Abstract
We will investigate the superconvergence for the semidiscrete finite element approximation of distributed convex optimal control problems governed by semilinear parabolic equations. The state and costate are approximated by the piecewise linear functions and the control is approximated by piecewise constant functions. We present the superconvergence analysis for both the control variable and the state variables.
1. Introduction
Finite element approximation of optimal control problems plays a very important role in numerical methods for these problems. There have been extensive studies on this aspect, for example, [1–14]. A systematic introduction of finite element method for PDEs and optimal control problems can be found in, for example, [15–18]. The superconvergence of nonlinear parabolic problem was studied in [19]. In [20], superconvergence was obtained for parabolic optimal control problems with convex control constraints, where the state partial differential equations are linear.
Optimal control problems governed by nonlinear parabolic state equations, a priori error estimates of finite element approximation, were studied in, for example, [21, 22]. In this paper, we will study the superconvergence of both the control variable and the state variables for this problem.
The model optimal control problem that we shall study in detail is the following convex optimal control problem: Here, the bounded open set is a convex polygon or has the smooth boundary . Let be a linear continuous operator from to and . Assume that , with being a symmetric matrix and, for any vector , there is a constant satisfying Here, denotes the admissible set of the control variable, which is defined by
In this paper, we adopt the standard notation for Sobolev spaces on with a norm given by a semi-norm given by We set . For , we denote
We denote by the Banach space of all integrable functions from into , with norm and the standard modification for , where . Similarly, one can define the spaces and . The details can be found in [23]. In addition, and denote general positive constants independent of .
The plan of the paper is as follows. In Section 2, we shall give a brief review on the finite element method and then construct the approximation scheme for the optimal control problem. In Section 3, we shall give some preliminaries and some intermediate error estimates. In Section 4, superconvergence results for both control and state variables were derived. In Section 5, we give a numerical example to demonstrate our theoretical results. In the last section we make a conclusion and state some future works.
2. Approximation for the Optimal Control Problem
To fix idea, we shall take the state space with and the control space with . Then the problem (1) can be restated as where , , and It follows from the assumption on that there is a positive constant such that , for all .
We make the following assumptions.(1) for any , for any , and .(2)Let , where is a smooth and convex function such that and . The function has the same property as .
It is well known (see, e.g., [24]) that the control problem (8)–(10) has a solution and that if a pair is the solution of (8)–(10), then there is a costate such that the triplet satisfies the following optimality conditions for : where is the adjoint operator of .
In the following we construct the finite element approximation for the optimal control problem (8)–(10). For ease of exposition we will assume that is a convex polygon. Let be a quasi-uniform (in the sense of [25]) partition of into triangles. And let be the maximum diameter of in . Moreover, we set where is the space of polynomial of degree less than or equal to 1.
For simplicity, in this paper we shall assume that . Now, the semidiscrete finite element approximation of the problem (8)–(10) is as follows: where and is an approximation of which will be defined below. The control problem (16)–(18) has a solution , and if a pair is the solution of (16)–(18), then there is a costate such that the triplet satisfies the following optimal conditions:
3. Some Preliminaries
First, we shall use some intermediate variables. For any , let be the solution of the following equations:
Then, for any , let be the solution of the following equations: Thus, we have
We define the standard -orthogonal projection , which satisfies, for any ,
Next, let us recall the elliptic projection , which satisfies, for any , Let
We have the approximation properties:
Now, we establish the following error estimates for the intermediate variables.
Lemma 1. Let be the solution of (12)–(14); for sufficiently small, there exists a positive constant which only depends on and , such that
Proof. Let and in (22)-(23), respectively; then we have the following error equations:
for any and . We shall estimate (31) and (32), respectively.
First, let in (33); we have
namely,
Now, we estimate the right side of (36). Using the continuity of and (29), we have
Combining (36)-(37), using the -Cauchy’s inequality and the assumption of and , we have
Notice that
then, integrating (38) in time and using Gronwall’s lemma, we have
Then, by choosing in (34), we have
namely,
Now, we estimate the right side of (42). From the assumption of the convex function , we have
and using the assumption of and -Cauchy’s inequality, we have
where we used the embedding . Combining (42)–(44) and from the assumption of and , we have
Notice that
then integrating (45) in time, using Gronwall’s lemma and (31), we have
which completes the proof of Lemma 1.
Lemma 2. For any , if the intermediate solution satisfies then, one has
Proof. From (22)-(23) and (24), we have the following error equations:
for any and . Using the definition of , the above equation can be restated as
Let in (52); we have
For the first term of (54), using the definition of , we have
Similarly, from the assumption of , we can obtain
Combining (54)–(56), using -Cauchy’s inequality, and from the assumption of and , we have
Notice that
Then, integrating (57) in time and using Gronwall’s lemma, we have
Let in (53); we have
Now we estimate the right side of (60):
where we used the assumption of :
where we used the definition of :
where we used the embedding and the assumption of :
where we used the assumption of and the definition of .
