A C0 interior penalty method for the Dirichlet control problem governed by biharmonic operator

Accepted Manuscript A C 0 interior penalty method for the dirichlet control problem governed by biharmonic operator Sudipto Chowdhury, Thirupathi Gudi PII: DOI: Reference:

S0377-0427(16)30598-2 http://dx.doi.org/10.1016/j.cam.2016.12.005 CAM 10929

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Journal of Computational and Applied Mathematics

Received date: 4 March 2016 Revised date: 14 October 2016 Please cite this article as: S. Chowdhury, T. Gudi, A C 0 interior penalty method for the dirichlet control problem governed by biharmonic operator, Journal of Computational and Applied Mathematics (2016), http://dx.doi.org/10.1016/j.cam.2016.12.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A C 0 INTERIOR PENALTY METHOD FOR THE DIRICHLET CONTROL PROBLEM GOVERNED BY BIHARMONIC OPERATOR SUDIPTO CHOWDHURY AND THIRUPATHI GUDI

Abstract. An energy space based Dirichlet boundary control problem governed by biharmonic equation is investigated and subsequently a C 0 -interior penalty method is proposed and analyzed. An abstract a priori error estimate is derived under the minimal regularity conditions. The abstract error estimate guarantees optimal order of convergence whenever the solution is sufficiently regular. Further an optimal order L2 -norm error estimate is derived. Numerical experiments illustrate the theoretical findings.

1. Introduction The optimal control problems constrained by partial differential equations appear in many applications in the modern scientific world. The optimal control problem consists of finding a control that minimizes a cost functional subject to a partial differential equation. The control can act on the equation either through an external force or through a boundary condition. In the former case the problem is said to be a distributed control problem and the later case is said to be a boundary control problem. In the case of boundary control problem, the control can act through either Dirichelt or Neumann or Robin boundary conditions. The numerical approximation of these class of problem thus vary depending on the type of the control. There are many notable results in this area, but it is difficult to cite all the results in this brief discussion but they can be found in the articles cited here and the references given therein. The finite element approximation and their error analysis with order of convergence for optimal control problems has begun with the works in [22] and [24]. The governing PDE in [22] and [24] is a second order linear problem and the control is distributed and Neumann, respectively. Therein, the control is discretized using piecewise constant functions and the state and the adjoint state are discretized by using continuous piecewise linear polynomials. In [11] and [49] continuous piecewise linear polynomials have been used to approximate the control space. We refer to [46] for the L∞ −norm estimate for the finite element approximation of the optimal control. An overview of the Neumann boundary control problems governed by linear state equations can be found in [41]. For the error analysis of the Neumann boundary control problems governed by semi linear state equations, we refer to [2] and [12]. In [1], Neumann boundary control problem is analyzed with graded mesh refinement. However it has been observed over the years that the error analysis of 1991 Mathematics Subject Classification. 65N30, 65N15. Key words and phrases. optimal control, finite element, error estimate, C 0 IP method, clamped plate, Cahn Hilliard boundary condition, biharmonic, Dirichlet control. 1

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S. CHOWDHURY AND T. GUDI

the numerical approximation of the Dirichlet boundary controls is subtle. In [13] an ultra weak formulation has been proposed for the Dirichlet boundary control problems where the state equation is semi-linear second order and the Dirichlet controls are square integrable on the boundary. In that article an error estimate for the finite element approximation of the optimal control in the L2 norm is derived, but this estimate is suboptimal. Subsequently in [42] an improved order error estimate for the optimal control in the L2 norm has been derived. Recently an energy space based Dirichlet optimal control problem with linear second order state equation is investigated in [47] and the error estimates for finite element approximation in the energy and the L2 norms are derived. But the method in [47] uses a harmonic extension for the trial and test functions of the control variable and hence the numerical scheme becomes computationally expensive. Recently in [16] an alternative energy space based approach for the Dirichlet boundary control problem is introduced to remedy the difficulty in [47] and subsequently optimal order error estimates in the energy and the L2 norms are derived. On the other hand, we refer to [19] for the finite element error analysis of Dirichlet control problems with L2 (∂Ω) controls in two and three dimensional curved domains. In [20] the finite element error estimates for a Dirichlet control problem governed by stationary Navier- Stokes equations is analyzed. In [35, 43, 50], superconvergence results are derived for the numerical approximation of the distributed optimal control problems. In [35] a variational discretization approach is proposed where the discretization of the control variable is implicitly done through the discretization of the adjoint state variable. But in [43] a fully discrete approach and subsequently a post processing of the discrete adjoint variable is introduced to achieve the super convergence result. In [50] a fully discretized approach is adapted along with a post processing to obtain a super convergence result for an interior optimal control problem governed by Stokes equation. Simultaneously there have been a lot of work on the state constrained optimal control problems, for the reference we cite to [18, 31, 48] and references there in. All the articles discussed above deals with the error analysis of the numerical approximations of optimal control problems governed by the second order state equations. There are comparatively less literature available for the finite element analysis of the optimal control problems with the governing state equation is of higher order. In [23], a control constrained distributed optimal control problem governed by a biharmonic equation is investigated and an optimal order error estimate is derived for the optimal control along with a superconvergence result. In [23], the state equation is the biharmonic equation with Dirichlet boundary conditions defined on a convex polygonal domain and is descretized by the mixed formulation. In [27], a C 0 IP scheme for a distributed optimal control problem on a general polygonal domain is proposed and analyzed and an optimal order a priori error estimate has been derived along with a superconvergence result. Furthermore, therein, a residual based reliable and efficient a posteriori error estimate is derived. Recently in [17], a general framework for the error analysis of discontinuous Galerkin methods for both distributed and Neumann control for the biharmonic equation is derived on general polygonal domains. Subsequently a residual based reliable and efficient a posteriori error estimate is derived. A general framework for the a posteriori error analysis of conforming finite element methods of an optimal

