝑇;𝑊^𝑚,𝑞(Ω)) the Banach space of all 𝐿^𝑠 integrable functions from (0,𝑇) into 𝑊^𝑚,𝑞(Ω) with norm ‖𝑣‖_𝐿^𝑠_(0,𝑇;𝑊^𝑚,𝑞_(Ω))=(∫^𝑇0‖𝑣‖^𝑠𝑊_{𝑚,𝑞(Ω)}𝑑𝑡)^1/𝑠 for 𝑠∈[1,∞) and the standard modification for 𝑠=∞. Similarly, one can define the spaces 𝐻¹(0,𝑇;𝑊^𝑚,𝑞(Ω)) and 𝐶^𝑘(0,𝑇;𝑊^𝑚,𝑞(Ω)). The details can be found in [17]. In addition, 𝑐 or 𝐶 denotes a general positive constant independent of the mesh size ℎ.

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 𝑊=𝐿²(0,𝑇;𝑉) with 𝑉=𝐻¹⁰(Ω) and the control space 𝑋=𝐿²(0,𝑇;𝑈) with 𝑈=𝐿²(Ω_𝑈). Let the observation space 𝑌=𝐿²(0,𝑇;𝐻) with 𝐻=𝐿²(Ω) and 𝑈_𝑎𝑑⊆𝑋 a convex subset.

We are interested in the following optimal control problem: min_{𝑢∈𝑈𝑎𝑑⊂𝑋}𝐽(𝑢,𝑦(𝑢))=12^𝑇0‖‖𝑦−𝑧_𝑑‖‖^20,Ω𝑑𝑡+^𝑇0‖𝑢‖^20,Ω_𝑈𝑑𝑡,(2.1) subject to 𝑦_𝑡−div𝐴∇𝑦_𝑡+𝐷∇𝑦+𝑦^𝑡0=𝐶𝑦0|(,𝑡,_𝑡=0𝜏o)n∇𝑦𝜕(Ω𝑥,×𝜏)[𝑑0𝜏,𝑇]=,𝑓+𝐵𝑢,inΩ×(0,𝑇],=𝑦⁰,inΩ,(2.2) where 𝑢 is control, 𝑦 is state, 𝑧_𝑑 is the observation, 𝑈_𝑎𝑑 is a closed convex subset, 𝑓(𝑥,𝑡)∈𝐿²(0,𝑇;𝐿²(Ω)), and 𝑧_𝑑 and 𝑦⁰∈𝐻¹(Ω) are some suitable functions to be specified later. 𝐵 is a linear bounded operator from 𝐿²(Ω_𝑈) to 𝐿²(Ω) independent of 𝑡. And 𝐴=𝐴(𝑥)=𝑎_𝑖,𝑗(⋅)_𝑛×𝑛∈𝐶^∞Ω^𝑛×𝑛,𝐷=𝐷(𝑥)=𝑑_𝑖,𝑗(⋅)_𝑛×𝑛∈𝐶^∞Ω^𝑛×𝑛,(2.3) such that there is a constant 𝑐>0 satisfying that for any vector 𝑋∈𝑅^𝑛 as follows: 𝑋^𝑡𝐴𝑋≥𝑐‖𝑋‖^2𝑅_𝑛,𝑋^𝑡𝐷𝑋≥𝑐‖𝑋‖^2𝑅_𝑛,(2.4)𝐶=𝐶(𝑥,𝑡,𝜏)=(𝑐_𝑖,𝑗(𝑥,𝑡,𝜏))_𝑛×𝑛∈(𝐶^∞(0,𝑇;𝐿²(Ω))^𝑛×𝑛).

Let 𝑓₁,(𝑢𝑓,₂𝑣)_𝑈=_Ω𝑓₁𝑓₂,∀𝑓₁,𝑓₂∈𝐻×𝐻,𝑎𝑐((𝑡𝑧,,𝜏𝜔;)𝑧,=𝜔()𝐴=∇=(𝑧𝐶,∇(𝑡_Ω𝑈𝜔,𝜏𝑢))𝑣,∇,𝑧,∀𝑑∇(𝑢𝜔(𝑧,)𝑣,,𝜔)∈)∀=𝑈𝑧,(×𝐷𝑤∇𝑈∈𝑧,,𝑉∇×𝜔𝑉).,(2.5) In the case that 𝑓₁∈𝑉 and 𝑓₂∈𝑉^∗, the dual pair (𝑓₁,𝑓₂) is understood as ⟨𝑓₁,𝑓₂⟩_𝑉×𝑉^∗.

Assume that there are constants 𝑐and 𝐶, such that for all 𝑡 and 𝜏 in [0,𝑇] as follows: ((𝑎b))||𝑎𝑎((𝑧𝑧,,𝑧𝑤))⩾||⩽𝑐‖𝐶𝑧‖‖𝑧^21,Ω‖,_1,Ω(c)||𝑐(𝑡,𝜏;𝑧,𝑤)||⩽𝐶‖𝑧‖‖𝑤_1,Ω‖_1,Ω,||𝑑(𝑧,𝑤)||⩽𝐶‖𝑧‖_1,Ω‖𝑤‖_1,Ω,‖𝑤‖_1,Ω.(2.6) for any 𝑧 and 𝑤 in 𝑉.

Then the weak form of the state equation reads as 𝑦_𝑡,𝑤+𝑎𝑦_𝑡,𝑤+𝑑(𝑦,𝑤𝑦)|+_𝑡=0^𝑡0𝑐(𝑡,𝜏;𝑦(𝜏),∀𝑤𝑤)𝑑∈𝜏𝑉=,𝑡(𝑓∈+(0𝐵,𝑢𝑇,𝑤],)=𝑦⁰.(2.7) It is well known (see, e.g., [1]) that the above weak formulation has at least one solution in  𝑦∈𝑊(0,𝑇)={𝑤∈𝐿^∞(0,𝑇;𝐻¹(Ω)),𝑤^𝑡∈𝐿²(0,𝑇;𝐻¹(Ω))}.

Therefore, the weak form of the control problem (2.1) and (2.2) reads as (OCP) 𝑦min_{𝑢∈𝑈𝑎𝑑}𝐽(𝑢,𝑦(𝑢)),_𝑡,𝑤+𝑎𝑦_𝑡,𝑤+𝑑(𝑦,𝑤)+^𝑡0𝑐(𝑡,𝜏;𝑦𝑦|(𝜏_𝑡=0),𝑤)𝑑𝜏=(𝑓+𝐵𝑢,𝑤)∀𝑤∈𝑉,𝑡∈(0,𝑇],=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 (𝑢,𝑦) for the minimization problem (2.8) such that 𝑢∈𝐿²(0,𝑇;𝐿²(Ω_𝑈)), 𝑦∈𝐿^∞(0,𝑇;𝐻¹(Ω)), and 𝑦_𝑡∈𝐿²(0,𝑇;𝐻¹(Ω)).

