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New a posteriori error estimates of mixed finite element methods for quadratic optimal control problems governed by semilinear parabolic equations with integral constraint
Boundary Value Problems volume 2013, Article number: 230 (2013)
In this paper, we investigate new and -posteriori error estimates of mixed finite element solutions for quadratic optimal control problems governed by semilinear parabolic equations. The state and the co-state are discretized by the order one Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in -norm and -norm for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the optimal control problem.
In this paper we consider quadratic optimal control problems governed by the semilinear parabolic equations
where the bounded open set is a convex polygon with the boundary ∂ Ω. . Let K be a closed convex set in the control space , , , , . For any , the function , for any , and . Assume that the coefficient matrix is a symmetric -matrix and there are constants satisfying for any vector , . We assume that the constraint on the control is an obstacle such that
Optimal control problems have been successfully utilized in scientific and engineering numerical simulation. Thus they must be solved by using some efficient numerical methods. Among these numerical methods, the finite element method was a good choice for solving partial differential equations. There have been extensive studies in convergence for finite element approximation of optimal control problems. A systematic introduction of the finite element method for optimal control problems can be found in [1–7].
Recently, an adaptive finite element method has been investigated extensively. It has become one of the most popular methods in the scientific computation and numerical modeling. Adaptive finite element approximation ensures a higher density of nodes in a certain area of the given domain, where the solution is more difficult to approximate, indicated by a posteriori error estimators. Hence it is an important approach to boost the accuracy and efficiency of finite element discretizations. There are lots of works concentrating on the adaptivity of many optimal control problems, for example, [8–12]. Note that all the above works aimed at the standard finite element method.
In many control problems, the objective functional contains the gradient of state variables. Thus, the accuracy of the gradient is important in numerical discretization of the coupled state equations. Mixed finite element methods are appropriate for the state equations in such cases since both the scalar variable and its flux variable can be approximated to the same accuracy by using such methods. 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.
Recently, in [13, 14] we did some primary work on a priori error estimates for nonlinear parabolic optimal control problems by mixed finite element methods. In , we considered a posteriori error estimates of triangular mixed finite element methods for semilinear elliptic optimal control problems. The state and the co-state were discretized by the Raviart-Thomas mixed finite element spaces and the control was approximated by piecewise constant functions. In , we derived a posteriori error estimates for linear parabolic optimal control problems by the lowest order Raviart-Thomas mixed finite element methods.
This paper is motivated by the idea of the article . We shall use the order one Raviart-Thomas mixed finite element to discretize the state and the co-state. Due to the limited regularity of the optimal control u in general, we therefore only consider a piecewise constant space. Then we derive a posteriori error estimates for the mixed finite element approximation of the optimal control problem. The estimators for the control, the state and the co-state variables are derived in the sense of -norm or -norm, which are different from the ones in .
In this paper, we adopt the standard notation for Sobolev spaces on Ω with a norm given by , a semi-norm given by . We set . For , we define , , and , . We denote by the Banach space of all integrable functions from J into with the norm for , and the standard modification for . Similarly, one can define the spaces and . The details can be found in .
The plan of this paper is as follows. In the next section, we shall construct the mixed finite element approximation and the backward Euler discretization for quadratic optimal control problems governed by semilinear parabolic equations (1.1)-(1.4). Then, we derive a posteriori error estimates for both the state and the control approximation in Section 3. Finally, we give a conclusion and some future work.
2 Mixed methods of optimal control problems
In this section we shall now discuss the mixed finite element approximation and the backward Euler discretization of quadratic semilinear parabolic optimal control problems (1.1)-(1.4). To fix the idea, we shall take the state spaces and , where V and W are defined as follows:
The Hilbert space V is equipped with the following norm:
Let , we recast (1.1)-(1.4) as the following weak form: find such that
It follows from  that optimal control problem (2.1)-(2.4) has a solution , and that if a triplet is the solution of (2.1)-(2.4), then there is a co-state such that satisfies the following optimality conditions:
where is the inner product of .
In , the expression of the control variable was given. Here, we adopt the same method to derive the following operator:
where denotes the integral average on Ω of the function z.