Notice that
then, combining (60)–(64), using -Cauchy’s inequality, and the assumption of and , we have
Integrating (66) in time and using Gronwall lemma, we have
which implies (50). Thus, we complete the proof of Lemma 2.
4. Superconvergence Properties
In this section, we will discuss the superconvergence for both the control variable and the state variables by using the results we have got before. Let be the average operator defined in [26]. Let In this paper, we assume that and are regular such that meas .
Let and be the solution of (9) and (17), respectively. Set Then the reduced problems of (8) and (16) read as respectively. It can be shown that where is the solution of (22)-(23) with .
In many applications, is uniform convex near the solution . The convexity of is closely related to the second order sufficient conditions of the control problem, which are assumed in many studies on numerical methods of the problem. For instance, in some applications, is convex; see [27] for examples. Thus if is uniform convex (e.g., ), which is frequently met, then, there is a constant , independent of , such that where and are solutions of (70) and (71), respectively, is the orthogonal projection of which is defined in (26). We shall assume the above inequality throughout this paper.
First, we are going to formulate the superconvergence result for the control variable.
Theorem 3. Let be the solution of (12)–(14) and let be the solution of (19)–(21). One assumes that the exact control and state solution satisfy Then, one has
Proof. Let in (14) and in (21) and then, add the two inequalities; we have
Hence,
For the second term of the right hand of (77), we divide it into four parts:
then, from (77)-(78), we have
Using the definition of and the assumption of , we have
From Taylor’s expansion of the function , there exists some value such that
where we used the assumption of and the approximation property (29).
Notice that
Obviously, . From (14), we have pointwise a.e. ; we choose and , so that . Hence, . Then,
From the assumption of , we have
Then, integrating (79) in time and combining Lemmas 1 and 2 and (79)–(84), we have
where we have used -Cauchy’s inequality which implies (75). Thus, we complete the proof of Theorem 3.
In the following, we shall establish the superconvergence results for the state variable and costate variable .
Theorem 4. Let be the solution of (12)–(14) and let be the solution of (19)–(21). One assumes that the exact control and state solution satisfy Then, one has
Proof. First, we have the following error equation from (12) and (19):
for any .
Using the definition of in (27), we have
for any .
We take in (90), and using the assumption of and , then
Now, we estimate the right hand of (91). From (29)-(30), (75), and using -Cauchy’s inequality, we have
Then, using the assumption of , we have
Therefore, inserting (92)–(94) in (91), we have
Notice that
then, integrating (95) in time, using Gronwall’s lemma, and from the result of Theorem 3, we can easily obtain that
which implies (87).
Then, from (13) and (20), we have the following error equation
for any . Using the definition of in (27), we have
We take , and using the assumption of and , then
Now, we estimate the right hand of (100). From the assumption of , we have
Using the definition of , we have
From the assumption of and the definition of , we can obtain
From the assumption of and -Cauchy’s inequality, we have
Therefore, inserting (101)–(104) in (100), we have
Notice that
then, integrating (105) in time, using Gronwall’s lemma and (97), we have
Thus, we complete the proof of Theorem 4.
5. Numerical Example
In this section, we carry out a numerical example to demonstrate our theoretical results. The optimal problem was solved numerically by a precondition projection algorithm; see, for instance, [28], with codes developed based on AFEPack [29]. In order to validate the superconvergence results, we shall consider the following full-discrete scheme. Let , , being the time-step, and , the integral part of . In the example, we choose the domain and .
We now shall consider the fully discrete approximation for semidiscrete problem (19)–(21) by using the backward Euler scheme in time. The scheme is as follows: find such that where is an approximation of defined above.
Example 5. The example is to solve the following 2D parabolic control problem: where
The dual equation of the state equation is
Table 1 shows the errors and on a sequence of uniformly refined meshes, where denotes and similarly for . We choose in our numerical example. The superconvergence phenomenon of can be observed clearly from Table 1.
6. Conclusion
In this paper, we present the superconvergence analysis for the semidiscrete finite element approximation of optimal control problems governed by semilinear parabolic equations. Here, the results seem to be new and detailed proof can be used in more areas. We will study some results of superconvergence for optimal control, such as superconvergence for optimal control problems governed by semilinear parabolic equations with mixed finite element method.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Science Foundation of China (11271145), Foundation for Talent Introduction of Guangdong Provincial University, Specialized Research Fund for the Doctoral Program of Higher Education (20114407110009), the Project of Department of Education of Guangdong Province (2012KJCX0036), and TianYuan Special Funds of the National Natural Science Foundation of China (Grant no. 11226313).