International Journal of Differential Equations
Volume 2018 (2018), Article ID 4753792, 11 pages
https://doi.org/10.1155/2018/4753792

Research Article

Finite Volume Element Approximation for the Elliptic Equation with Distributed Control

Quanxiang Wang,¹ Tengjin Zhao,² and Zhiyue Zhang²

¹College of Engineering, Nanjing Agricultural University, Nanjing 210031, China
²Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

Correspondence should be addressed to Zhiyue Zhang; zhangzhiyue@njnu.edu.cn

Received 1 February 2018; Revised 21 April 2018; Accepted 3 September 2018; Published 1 November 2018

Guest Editor: Omar Abu Arqub

Copyright © 2018 Quanxiang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we consider a priori error estimates for the finite volume element schemes of optimal control problems, which are governed by linear elliptic partial differential equation. The variational discretization approach is used to deal with the control. The error estimation shows that the combination of variational discretization and finite volume element formulation allows optimal convergence. Numerical results are provided to support our theoretical analysis.

1. Introduction

In recent years, the optimization with partial differential equation constraints (PDEs) has received a significant impulse. Because of wide applicability of the field, a lot of theoretical results have been developed. Generally, it is difficult to obtain the analytical solutions for optimal control problems with PDEs. Factually, only approximate solutions or numerical solutions can be expected. Therefore, many numerical methods have been proposed to solve the problems.

Finite element method is an important numerical method for the problems of partial differential equations and widely used in the numerical solution of optimal control problems. There are extensive studies in convergence of finite element approximation for optimal control problems. For example, priori error estimates for finite element discretization of optimal control problems governed by elliptic equations are discussed in many publications. In [1], a new approach to error control and mesh adaptivity is described for the discretization of the optimal control problems governed by elliptic partial differential equations. In [2], the error estimates for semilinear elliptic optimal controls in the maximum norm are presented. Chen and Liu present a priori error analysis for mixed finite element approximation of quadratic optimal control problems [3]. In [4], a priori error analysis for the finite element discretization of the optimal control problems governed by elliptic state equations is considered. Hou and Li investigate the error estimates of mixed finite element methods for optimal control problems governed by general elliptic equations and derive and error estimates for both the control and state variables [5].

The finite volume element method has been one of the most commonly used numerical methods for solving partial differential equations. The advantages of the method are that the computational cost is less than finite element method, and the mass conservation law is maintained. So it has been extensively used in computational fluid dynamics [6–12]. However, there are only a few published results on the finite volume element method for the optimal control problems. In [13], the authors discussed distributed optimal control problems governed by elliptic equations by using the finite volume element methods. The variational discretization approach is used to deal with the control and the error estimates are obtained in some norms. In [14], the authors considered the convergence analysis of discontinuous finite volume methods applied to distributed optimal control problems governed by a class of second-order linear elliptic equations.

In this paper, we will investigate the finite volume element method for the general elliptic optimal control problem with Dirichlet or Neumann boundary conditions. The variational discretization approach is used to deal with the control, which can avoid explicit discretization of the control and improve the approximation. In addition, we discuss the optimal control problems in polygonal domains with corner singularities. In this situation, the solution does not admit integrable second derivatives. The desired convergence results of finite volume element schemes cannot be expected. Two effective methods are proposed to compensate the negative effects of the corner singularities. The corresponding results will be reported in the future.

The rest of the paper is organized as follows. In Section 2, the model problem and the finite volume element schemes are introduced. Section 3 presents the error estimates of the finite volume element schemes. In Section 4, numerical results are supplied to justify the theoretical analysis. Brief conclusions are given in Section 5.

2. Problem Statement and Discretization

2.1. Model Problem

In this paper, we consider the following second-order elliptic partial differential equation:

where is a bounded convex polygon with boundary , is a symmetric and uniformly positive definite matrix, is a sufficient smooth function defined on , denotes the linear and continuous control operator, , and and have enough regularity so that this problem has a unique solution when we combine either homogeneous Dirichlet or Neumann boundary conditions on .