C 0 IP METHOD FOR DIRICHLET CONTROL PROBLEMS

3

control problem with control constraints is derived in [37]. The result therein is obtained with the help of appropriate auxiliary linear problems. We also refer to [33] and [40] for a posteriori error analysis of distributed optimal control problems. We refer to the monographs [34, 39, 53] for the theoretical and algorithmic aspects of PDE constrained optimal control problems. In this article, we propose an energy space based approach for the Dirichlet boundary control of a fourth order problem. This model problem can appear in plate bending process and in fluid flow governed by the Stokes equation (since the stream function formulation of the Stokes equation is the biharmonic equation). We derive the optimality system which will be useful for the numerical approximation of the solution. We propose a C 0 -interior penalty method for the numerical approximation of the optimality system. Note that C 0 interior penalty methods have become a very attractive alternatives in the recent past to approximate the solutions of fourth order problems [5, 6, 7, 8, 21, 28]. This is due to the fact that C 0 interior penalty methods are computationally less expensive than the classical conforming finite element methods for solving the fourth order problems. It turns out that the optimal control in this article satisfies a biharmonic equation with the boundary condition of the Cahn-Hilliard type, see (2.5). Hence even in convex polygonal domains, the optimal control will have regularity of H 2+α (Ω), (α > 0) only. Therefore the norms used in the analyses of [3, 5, 21, 44, 51] are not even well defined. In order to derive the energy norm estimates, we follow the error analysis introduced in [29] and the error analysis done in [7] for CahnHilliard type boundary conditions. Recently, it is shown in [36] that the C 0 interior penalty solution of the biharmonic problem has connection to the divergence-conforming solution of the Stokes problem. Therefore our results will also be useful in the context of control problems for the Stokes equation. In summery, the results in the article can be stated as follows: • Establish an optimal order energy norm error estimate for the optimal control under the minimum regularity assumption. Subsequently establish energy norm estimates for the optimal state and the adjoint state. • Derive an optimal order L2 norm error estimate for the optimal control, the optimal state and the optimal adjoint state. • Illustrate the theoretical results by numerical experiments. The rest of the article is organized as follows. In Section 2, we formulate the model optimal control problem and derive corresponding optimality system, then we discuss the equivalence of solving the model problem with a corresponding energy space based boundary control problem where the control is sought from H 3/2 (∂Ω) space (the definition of H 3/2 (∂Ω) is given therein). In Section 3, we introduce the general notation that is used in the rest of the article, define the finite element method and derive the discrete optimality system. In Section 4, we derive optimal order energy norm estimates under minimal regularity assumption for the optimal control and subsequently for optimal state and the adjoint state. In Section 5 Lemma 5.2 proves that the optimal control q ∈ H(div, Ω), followed by Lemma 5.3 which proves one relation between the optimal control q and the adjoint state φ on the boundary.

