Adaptive Mixed Finite Element Methods for Parabolic Optimal Control Problems

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 themixed 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.


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.

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 W m,p Ω for Sobolev spaces on Ω with a norm • m,p given by v p m,p ∞ 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: where the bounded open set Ω ⊂ R 2 is a convex polygon with the boundary ∂Ω.Let K be a closed convex set in U L 2 J; L 2 Ω , f ∈ L 2 J; L 2 Ω , J 0, T , y 0 x ∈ H 1 0 Ω .Furthermore, we assume that the coefficient matrix A x a i,j x 2×2 ∈ L ∞ Ω; R 2×2 is a symmetric 2 × 2-matrix and there is a constant c > 0 satisfying for any vector X ∈ R 2 , X t AX ≥ c X 2 R 2 .j is positive, g 1 , g 2 , and j are locally Lipschitz on L 2 Ω 2 , W, U, and that there is a c > 0 such that j u − j u , u − u ≥ c u − u 0 , for all u, u ∈ U. Now we will describe the mixed finite element discretization of parabolic optimal control problems 2.1 .Let The Hilbert space V is equipped with the following norm:

2.3
We recast 2.1 as the following weak form: find p, y, u ∈ V × W × K such that min u∈K⊂U T 0 g 1 p g 2 y j u dt , 2.4 Similar to 26 , the optimal control problems 2.4 -2.7 have a unique solution p, y, u , and a triplet p, y, u is the solution of 2.4 -2.7 if and only if there is a costate q, z ∈ V × W such that p, y, q, z, u satisfies the following optimality conditions: y t , w div p, w f u, w , ∀w ∈ W, 2.9 y x, 0 y 0 x , ∀x ∈ Ω, 2.10 where •, • U is the inner product of U. In the rest of the paper, we will simply write the product as •, • whenever no confusion should be caused.Let T h be regular triangulation of Ω.They are assumed to satisfy the angle condition which means that there is a positive constant C such that, for all τ ∈ T h , C −1 h 2 τ ≤ |τ| ≤ Ch 2 τ , where |τ| is the area of τ, h τ is the diameter of τ and h max h τ .In addition C or c denotes a general positive constant independent of h.
Let V h × W h ⊂ V × W denote the Raviart-Thomas space 27 of the lowest order associated with the triangulation T h of Ω. P k denotes the space of polynomials of total degree at most k.Let V τ {v ∈ P 2 0 τ x • P 0 τ }, W τ P 0 τ .We define

2.15
The mixed finite element discretization of 2.4 -2.7 is as follows

2.16
where y h 0 x ∈ W h is an approximation of y 0 .The optimal control problem 2.16 again has a unique solution p h , y h , u h , and a triplet p h , y h , u h is the solution of 2.16 if and only if there is a costate q h , z h ∈ V h × W h such that p h , y h , q h , z h , u h satisfies the following optimality conditions:

2.17
We now consider the fully discrete approximation for the semidiscrete problem.Let Δt > 0, N T/Δt ∈ Z, and t i iΔt, i ∈ Z. Also, let For i 1, 2, . . ., N, construct the finite element spaces Similarly, construct the finite element spaces K i h ∈ K h with the mesh T i h .Let h i τ denote the maximum diameter of the element τ i in T i h .Define mesh functions τ • and mesh size functions For ease of exposition, we will denote τ t and h τ t by τ and h τ , respectively.
The following fully discrete approximation scheme is to find

2.22
It follows that the optimal control problems 2.19 -2.22 have a unique solution p i h , y i h , u i h , i 1, 2, . . ., N, and that a triplet h satisfies the following optimality conditions:

2.30
For any function w ∈ C J; L 2 Ω , let

2.31
Then the optimality conditions 2.23 -2.29 can be restated as.

2.38
In the rest of the paper, we will use some intermediate variables.For any control function U h ∈ K, we first define the state solution p U h , y U h , q U h , z U h which satisfies

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 u ∈ K, let M 1 , M 2 be the inverse operators of the state equation 2.6 , such that p u M 1 u and y u M 2 u are the solutions of the state equations 2.6 .Similarly, for given

3.1
It can be shown that

3.2
It is clear that F and F h are well defined and continuous on K and K i h .Also the functional F h can be naturally extended on K. Then 2.4 and 2.19 can be represented as In many application, F • is uniform convex near the solution u.The convexity of F • 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, u → g 1 M 1 U and u → g 2 M 2 U are convex.Then there is a constant c > 0, independent of h, such that 3.5 Theorem 3.1.Let u and U h be the solutions of 3.3 and 3.4 , respectively.Assume that K i h ⊂ K.In addition, assume that F h U h | τ ∈ H s τ , for all τ ∈ T h , s 0, 1 , and there is a v h ∈ K i h such that 3.6 Then one has where

3.8
Proof.It follows from 3.3 and 3.4 that 3.9 Then it follows from assumptions 3.5 , 3.6 and Schwartz inequality that

3.10
It is not difficult to show where z U h is defined in 2.39 -2.44 .Thanks to 3.11 , it is easy to derive 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:

3.14
The following well-known stability results are presented in 28 .

3.15
where Now, we are able to derive the main result. where 3.17 Proof.Letting ϕ be the solution of 3.13 with G Y h − y U h , we infer

3.18
Then it follows from 2.39 -2.40 that where and after, δ is an arbitrary positive number, C δ is the constant dependent on δ −1 .Now, we are in the position of estimating the error P h − p U h 2 L 2 J;L 2 Ω .First, we derive from 2.32 -2.33 and 2.39 -2.40 the following useful error equations:

Choose v h P h − p U h and w h Y h − y U h as the test functions
and add the two relations of 3.20 -3.21 such that

3.22
Then, using -Cauchy inequality, we can find an estimate as follows:

3.23
Note that then, using the assumption on A, we verify that .

3.25
Integrating 3.25 in time and since Y h 0 − y U h 0 0, applying Gronwall's lemma, we can easily obtain the following error estimate:

3.26
Using the triangle inequality and 3.26 , we deduce that

3.27
Then letting δ be small enough, it follows from 3.18 -3.27 that
Similarly, letting ψ be the solution of 3.14 with G Z h − z U h , with the aid of 2.26 -2.27 , 2.42 -2.43 , we can conclude the following.