In addition, we use the following notations for the inner products and norms on , , and :

The corresponding weak formulation for (1) is

where

and

denotes either depending on the prescribed type of boundary conditions (homogeneous Neumann or Dirichlet).

Now, we consider the following optimal control problem for state variable and the control variable :

over all subject to elliptic state problem (3) and the control constraints

where is a given desired state and is a regularization parameter. We define the set of admissible control by

where is a nonempty, closed, and convex subset of ,.

From standard arguments for elliptic equations, we can obtain the following propositions.

Proposition 1. For fixed control , the state equation (3) admits a unique solution . Moreover, there is a constant , which does not depend on , such that

Proposition 2. Let be a nonempty, closed, bounded, and convex set, in and ; then the optimal control problem (6) admits a unique solution .

This proof follows standard techniques [15].

The adjoint state equation for is given by

where the equation is the weak formulation of the following elliptic problem:

with homogeneous Neumann or Dirichlet boundary conditions.

Proposition 3. The necessary and sufficient optimality conditions for (6) and (7) can be expressed as the variational inequality

Further, the variational inequality is equivalent to

where denotes the orthogonal projection in onto the admissible set of the control and is the adjoint operator of .

2.2. Discretization

Now we describe the finite volume element discretization of the optimal control problem (6).

We consider a quasi-uniform triangulation . Divide into a sum of finite number of small triangles such that they have no overlapping internal region and a vertex of any triangle does not belong to a side of any other triangle. At last, we can obtain a triangulation such that .

We then construct a dual mesh related to . Let be a node of a triangle, the adjacent nodes of , and the midpoint of . Choose the barycenter of triangle as the node of the dual mesh. Connect successively , ,..., , , to form a polygonal region , called a control volume. Figure 1 presents a sketch of a control volume.

Figure 1: Control volume with barycenter as internal point.

Let be the trial function space defined on the triangulation ,

and be the test function space defined on the dual mesh ,

In this way, we have

where are the standard node basis functions with the nodes and are the characteristic functions of the control volume .

Let and be the interpolation projections onto the trial function space and test function space , respectively. By the interpolation theory, we have for

Then the finite volume element schemes for (3), (10), and (13) are defined as follows:

where

3. Error Estimates

In order to present the error estimates, we first introduce some lemmas in preparation of the proof for the main convergence theorem.

3.1. Some Lemmas

According to [16], we have the following lemma, which indicates that the bilinear form is coercive on .

Lemma 4. is positive definite for small enough ; namely, there exist , such that for

We seldom have a symmetric bilinear form even though is symmetric. The following lemma is used to measure how far the bilinear form is from being symmetric [17].

Lemma 5. There exist positive constants , such that, for and , we have

Furthermore, we introduce the auxiliary functions and which are the solutions of the following problems:

For the problems, we can obtain the following results.

Lemma 6. Let and be the solution of (18) and (19) and , be the solution of (24). Then, we have

Proof. Combining (18) and (24), we have

By taking and using Lemma 4, we have

where Lemma 4 is used. At last, we can obtain (25) with Cauchy-Schwarz inequality. Equation (26) can be obtained similarly.

The results in [18] can easily be extended to cover the elliptic equations with homogeneous Neumann boundary conditions. Now we list the useful theoretical results in the following lemma.

Lemma 7. Let and be the solution of (4) and (10), respectively, and , be the solution of (24), , , , and . Then there exists a positive constant and such that for

3.2. Error Estimate

Theorem 8. Assume that and are the solutions of (6) and (20), respectively, , , , and . Then there exists a positive constant and such that for

Proof. Let us test (12) with , and (20) with , and sum up the two inequalities; we have

We further get

where

Combining the above equations, we can obtain

According to Lemmas 5, 6, and 7, we have

Using Lemmas 5 and 6, we conclude

Combining (34)–(37) and using Lemma 7, we can obtain the desirable result

Theorem 9. Assume that are the solutions of (6) and (11), respectively, and are the solutions of (18) and (19), respectively, , , , and . Then there exists a positive constant such that

