J Sci Comput

DOI 10.1007/s10915-015-0050-3

A Stabilized Mixed Finite Element Method for Elliptic

Optimal Control Problems

Hongfei Fu1 · Hongxing Rui2 ·

Jian Hou3 · Haihong Li1

Received: 14 September 2014 / Revised: 5 May 2015 / Accepted: 26 May 2015 © Springer Science+Business Media New York 2015

Abstract In this paper, we propose a new mixed finite element method, called stabilized mixed finite element method, for the approximation of optimal control problems constrained by a first-order elliptic system. This method is obtained by adding suitable elementwise least-squares residual terms for the primal state variable y and its flux σ . We prove the coercive and continuous properties for the newmixed bilinear formulation at both continuous and discrete levels. Therefore, the finite element function spaces do not require to satisfy the Ladyzhenkaya–Babuska–Brezzi consistency condition. Furthermore, the state and flux state variables can be approximated by the standard Lagrange finite element. We derive optimality conditions for such optimal control problems under the concept ofDiscretizationthen-Optimization, and then a priori error estimates in aweighted norm are discussed. Finally, numerical experiments are given to confirm the efficiency and reliability of the stabilized method.

Keywords Optimal control · Stabilized mixed finite element · LBB condition ·

A priori error estimates · Numerical experiments

Mathematics Subject Classification 49K20 · 49M25 · 65N15 · 65N30

B Hongfei Fu hongfeifu@upc.edu.cn

B Hongxing Rui hxrui@sdu.edu.cn

Jian Hou houjian@upc.edu.cn 1 College of Science, China University of Petroleum, Qingdao 266580, China 2 School of Mathematics, Shandong University, Jinan 250100, China 3 College of Petroleum Engineering, China University of Petroleum, Qingdao 266580, China 123

J Sci Comput 1 Introduction

Optimal control problems governed by partial differential equations (PDEs) are playing an increasingly important role in modern scientific and engineering applications. To the best of our knowledge, there is a growing number of publications for finite element approximations of optimal control problems. For detailed information about finite element methods for PDEs and optimal control problems, we refer the readers to [1–5].

In the last decades, classical mixed finite element methods have been found successful for solving optimal control problems governed by PDEs, seeRefs. [6–11], for example.However, it is well known that the choice ofmixed element spaces for the state variable and its fluxmust strictly satisfy LBB consistency condition, and a typical experience is to use the Raviart–

Thomas (RT) elements [12]. In this sense, some of the best known and widely used finite element spaces are thereby excluded. In particular, in high-dimensional problems, it is not a straightforward thing to construct such kinds of mixed finite element spaces. Meanwhile, the practical computation for classical mixed finite element methods is complicated. To conquer these difficulties appeared in using classical mixed finite element methods, some stabilization techniques have been proposed, for example, least-squares methods [13,14], H1-Galerkin methods [15], as well as splitting methods [16,17]. Recently, some of these methods have been applied to solve optimal control problems, see, Refs. [18–21].

In this paper, we propose another approach to circumvent the stability LBB condition. We call it stabilized mixed finite element method, and it is obtained by adding mesh-dependent least-squares terms to the dual-mixed bilinear formulation of the following first-order elliptic system ⎧ ⎨ ⎩ divσ = f + u, in , σ + ∇ y = 0, in , y = 0, on ∂. (1.1)

We prove that the new bilinear formulation is coercive and bounded at both continuous and discrete levels. Therefore, one can have a free choice of finite element function spaces. Then, we apply this stabilized method to solve the following type optimization problem

J (y, σ, u) = 1 2 ∫ (y − yd)2 + 1 2 ∫ (σ − σd)2 + γ 2 ∫ u2 (1.2) with ξ1 ≤ u ≤ ξ2, a.e. in . (1.3)

Here γ > 0 is a penalty parameter. It is used to change the relative importance of the terms appearing in the definition of the cost functional. Finally, we deduce both the continuous and discrete optimality system, and obtain a priori error estimates for both the control and states in the optimal control problem.

The rest of the paper is organized as follows: In Sect. 2, we formulate the optimal control problem and deduce the continuous stabilized optimality system. In Sect. 3, we aim to derive the corresponding discrete optimality system in the Discretization-then-Optimization approach. In Sect. 4, we introduce some intermediate variables associated with the optimal control, and then give corresponding intermediate error estimates in a weighted norm under control constrained by pointwise inequality. In Sect. 5, a priori error estimates for the optimal control problem is given. In Sect. 6, we conduct some numerical experiments to verify the theoretical analysis. In the last section, some concluding remarks are given.

In the following,we employ the usual notion for Lebesgue and Sobolev spaces, seeRef. [1] for details. Throughout, C and ε will denote a generic positive constant and small positive 123

J Sci Comput number which are independent of the discrete mesh parameters andmay have different values in different circumstances, respectively. 2 Optimality System

In this section, we first recall the continuous optimality system based on the idea of classical mixed formulation. Then, we propose the stabilized mixed weak formulation, and deduce the corresponding continuous optimality system.

Let be a bounded domain in Rd (d = 1, 2, 3) with Lipschitz boundary ∂. Define the state function spaces

V = H10 (), H = L2(),

W = H(div;) { τ ∈ L2()d : divτ ∈ L2() } , and the control function space

U = L2().

Furthermore, suppose the bounds ξ1, ξ2 ∈ R fulfill ξ1 < ξ2, and define the admissible control set Uad as follows