A LOCAL DISCONTINUOUS GALERKIN METHOD

1 2

A LOCAL DISCONTINUOUS GALERKIN METHOD FOR DIRICHLET BOUNDARY CONTROL PROBLEMS

3

† ¨ PETER BENNER∗ AND HAMDULLAH YUCEL

4 5 6 7 8

Abstract. In this paper, we consider Dirichlet boundary control of a convection-diffusion equation with L2 – boundary controls subject to pointwise bounds on the control posed on a two dimensional convex polygonal domain. We use the local discontinuous Galerkin method as a discretization method. We derive a priori error estimates for the approximation of the Dirichlet boundary control problem on a polygonal domain. Several numerical results are provided to illustrate the theoretical results.

9 10

Key words. Dirichlet boundary optimal control, local discontinuous Galerkin, convection–diffusion equation, a priori error estimate

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

AMS subject classifications. 49J20, 65N15, 65N30

1. Introduction. The investigation of optimal control problems governed by partial differential equations (PDEs) has seen a drastically increasing interest during the last decades. There have been extensive theoretical and numerical studies for finite element approximation of various optimal control problems, see, e.g., [31, 43] and the references therein. However, the works mentioned above are mainly contributions to distributed control. There are only a few contributions to boundary control problems reported in the literature, and most of them are also related to Neumann boundary control problems [2, 13, 24]. As to Dirichlet boundary control, there exist limited work in the literature, although it plays an important role in many applications such as the active boundary control of flows, modelled with the help of the Navier-Stokes equations; see, e.g. [23, 30]. The specific difficulties involved in the Dirichlet control problems result from the fact that they are not of variational type. To overcome this difficulty, the common approach is to use the concept of very weak solutions with L2 -boundary controls, see [9] for a more detailed discussion of this fact. It is also possible to define the state or control on different function spaces other than the L2 –space; see, e.g., [27, 33]. Moreover, the Dirichlet boundary controls were approximated by regularization based on Robin boundary controls [3, 6]. In this approach, the choice of the regularization parameter remains a delicate matter. A numerical approach to Dirichlet boundary control based on a discretization using the Nitsche method was proposed in [5] and further see [7] for a discussion and comparison of the different approaches for time–dependent problems. For a priori error analysis of the Dirichlet boundary control problems, there are few contributions in the literature. Casas and Raymond considered semilinear elliptic Dirichlet boundary control problems with pointwise bounds on two-dimensional convex polygonal domains in [15]. They obtained an error estimate for optimal control of order O(h1−1/s ), where s ≥ 2 depends on the smallest angle of the boundary polygon. Vexler provided O(h2 ) convergence for problems with finite dimensional control in [44]. May et al. in [38] considered a Dirichlet boundary control problem without control constraints on two-dimensional convex polygonal domains and obtained optimal error estimates in H −1/2 for the control. Moreover, Deckelnick et al. in [20] studied the finite element approximation of Dirichlet boundary control for elliptic PDEs on √ two- and three-dimensional curved domains. They obtained an error estimate of order O(h ln h) for the optimal control, based on a variational discretization ∗ Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany. ([email protected]). † Institute of Applied Mathematics, Middle East Technical University, 06800 Ankara, Turkey. ([email protected]). Corresponding author.

1

This manuscript is for review purposes only.

43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91

[29]. In [26], Gong and Yan established a mixed finite element approximation and obtained optimal and quasi-optimal error estimates for problems on polygonal and smooth domains, respectively. Using boundary element methods, Of et al. in [39] searched the control in H 1/2 such that the state belonging to H 1 satisfies the weak formulation. The regularities of the optimal state, control and adjoint which are limited by singularities due to corners of the domain and the presence of control constraints were discussed in [1]. Mateos has recently discussed several optimization procedures to solve finite element approximations of linearquadratic Dirichlet optimal control problems governed by an elliptic PDE with control and state constraints posed on a 2D or 3D Lipschitz domain in [37]. In the present work we consider Dirichlet boundary control of a convection–diffusion equation with L2 –boundary controls subject to pointwise bounds on the control. We include a convection term, due to future interest in considering similar problems, motivated by fluid mechanics considerations. The convection coefficient corresponds to the velocity field of the fluid in the fluid mechanics. Dirichlet boundary control problems governed by convection– diffusion equations were just studied in [19, 32]. In [32], the authors considered the numerical solution of an unsteady optimization problem by applying a semi-smooth Newton method. Recently, the symmetric interior penalty Galerkin method has been applied to approximate a Dirichlet boundary control problem without control constraints on a convex polygonal domain in [19]. In addition, some error estimates are derived. To the best of the authors’ knowledge no error analysis is available for Dirichlet boundary control of the convection– diffusion equations with control constraints. With the present paper we intend to fill this gap and derive a priori error estimates. Compared to the elliptic case, Dirichlet boundary control problems with convection term require more special attention in terms of both theoretical and numerical approaches, especially, when the convection dominates the diffusion. In this paper, we use the local discontinuous Galerkin (LDG) method as a discretization method. The LDG method is one of several discontinuous Galerkin methods which are being vigorously studied, especially as applied to convection–diffusion problems because of their applicability to a wide range of problems and their properties of local conservativity and higher degree of locality. In addition, they may be advantageous because of the ease with which the method handles hanging nodes, elements of general shapes, and local spaces of different types. Though these methods are known since the 1970s, much attention has been paid only in the past few years due to the availability of cheap computing resources. We would like to refer to [4, 28, 41] for details about discontinuous Galerkin methods. Moreover, discontinuous Galerkin methods have been studied in [8, 34, 45, 46, 47, 48] for optimal control problems. One of the reasons why we choose the LDG as a discretization method is that the second order state equation is rewritten into a mixed form. Then, the mixed type approach is adopted to approximate the control problem. When the state equation is understood in a very weak sense, the first order optimality condition involves the normal derivative of the adjoint state on the boundary of the domain. This complicates the problem both theoretically and numerically. However, by using such a kind of approach, the optimal control and adjoint are involved in a variational form in a natural sense. Moreover, the discontinuous Galerkin methods allows us the weak treatment of the boundary conditions. The rest of this paper is organized as follows: In the next section we present the Dirichlet boundary control problem and discuss the regularity of the solutions. Then, we propose a mixed variational scheme for the optimization problem. In Section 3, we establish the LDG scheme for approximation of the Dirichlet boundary control problem. A priori error estimates of the LDG approximation of the Dirichlet boundary control problem on the convex polygonal domain are derived in Section 4. Numerical results are given in Section 5 to verify our theoretical results. Conclusions and discussions are provided in the last section. 2


92 93 94 95 96

2. Model problem. Throughout the paper we adopt the standard notation W m,p (Ω) for Sobolev spaces on Ω with norm k · km,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 k · km,Ω and seminorm | · |m,Ω . It is noted that H 0 (Ω) = L2 (Ω) and H01 = {v ∈ H 1 (Ω) : v = 0 on ∂Ω}. The L2 –inner products on L2 (Ω) and L2 (Γ) are defined by (v, w) =

97

Z

Ω

v w dx

∀v, w ∈ L2 (Ω) and hv, wi =

Z

Γ

v w ds

∀v, w ∈ L2 (Γ),

101

respectively. In addition, c and C denote generic positive constants independent of the mesh size h and differ in various estimates. In this paper we consider the following Dirichlet boundary control problem governed by a convection–diffusion equation:

102

(1)

98 99 100

minimize u∈U ad

103

subject to

104

(2a) (2b)

105

1 ω ky − yd k20,Ω + kuk20,Γ 2 2

∇ · (−ε∇y + βy) + αy = f y=u

in Ω, on Γ,

110

where Ω is a convex polygonal domain in R2 with Lipschitz boundary Γ = ∂Ω. The velocity 2 field is denoted by β ∈ W 1,∞ (Ω) . We suppose that it satisfies incompressibility condition, that is, ∇ · β = 0. The diffusion and reaction terms are denoted by ε > 0 and α > 0, respectively. The regularization parameter ω is a positive constant. Further, the admissible control set U ad is specified by

111 112

(3)

113

where ua and ub are real numbers. It is natural to choose the control space as U := L2 (Γ). However, it avoids the choice of the associated state space to be H 1 (Ω), since the trace operator γ : H 1 (Ω) → L2 (Γ) is not surjective. To overcome this problem, we use a very weak formulation of the state equation (2), given by

106 107 108 109

114 115 116 117

U ad := {u ∈ L2 (Γ) : ua ≤ u(x) ≤ ub a.e. x ∈ Γ},

−ε(y, ∆v) − (y, β · ∇v) + (αy, v) = ( f , v) − εhu,

∂v iΓ ∂n

∀v ∈ H 2 (Ω) ∩ H01 (Ω),

118 119

(4)

120 121

where n is the unit outer normal to Γ. Then, the existence and uniqueness of a very weak solution of (4) is shown in the following lemma.

