The aim of this work is to study adaptive fully-discrete finite element methods for quadratic boundary optimal control problems governed by nonlinear parabolic equations. We derive a posteriori error estimates for the state and control approximation. Such estimates can be used to construct reliable adaptive finite element approximation for nonlinear quadratic parabolic boundary optimal control problems. Finally, we present a numerical example to show the theoretical results.
In this paper, we study the fully-discrete finite element approximation for quadratic boundary optimal control problems governed by nonlinear parabolic equations. Optimal control problems are very important models in engineering numerical simulation. They have various physical backgrounds in many practical applications. Finite element approximation of optimal control problems plays a very important role in the numerical methods for these problems. The finite element approximation of a linear elliptic optimal control problem is well investigated by Falk  and Geveci . The discretization for semilinear elliptic optimal control problems is discussed by Arada, Casas, and Tröltzsch in . Systematic introductions of the finite element method for optimal control problems can be found in [4-6].
As one of important kinds of optimal control problems, the boundary optimal control is widely used in scientific and engineering computing. The literature in this aspect is huge; see, e.g., [7-10]. For some quadratic boundary optimal control problems, Liu and Yan [11,12] investigated a posteriori error estimates and adaptive finite element methods. Alt and Mackenroth  were concerned with error estimates of finite element approximations to state constrained convex parabolic boundary optimal control problems. Arada et al. discussed the numerical approximation of boundary optimal control problems governed by semilinear elliptic equations with pointwise constraints on the control in . Although a priori error estimates and a posteriori error estimates of finite element approximation are widely used in numerical simulations, they have not yet been utilized in nonlinear parabolic boundary optimal control problems.
Adaptive finite element approximation is the most important method to boost accuracy of the finite element discretization. It ensures a higher density of nodes in a certain area of the given domain, where the solution is discontinuous or more difficult to approximate, using a posteriori error indicator. A posteriori error estimates are computable quantities in terms of the discrete solution that measure the actual discrete errors without the knowledge of exact solutions. They are essential in designing algorithms for mesh which equidistribute the computational effort and optimize the computation. Recently, in [15-18], we derived a priori error estimates, a posteriori error estimates and superconvergence for optimal control problems using mixed finite element methods.
In this paper, we adopt the standard notation for Sobolev spaces on Ω with a norm given by and a semi-norm given by . We set . For , we denote , , and , . We denote by the Banach space of all integrable functions from J into with the norm for , and the standard modification for . The details can be found in .
In this paper, we derive a posteriori error estimates for a class of boundary optimal control problems governed by a nonlinear parabolic equation. To our best knowledge, in the context of nonlinear parabolic boundary optimal control problems, these estimates are new. The problem that we are interested in is the following nonlinear quadratic parabolic boundary optimal control problem:
subject to the state equations
where the bounded open set is 2 regular convex polygon with boundary ∂Ω, , , , , and α is a positive constant. For any , the function , for any , and . We assume the coefficient matrix is a symmetric positive definite matrix, and there is a constant satisfying for any vector , . Here, K denotes the admissible set of the control variable defined by
The plan of this paper is as follows. In the next section, we present a finite element discretization for nonlinear quadratic parabolic boundary optimal control problems. A posteriori error estimates are established for the finite element approximation solutions in Section 3. In Section 4, we give a numerical example to prove the theoretical results.
2 Finite element methods for parabolic boundary optimal control
We shall now describe a finite element discretization of nonlinear quadratic parabolic boundary optimal control problem (1)-(4). Let , , . Let
Then quadratic parabolic boundary optimal control problem (1)-(4) can be restated as
where the inner product in or is indicated by , and B is a continuous linear operator from U to .
It is well known (see, e.g., ) that the optimal control problems have at least a solution , and that if a pair is the solution of (9)-(11), then there is a co-state such that the triplet satisfies the following optimality conditions:
where is the adjoint operator of B. In the rest of the paper, we shall simply write the product as whenever no confusion should be caused.
Let us consider the finite element approximation of control problem (9)-(11). Again, here we consider only n-simplex elements and conforming finite elements.
Let be a regular partition of Ω. Associated with is a finite dimensional subspace of such that are polynomials of m-order ( ) and . It is easy to see that . Let be a partition of ∂Ω into disjoint regular -simplices s, so that . Associated with is another finite dimensional subspace of such that are polynomials of m-order ( ) and . Let denote the maximum diameter of the element in , , and . In addition C or c denotes a general positive constant independent of h.
By the definition of a finite element subspace, the finite element discretization of (9)-(11) is as follows: compute such that
where , is an approximation of .
Again, it follows that optimal control problem (17)-(19) has at least a solution , and that if a pair is the solution of (17)-(19), then there is a co-state such that the triplet satisfies the following optimality conditions:
We now consider the fully discrete approximation for the semidiscrete problem. Let , , and let , . Also, let
For , we construct the finite element spaces with the mesh (similar to ). Similarly, we construct the finite element spaces with the mesh (similar to ). Let denote the maximum diameter of the element in . Define mesh functions , and mesh size functions , such that , , , . For ease of exposition, we denote , , , and by τ, s, , and , respectively.
Then the fully discrete finite element approximation of (17)-(19) is as follows. Compute , , such that
where , is an approximation of .
Now, it follows that optimal control problem (25)-(27) has at least a solution , , and that if a pair , , is the solution of (25)-(27), then there is a co-state , , such that the triplet satisfies the following optimality conditions:
For , let
For any function , let , . Then the optimality conditions (28)-(32) can be restated as follows:
In the rest of the paper, we shall use some intermediate variables. For any control function , we define that the state solution satisfies
Now we restate the following well-known estimates in .