Proof. Let {(𝑢^𝑛,𝑦^𝑛)}^∞𝑛=1 be a minimization sequence for the system (2.8), then the sequence {𝑢^𝑛}^∞𝑛=1 is bounded in 𝐿²(0,𝑇;𝐿²(Ω_𝑈)). Thus there is a subsequence of {𝑢^𝑛}^∞𝑛=1 (still denote by {𝑢^𝑛}^∞𝑛=1 ) such that 𝑢^𝑛 converges to 𝑢^∗ weakly in 𝐿²(0,𝑇;𝐿²(Ω_𝑈)). For the subsequence {𝑢^𝑛}^∞𝑛=1, we have 𝑦^𝑛𝑡,𝑤+𝑎𝑦^𝑛𝑡,𝑤+𝑑(𝑦^𝑛,𝑤)+^𝑡0𝑐(𝑡,𝜏;𝑦^𝑛(𝜏),𝑤(𝑡∀)𝑤)𝑑∈𝜏=𝑉,(𝑓𝑡∈+𝐵(0𝑢,^𝑛𝑇,𝑤].)(2.9) By setting 𝑤=𝑦^𝑛 and integrating from 0 to 𝑡 in (2.9), we give ‖𝑦^𝑛⩽(𝑡𝐶)‖^21,Ω‖‖+𝑦₀^𝑡0‖𝑦^𝑛‖^21,Ω𝑑𝜏‖‖_1,Ω+𝐶^𝑡0‖𝑓‖^2−1,Ω+‖𝑢^𝑛‖^20,Ω_𝑈𝑑𝑡+^𝑡0^𝜏0‖𝑦(𝑠)‖^21,Ω𝑑𝑠𝑑𝜏.(2.10) Applying Gronwall's inequality to (2.10) yields ‖𝑦^𝑛‖^2𝐿_{∞0,𝑇;𝐻1(Ω)}+‖𝑦^𝑛‖^2𝐿_{20,𝑇;𝐻1(Ω)}⩽𝐶‖‖𝑦⁰‖‖^21,Ω+^𝑇0‖𝑓‖^2−1,Ω+‖𝑢^𝑛‖^20,Ω_𝑈.(2.11) So {𝑢^𝑛}^∞𝑛=1 is a bounded set in 𝐿²(0,𝑇;𝐿²(Ω_𝑈)) and {𝑦^𝑛}^∞𝑛=1 is a bounded set in 𝐿^∞(0,𝑇;𝐻¹(Ω)). Thus 𝑢𝑦^𝑛⟶𝑢weaklyin𝐿²0,𝑇;𝐿²Ω_𝑈_𝑛,𝑦⟶𝑦weaklyin𝐿^∞0,𝑇;𝐻¹(Ω)_𝑛,(𝑇)⟶𝑦(𝑇)weaklyin𝐻¹(Ω).(2.12) Let 𝑊={𝑤;𝑤∈𝐿^∞(0,𝑇;𝐻¹(Ω)),𝑤^𝑡∈𝐿²(0,𝑇;𝐻¹(Ω))}.
By integrating time from 0 to 𝑇 in (2.9) and taking limit as 𝑛→∞, we obtain (𝑦(𝑇+),𝑤(𝑇))+𝑎(𝑦(𝑇),𝑤(𝑇))−^𝑇0𝑦,𝑤^𝑡+𝑎𝑦,𝑤^𝑡+𝑑(𝑦,𝑤)_𝑇0^𝑡0=𝑦𝑐(𝑡,𝜏;𝑦(𝜏),𝑤(𝑡))𝑑𝜏𝑑𝑡₀,𝑤(0)+𝑎𝑦⁰,𝑤(0)+^𝑇0(𝑓+𝐵𝑢,𝑤),∀𝑤∈𝑊.(2.13) Then, 𝑦_𝑡,𝑤+𝑎𝑦_𝑡,𝑤+𝑑(𝑦,𝑤)+^𝑡0𝑐(𝑡,𝜏;𝑦(𝜏),𝑤)𝑑𝜏=(𝑓+𝐵𝑢,𝑤),∀𝑤∈𝑉,𝑡∈(0,𝑇].(2.14) Furthermore, we have ^𝑇0𝑦^𝑡,𝑦_𝑡+𝑎𝑦_𝑡,𝑦_𝑡+𝑑𝑦,𝑦_𝑡+^𝑡0𝑐𝑡,𝜏;𝑦(𝜏),𝑦_𝑡(𝑡)𝑑𝜏𝑑𝑡=^𝑇0𝑓+𝐵𝑢,𝑦_𝑡.(2.15) Then, we get ^𝑇0‖‖𝑦_𝑡‖‖^21,Ω⩽𝐶^𝑇0‖𝑓‖^2−1,Ω+‖𝑢‖^20,Ω_𝑈+‖𝑦‖^21,Ω+^𝑡0‖𝑦‖^21,Ω𝑑𝜏.(2.16) This means 𝑦_𝑡∈𝐿²(0,𝑇;𝐻¹(Ω)). So (𝑢,𝑦) is one solution of (2.8).
Since ∫^𝑇0‖𝑦−𝑧_𝑑‖^20,Ω is a convex function on space 𝐿²(0,𝑇;𝐿²(Ω)) and (𝛼/2)∫^𝑇0‖𝑢‖^20,Ω_𝑈 is a strictly convex function on 𝑈, hence 𝐽(𝑢,𝑦(𝑢)) is a strictly convex function on 𝑈, 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,𝑢)∈𝐿²(0,𝑇;𝐻¹⁰(Ω))×𝐿²(0,𝑇;𝐿²(Ω_𝑈)) is the solution of the optimal control problem (2.8), if and only if there exists a costate 𝑝∈𝐿²(0,𝑇;𝐻¹⁰(Ω)) such that the triple (𝑦,𝑝,𝑢) satisfies the following optimality conditions: 𝑦_𝑡,𝑤+𝑎𝑦_𝑡,𝑤+𝑑(𝑦,𝑤)𝑦+|^𝑡0_𝑡=0𝑐(𝑡,𝜏;𝑦(𝜏),𝑤(𝑡∀))𝑤𝑑𝜏∈=𝑉,(𝑓𝑡+∈𝐵(𝑢0,,𝑤𝑇)],−𝑞,𝑝_𝑡=𝑦⁰;(2.17)−𝑎𝑞,𝑝_𝑡+𝑑(𝑞,𝑝)+^𝑇𝑡𝑐(𝜏,𝑡;𝑞𝑝|(𝑡)_𝑡=𝑇,𝑝(𝜏))𝑑𝜏=𝑦−𝑧_𝑑,𝑞∀𝑞∈𝑉,𝑡∈[0,𝑇),=0;(2.18)_𝑇0𝑢+𝐵^∗𝑝,𝑣−𝑢_𝑈𝑑𝑡⩾0,∀𝑣∈𝑈_𝑎𝑑,(2.19) where 𝐵^∗ is the adjoint operator of 𝐵.

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

Let Ω^ℎ be a polygonal approximation to Ω with boundary  𝜕Ω^ℎ. Let 𝑇^ℎ be a partitioning of Ω^ℎ into disjoint regular 𝑛-simplices 𝜏, so that Ω^ℎ=⋃_{𝜏∈𝑇ℎ}𝜏. Each element has at most one face on 𝜕Ω^ℎ, and 𝜏 and 𝜏′ have either only one common vertex or a whole edge or face if 𝜏 and 𝜏′∈𝑇^ℎ. We further require that 𝑃_𝑖∈𝜕Ω^ℎ⇒𝑃_𝑖∈𝜕Ω  where 𝑃_𝑖(𝑖=1,…,𝐽) is the vertex set associated with the triangulation 𝑇_ℎ. As usual, ℎ denotes the diameter of the triangulation 𝑇^ℎ. For simplicity, we assume that Ω is a convex polygon so that Ω=Ω^ℎ.

Associated with 𝑇^ℎ is a finite-dimensional subspace 𝑆^ℎ of 𝐶(Ω^ℎ), such that 𝜒|_𝜏 are polynomials of order 𝑚(𝑚≥1) for all 𝜒∈𝑆^ℎ and 𝜏∈𝑇^ℎ. Let 𝑉^ℎ={𝑣_ℎ∈𝑆_ℎ∶𝑣_ℎ(𝑃_𝑖)=0(𝑖=1,…,𝐽)},𝑊^ℎ=𝐿²(0,𝑇;𝑉^ℎ). Note that we do not impose a continuity requirement. It is easy to see that 𝑉^ℎ⊂𝑉,𝑊^ℎ⊂𝑊.

Let 𝑇^ℎ𝑈 be a partitioning of Ω^ℎ𝑈 into disjoint regular 𝑛-simplices 𝜏_𝑈, so that Ω^ℎ𝑈=⋃_{𝜏𝑈∈𝑇ℎ𝑈}𝜏_𝑈. 𝜏_𝑈 and 𝜏^𝑈 have either only one common vertex or a whole edge or face if 𝜏_𝑈 and 𝜏^𝑈∈𝑇^ℎ𝑈. We further require that 𝑃_𝑖∈𝜕Ω^ℎ𝑈⇒𝑃_𝑖∈𝜕Ω_𝑈, where 𝑃_𝑖(𝑖=1,…,𝐽) is the vertex set associated with the triangulation 𝑇^ℎ𝑈. For simplicity, we again assume that Ω_𝑈 is a convex polygon so that Ω_𝑈=Ω^ℎ𝑈.

Associated with 𝑇^ℎ𝑈 is another finite-dimensional subspace 𝑈^ℎ of 𝐿²(Ω^ℎ𝑈), such that 𝜒|_𝜏𝑈 are polynomials of order 𝑚(𝑚⩾0) for all 𝜒∈𝑈^ℎ and 𝜏_𝑈∈𝑇^ℎ𝑈. Here there is no requirement for the continuity. Let 𝑋^ℎ=𝐿²(0,𝑇;𝑈^ℎ). It is easy to see that 𝑋^ℎ⊂𝑋. Let ℎ_𝜏(ℎ_𝜏𝑈) denote the maximum diameter of the element 𝜏(𝜏_𝑈) in 𝑇^ℎ(𝑇^ℎ𝑈).

Due to the limited regularity of the optimal control 𝑢 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(Ω) denote all the 0-order polynomial over Ω. Therefore, we always take 𝑋^ℎ={𝑢∈𝑋∶𝑢(𝑥,𝑡)|_{𝑥∈𝜏𝑈}∈𝑃₀(𝜏_𝑈),forall𝑡∈[0,𝑇]}. 𝑈^ℎ𝑎𝑑 is a closed convex set in 𝑋^ℎ. For ease of exposition, in this paper, we assume that 𝑈^ℎ𝑎𝑑⊂𝑈_𝑎𝑑∩𝑋^ℎ.

Then a possible semi-discrete finite element approximation of (OCP) is as follows (OCP)^ℎ∶min_{𝑢ℎ∈𝑈ℎ𝑎𝑑}𝐽𝑢_ℎ,𝑦_ℎ=12^𝑇0‖‖𝑦_ℎ−𝑧_𝑑‖‖^20,Ω+^𝑇0‖‖𝑢_ℎ‖‖^20,Ω_𝑈,(2.20) such that 𝜕𝜕𝑦𝑡^ℎ,𝑤^ℎ+𝑎𝜕𝜕𝑦𝑡^ℎ,𝑤^ℎ+𝑑𝑦^ℎ,𝑤_ℎ+^𝑡0𝑐𝑡,𝜏;𝑦_ℎ(𝜏),𝑤_ℎ(𝑡)∀𝑤𝑑𝜏_ℎ=𝑓+𝐵𝑢_ℎ,𝑤_ℎ,𝑦∈𝑉^ℎ,𝑡∈(0,𝑇_ℎ],||_𝑡=0=𝑦^0ℎ,(2.21) where 𝑦_ℎ∈𝑊^ℎ and 𝑦^0ℎ∈𝑉^ℎ 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 (𝑦_ℎ,𝑢_ℎ)∈𝑊^ℎ×𝑈^ℎ𝑎𝑑.