122 123 124 125 126 127 128 129

L EMMA 1. If f ∈ L2 (Ω) and u ∈ L2 (Γ), then the very weak solution of (4) satisfies y ∈ Moreover, there exists a constant C independent of the given data f and u, such that for the corresponding solution y = y(u) we have (5) kyk0,Ω ≤ C k f k0,Ω + kuk0,Γ . L2 (Ω).

Proof. Fix given data ε, β and α, we can assume that the operator ∇ · (−ε∇ − β) + α is uniformly elliptic. Otherwise, a multiple c of the identity operator is added and the constant C is multiplied by the factor ecT . Then, the operator generates an analytic semi-group in L2 (Ω). For u = 0, the estimate (5) follows by standard semigroup arguments. To prove the case u 6= 0 3


131

we use the superposition principle for (2). Then, it is sufficient to consider the case f = 0. For each g ∈ L2 (Ω), let z ∈ H 2 (Ω) ∩ H01 (Ω) be the solution of

132

(6a)

133

(6b)

130

∇ · (−ε∇z − βz) + αz = g

z=0

in Ω, on Γ.

137

Then, we have ∂n z ∈ L2 (Γ) from [32, Thm. 3.2]. Also, according to [14, Lemma A.2], we can obtain ∂n z ∈ H 1/2 (Γ) by using z ∈ H 2 (Ω) and z = 0 on Γ. Let the continuous linear operator T : L2 (Ω) → L2 (Γ) be defined by T g := −∂n z. Letting y = T ∗ u, where T ∗ is the adjoint of T , we have

138

(7)

139

which verifies that y satisfies the very weak formulation (4). By (7), we obtain

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(y, g) = (y, ∇ · (−ε∇z − βz) + αz) = (T ∗ u, g) = (u, T g) = −hu, ∂n ziΓ ,

|(y, g)| ≤ Ckuk0,Γk∂n zk0,Γ ≤ Ckuk0,Γkgk0,Ω ,

140 141

which is the desired result.

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Now, we are ready to formulate the optimal control problem (1)-(2) by using the very weak formulation of the state equation

144

(8)

minimize u∈U ad

1 ω ky − yd k20,Ω + kuk20,Γ 2 2

145

subject to

146

(9)

147

Using standard arguments such as convexity of the cost functional (1), the weak sequential limit arguments, and Lemma 1 (see, e.g., [35]), it can be proved that the optimal control problem (8)-(9) admits a unique solution, and the solution can be characterized by the following first order optimality conditions: there exists (y, u, z) ∈ L2 (Ω) × L2 (Γ) × H01 (Ω) such that

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−ε(y, ∆v) − (y, β · ∇v) + (αy, v) = ( f , v) − εhu,

∂v iΓ ∂n

∀v ∈ H 2 (Ω) ∩ H01 (Ω).

(10a) (10b)

∇ · (−ε∇y + βy) + αy = f y=u

in Ω, on Γ,

155

(10c) (10d)

in Ω, on Γ,

156

(10e)

∇ · (−ε∇z − βz) + αz = y − yd z=0 ∂z hωu − , w − uiΓ ≥ 0 ∂n

152 153 154

Now, we turn to the regularity property of the optimal control u on the boundary Γ.

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w ∈ U ad .

T HEOREM 2. Assume that Ω is a convex domain. Let (y, u, z) ∈ L2 (Ω) × L2 (Γ) × H01 (Ω) be the solution of optimality system (10). For f , yd ∈ L2 (Ω) we then have u ∈ H 1/2 (Γ),

y ∈ H 1 (Ω),

and z ∈ H 2 (Ω) ∩ H01 (Ω).

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(11)

161

Proof. With the help of Lemma 1 and the assumptions f ∈ L2 (Ω) and u ∈ L2 (Γ) we conclude that y ∈ L2 (Ω). Thus, we obtain z ∈ H 2 (Ω) ∩ H01 (Ω) by yd ∈ L2 (Ω) and the convexity of the domain Ω. This implies that ∂n z ∈ H 1/2 (Γ); see, [14, 32]. From (10e) we obtain that u ∈ H 1/2 (Γ), and thus y ∈ H 1 (Ω); see [36].

162 163 164

4


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In order to define the local discontinuous Galerkin (LDG) scheme for the optimal control problem (1)-(2), we rewrite our state equation (2) as a system of first-order equations. Thus, we introduce an auxiliary variable: q = ∇y.

169

Then, the optimal control problem (1)-(2) can be rewritten as

170

(12)

165 166 167

minimize u∈U ad

1 ω ky − yd k20,Ω + kuk20,Γ 2 2

171

subject to

172 173

(13a) (13b)

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(13c)

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To obtain the weak formulation for the state equation (13), we multiply (13) by a piecewise smooth functions v and r, respectively, and integrate on Ω

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(14a)

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(14b)

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where

∇ · (βy − εq) + αy = f q = ∇y

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(εq − βy, ∇v) + (αy, v) − h(εq − βy) · n, viΓ = ( f , v)

(q, r) + (y, ∇ · r) = hu, r · niΓ

2 W := {w ∈ L2 (Ω) , ∇ · w ∈ L2 (Ω)},

v ∈ V,

r ∈ W,

V := L2 (Ω)

and the control u belongs to U ad ⊆ U = L2 (Γ). To ensure the well–definedness on Γ, we need to the following trace operator on W, introduced in [26, Lemma 2.1]. 1

L EMMA 3. Let the trace operator γ : W → H − 2 (Γ) such that γ(v) = v · n is defined in the sense that Z

186 187

on Γ.

y=u

180 181 182

in Ω, in Ω,

Γ

v · n w ds :=

Z

Ω

∇ · v w dx +

Z

Ω

v · ∇w dx

∀ w ∈ H 1 (Ω).

1

The operator γ is surjective from W to H − 2 (Γ) and continuous such that kv · nk− 1 ,Γ ≤ kvkW .

188

2

189 190 191 192 193

Before introducing the mixed formulation of the optimal control problem (12)–(13), we present the following well known estimate on the mixed variational problem (14). L EMMA 4. Let (y, q) ∈ W × V be the solution of (14). Under the condition (u, f ) ∈ H × L2 (Ω), we have (15) kyk0,Ω + kqkW ≤ C kuk 1 ,Γ + k f k0,Ω . 1 2

2

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Then, the weak formulation of the optimal control problem (12)-(13) can be expressed as follows: (16)

197

subject to

198

(17a)

199

(17b)

minimize u∈U ad

ω 1 ky − yd k20,Ω + kuk20,Γ 2 2


(q, r) + (y, ∇ · r) = hu, r · niΓ 5


v ∈ V,

r ∈ W.

203

It can be derived by standard techniques (see, e.g., [22, 35]) that the control problem (16)–(17) has a unique solution (y, q, u), and that a triple (y, q, u) is the solution of (16)–(17) if and only if there is an adjoint pair (z, p), such that (y, q, z, p, u) satisfies the optimality conditions:

204

(18a)

205

(18b) (18c)

200 201 202

206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244

(18d) (18e)


(q, r) + (y, ∇ · r) = hu, r · niΓ (βz − εp, ∇φ) + (αz, φ) + h(εp − βz) · n, φiΓ = (y − yd , φ) (p, ψ) − (z, ∇ · ψ) = 0 hωu + p · n, w − uiΓ ≥ 0

v ∈ V,

r ∈ W, φ ∈ V,

ψ ∈ W, w ∈ U ad .