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S. CHOWDHURY AND T. GUDI

This result will be used in proving the L2 norm error estimate for the optimal control. In Theorem 5.4 we derive the L2 norm estimate for the optimal control of order 2β under the regularity assumption for the optimal control of order 2 + β with β > 12 , which holds true if all the interior angles of the domain are less than 120o . Finally in Section 6 we perform some numerical experiments which supports the theoretical results. 2. Dirichlet Control Problem Let Ω ⊂ R2 be a polygonal domain with boundary ∂Ω. Assume that there is an integer m ≥ 1 such that the boundary ∂Ω is the union of line segments Γi ’s (1 ≤ i ≤ m) whose interiors in the induced topology are pairwise disjoint. Let D ⊆ Ω be a non empty open set. Denote the norm and the semi-norm on the corresponding Sobolev space H s (D), s > 0 by || · ||H s(D) and | · |H s(D) , respectively. Let || · || denotes the L2 (D) norm where the description of D will be clear from the context, if not then we will denote the L2 (D) norm by || · ||D . Let (u, v) denote the L2 (Ω) inner product for u and v ∈ L2 (Ω). We consider the following two spaces V and Q. From the space V , we will seek the adjoint state variable and from Q we will seek the state and the control variables: V = H02 (Ω), and Q = {p ∈ H 2 (Ω) :

∂p ∂n

= 0 on Γj , ∀ 1 ≤ j ≤ m}.

We define a bilinear form a : Q × Q → R by R P a(u, v) = Ω D 2 u : D 2 v, where D 2 u : D 2 v = i,j=1,2 uxi xj vxi xj .

The bilinear form a defined above is coercive on V and continuous on Q × Q, see [34]. Hence by the Lax-Milgram lemma [9, 14], for a given f ∈ L2 (Ω) and p ∈ Q there exists a unique uf ∈ V such that (2.1)

a(uf , v) = (f, v) − a(p, v) ∀v ∈ V.

Therefore u = uf + p is the weak solution of the following Dirichlet problem: ∆2 u = f in Ω, ∂u u = p, ∂n = 0 on ∂Ω. Hence we have the following definition of the solution operator S. Definition 2.1. For given f ∈ L2 (Ω), p ∈ Q we define the solution operator S : L2 (Ω) × Q → Q for (2.1) by S(f, p) = uf + p. Model Problem: For a given quadratic cost functional J, the Dirichlet boundary control problem consists of finding (u, q) ∈ Q × Q so that (2.2)

J(u, q) =

min

(w,p)∈Q×Q

J(w, p)

C 0 IP METHOD FOR DIRICHLET CONTROL PROBLEMS

5

subject to the condition that (w, p) ∈ Q × Q satisfies w = S(f, p), where f ∈ L2 (Ω) is a given function. Here the quadratic cost functional J is given by α 1 J(w, p) = kw − ud k2 + |p|2H 2 (Ω) , 2 2 where α > 0 is the regularization parameter for the optimal control problem and ud ∈ L2 (Ω) is the desired state. Proposition 2.2. There exists a unique solution (u, q) ∈ Q × Q for the above described Dirichlet optimal control problem. Furthermore there exists an adjoint state φ ∈ V , and the triplet (u, q, φ) ∈ Q × Q × V satisfies the following system, which is known as the optimality system:

(2.3) (2.4) (2.5)

u = uf + q, uf ∈ V, a(uf , v) = (f, v) − a(q, v) ∀v ∈ V, a(v, φ) = (u − ud , v) ∀v ∈ V, αa(q, p) = a(p, φ) − (u − ud , p) ∀p ∈ Q.

Proof. The proof will follow by the similar arguments as in [16].



Remark 2 .4 . From the trace theorem for polygonal domains we know that the first trace of Q into Πi=1,2...m H 3/2 (Γi ) is not surjective, but it is surjective onto a subspace of Πi=1,2...mH 3/2 (Γi ) [25]. From now on we denote that subspace by H 3/2 (∂Ω). For any p ∈ H 3/2 (∂Ω), its H 3/2 (∂Ω) semi-norm can be equivalently defined by the Dirichlet norm: |p|H 3/2 (∂Ω) := |up |H 2 (Ω) =

min

w∈Q,w=p on ∂Ω

|w|H 2 (Ω) ,

where the minimizer up ∈ Q satisfies the following equation: up = z + p,

z ∈ V,

a(z, v) = −a(p, v) ∀ v ∈ V, hence a(uq , v) = 0 ∀ v ∈ V. From (2.4)-(2.5), we obtain a(q, v) = 0 ∀ v ∈ V. Therefore for the optimal control q, we have q = uq and the minimum energy in (2.2) realizes with an equivalent H 3/2 (∂Ω) norm of the optimal control q.

