Research Article | Open Access
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 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.
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  and Geveci . Knowles was concerned with standard finite element approximation of parabolic time optimal control problems in . In  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 . 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 , 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. . Recently an adaptive finite element method for the estimation of distributed parameter in elliptic equation was discussed by Feng et al. . 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 , 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 for Sobolev spaces on with a norm given by , a seminorm given by . We set . For , we denote and .
We denote by the Banach space of all integrable functions from into with norm and the standard modification for . The details can be found in .
The parabolic optimal control problems that we are interested in are as follows: where the bounded open set is a convex polygon with the boundary . Let be a closed convex set in , , , . Furthermore, we assume that the coefficient matrix is a symmetric -matrix and there is a constant satisfying for any vector , . is positive, , , and are locally Lipschitz on , , , and that there is a such that , .
Now we will describe the mixed finite element discretization of parabolic optimal control problems (2.1). Let The Hilbert space is equipped with the following norm:
We recast (2.1) as the following weak form: find such that
Similar to , the optimal control problems (2.4)–(2.7) have a unique solution , and a triplet is the solution of (2.4)–(2.7) if and only if there is a costate such that satisfies the following optimality conditions: where is the inner product of . In the rest of the paper, we will simply write the product as whenever no confusion should be caused.
Let be regular triangulation of . They are assumed to satisfy the angle condition which means that there is a positive constant such that, for all , , where is the area of , is the diameter of and . In addition or denotes a general positive constant independent of .
Let denote the Raviart-Thomas space  of the lowest order associated with the triangulation of . denotes the space of polynomials of total degree at most . Let , . We define
The mixed finite element discretization of (2.4)–(2.7) is as follows: compute such that where is an approximation of . The optimal control problem (2.16) again has a unique solution , and a triplet is the solution of (2.16) if and only if there is a costate such that satisfies the following optimality conditions:
We now consider the fully discrete approximation for the semidiscrete problem. Let , , and , . Also, let For , construct the finite element spaces , (similar as ). Similarly, construct the finite element spaces with the mesh . Let denote the maximum diameter of the element in . Define mesh functions and mesh size functions such that , . For ease of exposition, we will denote and by and , respectively.
The following fully discrete approximation scheme is to find , such that It follows that the optimal control problems (2.19)–(2.22) have a unique solution , and that a triplet , , is the solution of (2.19)–(2.22) if and only if there is a costate such that satisfies the following optimality conditions:
For , let
For any function , let
In the rest of the paper, we will use some intermediate variables. For any control function , we first define the state solution which satisfies
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 , let , be the inverse operators of the state equation (2.6), such that and are the solutions of the state equations (2.6). Similarly, for given , , are the solutions of the discrete state equation (2.33). Let It can be shown that
In many application, is uniform convex near the solution . The convexity of 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, and are convex. Then there is a constant , independent of , such that
Proof. It follows from (3.3) and (3.4) that Then it follows from assumptions (3.5), (3.6) and Schwartz inequality that It is not difficult to show where 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:
The following well-known stability results are presented in .
Now, we are able to derive the main result.
Proof. Letting be the solution of (3.13) with , we infer
Then it follows from (2.39)-(2.40) that
where and after, is an arbitrary positive number, is the constant dependent on .
Now, we are in the position of estimating the error . First, we derive from (2.32)-(2.33) and (2.39)-(2.40) the following useful error equations: where , . Choose and as the test functions and add the two relations of (3.20)-(3.21) such that Then, using -Cauchy inequality, we can find an estimate as follows: Note that then, using the assumption on , we verify that Integrating (3.25) in time and since , applying Gronwall's lemma, we can easily obtain the following error estimate: Using the triangle inequality and (3.26), we deduce that Then letting be small enough, it follows from (3.18)–(3.27) that This proves (3.16).
Theorem 3.5. There is a constant , independent of , such that
Proof. Part I
Choose and as the test functions and add the two relations of (3.32)-(3.33), then we have Then, using the Cauchy inequality, we can find an estimate as follows: Note that then, using the assumption on , we can obtain that Integrating (3.41) in time and since , applying the Gronwall's lemma, we can easily obtain the following error estimate This implies (3.36).
Similarly, choose and as the test functions and add the two relations of (3.34)-(3.35), then we can obtain that Then, using the Cauchy inequality, we can find an estimate as follows: Note that then, using the assumption on , we verify that Integrating (3.46) in time and since , 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.
Theorem 3.6. Let and be the solutions of (2.8)–(2.14) and (2.32)–(2.38), respectively. In addition, assume that , and that there is a such that Then one has that, , where , , and 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 is equally distributed over the computational mesh. Assume that an a posteriori error estimator has the form of , where is the finite elements. At each iteration, an average quantity of all is calculated, and each is then compared with this quantity. The element is to be refined if is larger than this quantity. As represents the total approximation error over , 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.
Then we adopt the following mesh refinement strategy. All the triangles satisfying are divided into four new triangles in by joining the midpoints of the edges, where is the numbers of the elements of , is a given constant. In order to maintain the new triangulation 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 . The above procedure will continue until , where 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:
In our example, we choose the domain . Let be partitioned into as described Section 2. For the constrained optimization problem, where is a convex functional on and , the iterative scheme reads () where is a symmetric and positive definite bilinear form such that there exist constants and satisfying and the projection operator is defined. For given find such that The bilinear form provides suitable preconditioning for the projection algorithm. An application of (5.5) to the discretized parabolic optimal control problem yields the following algorithm: The main computational effort is to solve the four state and costate equations and to compute the projection . 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 efficiently. If one uses the 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, and , then where is the average of over .
In solving our discretized optimal control problem, we use the preconditioned projection gradient method with and a fixed step size . 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 .(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 and the function with
We assume that Thus, the control function is given by
In this example, the optimal control has a strong discontinuity, introduced by . The exact solution for the control is plotted in Figure 1. The control function is discretized by piecewise constant functions, whereas the state and the costate were approximation by the lowest-order Raviart-Thomas mixed finite elements. In Table 1, numerical results of , , and 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 , the accuracy has been improved obviously from the uniform meshes to the adaptive meshes in Table 1.
In Figure 2, the adaptive mesh for at is shown. In the computing, we use 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.
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.
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