It is noted that we decompose the adjoint equation into the mixed formulation by introducing p = −∇z. 3. LDG approximation. In this section we introduce the local discontinuous Galerkin (LDG) approximation of the optimal control problem (16)-(17). The LDG method can be considered as a mixed finite element method. However, the auxiliary variable q can be eliminated from the equations, which is usually not the case for classical methods. In the LDG method, the local conservativity holds because the conservation laws are weakly enforced element by element. In order to do that, suitable discrete approximations of the traces of the fluxes on the boundary elements are provided by the so-called numerical fluxes. These numerical fluxes enhance the stability of the method, and hence, the quality of its approximation. This is why the LDG method is strongly related to stabilized mixed finite elements. The stabilization is associated with the jump of the approximate solution across the element boundaries, see [4, 17] for details. We assume that the domain Ω is polygonal such that the boundary is exactly represented by boundaries of triangles. We denote {Th }h as a family of shape-regularSsimplicial triangulations of Ω. Each mesh Th consists of closed triangles such that Ω = K∈Th K holds. We assume that the mesh is regular in the following sense: for different triangles Ki , K j ∈ Th , i 6= j, the intersection Ki ∩ K j is either empty or a vertex or an edge, i.e., hanging nodes are not allowed. The diameter of an element K and the length of an edge E are denoted by hK and hE , respectively. We split the set of all edges Eh into the set Eh0 of interior edges and the set Eh∂ of boundary edges so that Eh = Eh0 ∪ Eh∂ . Let n denote the unit outward normal to Γ. The inflow and outflow parts of Γ are denoted by Γ− and Γ+ , respectively, Γ− = {x ∈ Γ : β · n < 0} ,

Γ+ = {x ∈ Γ : β · n ≥ 0} .

Similarly, the inflow and outflow boundaries of an element K are defined by ∂K − = {x ∈ ∂K : β · nK < 0} ,

∂K + = {x ∈ ∂K : β · nK ≥ 0} ,

where nK is the unit normal vector on the boundary ∂K of an element K. Let the edge E be a common edge for two elements K and K e . For a piecewise continuous scalar function y, there are two traces of y along E, denoted by y|E from inside K and ye |E from inside K e . The jump and average of y across the edge E are defined by: 1 [[y]] = y|E nK + ye |E nK e , {{y}} = y|E + ye |E , (19) 2 where nK (resp. nK e ) denotes the unit outward normal to ∂K (resp. ∂K e ). Similarly, for a piecewise continuous vector field q, the jump and average across an edge E are given by 1 (20) [[q]] = q|E · nK + qe |E · nK e , {{q}} = q|E + qe |E . 2 6


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For a boundary edge E ∈ K ∩ ∂Ω, we set {{q}} = q and [[y]] = yn, where n is the outward normal unit vector on Γ. Note that the jump in y is a vector and the jump in q is a scalar which only involves the normal component of q. To approximate the trial and test functions, we define the following discrete spaces: o n 2 2 (21a) Wh = w ∈ L2 (Ω) : w |K ∈ S1 (K) , ∀K ∈ Th , (21b) Vh = v ∈ L2 (Ω) : v |K ∈ S1 (K), ∀K ∈ Th , o n (21c) Uh = u ∈ L2 (Γ) : u |E ∈ S1 (E), ∀E ∈ Eh∂ ,

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where S1 (K) (resp. S1 (E)) is the local finite element space, which consists of linear polynomials in each element K (resp. on E). It is noted that for a given element K ∈ Th , the restrictions to K of y and of each of the components of q belong to the same local space; this renders the coding of these methods considerably simpler than that of the standard mixed methods, especially for high-degree polynomial local spaces. We also define Uhad = Uh ∩U ad . For ∀(v, r) ∈ Vh × Wh ) the approximate solution (yh , qh ) of the state solution (q, y) satisfies

259

(22a)

260

(22b)

261

where b qh , y, ˜ ybh denote numerical fluxes. They have to be suitably defined in order to ensure the stability of the method and to enhance its accuracy. We are now ready to introduce the expressions that define the numerical fluxes. Here, we follow the notation in [48]. The numerical traces for y associated with the diffusion and convection terms are characterized as (23)  E ∈ Γ− ,  uh , 0 {{yh }} + C12 · [[yh ]], E ∈ Eh , {{yh }} + D11 · [[yh ]], E ∈ Eh0 , and y˜h = ybh = uh , E ∈ Eh∂ ,  yh , E ∈ Γ+ ,

252 253 254 255 256

262 263 264 265

266

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(εqh − βyh, ∇v)K + (αyh , v)K − h(εb qh − βy) ˜ · n, vi∂K = ( f , v)K ,

(qh , r)K + (yh , ∇ · r)K = hb yh , r · ni∂K ,

respectively. The vector functions C12 and D11 are given by C12 · n =

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272

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277

D11 · n =

1 sign n · β , 2

where β˜ is a nonzero piecewise constant vector. We note that the numerical trace of y with respect to convection term is the classical upwinding trace. In addition, the numerical flux b qh is given by {{qh }} + C11 [[yh ]] − C12 [[qh ]], E ∈ Eh0 , b qh = (24) qh · n + C11(yh − uh ), E ∈ Eh∂ , where the parameter C11 is a positive number. Putting the numerical fluxes into (22) and summing over all elements, we obtain

∑

Z

K∈Th K

276

1 sign n · β˜ , 2

+

∑

(εqh − βyh) · ∇v dx −

Z

K∈Th K

αyh v dx +

∑

E∈Eh0

Z

E

∑

E∈Eh0

Z

E

ε ({{qh }} + C11 [[yh ]] − C12[[qh ]]) · [[v]] ds

{{yh }} + D11 · [[yh ]] β · [[v]] ds +

∑

Z

E∈Γ+ E

7


(n · β)yh v ds

278 279

280

− Z

∑

E∈Eh∂

Ω

Z

E

Z ε qh · n + C11yh v ds = f v dx − Ω

qh · r dx +

281 282

Z

∑

yh ∇ · r dx −

K∈Th K

∑

E

εC11 uh v ds −

∑−

E∈Γ

{{yh }} + C12 · [[yh ]] [[r]] ds =

E

E∈Eh0

E∈Eh∂

Z

Z

E

|β · n|uh v ds,

∑

E∈Eh∂

Z

E

uh r · n ds.

For simplicity, we define the following bi(linear) forms:

283

ah (q, r) :=

284

bh (y, r) := ch (y, v) :=

285

Z

Ω

Z

∑

Z

K∈Th K

K∈Th K

∑

E∈Eh0

∑

mh,1 (u, r) :=

287

q · r dx,

∑

−

286

E∈Eh∂

mh,2 (u, v) := −

288

F(v) :=

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Z

∑

Z

Z

E∈Eh0

Z

{{y}} + C12 · [[y]] [[r]] ds,

E

αy v − yβ · ∇v dx + E

εC11 [[y]] · [[v]] ds +

∑

Z

∑

Z

E∈Eh0

E

E∈Γ+ E

{{y}} + D11 · [[y]] β · [[v]] ds

(n · β)y v ds −

∑

E∈Eh∂

Z

E

εC11 y v ds,

u r · n ds,

∑

E∈Eh∂

Ω

∑

y ∇ · r dx −

Z

E

εC11 u v ds −

∑

Z

E∈Γ− E

|β · n|u v ds,

f v dx.

By applying integration by parts for bh (·, ·), we obtain bh (y, r) =

292

∑

K∈Th K

=−

293

=−

294 295

Z

y ∇ · r dx −

∑

Z

∑

Z

K∈Th K

K∈Th K

∑

E∈Eh0

∇y · r dx + ∇y · r dx +

Z

E

∑

Z

∑

Z

{{y}} + C12 · [[y]] [[r]] ds

K∈Th ∂K

E∈Eh0

E

yr · n ds −

∑

E∈Eh0

Z

E

{{y}} + C12 · [[y]] [[r]] ds

{{r}} − C12[[r]] · [[y]] ds +

∑

E∈Eh∂

Z

E

yr · n ds.

