RESIDUAL BASED A POSTERIORI ERROR ESTIMATION FOR DIRICHLET BOUNDARY CONTROL PROBLEMS

We study a residual–based a posteriori error estimate for the solution of Dirichlet boundary control problem governed by a convection diffusion equation on a two dimensional convex polygonal domain, using the local discontinuous Galerkin (LDG) method with upwinding for the convection term. With the usage of LDG method, the control variable naturally exists in the variational form due to its mixed finite element structure. We also demonstrate the application of our a posteriori error estimator for the adaptive solution of these optimal control problems.


Introduction
In this study, we investigate a numerical approximation of Dirichlet boundary control problems governed by a convection diffusion equation: subject to ∇ · (− ∇y + βy) + αy = f in Ω, where Ω is a convex polygonal domain in R 2 with Lipschitz boundary Γ = ∂Ω. We refer to u as the control on the Dirichlet boundary, to y as the state, and to (2) as the state equation. The velocity field is denoted by β ∈ W 1,∞ (Ω) 2 and we suppose that it satisfies incompressibility condition, that is, ∇ · β = 0. The constant coefficients > 0 and α > 0 are corresponding to diffusion and reaction terms, respectively. The regularization parameter ω is a positive constant. For the source function f and the desired state y d , we assume f, y d ∈ L 2 (Ω). By following the standard arguments in [21,26], Dirichlet boundary control problem (1)-(2) is equivalent to the following optimality system: ∇ · (− ∇z − βz) + αz = y − y d in Ω, z = 0 on Γ, ωu − ∂z ∂n = 0 on Γ, (3c) 1 Institute of Applied Mathematics, Middle East Technical University, 06800 Ankara, Turkey. Email: yucelh@metu.edu.tr where z is the adjoint variable and n is the unit outer normal to Γ. The adjoint equation (3b) is also a convection diffusion equation, but with convection term −β instead of β. The convergence properties of discretization methods applied to the optimal control problems can be substantially different from the convergence properties of discretization methods applied to a single convection dominated PDEs due to the transport of the information in the optimality system with opposite directions. In such kind of problems (1)- (2), the specific difficulty is that the control variable is not involved in the variational form of the standard finite element setting. To handle this difficulty, various numerical approaches have been proposed including very weak variational setting [7,9,13], modified cost functionals [11,14], approximating the Dirichlet boundary condition with a Robin boundary condition [4], weak boundary penalization (also called as Nitsche's penalty technique) [10], and a mixed formulation [15]. In this study, we employ local discontinuous Galerkin (LDG) method as a discretization technique since discontinuous Galerkin methods exhibit better convergence for the spatial discretization of optimal control problems governed by convection dominated PDEs [20] and the control variable is naturally involved in the variational form thanks to the mixed finite element structure of LDG and the weak enforcement of the boundary conditions. We would like to refer to [1,24] and references therein for details about local discontinuous Galerkin methods.
To obtain better accuracy with as few of degrees of freedom as possible, one particular way is adaptive finite element method. Although adaptive finite element method, contributed to the pioneer work of Babuska and Rheinboldt [2], has become a popular approach for the efficient solution of boundary and initial value problems for the PDEs, it is quite recent for constrained optimal control problems, initiated by Liu, Yan [23] and Becker, Kapp, Rannacher [3]. Adaptive finite element methods have been applied for various optimal control problems, see, e.g., [16,17,22,27,30,31] for control constrained problems, [5,18,29] for state constrained problems, and [6,19] for Neumann boundary control problems. However, there are a few studies for a posteriori error estimation of Dirichlet boundary control problems. Primal-dual weighted error estimates were derived in [28] for Dirichlet boundary control problem governed by a convection diffusion equation with control constraints. In [11,14], a residual-type a posteriori error analyses were carried out for Dirichlet boundary control governed by an elliptic equation with the control variable defined on an equivalent form of the norm in H 1 2 (Γ), using continuous finite element discretization. With the present paper, we intent to contribute a residual-based a-posteriori error estimates for the solution of Dirichlet boundary control problem governed by a convection diffusion equation, using the local discontinuous Galerkin method with upwinding for the convection term.
The rest of this paper is as follows: In the next section we present Dirichlet boundary control problem discretized by the local discontinuous Galerkin method. Section 3 is devoted to derivation of a residual-based a posteriori error estimator. Numerical results are given in Section 4 to illustrate the performance of the proposed error estimators.