It is well known (see [18]) that a pair (𝑦_ℎ,𝑢_ℎ)∈𝑊^ℎ×𝑈^ℎ𝑎𝑑 is a solution of (2.20)-(2.21), if and only if there exists a co-state 𝑝_ℎ∈𝑊^ℎ such that the triple (𝑦_ℎ,𝑝_ℎ,𝑢_ℎ) satisfies the following optimality conditions: 𝜕𝜕𝑦𝑡^ℎ,𝑤^ℎ+𝑎𝜕𝜕𝑦𝑡^ℎ,𝑤^ℎ+𝑑𝑦^ℎ,𝑤_ℎ+^𝑡0𝑐𝑡,𝜏;𝑦_ℎ(𝜏),𝑤_ℎ(𝑡)𝑑𝜏=𝑓∀+𝑤𝐵_ℎ𝑢_ℎ,𝑤_ℎ,𝑦_ℎ∈𝑉^ℎ,−𝑞|_𝑡=0=𝑦^0ℎ,(2.22)_ℎ,𝜕𝑝^ℎ𝜕𝑡−𝑎𝑞^ℎ,𝜕𝑝^ℎ𝜕𝑡+𝑑𝑞^ℎ,𝑝_ℎ+𝑝^𝑇𝑡_ℎ𝑐𝜏,𝑡;𝑞_ℎ,𝑝_ℎ(𝜏)𝑑𝜏=𝑦_ℎ−𝑧_𝑑,𝑞_ℎ,∀𝑞_ℎ∈𝑉^ℎ,|_𝑡=𝑇=0,(2.23)_𝑇0𝑢_ℎ+𝐵^∗𝑝_ℎ,𝑣_ℎ−𝑢_ℎ_𝑈𝑑𝑡⩾0,∀𝑣_ℎ∈𝑈^ℎ𝑎𝑑.(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_ℎ given by 𝜋_ℎ𝑤|_𝜏𝑈∶=∫^𝜏𝑈𝑤∫_𝜏𝑈1,∀𝜏^𝑈∈𝑇^ℎ𝑈.(2.25) Then, we have ∫_Ω𝑈𝑤=∫_Ω𝑈𝜋_ℎ𝑤 for any 𝑤∈𝐿²(0,𝑇;𝐿²(Ω_𝑈)),𝑡∈[0,𝑇] and (2.24) is equivalent to ^𝑇0𝑢_ℎ+𝜋_ℎ𝐵^∗𝑝_ℎ,𝑣_ℎ−𝑢_ℎ_𝑈𝑑𝑡⩾0,∀𝑣_ℎ∈𝑈^ℎ𝑎𝑑.(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 𝜋_ℎ be the standard Lagrange interpolation operator. For 𝑚=0 or 1, 𝑞>𝑛/2 and 𝑣∈𝑊^2,𝑞(Ω) as ||𝑣−𝜋_ℎ𝑣||_𝑚,𝑞,Ω⩽𝐶ℎ^2−𝑚|𝑣|_2,𝑞,Ω.(2.27)

Lemma 2.4 (see [20]). Let 𝜋_ℎ be the average interpolation operator defined in (2.25). For 𝑚=0 or 1, 1⩽𝑞⩽∞ and forall𝑣∈𝑊^1,𝑞(Ω^ℎ) as ||𝑣−𝜋_ℎ𝑣||_{𝑚,𝑞,𝜏}⩽_{⋂𝜏𝜏≠∅}𝐶ℎ^𝜏1−𝑚|𝑣|_{1,𝑞,𝜏′}.(2.28)

Lemma 2.5 (see [21]). For all 𝑣∈𝑊^1,𝑞(Ω),1⩽𝑞<∞ as ‖𝑣‖_{0,𝑞,𝜕𝜏}⩽𝐶ℎ^{𝜏−1/𝑞}‖𝑣‖_0,𝑞,𝜏+ℎ^{𝜏1−1/𝑞}|𝑣|_1,𝑞,𝜏.(2.29)

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

In this paper, the control constraints are given in an integral sense as follows: 𝑈_𝑎𝑑=𝑣∈𝑋;_Ω𝑈𝑣⩾0,𝑡∈[0,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,𝑢) and (𝑦_ℎ,𝑝_ℎ,𝑢_ℎ) be the solutions of (2.17)–(2.19) and (2.22)–(2.24). Then, we have ‖‖𝑢−𝑢_ℎ‖‖^2𝐿_{20,𝑇;𝐿2Ω𝑈}⩽𝐶𝜂²¹+𝐶‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω),(3.2) where 𝜂²¹=^𝑇0_𝜏𝑈_𝜏𝑈𝐵^∗𝑝_ℎ−𝑃_ℎ𝐵^∗𝑝_ℎ²𝑑𝑡,(3.3)𝑃_ℎ is the 𝐿²-projection from 𝐿²(Ω) to 𝑈^ℎ, and 𝑝(𝑢_ℎ) is defined by the following system: 𝜕𝜕𝑡𝑦=𝑢𝑓_ℎ+,𝜔𝐵𝑢_ℎ+𝑎𝜕𝜕𝑡𝑦𝑢^ℎ,𝜔+𝑑𝑦𝑢_ℎ,𝜔+^𝑡0𝑐𝑡,𝜏;𝑦𝑢_ℎ(𝜏),𝜔(𝑡)𝑑𝜏,𝜔,∀𝜔𝑦∈𝑢𝑉,_ℎ−𝑞,𝜕(0)=𝑦^ℎ0(𝑥),𝑥∈Ω,(3.4)𝜕𝑡𝑝𝑢^ℎ−𝑎𝑞,𝜕𝜕𝑡𝑝𝑢^ℎ+𝑑𝑞,𝑝𝑢_ℎ+^𝑇𝑡=𝑦𝑢𝑐𝜏,𝑡;𝑞(𝑡),𝑝𝑢_ℎ(𝜏)𝑑𝜏_ℎ−𝑧_𝑑,𝑞,∀𝑞∈𝑉.(3.5)