Proof. Using the triangle inequality, we have

From Lemma 6 and Theorem 8, we can obtain

Using Lemma 7, we can obtain the desired result

Similarly, we have

3.3. Error Estimate

Theorem 10. Assume that are the solutions of (6) and (11), respectively, and are the solutions of (18) and (19), respectively, , , , and . Then there exists a positive constant such that

Proof. Using the triangle inequality, we have

From Lemma 4, we can obtain

By using Lemma 7 and Theorems 8 and 9, we can obtain the desired result

Similarly, we have

Remark 11. In the case , the projection equations (13) and (20) become and , respectively. Using the above theorem, we can obtain the following error estimate:

3.4. Error Estimate

Theorem 12. Assume that are the solutions of (18), (19), and (20), respectively, , , , and . Then there exists a positive constant such that

Proof. Using the projection equations (13) and (20), we have

Similarly, we have

4. Numerical Experiments

In this section, we report some numerical results of finite volume element schemes for the elliptic optimal control problems. To illustrate the theoretical analysis, the following rate of convergence is defined:

where is the numerical solution with space step size and the analytical solution. The rate approaching the number 2 would indicate second-order accuracy in space.

4.1. Experiment 1

To validate the finite volume element schemes for the solution of elliptic optimal control problems, test example is needed for which the exact solutions are known in advance [15]. We consider the problems with homogeneous Neumann boundary condition,

subject to

where denotes unit square , =, is the outer unit normal vector, and . Under these settings, the optimal control is

The adjoint state is

and the associated state is

Then we can determine the function accordingly.

Errors of finite volume element schemes in , , and norm are computed. Data are listed in Tables 1–3. In Tables 1 and 3, errors in norm have optimal convergence order for both control and adjoint state. These results confirm our theoretical error analysis (44). In Table 2, due to additional smoothness of the state, the error is . The convergence results in Tables 1–3 demonstrate second-order accuracy in and norm for the control, state, and adjoint state.

Table 1: Errors of the control for different error norms.

error

error

error

1/8
1.7412E-01
-
4.4468E-02
-
2.3006
-

1/16
4.1870E-02
2.05
1.0374E-02
2.09
1.1313
1.02

1/32
1.0476E-02
2.00
2.5558E-03
2.02
5.6371E-01
1.00

1/64
2.6171E-03
2.00
6.3685E-04
2.00
2.8162E-01
1.00

Table 2: Errors of the state for different error norms.

error

error

error

1/8
1.1429E-03
-
3.7634E-04
-
5.3988E-03
-

1/16
2.1927E-04
2.38
8.5181E-05
2.14
1.1332E-03
2.25

1/32
5.1326E-05
2.09
2.0833E-05
2.03
2.7140E-04
2.06

1/64
1.2609E-05
2.03
5.1751E-06
2.01
6.7061E-05
2.02

Table 3: Errors of the adjoint state for different error norms.

error

error

error

1/8
1.7415E-02
-
4.4468E-02
-
2.3006
-

1/16
4.1867E-02
2.06
1.0373E-02
2.09
1.1313
1.02

1/32
1.0473E-02
1.99
2.5552E-03
2.02
5.6371E-01
1.00

1/64
2.6159E-03
2.00
6.3655E-04
2.01
2.8162E-01
1.00

Figure 2 depicts the development of the , , and error for the control, state, and adjoint state under uniform refinement of the mesh. From the figure, the expected order in and norm for the control is observed, and the order in norm is shown. Additionally, we observe convergence of order in and norm for state and adjoint state. Because of better smoothness of state, the order in norm is also observed.

Figure 2: The , , and error for the control, state, and adjoint state under uniform refinement of the mesh.

We perform a simulation with space size for this problem. Figure 3 presents the computed state, optimal control, and adjoint state. Examination of Figure 3 shows that the approximate solutions coincide with the true solutions. At the same time, the relationship between the control and adjoint state is preserved well.