296

Then, the LDG approximation of the state equation (2) reads as

297 298

(25a) (25b)

299

The LDG approximation scheme of the Dirichlet boundary control problem (1)–(2) is

300

(26)

301 302 303

ah (qh , r) + bh (yh , r) = mh,1 (uh , r) −bh (v, qh ) + ch (yh , v) = mh,2 (uh , v) + F(v)

minimize

uh ∈Uhad ,(yh ,qh )∈Vh ×Wh

∀r ∈ Wh , ∀v ∈ Vh .

1 ω kyh − yd k20,Ω + kuh k20,Γ 2 2

subject to (25). The system (26) has a solution since the cost functional (26) is continuous and Uhad is a nonempty compact set of Uh . With the help of the Lagrange multiplier method, we derive the 8


304

following first order discrete optimality system for the discrete control problem (26)

305

(27a) (27b)

306

308

(27c) (27d)

309

(27e)

307

310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326

ah (qh , r) + bh(yh , r) = mh,1 (uh , r) −bh(v, qh ) + ch(yh , v) = mh,2 (uh , v) + F(v) ah (ph , ψ) − bh (zh , ψ) = 0 bh (φ, ph ) + ch(φ, zh ) = (yh − yd , φ)

∀w ∈ Uhad ,

4. Error estimates. In this section we derive some error estimates for the LDG approximation of the Dirichlet boundary control problem on convex polygonal domains. The regularity of Dirichlet boundary control problems depends on the angles of the domain and the regularity of the given data (see [1, Thm. 3.4]). Therefore, the regularity results in Theorem 2 turn out to be more regular than required. Thus, we now state the regularity of the solutions for general convex polygonal domains; see, e.g., [15, 38]. L EMMA 5. Assume that Ω is a bounded convex polygonal domain with Lipschitz boundd ary Γ, f ∈ L2 (Ω) and yd ∈ Ls∗ , sd∗ > 2. Let (y, u, z) ∈ L2 (Ω) × L2 (Γ) × H01 (Ω) be the solution of the optimality system (10). Then, we have y ∈ W 1,s (Ω), u ∈ W 1−1/s,s(Γ), z ∈ W 2,s (Ω), 2 ≤ s ≤ s∗ , 2ωmax Ω where s∗ = min sd∗ , sΩ ∗ with s∗ = 2ωmax −π , and ωmax is the maximum interior angle of the polygonal domain Ω.

Now, we will summarize some known results, which will be used frequently in the rest of the paper. • Let Ψh : V → Vh and Φh : W → Wh be the special interpolation operators satisfying (see [12]) (y − Ψh y, v) = 0

∀v ∈ Vh ,

(28a)

328

(28b)

329

Then, the following approximation estimates hold

330

(29a)

332 333

∀ψ ∈ Wh , ∀φ ∈ Vh ,

hωuh + ph · n, w − uhi ≥ 0 where yh , qh , zh , ph , uh ∈ Vh × Wh × Vh × Wh × Uhad .

327

331

∀r ∈ Wh , ∀v ∈ Vh ,

∇ · (q − Φh q), v) = 0 ky − Ψhyk−s,r,Ω ≤ Ch1+s |y|1,r,Ω ,

∀v ∈ Vh .

s = 0, 1,

y ∈ W 1,r (Ω),

2 kq − Φh qks,r,Ω ≤ Ch1−s |q|1,r,Ω , s = 0, 1, q ∈ W 1,r (Ω) , (29c) k∇ · q − Φhq k−s,Ω ≤ Ch1+s |∇ · q|1,Ω , s = 0, 1, ∇ · q ∈ H 1 (Ω). (29b)

• Let Πh : L2 (Γ) → Uh be the L2 -projection such that (see [38, Section 3.1]) (u − Πhu, v) = 0

∀v ∈ Uh .

334

(30)

335

Then, we have the following approximation property for u ∈ W s,p (Γ)

336

(31)

337 338 339

ku − Πhuk0,s,Γ + hr kΠh ukr,s,Γ ≤ Chr kukr,s,Γ

0 ≤ r ≤ 1,

1 < s < ∞.

• For positive constants Ctr and Cinv independent of K ∈ Th and h, the trace and inverse inequalities are given as follow (see [11]): (32a)

340

(32b)

341

(32c)

kvk0,∂K ≤ Ctr kvk1,K

kvk0,∂K ≤ Ctr h−1 kvk20,K + hk∇vk20,K |v| j,K ≤ Cinv hi− j |v|i,K

9

1/2

∀v ∈ H 1 (K),

∀v ∈ H 1 (K),

∀v ∈ H 1 (K),


0 ≤ i ≤ j ≤ 2.

• There exists a constant C such that the local inverse inequalities (see, [16, p. 146])

342 343

(33a)

344

(33b)

1

kvk0,s,∂K ≤ Ch− s kvk0,s,K 1 ∂v k k0,s,∂K ≤ Ch− s k∇vk0,s,K ∂n

∀v ∈ W 1,s (K), ∀v ∈ W 1,s (K),

345 346

are valid for ∀K ∈ Th . Next, we define an approximation of the optimality condition (27e) as follows:

347

(34)

348

in which the auxiliary solution ph (u) ∈ Wh is the solution of the following system:

349

(35a) (35b)

350 351 352 353 354

′

hJh (u), v − uiΓ = hωu + ph(u) · n, v − uiΓ, ah (qh (u), r) + bh(yh (u), r) = mh,1 (u, r) −bh (v, qh (u)) + ch(yh (u), v) = mh,2 (u, v) + F(v)

∀r ∈ Wh , ∀v ∈ Vh ,

ah (ph (u), ψ) − bh(zh (u), ψ) = 0 ∀ψ ∈ Wh , d bh (φ, ph (u)) + ch(φ, zh (u)) = (yh (u) − y , φ) ∀φ ∈ Vh , where yh (u), qh (u), zh (u), ph (u) ∈ Vh × Wh ×Vh × Wh are auxiliary solutions for given u ∈ Uhad . Then, it is easy to see that (35c) (35d)

′

hJh (uh ), v − uhiΓ = hωuh + ph · n, v − uhiΓ ,

355

(36)

356 357

where we use the notation ph = ph (uh ) for simplicity. Further, let (zh (y), ph (y)) ∈ Vh × Wh be the solution of the following auxiliary system:

358

(37a)

ah (ph (y), ψ) − bh(zh (y), ψ) = 0

bh (φ, ph (y)) + ch (φ, zh (y)) = (y − y , φ) d

∀ψ ∈ Wh , ∀φ ∈ Vh .

359

(37b)

360

Here, (zh (y), ph (y)) is an LDG approximation of (z, p). Before deriving a priori error estimates for the optimal control problem, we present the following a priori error estimates.

361 362 363 364

L EMMA 6. Let y, q ∈ V × W be the solutions of (17) and let yh , qh ∈ Vh × Wh be the solutions of the discretized problem (25). Assume that y ∈ H r+1 , then |y − yh|1,Ω + kq − qhk0,Ω ≤ Chmin{r,k} kykr+1,Ω ,

365

(38)

366 367

where k is the polynomial degree of the ansatz functions. Moreover, we have the following L2 -error bound

368

(39)

369

Further, for 2 ≤ s < ∞, we have

370

(40)

371

Proof. We refer to [18] for the proof of (38) and (39), while the Ls -error estimate can be derived as done in [21].

372 373 374 375 376 377

ky − yhk0,Ω ≤ Chmin{r,k}+1 kykr+1,Ω . kq − qhk0,s,Ω ≤ Chkqk1,s,Ω.

Lemma 5 shows that the solution y does not belong to H 2 (Ω), i.e., y ∈ W 1,s (Ω). Therefore, we need to derive error estimates under weaker assumptions on the regularity of the solution. We note that the generic constant C is positive and independent of the mesh size h, and also differs in the various estimates. In other words, it just replaces ”const.” in all the following derivations where it occurs. 10


379 380

L EMMA 7. Let y, q and yh (u), qh (u) be the solutions of (18) and (35), respectively. Assume that the conditions of Lemma 5 are satisfied and Ω is a convex polygonal domain. Then, we have

381

(41)

378

ky − yh(u)k0,Ω ≤ Ch. Proof. For g ∈ L2 (Ω), consider the following convection–diffusion equation: ∇ · − ε∇φ + βφ + αφ = g in Ω,

382 383

φ=0

384 385

Hence, the following regularity result holds

386

(42)

387

provided that φ ∈ is

388

kφk2,Ω ≤ Ckgk0,Ω H01 (Ω) ∩ H 2 (Ω).