Proof. From (2.19), we have 𝑢,𝑢−𝑢_ℎ_𝑈⩽−𝐵^∗𝑝,𝑢−𝑢_ℎ_𝑈.(3.6) Then, by (2.24) and (3.6), we have ‖‖𝑢−𝑢_ℎ‖‖^2𝐿_{20,𝑇;𝐿2Ω𝑈}==−^𝑇0_𝑇0𝑢,𝑢−𝑢_ℎ_𝑈−𝑢_ℎ,𝑢−𝑢_ℎ_𝑈𝑑𝑡=^𝑇0−𝐵^∗𝑝+𝑢_ℎ,𝑢−𝑢_ℎ_𝑈𝑑𝑡+𝐵^∗𝑝_ℎ+𝑢_ℎ,𝑢−𝑣_ℎ_𝑈𝑑𝑡−^𝑇0𝐵^∗𝑝_ℎ+𝑢_ℎ,𝑣_ℎ−𝑢_ℎ_𝑈𝑑𝑡_𝑇0⩽inf_{𝑣ℎ∈𝑈ℎ𝑎𝑑}𝐵^∗𝑝_ℎ−𝐵^∗𝑝𝑢_ℎ,𝑢−𝑢_ℎ_𝑈𝑑𝑡+^𝑇0𝐵^∗𝑝𝑢_ℎ−𝐵^∗𝑝,𝑢−𝑢_ℎ_𝑈𝑑𝑡^𝑇0𝐵^∗+𝑝_ℎ+𝑢_ℎ,𝑣_ℎ−𝑢_𝑈𝑑𝑡_𝑇0=𝐼₁𝐵^∗𝑝_ℎ−𝑝𝑢_ℎ,𝑢−𝑢_ℎ_𝑈𝑑𝑡+^𝑇0𝐵^∗𝑝𝑢_ℎ−𝑝,𝑢−𝑢_ℎ_𝑈𝑑𝑡+𝐼₂+𝐼₃.(3.7) Next, we will estimate 𝐼₁,𝐼₂,and𝐼₃, respectively.
(1) We first estimate 𝐼₁. Let 𝑃_ℎ be the 𝐿²-projection from 𝐿²(Ω) to 𝑈^ℎ.
We have _Ω𝑈𝑃_ℎ𝑣−𝑣𝜙=0,∀𝜙∈𝑋^ℎ,𝑣∈𝑈_𝑎𝑑,𝑡∈(0,𝑇].(3.8) Since 𝑣∈𝑈_𝑎𝑑, so ∫_Ω𝑈𝑃_ℎ𝑣⩾0, then 𝑃_ℎ𝑣∈𝑈^ℎ𝑎𝑑. So that we can take 𝑣_ℎ=𝑃_ℎ𝑢 in 𝐼₁.
For given 𝑡∈(0,𝑇], let 𝑢_ℎ=𝑃_ℎ−𝐵^∗𝑝_ℎ+max0,∫^Ω𝑈𝐵^∗𝑝_ℎ∫_Ω𝑈1.(3.9) We have 𝑢_ℎ∈𝑋^ℎ. We will show that 𝑢_ℎ is the solution of the variational inequality in (2.24) assuming 𝑝_ℎ is known.
Since ∫_Ω𝑈[𝑃_ℎ(−𝐵^∗𝑝_ℎ+max{0,∫_Ω𝑈𝐵^∗𝑝_ℎ/∫_Ω𝑈1})−(−𝐵^∗𝑝_ℎ+max{0,∫_Ω𝑈𝐵^∗𝑝_ℎ/∫_Ω𝑈1})]=0, we have _Ω𝑈𝑢_ℎ=−_Ω𝑈𝐵^∗𝑝_ℎ+_Ω𝑈max0,∫^Ω𝑈𝐵^∗𝑝_ℎ∫_Ω𝑈1=−∫^Ω𝑈0,𝐵^∗𝑝_ℎ,∫∫_Ω𝑈Ω𝑈𝐵^∗𝑝_ℎ<0,𝐵^∗𝑝_ℎ⩾0.(3.10) Thus, ∫_Ω𝑈𝑢_ℎ⩾0, we have 𝑢_ℎ∈𝑈^ℎ𝑎𝑑. Note that for all 𝑣_ℎ∈𝑈^ℎ𝑎𝑑,𝑡∈(0,𝑇], we have 𝑢_ℎ+𝐵^∗𝑝_ℎ,𝑣_ℎ−𝑢_ℎ_𝑈=_Ω𝑈𝑃^ℎ−𝐵^∗𝑝_ℎ+max0,∫^Ω𝑈𝐵^∗𝑝_ℎ−−𝐵_∗∫_Ω𝑈1𝑝_ℎ+max0,∫^Ω𝑈𝐵^∗𝑝_ℎ∫_Ω𝑈1+max0,∫^Ω𝑈𝐵^∗𝑝_ℎ∫_Ω𝑈1𝑣^ℎ=_Ω𝑈−𝑢_ℎmax0,∫^Ω𝑈𝐵^∗𝑝_ℎ∫_Ω𝑈1𝑣^ℎ−𝑢_ℎ.(3.11) If ∫_Ω𝑈𝐵^∗𝑝_ℎ<0, then 𝑢_ℎ+𝐵^∗𝑝_ℎ,𝑣_ℎ−𝑢_ℎ_𝑈=_Ω𝑈0⋅𝑣_ℎ−𝑢_ℎ=0.(3.12) If ∫_Ω𝑈𝐵^∗𝑝_ℎ⩾0, since _Ω𝑈𝑢_ℎ=_Ω𝑈−𝐵^∗𝑝_ℎ+max0,∫^Ω𝑈𝐵^∗𝑝_ℎ∫_Ω𝑈1=0.(3.13) we have 𝑢_ℎ+𝐵^∗𝑝_ℎ,𝑣_ℎ−𝑢_ℎ_𝑈=∫^Ω𝑈𝐵^∗𝑝_ℎ∫_Ω𝑈1_Ω𝑈𝑣_ℎ−𝑢_ℎ=∫^Ω𝑈𝐵^∗𝑝_ℎ∫_Ω𝑈1⋅_Ω𝑈𝑣_ℎ⩾0.(3.14) From (3.11)–(3.14), we obtain 𝑢_ℎ+𝐵^∗𝑝_ℎ,𝑣_ℎ−𝑢_ℎ_𝑈⩾0,∀𝑣_ℎ∈𝑈^ℎ𝑎𝑑.(3.15) So 𝑢_ℎ=𝑃_ℎ(−𝐵^∗𝑝_ℎ+max{0,∫_Ω𝑈𝐵^∗𝑝_ℎ/∫_Ω𝑈1}) is the solution of the variational inequality in (2.24) assuming 𝑝_ℎ is known.
Then, 𝐼₁⩽=^𝑇0𝐵^∗𝑝_ℎ+𝑢_ℎ,𝑃_ℎ𝑢−𝑢_𝑈𝑑𝑡_𝑇0_𝜏𝑈_𝜏𝑈𝑃^ℎ−𝐵^∗𝑝_ℎ+max0,∫^Ω𝑈𝐵^∗𝑝_ℎ∫_Ω𝑈1+𝐵^∗𝑝_ℎ𝑃^ℎ𝑢−𝑢𝑑𝑡.(3.16) Since ∫_𝜏𝑈(𝑃_ℎ𝑢−𝑢)=0, we have 𝐼₁⩽^𝑇0_𝜏𝑈_𝜏𝑈=−𝑃_ℎ𝐵^∗𝑝_ℎ+𝐵^∗𝑝_ℎ𝑃_ℎ𝑢−𝑢𝑑𝑡_𝑇0_𝜏𝑈_𝜏𝑈⩽𝐶(𝛿)_𝑇0−𝑃_ℎ𝐵^∗𝑝_ℎ+𝐵^∗𝑝_ℎ𝑃_ℎ𝑢−𝑢_ℎ−𝑢−𝑢_ℎ𝑑𝑡_𝜏𝑈_𝜏𝑈−𝑃_ℎ𝐵^∗𝑝_ℎ+𝐵^∗𝑝_ℎ²𝑑𝑡+𝛿‖‖𝑢−𝑢^ℎ‖‖^2𝐿_{20,𝑇;𝐿2Ω𝑈}=𝐶𝜂²¹+𝛿‖‖𝑢−𝑢_ℎ‖‖^2𝐿_{20,𝑇;𝐿2Ω𝑈}.(3.17)(2) Consider 𝐼₂=^𝑇0𝐵^∗𝑝_ℎ−𝑝𝑢_ℎ,𝑢−𝑢_ℎ_𝑈𝑑𝑡⩽𝐶‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω)+𝛿‖‖𝑢−𝑢_ℎ‖‖^2𝐿_{20,𝑇;𝐿2Ω𝑈}.(3.18)(3) By (3.4) and (2.17), we have for 𝑡∈(0,𝑇]𝜕𝜕𝑡+𝑦−𝑦𝑢^ℎ,𝜔+𝑎𝜕𝜕𝑡𝑦−𝑦𝑢^ℎ,𝜔+𝑑𝑦−𝑦𝑢_ℎ,𝑤_𝑡0𝑐𝑡,𝜏;𝑦−𝑦𝑢_ℎ(𝜏),𝜔(𝑡)𝑑𝜏=𝐵𝑢−𝑢_ℎ,𝜔,∀𝜔∈𝑉,(3.19) and from (3.5) and (2.18), we have −𝑞,𝜕+𝜕𝑡𝑝−𝑝𝑢^ℎ−𝑎𝑞,𝜕𝜕𝑡𝑝−𝑝𝑢^ℎ+𝑑𝑞,𝑝−𝑝𝑢_ℎ_𝑇𝑡𝑐𝜏,𝑡;𝑞(𝑡),𝑝−𝑝𝑢_ℎ(𝜏)𝑑𝜏=𝑦−𝑦𝑢_ℎ,𝑞,∀𝑞∈𝑉.(3.20) Then, from (3.19), (3.20), and integrating by part we have 𝐼₃==^𝑇0𝐵^∗𝑝𝑢_ℎ−𝑝,𝑢−𝑢_ℎ_𝑈𝑑𝑡=^𝑇0𝑝𝑢_ℎ−𝑝,𝐵𝑢−𝑢_ℎ_𝑈𝑑𝑡_𝑇0+𝜕𝜕𝑑𝑡𝑦𝑦−−𝑦𝑦𝑢𝑢_ℎℎ,𝑝𝑢_ℎ−𝑝+𝑎𝜕𝜕𝑡𝑦−𝑦𝑢^ℎ,𝑝𝑢_ℎ−𝑝=,𝑝𝑢_ℎ−𝑝+^𝑡0𝑐𝑡,𝜏;𝑦−𝑦𝑢_ℎ(𝜏),𝑝𝑢_ℎ−𝑝(𝑡)𝑑𝜏𝑑𝑡_𝑇0−+𝑑𝑦−𝑦𝑦−𝑦𝑢^ℎ𝑢_ℎ,𝜕𝜕𝑡𝑝𝑢^ℎ−𝑝−𝑎𝑦−𝑦𝑢_ℎ,𝜕𝜕𝑡𝑝𝑢^ℎ−𝑝=,𝑝𝑢_ℎ−𝑝+^𝑇𝑡𝑐𝜏,𝑡;𝑦−𝑦𝑢_ℎ(𝑡),𝑝𝑢_ℎ−𝑝(𝜏)𝑑𝜏𝑑𝑡_𝑇0−𝑦−𝑦𝑢_ℎ,𝑦−𝑦𝑢_ℎ𝑑𝑡⩽0.(3.21) Following from (3.17)–(3.21), let 𝛿 be small enough as ‖‖𝑢−𝑢_ℎ‖‖^2𝐿_{20,𝑇;𝐿2Ω𝑈}⩽𝐶𝜂²¹+𝐶‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω).(3.22) This completes the proof.

Lemma 3.2. Let (𝑦,𝑝,𝑢) and (𝑦_ℎ,𝑝_ℎ,𝑢_ℎ) be the solutions of (2.17)–(2.19), and (2.22)–(2.24) respectively. Then, there hold the a posteriori error estimates as ‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^2𝐿∞_{0,𝑇;𝐻}¹_(Ω)+‖‖‖𝜕𝜕𝑡𝑦^ℎ−𝑦𝑢_ℎ‖‖‖^2𝐿_{20,𝑇;𝐻1(Ω)}+‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^2𝐿∞_{0,𝑇;𝐻}¹_(Ω)+‖‖‖𝜕𝜕𝑡𝑝^ℎ−𝑝𝑢_ℎ‖‖‖^2𝐿_{20,𝑇;𝐻1(Ω)}⩽𝐶⁶_𝑖=2𝜂^2𝑖,(3.23) where 𝜂²²=^𝑇0_𝜏ℎ^2𝜏_𝜏𝜕𝜕𝑝𝑡^ℎ−div𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡+div𝐷^∗+_𝑇𝑡∇𝑝_ℎdiv𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)𝑑𝜏+𝑦_ℎ−𝑧_𝑑²𝜂𝑑𝜏𝑑𝑡,₂₃=^𝑇0_𝜏ℎ_𝑙_𝜕𝜏−𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡⋅𝑛+𝐷^∗𝜂∇𝑝_ℎ⋅𝑛+^𝑇𝑡𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)⋅𝑛𝑑𝜏²𝑑𝑙𝑑𝑡,₂₄=^𝑇0_𝜏ℎ^2𝜏_𝜏𝜕𝜕𝑡𝑦^ℎ−div𝐴∇𝜕𝑦^ℎ−_𝑡0𝜕𝑡−div𝐷∇𝑦^ℎdiv𝐶(𝑡,𝜏)∇𝑦_ℎ(𝜏)𝑑𝜏−𝑓−𝐵𝑢_ℎ𝜂²𝑑𝑡,₂₅=^𝑇0_𝜏ℎ_𝑙_𝜕𝜏𝐴∇𝜕𝜕𝑦𝑡^ℎ⋅𝑛+𝐴∇𝑦^ℎ⋅𝑛+^𝑡0𝐶(𝑡,𝜏)∇𝑦_ℎ𝜂(𝜏)⋅𝑛𝑑𝜏²𝑑𝑙𝑑𝑡,₂₆=‖‖𝑦^ℎ0−𝑦₀‖‖^21,Ω,(3.24) where 𝑙 is a face of an element 𝜏, ℎ_𝑙 is the maximum diameter of 𝑙,  and  [∇𝑝_ℎ⋅𝑛] and [∇𝑦_ℎ⋅𝑛] are the normal derivative jumps over the interior face 𝑙 defined by ∇∇𝑝𝑦^ℎ_ℎ⋅𝑛_𝑙=∇𝑝_ℎℎ|_𝜏^1𝑙−∇𝑝_ℎ|_𝜏^2𝑙⋅𝑛,⋅𝑛_𝑙=∇𝑦_ℎ|_𝜏^1𝑙−∇𝑦_ℎ|_𝜏^2𝑙⋅𝑛,(3.25) where 𝑛 is the unit normal vector on 𝑙=𝜏^1𝑙∩𝜏^2𝑙 outwards 𝜏^1𝑙. For later convenience, one can define [∇𝑝_ℎ⋅𝑛]_𝑙=0 and [∇𝑦_ℎ⋅𝑛]_𝑙=0 when 𝑙⊂𝜕Ω.