Figure 3: Numerical results of Experiment 1: optimal state, optimal control, and corresponding adjoint state.

4.2. Experiment 2

Now, we consider the optimal control problem with homogeneous Dirichlet boundary condition and control constraint,

subject to

where denotes the unit circle, , , and .

The exact solution of the problem is not known in advance. So we use the numerical results computed on a grid with as reference solutions. The , , and errors for state, control, and adjoint state of the above problems have been computed. They are displayed in Tables 4–6 for the finite volume element schemes. Examination of the tables shows that the error measures of the schemes diminish approximately quadratically for the error in and norm and linearly for the error in norm, which are consistent with our theoretical analysis.

Table 4: Errors of the control for different error norms.

error

error

error

1/8
4.2814E-03
-
2.7687E-03
-
2.2775E-02
-

1/16
1.2186E-03
1.81
7.7173E-04
1.84
1.2642E-02
0.85

1/32
2.9931E-04
2.02
1.8098E-04
2.09
5.4512E-03
1.21

1/64
7.6218E-05
1.97
4.0938E-05
2.14
2.2949E-03
1.24

Table 5: Errors of the state for different error norms.

error

error

error

1/8
5.2997E-04
-
3.8069E-04
-
2.9536E-03
-

1/16
1.3047E-04
2.02
1.0869E-04
1.81
1.7383E-03
0.76

1/32
3.2119E-05
2.02
2.7098E-05
2.00
7.6992E-04
1.17

1/64
7.9037E-06
2.02
6.2918E-06
2.10
3.3199E-04
1.21

Table 6: Errors of the adjoint state for different error norms.

error

error

error

1/8
5.9019E-04
-
3.9236E-04
-
3.0287E-03
-

1/16
1.6144E-04
1.87
1.1101E-04
1.82
1.7719E-03
0.77

1/32
4.2371E-05
1.93
2.8422E-05
1.97
8.0665E-04
1.13

1/64
1.0588E-05
2.00
6.9125E-06
2.04
3.6516E-04
1.14

In Figure 4, the development of the , , and error for control, state, and adjoint state under uniform refinement of the mesh is shown. Here, the expected order in and norm for the control is observed. Again, we observe convergence of order in and norm for state and adjoint state, which is consistent with our expectation of the order of convergence. The errors in norm confirm our error estimation (11). Figure 5 displays the numerical solution computed by the finite volume element schemes with . The results are nearly the same as those in [19]. The relationship between the control and adjoint state is also preserved well.

Figure 4: The , , and error for the control, state, and adjoint state under uniform refinement of the mesh.

Figure 5: Numerical results of Experiment 2: optimal state, optimal control, and corresponding adjoint state.

4.3. Experiment 3

Now we consider the optimal control problem (59) with denoting an -shaped domain, . Further, we set and .

In this situation, the solution does not admit integrable second derivatives. The desired convergence results of finite volume element schemes cannot be expected. So we only present the numerical solutions of the finite volume element schemes in Figure 6, which are nearly the same as those in [19]. On one hand, the desired convergence results may be obtained by using graded meshes and postprocessing [20], which will need more computational cost. On the other hand, we can modify finite volume element schemes near the corner to obtain the second-order accuracy. The related results will be reported in the future.

Figure 6: Numerical results of Experiment 3: optimal state, optimal control, and corresponding adjoint state.

5. Conclusions

In this article, we have investigated the finite volume element discretizations of optimal control problems governed by linear elliptic partial differential equations and subject to pointwise control constraints. Optimal order , , and error estimates for the considered problems are obtained and numerical experiments validate the theoretical results. In addition, we discuss the optimal control problems in polygonal domains with corner singularities. Two effective methods are proposed to compensate the negative effects of the corner singularities. The corresponding results will be reported in the future.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project is partially supported by the Fundamental Research Funds for the Central Universities (Nos. KYZ201565 and KJQN201839) and the National Natural Science Foundation of China (Nos. 11701283, 11426134, and 11471166).

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