Then, the mixed weak formulation of the above problem

∇ · (−εψ + βφ) + αφ, v = (g, v)

389

(ψ, w) + (φ, ∇ · w) = 0

390 391

where (φ, ψ) ∈ V × W. We have that (see, e.g., [42])

392

(43)

393 394 395 396 397 398 399 400

on Γ.

v ∈ V,

w ∈ W,

ε3/2 kφk2,Ω + ε1/2 kφk1,Ω + kφk0,Ω ≤ Ckgk0,Ω.

Setting θ = −εψ + βφ and using the interpolation operators defined in (28), we obtain (y − yh(u), g) = y − yh(u), ∇ · θ + (y − yh(u), αφ) = y − yh(u), ∇ · θ + (y − yh(u), αφ) + (ψ, q − qh (u)) + φ, ∇ · (q − qh(u)) = y − yh(u), ∇ · (θ − Φh θ) + (y − yh(u), α(φ − Ψh φ)) (44) + ψ − Φh ψ, q − qh(u) + φ − Ψh φ, ∇ · (q − qh(u)) .

From the definition of operators the Ψh and Φh in (28), the Cauchy–Schwarz inequality, and the estimates in (29), (43), and (15), we obtain the following estimates: y − yh(u), ∇ · (θ − Φh θ) = y − Ψh y, ∇ · (θ − Φh θ) + Ψh y − yh(u), ∇ · (θ − Φh θ) | {z } =0

401

≤ ky − Ψhyk0,Ω k∇ · (θ − Φhθ)k0,Ω

402 403

(45)

404

≤ Ch2 |y|1,Ω |∇ · θ|1,Ω

≤ Ch2 k f k0,Ω + kuk 1 ,Γ kgk0,Ω ≤ Chkgk0,Ω, 2

(y − yh (u), α(φ − Ψh φ)) = (y − Ψh y, α(φ − Ψh φ)) + (Ψh y − yh(u), α(φ − Ψh φ)) | {z }

405

=0

406

≤ Cαkφ − Ψh φk0,Ω ky − Ψhyk0,Ω ≤ Cαh2 |φ|1,Ω |y|1,Ω ≤ Chkgk0,Ω.

407

(46)

408 409

Moreover, the Cauchy–Schwarz inequality and the estimates in (29), (38), (43), and (15) give us ψ − Φhψ, q − qh (u) ≤ kψ − Φhψk0,Ω kq − qh (u)k0,Ω ≤ Ch|ψ|1,Ω kyk1,Ω ≤ Chkgk0,Ω k f k0,Ω + kuk 1 ,Γ ≤ Chkgk0,Ω, (47)

410 411 412

2

11


413

φ − Ψh φ, ∇ · (q − qh(u)) ≤ kφ − Ψhφk0,Ω kq − qh(u)kW ≤ Ch|φ|1,Ω k f k0,Ω + kuk 1 ,Γ ≤ Chkgk0,Ω.

414 415

(48)

416

Inserting (45)-(48) into (44), we obtain

417

ky − yh(u)k0,Ω =

2

(y − yh (u), g) ≤ Ch, kgk0,Ω g∈L2 (Ω),g6=0 sup

418

which is the desired result.

419

R EMARK 8. In this paper, since we use a kind of mixed finite element method, the first order optimality condition does not involve the normal derivative of the adjoint on the boundary. Therefore, the usage of linear discontinuous finite elements for each variable does not cause any difficulty or problem. When discontinuous finite elements are used for the dicretization of the control, the computation of the projection operator is more efficient due to the elementwise computation. Further, the results obtained in [25] show that the full discretization and variational discretization exhibit similar convergence rates.

420 421 422 423 424 425 426 427 428 429 430 431

Now, we are ready to derive an a priori error estimate for the LDG approximation of the Dirichlet boundary control problem. L EMMA 9. Let (y, q, z, p, u) and yh ,qh , zh , ph , uh be the solutions of (18) and (27), respectively. Let yh (u), qh (u), zh (u), ph (u) be the auxiliary solution in (35). Assume that the conditions of Lemma 5 and 7 are satisfied. Then, we have 1

ku − uhk0,Γ + kyh (u) − yhk0,Ω ≤ Ch1− s .

(49)

′

Proof. By the definition of Jh , we have

432

′

′

hJh (u) − Jh(uh ), u − uhiΓ = ωhu − uh, u − uhiΓ + h(ph (u) − ph ) · n, u − uhiΓ .

433

(50)

434

Then, the auxiliary solutions in (35) yield h(ph (u) − ph) · n, u − uhiΓ = ah (qh (u) − qh, ph (u) − ph) + bh(yh (u) − yh , ph (u) − ph)

435 436

= bh (zh (u) − zh , qh (u) − qh ) − ch(yh (u) − yh, zh (u) − zh) + yh (u) − yh, yh (u) − yh

437

= kyh (u) − yhk20,Ω − mh,2(u − uh, zh (u) − zh).

438 439

(51)

440

Let uI = Πh u ∈ Uh be the L2 -projection of u defined in (30). By (50) and (51), we obtain ωku − uhk20,Γ + kyh(u) − yhk20,Ω

441

′

443 444 445 446 447 448

′

= hJh (u) − Jh(uh ), u − uhiΓ + mh,2(u − uh, zh (u) − zh )

442

= hωu + ph(u) · n, u − uhiΓ − hωuh + ph · n, u − uhiΓ + mh,2 (u − uh, zh (u) − zh) = hωu + p · n, u − uhiΓ + h(ph (u) − p) · n, u − uhiΓ + hωuh + ph · n, uh − uI iΓ

+ hωuh + ph · n, uI − uiΓ + mh,2 (u − uh, zh (u) − zh), where zh (u), ph (u) is the solution of the auxiliary system (35). With the help of the optimality conditions (18e) and (27e) with uI ∈ Uhad , we obtain

(52)

449

(53a)

450

(53b)

hωu + p · n, u − uhiΓ ≤ 0,

hωuh + ph · n, uh − uI iΓ ≤ 0. 12


451

Then, it follows from (53) that

452

ωku − uhk20,Γ + kyh(u) − yhk20,Ω = h(ph (u) − p) · n, u − uhiΓ + hωuh + ph · n, uI − uiΓ {z } | {z } | M1

453

(54)

M2

+ mh,2(u − uh, zh (u) − zh ) . | {z } M3

454 455

Bound for M1 : M1 can be split according to

456

(55)

457 458

where (zh (y), ph (y)) is the auxiliary solution of (37). With the help of the auxiliary solutions in (35) and (37), of Young’s inequality, and of the estimate (41), we obtain

459

h(ph (u) − p(y)) · n, u − uhiΓ = ah (qh (u) − qh , ph (u) − ph (y)) + bh(yh (u) − yh, ph (u) − ph (y)) = yh (u) − y, yh (u) − yh − mh,2 (u − uh, zh (u) − zh(y)) = yh (u) − y, yh (u) − yh + hεC11 (u − uh), zh (u) − zh (y)iΓ + h|β · n| (u − uh), zh (u) − zh(y)iΓ−

460 461 462

M1 = h(ph (u) − ph (y)) · n, u − uhiΓ + h(ph (y) − p) · n, u − uhiΓ ,

≤C(δ) ky − yh(u)k20,Ω + δkyh(u) − yh k20,Ω + C(δ) εC11 + kβkL∞(Γ) kzh (u) − zh (y)k20,Γ + δku − uhk20,Γ

463 464

≤C(δ)h2 + δkyh(u) − yh k20,Ω + C(δ) εC11 + kβkL∞(Γ) kzh (u) − zh (y)k20,Γ + δku − uhk20,Γ ,

465 466 467

(56)

471

where δ denotes an arbitrary small positive number. The same notation will be used frequently in the rest of the paper. An application of the local inverse inequality (33) and Lemma 6 yields 1 kzh (u) − zh(y)k0,Γ ≤ Ch− 2 kzh (u) − zk0,Ω + kz − zh(y)k0,Ω