Proof. Let 𝑅𝑢_ℎ,𝑣=−+𝑣,𝜕𝜕𝑡𝑝^ℎ−𝑝𝑢_ℎ−𝑎𝑣,𝜕𝜕𝑡𝑝^ℎ−𝑝𝑢_ℎ+𝑑𝑣,𝑝_ℎ−𝑝𝑢_ℎ_𝑇𝑡𝑐𝜏,𝑡;𝑣(𝑡),𝑝_ℎ−𝑝𝑢_ℎ(𝜏)𝑑𝜏,(3.26) and 𝜋_ℎ the average interpolation operator defined as in (2.25) and 𝑒=𝑝_ℎ−𝑝(𝑢_ℎ). Then, it follows from (2.23) and (3.5) that −𝑞_ℎ,𝜕𝜕𝑡𝑝^ℎ−𝑝𝑢_ℎ−𝑎𝑞_ℎ,𝜕𝜕𝑡𝑝^ℎ+−𝑝𝑢_ℎ+𝑑𝑞_ℎ,𝑝_ℎ−𝑝𝑢_ℎ_𝑇𝑡𝑐𝜏,𝑡;𝑞_ℎ(𝑡),𝑝_ℎ−𝑝𝑢_ℎ(𝜏)𝑑𝜏=𝑦_ℎ−𝑦𝑢_ℎ,𝑞_ℎ,∀𝑞_ℎ∈𝑉^ℎ.(3.27) We have =𝑅−𝑢_ℎ𝑣−,𝑣𝜋_ℎ𝑣,𝜕𝜕𝑡𝑝^ℎ−𝑝𝑢_ℎ−𝑎𝑣−𝜋_ℎ𝑣,𝜕𝜕𝑡𝑝^ℎ+−𝑝𝑢_ℎ+𝑑𝑣−𝜋_ℎ𝑣,𝑝_ℎ−𝑝𝑢_ℎ_𝑇𝑡𝑐𝜏,𝑡;𝑣−𝜋_ℎ𝑣(𝑡),𝑝_ℎ−𝑝𝑢_ℎ(𝜏)𝑑𝜏−𝜋_ℎ𝑣,𝜕𝜕𝑡𝑝^ℎ−𝑎𝜋_ℎ−𝑝𝑢_ℎ𝑣,𝜕𝜕𝑡𝑝^ℎ=−𝑣−𝜋_ℎ−𝑝𝑢_ℎ+𝑑𝜋_ℎ𝑣,𝑝_ℎ−𝑝𝑢_ℎ+^𝑇𝑡𝑐𝜏,𝑡;𝜋_ℎ𝑣(𝑡),𝑝_ℎ−𝑝𝑢_ℎ(𝜏)𝑑𝜏𝑣,𝜕𝑝^ℎ𝜕𝑡−𝑎𝑣−𝜋^ℎ𝑣,𝜕𝑝^ℎ𝜕𝑡+𝑑𝑣−𝜋^ℎ−𝑦𝑢𝑣,𝑝_ℎ+^𝑇𝑡𝑐𝜏,𝑡;𝑣−𝜋_ℎ𝑣(𝑡),𝑝_ℎ(𝜏)𝑑𝜏_ℎ=_𝜏−𝑧_𝑑,𝑣−𝜋_ℎ𝑣+𝑦_ℎ−𝑦𝑢_ℎ,𝜋_ℎ𝑣_𝜏−𝜕𝜕𝑝𝑡^ℎ+div𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡−div𝐷^∗∇𝑝_ℎ−^𝑇𝑡div𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)𝑑𝜏−𝑦_ℎ+𝑧_𝑑×𝑣−𝜋_ℎ𝑣+_𝜏_𝜕𝜏−𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡⋅𝑛+𝐷^∗×𝑣−𝜋_ℎ∇𝑝_ℎ⋅𝑛+^𝑇𝑡𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)⋅𝑛𝑑𝜏⩽𝑣+𝑦_ℎ−𝑦𝑢_ℎ,𝑣_𝜏_𝜏ℎ^2𝜏−𝜕𝜕𝑝𝑡^ℎ+div𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡−div𝐷^∗−_𝑇𝑡∇𝑝_ℎdiv𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)𝑑𝜏−𝑦_ℎ+𝑧_𝑑²+_𝜏_𝜕𝜏ℎ_𝑙−𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡⋅𝑛+𝐷^∗∇𝑝_ℎ⋅𝑛+^𝑇𝑡𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)⋅𝑛𝑑𝜏²^1/2×‖𝑣‖_1,Ω+𝑦_ℎ−𝑦𝑢_ℎ,𝑣.(3.28) Taking 𝑣=𝑝_ℎ−𝑝(𝑢_ℎ) in (3.28) and from (2.6), we have −12𝑑𝑑𝑡‖‖𝑝^ℎ−𝑝(𝑢_ℎ)‖‖^20,Ω−12𝑑𝑑𝑡𝑎𝑝^ℎ−𝑝𝑢_ℎ,𝑝_ℎ−𝑝𝑢_ℎ+𝑐‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^21,Ω⩽_𝜏_𝜏ℎ^2𝜏−𝜕𝜕𝑝𝑡^ℎ+div𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡−div𝐷^∗−𝑦_ℎ∇𝑝_ℎ−^𝑇𝑡div𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)𝑑𝜏+𝑧_𝑑²+_𝜏_𝜕𝜏ℎ_𝑙−𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡⋅𝑛+𝐷^∗∇𝑝_ℎ⋅𝑛²^1/2×−‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖_1,Ω+𝑦_ℎ−𝑦𝑢_ℎ,𝑝_ℎ−𝑝𝑢_ℎ_𝑇𝑡𝑐𝜏,𝑡;𝑝_ℎ−𝑝𝑢_ℎ(𝑡),𝑝_ℎ−𝑝𝑢_ℎ(𝜏)𝑑𝜏.(3.29) Integrating time from 𝑡 to 𝑇 in (3.29) and by Schwartz inequality, Lemmas 2.4 and 2.5, we have 12‖‖𝑝^ℎ⩽−𝑝𝑢_ℎ‖‖^20,Ω+𝑐‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^21,Ω+𝑐^𝑇𝑡‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^21,Ω𝑑𝜏_𝑇𝑡_𝜏ℎ^2𝜏×_𝜏𝜕𝜕𝑝𝑡^ℎ−div𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡+div𝐷^∗+_𝑇𝜏∇𝑝_ℎdiv𝐶^∗(𝑠,𝜏)∇𝑝_ℎ(𝑠)𝑑𝑠+𝑦_ℎ−𝑧_𝑑²+𝑑𝜏_𝑇𝑡_𝜏ℎ_𝑙_𝜕𝜏−𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡⋅𝑛+𝐷^∗+𝛿_𝑇𝑡∇𝑝_ℎ⋅𝑛+^𝑇𝜏𝐶^∗(𝑠,𝜏)∇𝑝_ℎ(𝑠)⋅𝑛𝑑𝑠²𝑑𝜏‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^21,Ω𝑑𝜏+𝐶^𝑇𝑡‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^20,Ω𝑑𝜏+𝐶^𝑇𝑡^𝑇𝜏‖‖𝑝_ℎ−𝑝𝑢_ℎ(𝑠)‖‖^21,Ω𝑑𝑠𝑑𝜏.(3.30) Letting 𝛿 be small enough, we have ^𝑇𝑡⩽‖‖𝑝𝐶_ℎ−_𝑇𝑡𝑝(𝑢_ℎ)‖‖^21,Ω𝑑𝜏_𝜏ℎ^2𝜏_𝜏𝜕𝜕𝑝𝑡^ℎ−div𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡+div𝐷^∗+_𝑇𝜏∇𝑝_ℎdiv𝐶^∗(𝑠,𝜏)∇𝑝_ℎ(𝑠)𝑑𝑠+𝑦_ℎ−𝑧_𝑑²+𝐶_𝑇𝑡𝑑𝜏_𝜏ℎ_𝑙_𝜕𝜏−𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡⋅𝑛+𝐷^∗+∇𝑝_ℎ⋅𝑛_𝑇𝜏+𝐶_𝑇𝑡𝐶^∗(𝑠,𝜏)∇𝑝_ℎ(𝑠)⋅𝑛𝑑𝑠²𝑑𝜏‖‖𝑦_ℎ−𝑦(𝑢_ℎ)‖‖^20,Ω𝑑𝜏+𝐶^𝑇𝑡^𝑇𝜏‖‖𝑝_ℎ−𝑝𝑢_ℎ(𝑠)‖‖^21,Ω𝑑𝑠𝑑𝜏.(3.31) Then, from Gronwall inequality and (3.28)–(3.31) we have ‖‖𝑝_ℎ−𝑝(𝑢_ℎ)‖‖^2𝐿_{20,𝑇;𝐻1(Ω)}⩽𝐶𝜂²²+𝐶𝜂²³+𝐶‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω).(3.32) Similarly, ‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^2𝐿∞_(0,𝑇;𝐻¹_(Ω))⩽𝐶𝜂²²+𝜂²³+‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω)+𝐶_𝑇0^𝑇𝑡⩽𝐶𝜂₂₂‖‖𝑝_ℎ−𝑝𝑢_ℎ(𝜏)‖‖^21,Ω𝑑𝜏𝑑𝑡+𝐶𝜂²³+𝐶‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω).(3.33) In the same way of getting (3.32),by setting 𝑣=(𝜕/𝜕𝑡)(𝑝_ℎ−𝑝(𝑢_ℎ)) in (3.28), we have ‖‖‖𝜕𝜕𝑡𝑝^ℎ−𝑝𝑢_ℎ‖‖‖^2𝐿_{20,𝑇;𝐻1(Ω)}⩽𝐶𝜂²²+𝐶𝜂²³+𝐶‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω).(3.34) Similarly analysis for ‖𝑦_ℎ−𝑦(𝑢_ℎ)‖_𝐿^∞_(0,𝑇;𝐻¹_(Ω)), we let 𝑄𝑢_ℎ,𝑣=+𝜕𝜕_𝑡𝑦_ℎ−𝑦𝑢_ℎ,𝑣+𝑎𝜕𝜕_𝑡𝑦_ℎ−𝑦𝑢_ℎ,𝑣+𝑑𝑦_ℎ−𝑦𝑢_ℎ,𝑣_𝑡0𝑐𝑡,𝜏;𝑦_ℎ−𝑦𝑢_ℎ(𝜏),𝑣(𝑡)𝑑𝜏.(3.35) From (2.22) and (3.4), we obtain 𝜔^ℎ+,𝜕𝜕𝑡_𝑡0𝑦^ℎ−𝑦𝑢_ℎ+𝑎𝜕𝜕𝑡𝑦^ℎ−𝑦𝑢_ℎ,𝜔_ℎ+𝑑𝑦^ℎ−𝑦𝑢_ℎ,𝜔_ℎ𝑐𝑡,𝜏;𝑦_ℎ−𝑦𝑢_ℎ(𝜏),𝜔_ℎ(𝑡)𝑑𝜏=0,∀𝜔_ℎ∈𝑉^ℎ.(3.36) We have =𝑄𝜕𝑢_ℎ,𝑣𝜕𝑡𝑦^ℎ−𝑦𝑢_ℎ,𝑣−𝜋_ℎ𝑣+𝑎𝜕𝜕𝑡𝑦^ℎ+_𝑡0−𝑦𝑢_ℎ,𝑣−𝜋_ℎ𝑣+𝑑𝑦_ℎ−𝑦𝑢_ℎ,𝑣−𝜋_ℎ𝑣=𝜕𝑦𝑐_ℎ𝑡,𝜏;𝑦_ℎ−𝑦𝑢_ℎ(𝜏),𝑣−𝜋_ℎ𝑣(𝑡)𝑑𝜏𝜕𝑡,𝑣−𝜋^ℎ𝑣+𝑎𝜕𝑦^ℎ𝜕𝑡,𝑣−𝜋^ℎ+_𝑡0𝑣+𝑑𝑦_ℎ,𝑣−𝜋_ℎ𝑣=_𝜏𝑐𝑡,𝜏;𝑦_ℎ(𝜏),𝑣−𝜋_ℎ𝑣(𝑡)𝑑𝜏−𝑓+𝐵𝑢_ℎ,𝑣−𝜋_ℎ𝑣_𝜏𝜕𝜕𝑦𝑡^ℎ−div𝐴∇𝜕𝜕𝑦𝑡^ℎ−div𝐷∇𝑦^ℎ−^𝑡0div𝐶(𝑡,𝜏)∇𝑦_ℎ𝑑𝜏−𝑓−𝐵𝑢_ℎ𝑣−𝜋^ℎ+_𝜏𝑣_𝜕𝜏𝐴∇𝜕𝜕𝑦𝑡^ℎ⋅𝑛+𝐷∇𝑦^ℎ⋅𝑛+^𝑡0𝐶(𝑡,𝜏)∇𝑦_ℎ⋅𝑛𝑑𝜏𝑣−𝜋_ℎ⩽_𝜏𝑣_𝜏ℎ^2𝜏𝜕𝜕𝑡𝑦^ℎ−div𝐴∇𝜕𝑦^ℎ𝜕𝑡−div𝐷∇𝑦^ℎ−^𝑡0div𝐶(𝑡,𝜏)∇𝑦_ℎ𝑑𝜏−𝑓−𝐵𝑢_ℎ²+_𝜏_𝜕𝜏ℎ_𝑙𝐴∇𝜕𝜕𝑦𝑡^ℎ⋅𝑛+𝐷∇𝑦^ℎ⋅𝑛+^𝑡0𝐶(𝑡,𝜏)∇𝑦_ℎ⋅𝑛𝑑𝜏²^1/2‖𝑣‖_1,Ω.(3.37) By setting 𝑣=𝑦_ℎ−𝑦(𝑢_ℎ) and Swartz inequality, we have 12𝑑𝑑𝑡‖‖𝑦^ℎ−𝑦𝑢_ℎ‖‖^20,Ω+12𝑑𝑑𝑡𝑎𝑦^ℎ−𝑦𝑢_ℎ,𝑦_ℎ−𝑦𝑢_ℎ+𝑐‖‖𝑦_ℎ−𝑦(𝑢_ℎ)‖‖^21,Ω⩽_𝜏_𝜏ℎ^2𝜏𝜕𝜕𝑦𝑡^ℎ−div𝐴∇𝜕𝜕𝑦𝑡^ℎ−div𝐷∇𝑦^ℎ−^𝑡0div𝐶(𝑡,𝜏)∇𝑦_ℎ𝑑𝜏−𝑓−𝐵𝑢_ℎ²+_𝜏_𝜕𝜏ℎ_𝑙𝐴∇𝜕𝜕𝑦𝑡^ℎ⋅𝑛+𝐷∇𝑦^ℎ⋅𝑛+^𝑡0𝐶(𝑡,𝜏)∇𝑦_ℎ⋅𝑛𝑑𝜏²+𝛿‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^21,Ω−^𝑡0𝑐𝑡,𝜏;𝑦_ℎ−𝑦𝑢_ℎ(𝜏),𝑦_ℎ−𝑦𝑢_ℎ(𝑡)𝑑𝜏.(3.38) Integrating time from 0 to 𝑡 in (3.38), we obtain ‖‖⩽𝑦_ℎ𝐶−𝑦_𝑡0𝑢_ℎ‖‖^21,Ω+𝑐^𝑡0‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^21,Ω𝑑𝜏_𝜏ℎ^2𝜏_𝜏−𝜕𝜕𝑦_𝜏0𝑡^ℎ−div𝐴∇𝜕𝜕𝑦𝑡^ℎ−div𝐷∇𝑦^ℎdiv𝐶(𝜏,𝑠)∇𝑦_ℎ𝑑𝑠−𝑓−𝐵𝑢_ℎ²+𝑑𝜏_𝑡0_𝜏ℎ_𝑙_𝜕𝜏𝐴∇𝜕𝜕𝑦𝑡^ℎ⋅𝑛+𝐷∇𝑦^ℎ⋅𝑛+^𝜏0𝐶(𝜏,𝑠)∇𝑦_ℎ⋅𝑛𝑑𝑠²+𝛿_𝑡0𝑑𝑡‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^21,Ω𝑑𝜏+𝐶^𝑡0^𝜏0‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^21,Ω𝑑𝑠𝑑𝜏+𝐶‖‖𝑦₀−𝑦^ℎ0‖‖^21,Ω.(3.39) Since 𝛿 is small enough, then from (3.39) and Gronwall inequality, we have ^𝑡0⩽‖‖𝐶𝑦_ℎ𝑡0−𝑦𝑢_ℎ‖‖^21,Ω𝑑𝑡_𝜏ℎ^2𝜏_𝜏−𝜕𝜕𝑦𝑡^ℎ_𝜏0−div𝐴∇𝜕𝜕𝑦𝑡^ℎ−div𝐷∇𝑦^ℎdiv𝐶(𝜏,𝑠)∇𝑦_ℎ𝑑𝑠−𝑓−𝐵𝑢_ℎ²+𝐶_𝑡0𝑑𝜏_𝜏ℎ_𝑙_𝜕𝜏𝐴∇𝜕𝜕𝑦𝑡^ℎ⋅𝑛+𝐷∇𝑦^ℎ⋅𝑛+^𝜏0𝐶(𝜏,𝑠)∇𝑦_ℎ⋅𝑛𝑑𝑠²𝑑𝑡+𝐶‖‖𝑦₀−𝑦^ℎ0‖‖^21,Ω.(3.40) Then, ‖‖‖‖𝑦𝑦_ℎℎ−𝑦𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐻}¹_(Ω)⩽𝐶𝜂²⁴+𝜂²⁵+𝜂²⁶,−𝑦𝑢_ℎ‖‖^2𝐿∞_{0,𝑇;𝐻}¹_(Ω)⩽𝐶𝜂²⁴+𝜂²⁵+𝜂²⁶.(3.41) In the same way of getting (3.34), we can similarly obtain ‖‖‖𝜕𝜕𝑡𝑦^ℎ−𝑦𝑢_ℎ‖‖‖^2𝐿_{20,𝑇;𝐻1(Ω)}⩽𝐶𝜂²⁴+𝜂²⁵+𝜂²⁶.(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,𝑢) and (𝑦_ℎ,𝑝_ℎ,𝑢_ℎ) be the solutions of(2.17)–(2.19) and (2.22)–(2.24) respectively. Then, there hold the a posteriori error estimates as ‖‖𝑢−𝑢_ℎ‖‖^2𝐿_{20,𝑇;𝐿2Ω𝑈}+‖‖𝑦−𝑦_ℎ‖‖^2𝐿_{∞0,𝑇;𝐻1(Ω)}+‖‖‖𝜕𝜕𝑡𝑦−𝑦^ℎ‖‖‖^2𝐿_{20,𝑇;𝐻1(Ω)}+‖‖𝑝−𝑝_ℎ‖‖^2𝐿_{∞0,𝑇;𝐻1(Ω)}+‖‖‖𝜕𝜕𝑡𝑝−𝑝^ℎ‖‖‖^2𝐿_{20,𝑇;𝐻1(Ω)}⩽𝐶⁶_𝑖=1𝜂^2𝑖,(3.43) where 𝜂²¹ is defined in Lemma 3.1.

