International Journal of Differential Equations

Volume 2018, Article ID 4753792, 11 pages

https://doi.org/10.1155/2018/4753792

## Finite Volume Element Approximation for the Elliptic Equation with Distributed Control

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

Correspondence should be addressed to Zhiyue Zhang; nc.ude.unjn@euyihzgnahz

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) iswhereand 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 constraintswhere is a given desired state and is a regularization parameter. We define the set of admissible control bywhere 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 bywhere 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 inequalityFurther, the variational inequality is equivalent towhere 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.