472

Putting (57) into (56), we obtain

473

(58)

474 475

Now, we will derive a bound for the second term in the right-hand side of (55). By the interpolation operator Φh (28) and the Cauchy-Schwarz inequality, we have that

468 469 470

476 477 478 479 480

481

h(ph (u) − py (y)) · n, u − uhiΓ ≤ C(δ)h2 + δkyh(u) − yhk20,Ω + δku − uhk20,Γ .

h(ph (y) − p) · n, u − uhiΓ = h(ph (y) − Φh p) · n, u − uhiΓ + h(Φh p − p) · n, u − uhiΓ

≤ C kΦh p − pk0,Γku − uhk0,Γ + C kph (y) − Φh pk0,Γ ku − uhk0,Γ . 2 Recall that from Lemma 5 we have z ∈ W 2,s (Ω) and so p ∈ W 1,s (Ω) , s ≥ 2. Therefore,

(59)

1

r · n ∈ W 1− s (Γ) is well-defined on Γ. Then, using the local inverse inequality (33) and the interpolation error estimate (29), we obtain ! 1s

∑ kΦhp − pks0,s,E

kΦh p − pk0,Γ ≤ CkΦh p − pk0,s,Γ ≤ C

482

483

3

3

≤ Ch 2 kzk2,Ω ≤ Ch 2 .

(57)

(60)

∑

≤C

E⊂K∩Γ

≤C

∑

E⊂K∩Γ

h

h

−1

E⊂Γ

kΦh p − pks0,s,K

s−1

kzks2,s,K

! 1s

13

! 1s

≤C

∑

E⊂K∩Γ

1

≤ Ch1− s kzk2,s,Ω .


h

s−1

|p|s1,s,K

! 1s

484

Moreover, the estimates (33), (29), (60), and (40) yield 1

kph (y) − Φh pk0,Γ ≤ Ckph (y) − Φh pk0,s,Γ ≤ Ch− s kph (y) − Φh pk0,s,Ω

485

1

≤ Ch− s (kph (y) − pk0,s,Ω + kp − Φhpk0,s,Ω ) 1 1 ≤ Ch− s hkzk2,s,Ω + h1− s kzk2,s,Ω

486 487

1

1

≤ Ch1− s kzk2,s,Ω ≤ Ch1− s .

488

(61)

489

Then, it follows from (59)-(61) that

490

(62)

491

Combining the result of (58) and (62), we obtain

492

(63)

493 494

Bound for M2 : Now, we consider the second term in the right hand side of (54). It is easy to see that

495

1

h(ph (y) − p) · n, u − uhiΓ ≤ C(δ)h2(1− s ) + δku − uhk20,Γ .

1 M1 ≤ C(δ)h2(1− s ) + δ ku − uhk20,Γ + kyh(u) − yhk20,Ω .

M2 = hωu − p · n, Πhu − uiΓ + hωuh − ωu, Πhu − uiΓ

+h(p − ph(y)) · n, Πh u − uiΓ + h(ph (y) − ph ) · n, Πh u − uiΓ,

496

(64)

497

where Πh is the L2 -projection operator (30). It follows from the definition of Πh , the Young’s inequality, the estimate (31), and boundedness of the solutions that

498

hωu − p · n, Πhu − uiΓ = hωu − ωΠhu + Πh(p · n) − p · n, Πhu − uiΓ + hΠh (ωu − p · n), Πhu − uiΓ | {z }

499 500

≤C

501

=0 2 2 2 ωku − Πhuk0,Γ + kΠh(p · n) − p · nk0,Γ + ku − Πhuk0,Γ

1 1 ≤ Ch2(1− s ) ωkuk21− 1 ,Γ + kpk21− 1 ,Γ + kuk21− 1 ,Γ ≤ Ch2(1− s ) .

502 503

(65)

504

Next, Young’s inequality and the estimate (31) give us

s

s

hω(uh − u), Πhu − uiΓ ≤ C(δ) kΠh u − uk20,Γ + δωku − uhk20,Γ

505 506

s

(66)

1

≤ C(δ) h2(1− s ) kuk21− 1 ,Γ + δωku − uhk20,Γ . s

507

We can derive from (31), (60), (61) and boundedness of the solution kuk0,Γ that

508

(67)

509 510

Now, we find a bound for the last term of (64). Let yh (Πh u), qh (Πh u) ∈ Vh × Wh be the solution pair of the following auxiliary problem:

511

(68a)

512

(68b)

1

h(p − ph (y)) · n, Πh u − uiΓ ≤ kp − ph(y)k0,Γ kΠh u − uk0,Γ ≤ Ch2(1− s ) .

ah (qh (Πh u), r) + bh(yh (Πh u), r) = mh,1 (Πh u, r) −bh (v, qh (Πh u)) + ch (yh (Πh u), v) = mh,2 (Πh u, v) + F(v) 14


∀r ∈ Wh , ∀v ∈ Vh .

513

With the help of the auxiliary solutions in (35) and (68), and the estimate (31), we obtain h(ph (y) − ph ) · n, Πh u − uiΓ = ah (qh (Πh u) − qh(u), ph (y) − ph ) + bh(yh (Πh u) − yh(u), ph (y) − ph ) = y − yh, yh (Πh u) − yh(u) − mh,2(Πh u − u, zh(y) − zh )

514 515 516

≤ C(δ)kyh (Πh u) − yh(u)k20,Ω + δky − yhk20,Ω + C(δ) εC11 + kβkL∞(Γ) kzh (y) − zh k20,Γ + δkΠhu − uk20,Γ

517 518 519

≤ C(δ)kyh (Πh u) − yh(u)k20,Ω + δky − yhk20,Ω 1 + C(δ) εC11 + kβkL∞(Γ) kzh (y) − zh k20,Γ + δh2(1− s ) kuk21− 1 ,Γ .

520 521

(69)

522

Let (φ, ψ) ∈ V × W satisfy the following mixed system:

s

∇ · (εψ − βφ) + αφ = yh (Πh u) − yh(u) in Ω,

523 524

ψ + ∇φ = 0 φ=0

in Ω, on Γ.

525

(70)

526

Then, the following stability result holds (see e.g., ([42]))

527

(71)

528

For (φh , ψh ) ∈ Vh × Wh , the LDG approximation of the system (70) is

529

ε3/2 kφk2,Ω + ε1/2 kφk1,Ω + kφk0,Ω ≤ Ckyh (Πh u) − yh(u)k0,Ω .

ah (ψh , r) − bh(φh , r) = 0 bh (v, ψh ) + ch(v, φh ) = (yh (Πh u) − yh(u), v)

r ∈ Wh , v ∈ Vh .

530

(72)

531 532

Choosing v = yh (Πh u) − yh (u) and using the definitions of the auxiliary solutions with r = qh (Πh u) − qh(u), we obtain kyh (Πh u) − yh(u)k20,Ω = bh (yh (Πh u) − yh(u), ψh ) + ch(yh (Πh u) − yh(u), φh ) = bh (yh (Πh u) − yh(u), ψh ) + ch(yh (Πh u) − yh(u), φh )

533 534

+ah(ψh , qh (Πh u) − qh(u)) − bh(φh , qh (Πh u) − qh(u)) = mh,1 (Πh u − u, ψh) + mh,2(Πh u − u, φh)

535 536 537

≤ ku − Πhuk0,Γ kψh k0,Γ + εC11 ku − Πhuk0,Γ kφh k0,Γ +kβkL∞,Γ ku − Πhuk0,Γ kφh k0,Γ .