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S. CHOWDHURY AND T. GUDI

3. Quadratic C 0 Interior Penalty Method Let Th be a simplicial and regular triangulation of Ω. We denote the set of all interior edges by Ehi , the set of boundary edges by Ehb , and let Eh = Ehi ∪ Ehb , i.e., Eh is the collection of all edges in the triangulation Th . Let hT be the diameter of T , h = max{hT : T ∈ Th } and the length of e is denoted by |e|. The finite element spaces are defined by Vh = {vh ∈ H01 (Ω) : vh |T ∈ P2 (T ), ∀T ∈ Th }, Qh = {ph ∈ H 1 (Ω) : ph |T ∈ P2 (T ), ∀T ∈ Th },

(3.1) (3.2)

where P2 (D) is the space of polynomials of degree less than or equal to two restricted to the set D. In the error analysis, we need the jump of the normal derivative and the average of the second order normal derivative of discrete functions across the inter element boundaries. For any e ∈ Ehi , let T+ and T− denote the two triangles sharing e. Let ne denotes the unit normal on e pointing from T− to T+ and hence the unit normal pointing from T+ to T− is −ne . We denote p|T+ by p+ and p|T− by p− . Then according to this notation for any p ∈ Qh , define the jump of the normal derivative of p on e by   ∂p = (∇p+ − ∇p− ) · ne . ∂ne Also for any p ∈ Qh , we define the average inter element boundaries by  2  ∂ p = ∂n2e

of the second order normal derivative across the 1 ∂ 2 p+ ∂ 2 p− + ). ( 2 ∂n2e ∂n2e

If e ∈ Ehb , then there is only one element T∈ Th such that e = ∂T∩ ∂Ω. Then the jump of the normal derivative and the average of the second order normal derivative across the edge e is defined by   ∂p = −∇p− · ne , ∂ne and





=

e∈Eh

e

∂2p ∂n2e

∂ 2 p− . ∂n2e

We define a mesh dependent bilinear form ah (·, ·) on Qh × Qh by XZ XZ 2 2 {{∂ 2 ph /∂n2e }}[[∂rh /∂ne ]] D ph : D rh + ah (ph , rh ) = T ∈Th

(3.3)

+

T

XZ

e∈Eh

e

{{∂

2

rh /∂n2e }}[[∂ph /∂ne ]]

X σ Z + [[∂ph /∂ne ]][[∂rh /∂ne ]], |e| e e∈E h

where without loss of generality we can assume the penalty parameter σ ≥ 1.

C 0 IP METHOD FOR DIRICHLET CONTROL PROBLEMS

7

We define below some mesh dependent norms and semi-norms on Qh that will be used in our analysis. Define X σ X |ph |2H 2 (T ) + (3.4) k[[∂ph /∂ne ]]k2e ∀ph ∈ Qh . kph k2h = |e| e∈E T ∈T h

h

We note that (3.4) defines a norm on Vh whereas it defines a semi-norm on Qh . On Qh the energy norm is denoted by |k · k|h which is defined as |kph k|2h = kph k2h + kph k2Ω

(3.5)

∀ph ∈ Qh .

We define one more mesh dependent norm on Vh (resp. semi-norm on Qh ) by X σ X X |e|k{{∂ 2 ph /∂n2e }}k2e + |ph |2H 2 (T ) + (3.6) kph k2Qh = k[[∂ph /∂ne ]]k2e |e| e∈E e∈E T ∈T h

h

h

∀ph ∈ Qh .

We note that (3.6) defines a semi-norm on Qh , but it is a norm on Vh . This norm is introduced in [7]. It is clear that with the help of a trace inequality for finite element spaces, we can show that there exist positive constants C and c such that also

ckph kh ≤ kph kQh ≤ Ckph kh ah (ph , ph ) ≥ ckph k2h

∀ph ∈ Qh ,

∀ph ∈ Qh ,

|ah (xh , yh )| ≤ Ckxh kh kyh kh

For reference we refer the reader to [5].

∀xh , yh ∈ Qh .

For f ∈ L2 (Ω), ph ∈ Qh , let vh (f, ph ) ∈ Vh be the unique solution of the following equation:

(3.7)

ah (vh (f, ph ), wh ) = (f, wh ) − ah (ph , wh ) ∀wh ∈ Vh .