Let be regular triangulations of Ω. is the diameter of τ and . Let denote the order one Raviart-Thomas space associated with the triangulations of Ω. denotes the space of polynomials of total degree at most k. Let , . We define
Let and . The mixed finite element discretization of (2.1)-(2.4) is as follows: compute such that
where is an approximation of . Optimal control problem (2.13)-(2.16) again has a solution , and that if a triplet is the solution of (2.13)-(2.16), then there is a co-state such that satisfies the following optimality conditions:
Next, we define the standard -orthogonal projection , which satisfies: for any ,
Similar to (2.12), for variational inequality (2.23), we have the following conclusion . Assume that is known in variational inequality (2.23). The solution of the variational inequality is
Now we consider the fully discrete approximation for the above semidiscrete problem. Let , , and , . Also, let
The following fully discrete approximation scheme is to find , , such that
It follows that optimal control problem (2.27)-(2.30) has a solution , , and that if a triplet , , is the solution of (2.27)-(2.30), then there is a co-state such that satisfies the following optimality conditions:
For , let
For any function , let
Then optimality conditions (2.31)-(2.37) satisfy
Similar to (2.26), the solution of variational inequality (2.44) is
In the rest of the paper, we shall use some intermediate variables. For any control function , we first define the state solution satisfying
For , we shall write
are bounded functions in .
Let be the orthogonal -projection into  which satisfies:
Let be the Raviart-Thomas projection operator  which satisfies: for any ,
where denote the set of element sides in . We have the commuting diagram property
where and after, I denotes an identity matrix.
Further, the interpolation operator satisfies a local error estimate
The following lemmas are important in deriving a posteriori error estimates of residual type.
Lemma 2.1 Let be the average interpolation operator defined in . For or 1, and ,
Lemma 2.2 Let be the standard Lagrange interpolation operator . Then, for or 1, and ,
3 A posteriori error estimates
In this section we study new and -posteriori error estimates for the mixed finite element approximation to the semilinear parabolic optimal control problems. Let
It can be shown that (see  for some detail discussions)
It is clear that S and are well defined and continuous on K and . Also, the functional can be naturally extended on K. Then (2.1) and (2.27) can be represented as
In many applications, is uniform convex near the solution u. 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, there is , independent of h, such that
Firstly, let us derive a posteriori error estimates for the control u.
Theorem 3.1 Let u and be the solutions of (3.6) and (3.7), respectively. Assume that , (), and there is such that
Then we have
Proof It follows from (3.6) and (3.7) that
Then it follows from assumptions (3.8), (3.9), and the Schwarz inequality that
It is not difficult to show
where is defined in (2.46)-(2.51). Thanks to (3.14), it is easy to derive
Then, by estimates (3.13) and (3.15), we can prove the requested result (3.10). □
To estimate the error , we need the following well-known stability results for the following dual equations:
Next, we estimate the errors and .
Theorem 3.2 Let and be the solutions of (2.38)-(2.44) and (2.46)-(2.51), respectively. Then we have
Proof We define as follows:
Then from (3.20) we deduce that
Combining (3.20)-(3.21) and the definitions of and , we can get the following equality:
Let κ be the solution of (3.16) with , we infer that
Furthermore, using (2.38)-(2.40), (2.46)-(2.48) and (2.56)-(2.58), we can obtain that
When , ,
Similar to Theorem 3.2 of reference , we have derived the following estimate:
This proves (3.19). □
Now, we are in a position to estimate the errors and .
Theorem 3.3 Let and be the solutions of (2.38)-(2.44) and (2.46)-(2.51), respectively. Then we have the following error estimate:
where are defined in Theorem 3.2, and
Proof We first define as follows:
Then from (2.34) and (3.29) we deduce that
Now, we let
Combining (2.41), (3.30) and the definitions of , , and , we get
Let ϖ be the solution of (3.17) with . Then it follows from (2.41)-(2.43), (2.49)-(2.51) and (2.56)-(2.58) that
To prove (3.28), the first step is to estimate . Let , when , by Lemmas 2.1, 2.2 and 3.1, we have
Now we estimate . Let again. Similarly, when ,
Next we estimate , . It follows from Lemma 3.1 that
Furthermore, we estimate , . It follows from Lemma 3.1 that
Hence, from (3.33)-(3.40) we have that when , ,
Then it follows from (3.41)-(3.42) that
Similar to (3.27), we can prove that
The triangle inequality and (3.43) yield (3.28). □
Let and be the solutions of (2.5)-(2.11) and (2.38)-(2.44), respectively. We decompose the errors as follows:
From (2.5)-(2.11) and (2.38)-(2.44), we derive the error equations:
for any , .
Theorem 3.4 Let and be the solutions of (2.5)-(2.11) and (2.46)-(2.51), respectively. There is a constant , independent of h, such that
Proof Part I. Choosing and as the test functions and adding the two relations of (3.45)-(3.46), we have
Then, using the ϵ-Cauchy inequality, we find an estimate as follows:
then, using the assumption on A, we obtain that
Integrating (3.53) in time and since , applying Gronwall’s lemma, we easily obtain the following error estimate:
This implies (3.49).
Part II. Similarly, choosing