538

(73)

539

Let ψI = Φh ψ be the projection of ψ as defined in (28). Using the local inverse inequality (33), the estimates in (29), Lemma 6, and the stability estimate (71), we obtain

540

kψh k0,Γ ≤ kψh − ψI k0,Γ + kψI − ψk0,Γ + kψk0,Γ

541

1

1

≤ Ch− 2 kψh − ψI k0,Ω + Ch− 2 kψI − ψk0,Ω + Ckψk1,Ω

542

1

1

≤ Ch− 2 kψh − ψI k0,Ω + Ch 2 kψk1,Ω + Ckψk1,Ω

543

1

≤ Ch− 2 (kψh − ψk0,Ω + kψ − ψI k0,Ω ) + Ckψk1,Ω

544

1

545 546

(74)

≤ Ch− 2 h (kφk2,Ω + kψk1,Ω) + Ckψk1,Ω

≤ C (kφk2,Ω + kψk1,Ω) ≤ Ckyh (Πh u) − yh(u)k0,Ω . 15


547

Using the same arguments as in (74), the following estimate is derived

548

(75)

549 550

Then, an application of Young’s inequality in (73) with the inequalities (74) and (75), and the estimate (31) yields

551

(76)

552

Thus, (69), (76), (57), and Lemma 7 imply that

kφh k0,Γ ≤ Ckyh (Πh u) − yh(u)k0,Ω .

1

kyh (Πh u) − yh(u)k0,Ω ≤ Cku − Πhuk0,Γ ≤ Ch1− s kuk1− 1 ,Γ . s

1

h(ph (y) − ph) · n, Πh u − uiΓ ≤ Ch2(1− s ) + δky − yh(u)k20,Ω + δkyh(u) − yhk20,Ω

553

1

≤ Ch2(1− s ) + Cδh2 + δkyh(u) − yh k20,Ω

554

1

≤ Ch2(1− s ) + δkyh(u) − yh k20,Ω .

555

(77)

556

Combination of (65)-(67) and (77) results in

557

(78)

558

Bound for M3 : Last, we derive a bound for the third term in the right-hand side of (54). Young’s inequality gives us (79) M3 ≤ δku − uhk20,Γ + C(δ) εC11 + kβkL∞(Γ) kzh (u) − zh k20,Γ .

559 560

1 M2 ≤ Ch2(1− s ) + δ ku − uhk20,Γ + kyh(u) − yh k20,Ω .

563

With the help of the local inverse inequality (33) and Lemma 6 we obtain 3 3 1 (80) kzh (u) − zh k0,Γ ≤ Ch− 2 kzh (u) − zk0,Ω + kz − zhk0,Ω ≤ Ch 2 kzk2,Ω ≤ Ch 2 .

564

(81)

565

Summing up, the result (49) follows from (63), (78), and (81).

566 567

Now, we can present our main estimate for the LDG approximation of the Dirichlet boundary control problem. T HEOREM 10. Let (y, q, z, p, u) and yh , qh , zh , ph , uh be the solutions of (18) and (27), respectively. Assume that the conditions of Lemma 5 are satisfied. Then, we have

561 562

568 569 570

Then, it is easy see that

(82)

M3 ≤ δku − uhk20,Γ + C(δ) εC11 + kβkL∞(Γ) h3 .

1

ku − uhk0,Γ + ky − yhk0,Ω + kz − zhk0,Ω ≤ Ch1− s . Proof. From Lemma 7 and Lemma 9, we have

571

1

ky − yhk0,Ω ≤ ky − yh(u)k0,Ω + kyh (u) − yhk0,Ω ≤ Ch1− s .

572

(83)

573

We can split kz − zh k0,Ω according to

574

(84)

575 576

Using (27), (37), and (83) with the discrete Babu˘ska–Brezzi condition (see, e.g., [40]), we obtain

577

(85)

578

Finally, combining (83)-(85) with Lemmas 6 and 9, the desired result is obtained.

579

R EMARK 11. Although Lemma 5 shows that we can expect a better convergence for the adjoint z, the error estimates we obtained in Theorem 10 reflect the worst cases we can expect for the Dirichlet optimal boundary problems defined on convex polygonal domains.

580 581

kz − zh k0,Ω ≤ kz − zh(y)k0,Ω + kzh(y) − zh k0,Ω .

1

kzh (y) − zh k0,Ω ≤ Cky − yhk0,Ω ≤ Ch1− s .

16


582 583 584 585 586 587 588 589

5. Numerical Experiments. In this section we carry out some numerical experiments to support our theoretical findings. Although we derived the a priori error estimates for control constrained problems, we also present numerical results for the unconstrained case with analytic solution. The state, the adjoint, and the control variables are discretized using piecewise discontinuous linear polynomials. Discretized control constraint problems are solved by the primal dual active set algorithm as a semismooth Newton step; see, e.g., [10]. The algorithm is terminated when two consecutive active sets coincide. Further, we use the following formula to show the experimental order of convergence:

rate =

590

1 ke(h)k0,Ω ln , ln 2 ke(h/2)k0,Ω

591

where e(h) denotes error on the triangulation with mesh size h.

592 593

5.1. Example 1. Our first example, a modified form of the elliptic problem in [38], is an unconstrained problem with analytical solutions. The data of the problem are chosen as Ω = [0, 1] × [0, 1],

594

595 596

β = (1, 1)T ,

α = 1,

ω = 1.

The source function f (x1 , x2 ) and the desired state yd (x1 , x2 ) are chosen so that the analytical solutions of the state y, adjoint z, and control u are given by

y(x1 , x2 ) = −

597

1 (x1 (1 − x1) + x2(1 − x2)) , ω

1 x1 x2 (1 − x1)(1 − x2 ), ω 1 u(x1 , x2 ) = − (x1 (1 − x1) + x2(1 − x2)) , ω z(x1 , x2 ) =

598 599

600

ε = 1,

respectively.

Fig. 1: Example 5.1: Computed solutions of state y, adjoint z, and control u (from left to right). 17


# elements 32 128 512 2048 8192

ky − yhk0,Ω 1.20e-02 4.56e-03 1.94e-03 8.98e-04 4.35e-04

rate 1.3994 1.2337 1.1097 1.0474

kz − zhk0,Ω 1.79e-03 4.81e-04 1.17e-04 2.75e-05 9.30e-06

rate 1.8929 2.0360 2.0956 1.5619

ku − uhk0,Γ 2.26e-02 1.03e-02 4.98e-03 2.47e-03 1.24e-03

rate 1.1340 1.0457 1.0097 1.0001

Table 1: Example 5.1: Error of state y, adjoint z, and control u.

606 607

The mesh is constructed by first dividing Ω into N × N uniform squares and then dividing each square into two triangles. Figure 1 shows the computed solutions of the state y, adjoint z, and control u on the fine mesh with 8192 elements. In Table 1, we present both L2 (Ω)-errors for the state y and adjoint z, as well as the L2 (Γ)-error of the control u. Since the domain is the unit square with the largest interior angle π2 , the expected convergence rate of the control is ku − uhkL2 (Γ) ≤ C h; see, e.g., Theorem 10 and [38]. Therefore, our numerical results are consistent with the theoretical ones.

608 609

5.2. Example 2. Our second example defined in the unit square Ω = [0, 1] × [0, 1] has been adopted from [15]. The rest of the data are

601 602 603 604 605

610

611

612

f = 0,

yd =

1 , (x21 + x22 )1/3

β = (1, 1)T ,

α = 1,

ω = 1.

The control set is given by

U ad = {u ∈ L2 (Γ) : −0.75 ≤ u(x) ≤ 0 a.e. x ∈ Γ}.

Fig. 2: Example 5.2: Computed solutions of state y, adjoint z, and control u with ε = 1. 18


# elements 32 128 512 2048 8192 32768

ky − yhk0,Ω 9.13e-02 4.08e-02 1.91e-02 9.24e-03 4.41e-03 1.92e-03

rate 1.1619 1.0973 1.0458 1.0664 1.1979

kz − zhk0,Ω 2.42e-02 1.31e-02 6.73e-03 3.37e-03 1.65e-03 7.36e-04

ku − uhk0,Γ 2.89e-01 1.62e-01 8.75e-02 4.58e-02 2.28e-02 9.94e-03

rate 0.8785 0.9659 0.9984 1.0326 1.1605

rate 0.8299 0.8915 0.9360 1.0039 1.1984

Table 2: Example 5.2: Error of state y, adjoint z, and control u with ε = 1.