As in the continuous case, we introduce the discrete solution operator Sh . Definition 3.1. Def ine the discrete solution operator Sh : L2 (Ω)×Qh → Qh by Sh (f, ph ) = vh + ph , where vh = vh (f, ph ), as in (3.7). Discrete system. A C 0 IP discretization of the continuous optimality system consists of finding uh ∈ Qh , φh ∈ Vh and qh ∈ Qh such that uh = uhf + qh ,

(3.8) (3.9) (3.10)

ufh ∈ Vh ,

ah (uhf , vh ) = (f, vh ) − ah (qh , vh ) ∀vh ∈ Vh ,

ah (φh , vh ) = (uh − ud , vh ) ∀vh ∈ Vh ,

αah (qh , ph ) = ah (φh , ph ) − (uh − ud , ph ) ∀p ∈ Qh .

It is easy to check that if f = ud = 0, then uhf = qh = φh = 0. This implies that the discrete system is uniquely solvable. For ph ∈ Qh , we define uhph ∈ Qh as follows: (3.11)

uhph = wh + ph ,

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S. CHOWDHURY AND T. GUDI

where wh ∈ Vh solves the following equation: ah (wh , vh ) = −ah (ph , vh )

∀ vh ∈ Vh .

In the following section, in order to derive the energy norm error estimate, we will use a ˜ h . The smoothing operator Eh which maps Qh into a conforming finite element space Q ˜ h is briefly given as follows. Let Wh be the Hsieh-Clough-Tocher macro construction of Q ¯ and on finite element space associated with Th , [14]. Functions in Wh belong to C 1 (Ω), each triangle they are piecewise cubic polynomials with respect to the partition obtained by connecting the centroid of the triangle to its vertices. Such functions are determined by their derivatives up to first order at the vertices and their normal derivatives at the midpoint of ˜ h be defined by Q ˜ h = Q ∩ Wh . Now the smoothing operator Eh : Qh → Q ˜h the edges. Let Q which is also known as enriching operator is defined as follows. Given any ph ∈ Qh , we can define the macro finite element function Eh (ph ) by specifying its degrees of freedom (dofs), which are either its values at the vertices of Th , the values of its first order partial derivatives at the vertices, or the values of its normal derivatives at the midpoints of the edges of Th . Let xi be a degree of freedom in Th , if xi is a corner point of Ω then we define ∇Eh (ph )(xi )=0, ∂ and if xi ∈ ∂Ω but xi is not a corner point of Ω then we define ∂n Eh (ph )(xi )=0, otherwise we assign these dofs of Eh (ph ) by averaging. For a detailed discussion on the definition of this operator and its approximation properties that are used in the following section, we refer the reader to [7]. 4. Energy Norm Estimate In this section, we discuss the error estimates between the solutions of (2.5) and (3.8) with respect to the energy norm defined in (3.5). The energy norm estimate is derived under the minimal regularity assumption on the optimal control, i.e. the domain is assumed to be only polygonal, no more assumptions on the interior angles of the domain are made. In the following theorem we will prove an optimal order error estimate for the optimal control variable and in the subsequent theorem we prove error estimate for the optimal state variable and the adjoint state variable. We will use some of the error analysis developed in [29]. We prove the following theorem on an optimal order energy norm error estimate for the optimal control q. Theorem 4.1. For the optimal control q, the following optimal order estimate holds: X h2T ku − ud kT , k|q − qh |kh ≤ Chmin(γ,1) (kqkH 2+γ1 (Ω) + kφkH 2+γ2 (Ω) + kf k) + T ∈Th

where γ = min{γ1 , γ2} is the minimum of the regularity index between the optimal control q and the adjoint state φ. Also qh is the discrete optimal control and the generic constant C depends only on the shape regularity of the triangulation. Proof. Let vh ∈ Qh be arbitrary and ψ = qh − vh . The bilinear form defined in (3.3) can be shown to be coercive with respect to the mesh dependent semi-norm defined in (3.4) over

C 0 IP METHOD FOR DIRICHLET CONTROL PROBLEMS

9

Qh × Qh , [5]. Using this fact, we find ckqh − vh k2h ≤αah (qh − vh , qh − vh ) =αah (qh − vh , ψ)

=αa(q, Eh (ψ)) − αah (vh , Eh (ψ)) + ah (φh , ψ) − (uh − ud , ψ) − ah (φ, Eh (ψ)) + (u − ud , Eh (ψ)) − αah (vh , ψ − Eh (ψ))

≤αa(q, Eh (ψ)) − αah (vh , Eh (ψ)) + ah (φh , ψ) − (uh − ud , ψ) − ah (Ih (φ), ψ)

+ (u − ud , ψ) + ah (Ih (φ), ψ) − (u − ud , ψ) − a(φ, Eh (ψ)) + (u − ud , Eh (ψ))

Hence,

− αah (vh , ψ − Eh (ψ)).

ckqh − vh k2h + kqh − vh k2 ≤αa(q, Eh (ψ)) − αah (vh , Eh (ψ)) + ah (φh , ψ) − ah (Ih (φ), ψ) + (uf − uhf , ψ) + (q − vh , ψ) − (u − ud , ψ − Eh (ψ))

+ ah (Ih (φ), ψ) − a(φ, Eh (ψ)) − αah (vh , ψ − Eh (ψ)).