Lemma 2.1Let be the Clément-type interpolation operator defined in. Then for any and all elementτ,
wherelis the edge of the element.
For , we write
are bounded functions in .
3 A posteriori error estimates
In this section we obtain a posteriori error estimates for nonlinear quadratic parabolic boundary optimal control problems. Firstly, we estimate the error .
Theorem 3.1Let and be the solutions of (41)-(44) and (36)-(40), respectively. Then
wherelis a face of an elementτ, is the size of the facel, is theA-normal derivative jump over the interior faceldefined by
wherenis the unit normal vector on outwards .
Proof Let , and let be the Clément-type interpolator of defined in Lemma 2.1. Note that
Using equations (36) and (41), we infer that
Let us bound each of the terms on the right-hand side of (49). By Lemma 2.1 we have
Next, using Lemma 2.1, we get
For - , the Schwarz inequality implies
Finally, add inequalities (49)-(55) to obtain
This completes the proof. □
Analogously to Theorem 3.1, we show the following estimates.
Theorem 3.2Let and be the solutions of (41)-(44) and (36)-(40), respectively. Then
where - are defined in Theorem 3.1, lis a face of an elementτ, is theA-normal derivative jump over the interior faceldefined by
wherenis the unit normal vector on outwards .
Proof Let , and let , where is the Clément-type interpolator defined in Lemma 2.1. Note that , then we obtain
Using equations (38) and (43), we obtain
Now let us bound each of the terms on the right-hand side of (58). By Lemma 2.1 we have
Next, using Lemma 2.1, we get
The Schwarz inequality implies
Next, for , we obtain
Finally, add inequalities (58)-(64) and combine Theorem 3.1 to obtain
This completes the proof. □
For given , let M be the inverse operator of the state equation (12) such that is the solution of the state equation (12). Similarly, for given , is the solution of the discrete state equation (36). Let
It is clear that S and are well defined and continuous on K and . Also, the functional can be naturally extended on K. Then (9) and (25) can be represented as
It can be shown that
where is the solution of equations (41)-(43).
In many applications, is uniform convex near the solution u (see, e.g., ). The convexity of is closely related to the second-order sufficient conditions of the control problems, which are assumed in many studies on numerical methods of the problems. If is uniformly convex, then there is a such that
where u and are the solutions of (66) and (67), respectively. We assume the above inequality throughout this paper.
In order to have sharp a posteriori error estimates, we divide ∂Ω into some subsets:
Then it is clear that three subsets do not intersect each other, and , .
Let be the solution of (41)-(44). We establish the following error estimate, which can be proved similarly to the proofs given in .
Theorem 3.3Letuand be the solutions of (66) and (67), respectively. Then
Proof It follows from the inequality (68) that
It is easy to see that
Since is piecewise constant, if is not empty. If , there exists and such that , and . For example, one can always find such a required β from one of the shape functions on s. Hence, , where as and otherwise . Then it follows from (40) that
Note that on , , and from (72) we have that
Let be the reference element of s, , and be a part mapped from . Note that , are both norms on . In such a case, for the function β fixed above, it follows from the equivalence of the norm in the finite-dimensional space that
where the constant C can be made independent of β since it is always possible to find the required β from the shape functions on s so that
It follows from the definition of that on . Note that , we have that
It is easy to show that
Therefore, (69) follows from (70)-(72) and (76)-(78). □
Hence, we combine Theorems 3.1-3.3 to conclude the following.
Theorem 3.4Let and be the solutions of (12)-(16) and (36)-(40), respectively. Then
where , and are defined in Theorems 3.1-3.3, respectively.
Proof From (12)-(15) and (41)-(44), we obtain the error equations
for all and . Thus it follows from (80)-(81) that
By using the stability results in , we can prove that
Finally, combining Theorems 3.1-3.3 and (84)-(85) leads to (79). □
4 Numerical example
In the section, we use a posteriori error estimates presented in our paper as an indicator for the adaptive finite element approximation. The optimization problem is solved numerically by a preconditioned projection algorithm, with codes developed based on AFEPACK. The optimal control problem is
In the example, we choose the domain and . Let Ω be partitioned into as described in Section 2. We use as the control mesh refinement indicator and - as the states and co-states.
For the constrained optimization problem , where is a convex functional on U, 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 as follows. For given , find such that
The bilinear form provides suitable preconditioning for the projection algorithm. An application of (86) to the discretized nonlinear parabolic boundary optimal control problem yields the following algorithm:
The main computational effort is to solve the state and co-state equations and to compute the projection . In this paper we use a fast algebraic multigrid solver to solve the state and co-state equations. Then it is clear that the key to saving computing time is finding how to compute efficiently. For the piecewise constant elements, and , then
where is the average of over T. In solving our discretized optimal control problem, we use the preconditioned projection gradient method (89)-(92) with and a fixed step size . In the numerical simulation, we use a piecewise linear finite element space for the approximation of y and p, and a piecewise constant for u.
It can be clearly seen from Table 1 that on the adaptive meshes one may use less degree of freedom to produce a given control error reduction. Then it is clear that these a posteriori error estimates are very good for the parabolic boundary optimal control, and the adaptive finite element method is more efficient.
Table 1. Comparison of uniform mesh and adaptive mesh
The author declares that he has no competing interests.
ZL participated in the design of all the study and drafted the manuscript.
This work is supported by National Science Foundation of China (11201510), Mathematics TianYuan Special Funds of the National Natural Science Foundation of China (11126329), China Postdoctoral Science Foundation funded project (2011M500968), Natural Science Foundation Project of CQ CSTC (cstc2012jjA00003), and Natural Science Foundation of Chongqing Municipal Education Commission (KJ121113). The author expresses his thanks to the referees for their helpful suggestions, which led to improvements of the presentation.
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