613 614 615 616 617 618 619 620 621 622 623 624

We have no explicit expression of the analytic solution, therefore we solve this problem numerically using 131072 elements, and we use this solution as a reference solution to make a comparison with other solutions on coarser meshes. Figure 2 displays computed numerical solutions on the reference mesh with ε = 1. It was shown in [15] that yd ∈ Ls (Ω) for all 2 ≤ s < sd∗ = 3 since yd has a singularity at the 3 1 boundary Γ. Consequently, the optimal solution has reduced regularity (y, u, z) ∈ H 2 − s (Ω) × W 1−1/s,s (Γ) × W 2,s (Ω) with 2 ≤ s < 3. Hence, we expect that the errors of the optimal solutions, i.e., y, u, z, converge with the rates ≈ 32 − s1d = 76 , ≈ 1 − s1d = 32 , and ≈ 2 − s1d − s1Ω = ∗

∗

∗

∗

respectively, where sΩ ∗ = ∞. The numerical results in Table 2 show that the convergence order of the optimal control is better than the theoretical ones. On the other hand, although the convergence rate of the adjoint z is lower than we expected, the results are consistent with the ones obtained in [15, 26]. 5 3,

ε = 0.1

ε = 0.01

ε = 0.001

0

0

0

−0.2

−0.2

−0.2

−0.4

−0.4

−0.4

−0.6

−0.6

−0.6

−0.8 1

−0.8 1

−0.8 1

1 0.5 x

2

0.5

0.5 0

0

x

1

x 1

2

1 0.5

0.5 0

0

x

x 1

2

0.5 0

0

x

1

Fig. 3: Example 5.2: Computed solutions of control u for ε = 10−1, 10−2 , and 10−3 (from left to right).

# elements 2048 8192 32768

ky − yhk0,Ω 5.23e+00 4.11e+00 2.69e+00

rate 0.3492 0.6093

kz − zhk0,Ω 1.38e+00 1.07e+00 6.39e-01

rate 0.3726 0.7439

ku − uhk0,Γ 7.33e-01 7.09e-01 6.87e-01

rate 0.0484 0.0456

Table 3: Example 5.2: Error of state y, adjoint z, and control u with ε = 0.001.

625

Next, we test our example for small values of the ε. Figure 3 shows numerical solutions 19


626 627 628 629 630 631 632 633 634 635

636 637 638

of the control u for ε = 10−1 , 10−2, and 10−3 . Although there exist some singularities on the boundary, we still have good convergence results for the smaller ε; see Table 3. When ε is too small, the problem becomes convection dominated and the stability of the state depends on the ε. Since the convergence rate of the control u also depends on the data of the state y, the convergence rate cannot be expected for the control. However, the LDG approximations produce some convergence rates. Further, our results satisfy the control constraints even for the smaller values of ε. 5.3. Example 3. The next example, adapted from [32], is given in a unit square Ω = [0, 1] × [0, 1] with unknown solutions. The data are given as follow: 2, x1 < 0.25, f= β = (0, 10)T , α = 0, ω = 0.1, −35, otherwise, 2 − 2x1, x1 < 0.50, d y = 2 − 2x2, otherwise. In addition, the control constraints are given by U ad = {u ∈ L2 (Γ) : −0.75 ≤ u(x) ≤ −0.25 a.e. x ∈ Γ}.

639

Fig. 4: Example 5.3: Computed solutions of state y, adjoint z, and control u with ε = 1.

# elements 32 128 512 2048 8192 32768

ky − yhk0,Ω 3.74e-01 2.14e-01 1.13e-01 5.69e-02 2.79e-02 1.25e-02

rate 0.8029 0.9275 0.9851 1.0285 1.1593

kz − zhk0,Ω 2.97e-02 1.70e-02 8.94e-03 4.51e-03 2.21e-03 9.90e-04

rate 0.8016 0.9293 0.9857 1.0288 1.1593

ku − uhk0,Γ 2.94e-01 1.74e-01 8.33e-02 4.15e-02 2.02e-02 8.24e-03

rate 0.7512 1.0663 1.0042 1.0425 1.2908


640 641 642 643 644

We solve the problem numerically on a fine mesh with 131072 elements and use it as a reference solution to make a comparison with other solutions on coarser meshes. Tables 4 and 5 show L2 -errors of the state y, adjoint z, and control u for ε = 1 and ε = 0.01, respectively. We observe that the obtained convergence rates are in agreement with the theoretically predicted ones. 20


# elements 32 128 512 2048 8192 32768

ky − yhk0,Ω 3.60e+00 3.33e+00 3.00e+00 2.51e+00 1.83e+00 9.86e-01

rate 0.1155 0.1470 0.2587 0.4556 0.8925

kz − zhk0,Ω 2.05e-01 1.90e-01 1.70e-01 1.41e-01 1.03e-01 5.66e-02

rate 0.1081 0.1580 0.2682 0.4520 0.8692

ku − uhk0,Γ 6.02e-01 4.18e-01 3.19e-01 2.13e-01 1.21e-01 5.49e-02

rate 0.5261 0.3881 0.5853 0.8171 1.1374

Table 5: Example 5.3: Error of state y, adjoint z, and control u with ε = 0.01.

645 646 647 648 649 650

5.4. Example 4. Our last example, modified from [38], is defined in a polygonal domain with maximum interior angle θ = 56 π as shown in Figure 5. The remaining data of the problem are given by −1, 0 ≤ x2 < 0.5, yd = f = 1, β = (1, 0)T , α = 2, ω = 1. 1, 0.5 ≤ x2 < 1, We choose the control set as U ad = {u ∈ L2 (Γ) : 0 ≤ u(x) ≤ 0.5 a.e. x ∈ Γ}. y 1

Ω

θ 1

x

Fig. 5: Example 5.4: Domain with θ = 65 π.

Fig. 6: Example 5.4: Computed solutions of state y, adjoint z, and control u with ε = 1. 21


# elements 12 48 192 768 3072 12288

ky − yhk0,Ω 6.14e-02 3.48e-02 1.76e-02 8.62e-03 4.19e-03 1.86e-03

kz − zhk0,Ω 1.28e-02 1.15e-02 6.43e-03 3.29e-03 1.62e-03 7.24e-04

rate 0.8192 0.9862 1.0260 1.0423 1.1679

ku − uhk0,Γ 1.67e-01 1.07e-01 5.86e-02 2.87e-02 1.32e-02 5.21e-03

rate 0.1467 0.8435 0.9684 1.0247 1.1583

rate 0.6428 0.8695 1.0310 1.1145 1.3456


651 652 653 654 655 656 657

The reference solutions have been computed on a fine mesh with 49152 elements, see Figure 6 with ε = 1. In this example, the optimal adjoint belongs to W 2,s (Ω) for 2 ≤ s ≤ 5 2θ Ω sΩ ∗ , where s∗ = 2θ−π = 2 . Under this configuration, the expected convergence rates are 1.1, 0.6 and 1.2 for the optimal state, control and adjoint, respectively. The obtained results are presented in Table 6 with ε = 1. It is shown that the orders of convergence for the state and adjoint are in reasonable agreement with the theoretically predicted ones, while it seems to be better for the control as in the previous examples. ε = 0.1

ε = 0.001

ε = 0.01

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2 0.1

0.1

0.1 −2

0 −1 0

0

x2

−1

0 0

0.5 1 1

−2

0

−1 0

0

0.5

−2

x1

1

1

0

0.5

x1

1

x

x

2

1

x

1

2

Fig. 7: Example 5.4: Computed solutions of control u for ε = 10−1 , 10−2, 10−3 (from left to right).

658 659 660

Figure 7 displays computed solutions of the control u for ε = 10−1 , 10−2, and 10−3 . We observe that the control bounds are satisfied well. As the diffusion parameter ε becomes smaller, some oscillations emerge on the boundary.

669

6. Conclusions. In this paper we considered the Dirichlet boundary control problem for a convection–diffusion equation with L2 -boundary controls subject to pointwise bounds on the control, discretized by the LDG method. By the help of the LDG method, we imposed the Dirichlet boundary condition into the weak form naturally. In addition, we derived a priori estimates for the optimal solutions in the convex polygonal domain. Some numerical results are provided to illustrate the theoretical results even for the small values of the diffusion parameter ε. When the problem becomes convection dominated, some oscillations emerge on the boundary. Therefore, a posteriori error estimates and related adaptive discontinuous Galerkin methods can be addressed as the future work.

670

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661 662 663 664 665 666 667 668

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24