Proof. First,  from (3.27) and (3.36), and [2], we have the following stability results: ‖‖𝑦−𝑦𝑢_ℎ‖‖^2𝐿∞_{0,𝑇;𝐻}¹_(Ω)+‖‖‖𝜕𝜕𝑡𝑦−𝑦𝑢^ℎ‖‖‖^2𝐿_{20,𝑇;𝐻1(Ω)}‖‖𝑝−𝑝𝑢_ℎ⩽𝐶‖‖𝑢−𝑢_ℎ‖‖^2𝐿_{20,𝑇;𝐿2Ω𝑈},‖‖^2𝐿∞_{0,𝑇;𝐻}¹_(Ω)+‖‖‖𝜕𝜕𝑡𝑝−𝑝𝑢^ℎ‖‖‖^2𝐿_{20,𝑇;𝐻1(Ω)}⩽𝐶‖‖𝑦−𝑦𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω)⩽𝐶‖‖𝑢−𝑢_ℎ‖‖^2𝐿_{20,𝑇;𝐿2Ω𝑈}.(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(Ω))-Norm

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

We will introduce the following dual equations.

For given 𝐹∈𝐿²(0,𝑇;𝐿²(Ω)), we have 𝜕𝜙𝜕𝑡−div𝐴∇𝜕𝜕𝜙𝑡−div(𝐷∇𝜙)−^𝑡0𝜙div(𝐶(𝑡,𝜏)∇𝜙(𝜏))𝑑𝜏=𝐹,(𝑥,𝑡)∈Ω×(0,𝑇],𝜕𝜙Ω(𝑥,=0)0=,0𝑡,∈𝑥(0∈,𝑇Ω],,(4.1) and its dual equation −𝜕𝜓𝜕𝑡+div𝐴^∗∇𝜕𝜓𝜕𝑡−div𝐷^∗∇𝜓𝜓−^𝑇𝑡div𝐶^∗(𝜏,𝑡)∇𝜓(𝜏)𝑑𝜏=𝐹,(𝑥,𝑡)∈Ω×(0,𝑇],𝜓𝜕Ω(𝑥,=𝑇0),=𝑡0,∈(𝑥0,∈𝑇Ω],.(4.2) From [1, 2], we have the following stability results.

Lemma 4.1. Assume that Ω is a convex domain. Let 𝜙 and 𝜓 be the solution of (4.1) and (4.2), respectively. Then, ‖‖∇𝜙𝜙‖‖_𝐿^∞_(0,𝑇;𝐿²_{(Ω))𝐿2(0,𝑇;𝐿2(Ω))}⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),‖‖𝐷₂⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),‖‖‖𝜕𝜙‖‖_{𝐿2(0,𝑇;𝐿2(Ω))}⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),𝜕𝑡𝜙‖‖‖_𝐿²_(0,𝑇;𝐿²_(Ω))‖𝜓‖_{𝐿∞(0,𝑇;𝐿2(Ω))}⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),‖∇𝜓‖_{𝐿2(0,𝑇;𝐿2(Ω))}⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),‖‖𝐷₂⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),‖‖‖𝜕𝜓‖‖_{𝐿2(0,𝑇;𝐿2(Ω))}⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),𝜕𝑡𝜓‖‖‖_𝐿²_(0,𝑇;𝐿²_(Ω))⩽𝐶‖𝐹‖_𝐿²_(0,𝑇;𝐿²_(Ω)),(4.3) where 𝐷²𝜙=𝜕²𝜙/𝜕𝑥_𝑖𝜕𝑥_𝑗,1⩽𝑖,𝑗⩽𝑛, and 𝐷²𝜓 is defined similarly.

Using Lemmas 3.1 and 4.1, we have the following upperbounds.

Lemma 4.2. Let (𝑦,𝑝,𝑢) and (𝑦_ℎ,𝑝_ℎ,𝑢_ℎ) be the solutions of (2.17)–(2.19) and (2.22)–(2.24), respectively. Then, there hold the a posteriori error estimates as ‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω)+‖‖𝑝_ℎ−𝑝(𝑢_ℎ)‖‖^2𝐿_{20,𝑇;𝐿2(Ω)}⩽𝐶⁵_𝑖=2𝜉^2𝑖+𝜂²⁶,(4.4) where 𝜂²⁶ is defined in Lemma 3.2, and 𝜉²²=^𝑇0_𝜏ℎ^4𝜏_𝜏𝜕𝜕𝑝𝑡^ℎ−div𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡+div𝐷^∗+_𝑇𝑡∇𝑝_ℎdiv𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)𝑑𝜏+𝑦_ℎ−𝑧_𝑑𝜉²𝑑𝑡,₂₃=^𝑇0_𝜏ℎ^3𝑙_𝜕𝜏−𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡⋅𝑛+𝐷^∗𝜉∇𝑝_ℎ⋅𝑛+^𝑇𝑡𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)⋅𝑛𝑑𝜏²𝑑𝑙𝑑𝑡,₂₄=^𝑇0_𝜏ℎ^4𝜏_𝜏−𝜕𝜕𝑦𝑡^ℎ_𝑡0−div𝐴∇𝜕𝜕𝑦𝑡^ℎ−div𝐷∇𝑦^ℎdiv𝐶(𝑡,𝜏)∇𝑦_ℎ(𝜏)𝑑𝜏−𝑓−𝐵𝑢_ℎ𝜉²𝑑𝑡,₂₅=^𝑇0_𝜏ℎ^3𝑙_𝜕𝜏𝐴∇𝜕𝜕𝑦𝑡^ℎ⋅𝑛+𝐷∇𝑦^ℎ⋅𝑛+^𝑡0𝐶(𝑡,𝜏)∇𝑦_ℎ(𝜏)⋅𝑛𝑑𝜏²𝑑𝑙𝑑𝑡.(4.5)