Theorem 3.4. Let Y h , P h , Z h , Q h , U h and y U h , p U h , z U h , q U h
, U h be the solutions of 2.32 -2.38 and 2.39 -2.44 .Then, where

3.30
Let p, y, q, z, u and P h , Y h , Q h , Z h , U h be the solutions of 2.8 -2.14 and 2.32 -2.38 , respectively.We decompose the errors as follows:

3.35
Theorem 3.5.There is a constant C > 0, independent of h, such that

Mathematical Problems in Engineering
Proof.Part I Choose v 1 and w r 1 as the test functions and add the two relations of 3.32 -3.33 , then we have

3.38
Then, using the Cauchy inequality, we can find an estimate as follows: then, using the assumption on A, we can obtain that Integrating 3.41 in time and since r 1 0 0, applying the Gronwall's lemma, we can easily obtain the following error estimate

Part II
Similarly, choose v 2 and w r 2 as the test functions and add the two relations of 3.34 -3.35 , then we can obtain that

3.43
Then, using the Cauchy inequality, we can find an estimate as follows: then, using the assumption on A, we verify that Integrating 3.46 in time and since r 2 T 0, applying the Gronwall's lemma, we can easily obtain the following error estimate 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, q, z, u and P h , Y h , Q h , Z h , U h be the solutions of 2.8 -2.14 and 2.32 -2.38 , respectively.In addition, assume that j U h Z h | τ ∈ H s τ , for all τ ∈ T h , s 0, 1 , and that there is a

3.48
Then one has that, for all t ∈ 0, T ,

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 η is equally distributed over the computational mesh.Assume that an a posteriori error estimator η has the form of η 2 τ i η 2 τ i , where τ i is the finite elements.At each iteration, an average quantity of all η 2 τ i is calculated, and each η 2 τ i is then compared with this quantity.The element τ i is to be refined if η 2 τ i is larger than this quantity.As η 2 τ i represents the total approximation error over τ i , 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.

Mathematical Problems in Engineering
Starting from initial triangulations T h 0 of Ω, we construct a sequence of refined triangulation T h j as follows.Given T h j , we compute the solutions P h , Y h , Q h , Z h , U h of the system 2.32 -2.38 and their error estimator

4.1
Then we adopt the following mesh refinement strategy.All the triangles τ ∈ T h j satisfying η 2 τ ≥ αE j /n are divided into four new triangles in T h j 1 by joining the midpoints of the edges, where n is the numbers of the elements of T h j , α is a given constant.In order to maintain the new triangulation T h j 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 T h j 1 .The above procedure will continue until E j ≤ tol, where tol is a given tolerance error.

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: In our example, we choose the domain Ω 0, 1 × 0, 1 .Let Ω be partitioned into T h as described Section 2. For the constrained optimization problem, min u∈K F u , 5.4 where F u is a convex functional on U and K {u ∈ L 2 Ω : u ≥ 0 a.e. in Ω × J}, the iterative scheme reads n 0, 1, 2, . . .
where b •, • is a symmetric and positive definite bilinear form such that there exist constants c 0 and c 1 satisfying and the projection operator The bilinear form b •, • provides suitable preconditioning for the projection algorithm.An application of 5.5 to the discretized parabolic optimal control problem yields the following algorithm: − z nt , w h div q n , w h z n T , w h T T 0 y n − y d , w h , ∀w h ∈ W h , u n 1 P b K u n 1/2 , u n 1/2 , u n ∈ U h .

5.8
The main computational effort is to solve the four state and costate equations and to compute the projection P b K u n 1/2 .In this paper we use a fast algebraic multigrid solver to solve the

5.14
Thus, the control function is given by u x 1 , x 2 max u 0 − z, 0 .

5.15
In this example, the optimal control has a strong discontinuity, introduced by u 0 .The exact solution for the control u is plotted in Figure 1.The control function u is discretized by piecewise constant functions, whereas the state y, p and the costate z, q were approximation by the lowest-order Raviart-Thomas mixed finite elements.In Table 1, numerical results of u, 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 u, the accuracy has been improved obviously from the uniform meshes to the adaptive meshes in Table 1.In Figure 2, the adaptive mesh for u at t 0.25 is shown.In the computing, we use η 1 − η 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.

Figure 1 :
Figure 1: The profile of the control solution u at t 0.25.

Figure 2 :
Figure 2: The adaptive meshes of the control solution u at t 0.25.
is the solution of 2.19 -2.22 if and only if there is a costate Theorem 3.3.Let Y h , P h , Z h , Q h , U h and y U h , p U h , z U h , q U h , U h be the solutions of 2.32 -2.38 and 2.39 -2.44 .Then,

Table 1 :
Numerical results on uniform and adaptive meshes.