Discretization of Model Problem
Throughout the paper we adopt the standard notation W m,p (Ω) for Sobolev spaces on Ω with norm · m,p,Ω and seminorm | · | m,p,Ω for m ≥ 0 and 1 ≤ p ≤ ∞. We denote W m,2 (Ω) by H m (Ω) with norm · m,Ω and seminorm | · | m,Ω . The L 2 -inner products on L 2 (Ω) and L 2 (Γ) are defined by respectively. In addition, C denotes a generic positive constant independent of the mesh size h and differs in various estimates. The LDG method, one of several discontinuous Galerkin methods, can be considered as a mixed finite element method. As in mixed finite element methods, we rewrite the optimality system as a system of first order equations and discretize it by introducing auxiliary variables. Introducing q = 1 2 ∇y and p = − 1 2 ∇z, the optimality system (3) can be rewritten as a system of the first order equations: We assume that the domain Ω is polygonal such that the boundary is exactly represented by boundaries of triangles. We denote {T h } h as a family of shape-regular simplicial triangulations of Ω. Each mesh T h consists of closed triangles such that Ω = K∈T h K holds. The set of all edges E h consists of the set E 0 h of interior edges and the set E ∂ h of boundary edges so that The diameter of an element K and the length of an edge E are denoted by h K and h E , respectively, and also h = max where n K (resp. n K e ) denotes the unit outward normal to ∂K (resp. ∂K e ). For a boundary edge E ∈ K ∩ ∂Ω, we set { {q} } = q and [[y]] = yn, where n is the outward normal unit vector on Γ.
To obtain weak formulation for the state equation in (4a)-(4b), we multiply it by piecewise smooth test functions v and r, respectively, and integrate by parts over the element Next, we seek to approximate the state solutions (y, q) with functions (y h , q h ) in the following finite element where S 1 (K) (resp. S 1 (E)) is the local finite element space, which consists of linear polynomials in each element where q h , y h , y h denote numerical fluxes. They have to be suitably defined in order to ensure the stability of the method and to enhance its accuracy. The numerical traces of y associated with the diffusion and convection terms are characterized as done in [31] respectively. We note that the numerical trace of y with respect to convection term is the classical upwinding trace. In addition, the numerical flux q h is given by In the numerical implementations, C 11 is chosen as C 11 = √ /h E for each E ∈ E h , whereas we take C 12 normal to the edges and modulus 1/2, i.e., C 12 · n E = 1 2 , and the vector function D 11 is given by D 11 · n = 1 2 sign n · β . Employing optimize-discretize approach, see, e.g., [26], to solve the Dirichlet boundary control problem (1)-(2), we obtain discrete optimality system for where We refer to [12,31] for derivation of (bi)-linear forms and references therein. Further, it is easy show that the continuous solution y, q, z, p, u satisfies the following optimality system:

A Posteriori Error Estimates
In this section we derive a residual-based a posteriori error estimate for Dirichlet boundary control problem governed by a convection diffusion equation, discretized by the local discontinuous Galerkin (LDG). We first give some known results, which will be needed throughout of the paper.
To prove our reliability result, we need the auxiliary solution y which solves the following system: Then, with the help of the bilinear forms (11) and (15) and the corresponding coercivity property of the bilinear forms, we obtain where the constant C is dependent on . Now, we derive a residual based a posteriori error estimate for the control u, the state y, and the adjoint z.
Theorem 3.1. Let y, q, z, p, u and y h , q h , z h , p h , u h be the solutions of (4) and (10), respectively. Then, it holds that The optimality systems (11) and (15) give us From (20), we have Similarly, we can deduce Combining (21) and (22) we obtain From the optimality conditions (4e), (3c), and definition of m 2 (·, ·), we have We first find a bound for the first term in (24) by using trace, inverse, and Young's inequalities as follows Next, the estimate in (13), the bound (14) with g = y h − y[u h ], and Young's inequality give us where for any small δ > 0. We note that it is assumed that i h z approximates z h . Putting (25) - (27) into (24), we obtain Now, define the following functional Then, the following Galerkin orthogonality property holds By the definitions of (bi)-linear forms in (10) and then applying integration by parts, we obtain To obtain an estimate for y[u h ] − y h 0,Ω , we apply duality argument as follows where (32), and using the operators defined in (12), the definition of A(·, ·), and the Galerkin orthogonality (30), we obtain By the residual (31) and Cauchy-Schwarz inequality, we get The estimates in (13), regularity estimate in (14), Young's inequality, and the assumption β = O(1) in (14) yield Then, combining (16a) and (35), we obtain Now, we derive an upper bound for z[u h ] − z h 0,Ω by following the procedure as in the derivation of (31).
Analogously, we apply the duality argument to find an estimate for z[u h ] − z h 0,Ω . It is easy see that (φ, ψ) satisfies Setting (38), and using the operators defined in (12) and the definition of A * (·, ·), we obtain Using the residual (39) with the estimates (13), regularity estimate in (14), and Young's inequality, we get Combination of (16) and (40) yields In conclusion, the desired result is obtained by combining (28), (36), and (41).

Numerical Experiments
An adaptive procedure consists of successive loops of the following sequence: The SOLVE step is the numerical solution of the optimal control problem with respect to the given triangulation T h using the LDG discretization with upwind for the convection. The ESTIMATE step requires the computation of the residual-based a posteriori error estimators. We use a bulk criterion in the MARK step to specify the elements in T h by using the a posteriori error estimator and by choosing subsets M K ⊂ T h such that the bulk criterion is satisfied for a given marking parameter Θ with 0 < Θ < 1: Θ η K is the a posteriori error estimator derived in Section 3. Finally, in the REFINE step, the marked elements are refined by longest edge bisection, whereas the elements of the marked edges are refined by bisection strategy. The adaptive procedure repeats until a given complexity #vertices, i.e., the number of vertices, is satisfied. Now, we test the performance of the residual-based a posteriori error estimators (18) for the following data: Ω = [0, 1] 2 , β = [1, 1] T , α = 2, ω = 0.1.
The source function f and the desired state y d are computed by using the following analytical solutions . Figure 1 illustrates the performance of the error estimator (18) in terms of the number of vertices for the marking parameter θ = 0.60 and the diffusion parameter = 10 −2 on adaptively and uniformly refined meshes. In Table 1, we display global errors with = 10 −3 on adaptively and uniformly refined meshes. Due to oscillation on the inflow boundary, we do not obtain stable solutions for the control on uniformly refined meshes. However, more stable solutions are obtained on adaptively refined meshes, see Figure 2. In conclusion, the obtained results show that adaptive refinements lead to better approximate solutions than uniform refinements. Table 1. Global errors of the state, adjoint, and control in L 2 -norm on adaptively and uniformly refined meshes with = 10 −3 and θ = 0.5 in the bulk criterion.