Proof. We first estimate ‖𝑝_ℎ−𝑝(𝑢_ℎ)‖^2𝐿_{2(0,𝑇;𝐿2(Ω))}.
Let 𝜙 be the solution of (4.1) with 𝐹=𝑝_ℎ−𝑝(𝑢_ℎ),  and  𝜙_𝐼=𝜋_ℎ𝜙 the interpolation of 𝜙 in Lemma 2.3.
From (4.1), (3.27), and by integrating by parts we obtain ‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω)==^𝑇0𝐹(𝑡),𝑝_ℎ−𝑝𝑢_ℎ(𝑡)𝑑𝑡_𝑇0−𝜕𝜕𝑡𝑝^ℎ−𝑝𝑢_ℎ,𝜙−𝑎𝜙,𝜕𝜕𝑡𝑝^ℎ+𝑑𝜙,𝑝_ℎ−𝑝𝑢_ℎ=−𝑝𝑢_ℎ+^𝑡0𝑐𝑡,𝜏;𝜙(𝜏),𝑝_ℎ−𝑝𝑢_ℎ(𝑡)𝑑𝜏𝑑𝑡_𝑇0−+𝑑𝜕𝜕𝑡𝜙𝑝−_ℎ−𝜙_𝐼𝑝𝑢_ℎ,𝜙−𝜙_𝐼−𝑎𝜙−𝜙^𝐼,𝜕𝜕𝑡𝑝^ℎ−𝑝𝑢_ℎ−𝜕,𝑝_ℎ−𝑝𝑢_ℎ+^𝑡0𝑐𝑡,𝜏;𝜙−𝜙_𝐼(𝜏),𝑝_ℎ−𝑝𝑢_ℎ(𝑡)𝑑𝜏𝜕𝑡𝑝^ℎ−𝑝𝑢_ℎ,𝜙_𝐼−𝑎𝜙^𝐼,𝜕𝜕𝑡𝑝^ℎ+𝑑𝜙_𝐼−𝑝𝑢_ℎ=,𝑝_ℎ−𝑝𝑢_ℎ+^𝑡0𝑐𝑡,𝜏;𝜙_𝐼(𝜏),𝑝_ℎ−𝑝𝑢_ℎ(𝑡)𝑑𝜏𝑑𝑡_𝑇0−𝜕𝜕𝑝𝑡^ℎ,𝜙−𝜙^𝐼−𝑎𝜙−𝜙^𝐼,𝜕𝑝^ℎ+𝑑𝜙−𝜙_𝐼𝜕𝑡,𝑝_ℎ+^𝑡0𝑐𝑡,𝜏;𝜙−𝜙_𝐼(𝜏),𝑝_ℎ(𝑡)𝑑𝜏+𝜕𝑝𝑢^ℎ+𝑎𝜙−𝜙_𝐼𝜕𝑡,𝜙−𝜙^𝐼,𝜕𝑝𝑢^ℎ𝜕𝑡−𝑑𝜙−𝜙^𝐼−𝜕,𝑝𝑢_ℎ−^𝑡0𝑐𝑡,𝜏;𝜙−𝜙_𝐼(𝜏),𝑝𝑢_ℎ(𝑡)𝑑𝜏𝜕𝑡𝑝^ℎ−𝑝𝑢_ℎ,𝜙_𝐼−𝑎𝜙^𝐼,𝜕𝜕𝑡𝑝^ℎ+𝑑𝜙_𝐼−𝑝𝑢_ℎ=,𝑝_ℎ−𝑝𝑢_ℎ+^𝑡0𝑐𝑡,𝜏;𝜙_𝐼(𝜏),𝑝_ℎ−𝑝𝑢_ℎ(𝑡)𝑑𝜏𝑑𝑡_𝑇0−𝜕𝜕𝑝𝑡^ℎ,𝜙−𝜙^𝐼−𝑎𝜙−𝜙^𝐼,𝜕𝑝^ℎ+𝑑𝜙−𝜙_𝐼𝜕𝑡+𝜕𝑝𝑢_ℎ,𝑝_ℎ+^𝑇𝑡𝑐𝜏,𝑡;𝜙−𝜙_𝐼(𝑡),𝑝_ℎ(𝜏)𝑑𝜏𝜕𝑡,𝜙−𝜙^𝐼+𝑎𝜙−𝜙^𝐼,𝜕𝑝𝑢^ℎ−𝑑𝜙−𝜙_𝐼𝜕𝑡−𝜕,𝑝𝑢_ℎ−^𝑇𝑡𝑐𝜏,𝑡;𝜙−𝜙_𝐼(𝑡),𝑝𝑢_ℎ(𝜏)𝑑𝜏𝜕𝑡𝑝^ℎ−𝑝𝑢_ℎ,𝜙_𝐼−𝑎𝜙^𝐼,𝜕𝜕𝑡𝑝^ℎ+𝑑𝜙_𝐼−𝑝𝑢_ℎ=,𝑝_ℎ−𝑝𝑢_ℎ+^𝑇𝑡𝑐𝜏,𝑡;𝜙_𝐼(𝑡),𝑝_ℎ−𝑝𝑢_ℎ(𝜏)𝑑𝜏𝑑𝑡_𝑇0−𝜕𝜕𝑝𝑡^ℎ,𝜙−𝜙^𝐼−𝑎𝜙−𝜙^𝐼,𝜕𝑝^ℎ𝜕𝑡+𝑑𝜙−𝜙^𝐼−,𝑝_ℎ+^𝑇𝑡𝑐𝜏,𝑡;𝜙−𝜙_𝐼(𝑡),𝑝_ℎ(𝜏)𝑑𝜏𝑑𝑡_𝑇0=𝑦𝑢_ℎ−𝑧_𝑑,𝜙−𝜙_𝐼𝑑𝑡+^𝑇0𝑦_ℎ−𝑦𝑢_ℎ,𝜙_𝐼𝑑𝑡_𝑇0_𝜏_𝜏−𝜕𝜕𝑝𝑡^ℎ+div𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡−div𝐷^∗+_𝑇𝑡∇𝑝_ℎdiv𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)𝑑𝜏+𝑦_ℎ−𝑧_𝑑+𝜙−𝜙^𝐼𝑑𝑡_𝑇0_𝜏_𝜕𝜏−𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡⋅𝑛+𝐷^∗×𝜙−𝜙_𝐼∇𝑝_ℎ⋅𝑛+^𝑇𝑡𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)⋅𝑛𝑑𝜏=𝐽₁𝑑𝑙𝑑𝑡+^𝑇0𝑦_ℎ−𝑦𝑢_ℎ,𝜙𝑑𝑡+𝐽₂+𝐽₃.(4.6) It follows from Lemmas 2.3, 2.5, and 4.1 that 𝐽₁⩽𝐶(𝛿)^𝑇0_𝜏ℎ^4𝜏_𝜏𝜕𝜕𝑝𝑡^ℎ−div𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡+div𝐷^∗+_𝑇𝑡∇𝑝_ℎdiv𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)𝑑𝜏+𝑦_ℎ−𝑧_𝑑+𝛿_𝑇0²𝑑𝑡𝐽₂||𝜙||^22,Ω𝑑𝑡⩽𝐶(𝛿)𝜉²²+𝛿‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω),⩽𝐶(𝛿)^𝑇0_𝜏ℎ^3𝑙_𝜕𝜏−𝐴^∗∇𝜕𝑝^ℎ𝜕𝑡⋅𝑛+𝐷^∗+𝛿_𝑇0∇𝑝_ℎ⋅𝑛+^𝑇𝑡𝐶^∗(𝜏,𝑡)∇𝑝_ℎ(𝜏)⋅𝑛𝑑𝜏²𝑑𝑙𝑑𝑡⩽𝐶(𝛿)𝜉||₂₃𝜙||^20,Ω𝑑𝑡+𝛿‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω).(4.7) By Schwartz inequality, we have 𝐽₃⩽𝐶(𝛿)‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω)+𝛿‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω).(4.8) Letting 𝛿 be small enough, it follows from (4.6)–(4.8) that ‖‖𝑝_ℎ−𝑝𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω)⩽𝐶³_𝑖=2𝜉^2𝑖+𝐶‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω).(4.9) Next, we estimate ‖𝑦_ℎ−𝑦(𝑢_ℎ)‖^2𝐿_{2(0,𝑇;𝐿2(Ω))}. Similarly let 𝜓 be the solution of (4.2) with 𝐹=𝑦_ℎ−𝑦(𝑢_ℎ),  and  𝜓_𝐼=𝜋_ℎ𝜓 the interpolation of 𝜓 in Lemma 2.3.
Then, it follows from (3.36) and integrating by parts that ‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω)==^𝑇0𝐹(𝑡),𝑦_ℎ−𝑦𝑢_ℎ(𝑡)𝑑𝑡_𝑇0𝜕𝜕𝑡𝑦^ℎ−𝑦𝑢_ℎ,𝜓+𝑎𝜕𝜕𝑡𝑦^ℎ+−𝑦𝑢_ℎ,𝜓+𝑑𝑦_ℎ−𝑦𝑢_ℎ,𝜓_𝑇𝑡+𝑎𝑦_ℎ0𝑐𝜏,𝑡;𝑦_ℎ−𝑦𝑢_ℎ(𝑡),𝜓(𝜏)𝑑𝜏𝑑𝑡=−𝑦₀,𝜓(0)+𝑦^ℎ0−𝑦₀,𝜓(0)_𝑇0𝜕𝜕𝑡𝑦^ℎ−𝑦𝑢_ℎ,𝜓−𝜓_𝐼+𝑎𝜕𝜕𝑡𝑦^ℎ+𝑑𝑦_ℎ−𝑦𝑢_ℎ,𝜓−𝜓_𝐼+−𝑦𝑢_ℎ,𝜓−𝜓_𝐼+^𝑇𝑡𝑐𝜏,𝑡;𝑦_ℎ−𝑦𝑢_ℎ(𝑡),𝜓−𝜓_𝐼(𝜏)𝑑𝜏𝑑𝑡_𝑇0𝜕𝜕𝑡𝑦^ℎ−𝑦𝑢_ℎ,𝜓_𝐼+𝑎𝜕𝜕𝑡𝑦^ℎ+𝑑𝑦_ℎ−𝑦𝑢_ℎ,𝜓_𝐼+𝑎𝑦_ℎ0−𝑦𝑢_ℎ,𝜓_𝐼+^𝑇𝑡𝑐𝜏,𝑡;𝑦_ℎ−𝑦𝑢_ℎ(𝑡),𝜓_𝐼(𝜏)𝑑𝜏𝑑𝑡=−𝑦₀,𝜓(0)+𝑦^ℎ0−𝑦₀,𝜓(0)_𝑇0𝜕𝜕𝑦𝑡^ℎ,𝜓−𝜓^𝐼+𝑎𝜕𝜕𝑦𝑡^ℎ,𝜓−𝜓^𝐼+𝑑𝑦^ℎ+_𝑡0,𝜓−𝜓_𝐼−𝑐𝑡,𝜏;𝑦_ℎ(𝜏),𝜓−𝜓_𝐼(𝑡)𝑑𝜏𝑑𝑡_𝑇0=𝑓+𝐵𝑢_ℎ,𝜓−𝜓_𝐼𝑑𝑡+𝑎𝑦^ℎ0−𝑦₀,𝜓(0)+𝑦^ℎ0−𝑦₀,𝜓(0)_𝑇0_𝜏_𝜏𝜕𝜕𝑦𝑡^ℎ−div𝐴∇𝜕𝜕𝑦𝑡^ℎ−div𝐷∇𝑦^ℎ−_𝑡0div𝐶(𝜏,𝑡)∇𝑦_ℎ(𝜏)𝑑𝜏−𝑓−𝐵𝑢_ℎ+𝜓−𝜓^𝐼𝑑𝑡_𝑇0_𝜏_𝜕𝜏𝐴∇𝜕𝜕𝑦𝑡^ℎ⋅𝑛+𝐷∇𝑦^ℎ⋅𝑛−^𝑡0𝐶(𝑡,𝜏)∇𝑦_ℎ+𝑎𝑦_ℎ0(𝜏)⋅𝑛𝑑𝜏𝜓−𝜓_𝐼𝑑𝑙𝑑𝑡−𝑦₀,𝜓(0)+𝑦^ℎ0−𝑦₀,𝜓(0)=𝐷₁+𝐷₂+𝐷₃+𝐷₄.(4.10) Similarly, it follows from Lemmas 2.3,  2.5 and 4.1 that 𝐷₁⩽𝐶(𝛿)^𝑇0_𝜏ℎ^4𝜏_𝜏−𝜕𝜕𝑦_𝑡0𝑡^ℎ−div𝐴∇𝜕𝜕𝑦𝑡^ℎ−div𝐷∇𝑦^ℎdiv𝐶(𝜏,𝑡)∇𝑦_ℎ(𝜏)𝑑𝜏−𝑓−𝐵𝑢_ℎ+𝛿_𝑇0²𝑑𝑡𝐷₂||𝜓||^22,Ω𝑑𝑡⩽𝐶𝜉²⁴+𝛿‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω),⩽𝐶(𝛿)^𝑇0_𝜏ℎ^3𝑙_𝜕𝜏𝐴∇𝜕𝜕𝑦𝑡^ℎ⋅𝑛+𝐷∇𝑦^ℎ⋅𝑛−^𝑡0𝐶(𝑡,𝜏)∇𝑦_ℎ+𝛿_𝑇0(𝜏)⋅𝑛²𝑑𝑙𝑑𝑡||𝜓||^22,Ω𝑑𝑡⩽𝐷𝐶₃𝜉²⁵+𝛿‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω),+𝐷₄⩽𝐶𝜂²⁶+𝛿‖‖𝑦_ℎ−𝑦(𝑢_ℎ)‖‖^2𝐿_{20,𝑇;𝐿2(Ω)}.(4.11) Letting 𝛿 be small enough, then from (4.10)–(4.11), we have ‖‖𝑦_ℎ−𝑦𝑢_ℎ‖‖^2𝐿2_{0,𝑇;𝐿}²_(Ω)⩽𝐶𝜉²⁴+𝜉²⁵+𝜂²⁶.(4.12) The desired results (4.4) follows from (4.9)–(4.12).This completes the proof.

Using Lemmas 3.1 and 4.2, we have the following upper bounds.

Theorem 4.3. Let (𝑦,𝑝,𝑢) and (𝑦_ℎ,𝑝_ℎ,𝑢_ℎ) be the solutions of (2.17)–(2.19) and (2.22)–(2.24), respectively. Then, there hold the a posteriori error estimates as ‖‖𝑢−𝑢_ℎ‖‖^2𝐿_{20,𝑇;𝐿2Ω𝑈}+‖‖𝑦−𝑦_ℎ‖‖^2𝐿_{20,𝑇;𝐿2(Ω)}+‖‖𝑝−𝑝_ℎ‖‖^2𝐿_{20,𝑇;𝐿2(Ω)}⩽𝐶𝜂²¹+⁵_𝑖=2𝜉^2𝑖+𝜂²⁶.(4.13)

Proof. By triangle inequality, (3.44) and (3.38), Lemmas 3.1 and 4.2, we can easily prove (4.13) in the same way of getting (3.43).
This completes the proof.

5. Conclusion

In this paper, we study the semi-discrete adaptive finite element method for optimal control problem governed by a linear quasiparabolic Integrodifferential equation. We extend the existing methods in studying adaptive finite element approximation of optimal control governed by a parabolic Integrodifferential equation to the control governed by a quasiparabolic Integrodifferential equation. After presenting the weak form and the existence and uniqueness of the solution for the optimal control problem, the a posteriori error estimates for semi-discrete finite element approximations in 𝐿^∞(0,𝑇;𝐻¹(Ω))-norm and 𝐿²(0,𝑇;𝐿²(Ω))-norm are derived. The work will pave a way to derive the a posteriori error estimates of full discrete finite element approximations of this optimal control problem

Acknowledgments

This research was supported by Science and Technology Development Planning project of Shandong Province (no. 2011GGH20118), Shandong Province Natural Science Foundation (no. ZR2009AQ004), and National Natural Science Foundation of China (no. 11071141).