(4.1)

In the above equation, we take vh = Ih (q) (and hence ψ = qh − Ih (q)). Note that (4.2)

ah (φh , ψ) − ah (Ih (φ), ψ) =ah (φh − Ih (φ), ψ − uhψ ) + ah (φh − Ih (φ), uhψ ) =ah (φh − Ih (φ), ψ − uhψ ),

and

ψ − uhψ = qh − Ih (q) − uhqh + uhIh(q) . From the discrete optimality condition, we have qh = uhqh . Hence we obtain ψ − uhψ = uhIh(q) − Ih (q). Let wh = uhIh (q) − Ih (q). In order to estimate wh , we will use a similar techniques as in [29, Lemma 4.1]. Define, (4.3) Note that wh ∈ Vh and

Vh = {vh ∈ Qh : vh = 0 on ∂Ω}.

ckwh k2h ≤ ah (wh , wh )

= −ah (Ih (q), wh )

(4.4)

= −ah (Ih (q), wh ) + ah (Ih (q), Eh (wh )) − ah (Ih (q), Eh (wh )) + a(q, Eh (wh )) ≤ C|ah (Ih (q), wh − Eh (wh ))| + Ckq − Ih (q)kh kwh kh .

Now by using the bubble function technique and an integration by parts formula as in [29, Lemma 4.1] and [29, eq.no. (4.16),(4.19)], we find (4.5)

|ah (Ih (q), wh − Eh (wh ))| ≤ Ckq − Ih (q)||h kwh ||h .

From (4.4) and (4.5), we obtain (4.6)

kwh kh ≤ Ckq − Ih (q)kh .

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S. CHOWDHURY AND T. GUDI

We rewrite the term ah (φh − Ih (φ), wh ) in (4.1) as

ah (φh − Ih (φ), wh ) =ah (φh − φ + φ − Ih (φ), wh )

=ah (φh , wh ) − ah (φ, wh ) + ah (φ − Ih (φ), wh ).

(4.7)

Since the adjoint state φ satisfies the biharmonic equation with clamped plate boundary condition, it also satisfies its corresponding discrete formulation [5]. Hence it satisfies the following equation: ah (φ, wh ) = (u − ud , wh ).

(4.8)

Therefore from (4.7) and (4.8), we obtain ah (φh − Ih (φ), wh ) =(uh − ud , wh ) − (u − ud , wh ) + ah (φ − Ih (φ), wh ) =(uh − u, wh) + ah (φ − Ih (φ), wh )

=(uhf − uf , wh ) − (q − qh , wh ) + ah (φ − Ih (φ), wh )

=(uhf − uf , wh ) − (q − Ih (q), wh ) − (Ih (q) − qh , wh ) + ah (φ − Ih (φ), wh )

≤(uhf − uf , wh ) − (q − Ih (q), wh ) + δ||Ih (q) − qh ||2 + (1/δ)kwh k2 + ah (φ − Ih (φ), wh ).

Since ah (φh − Ih (φ), ψ) = ah (φh − Ih (φ), wh ), we have (4.9)

ah (φh − Ih (φ), ψ) ≤(uhf − uf , wh ) − (q − Ih (q), wh ) + δkIh (q) − qh k2 + (1/δ)kwh k2 + ah (φ − Ih (φ), wh).

We rewrite the term ah (Ih (φ), ψ) − a(φ, Eh (ψ)) in (4.1) as

ah (Ih (φ), ψ) − a(φ, Eh (ψ)) =ah (Ih (φ), ψ) − ah (Ih (φ), Eh (ψ)) + ah (Ih (φ), Eh (ψ)) − a(φ, Eh (ψ)) (4.10)

=ah (Ih (φ), ψ − Eh (ψ)) + ah (Ih (φ) − φ, Eh (ψ)).

Hence from (4.1) and (4.10), we find ckqh − vh k2h + kqh − vh k2 ≤αa(q, Eh (ψ)) − αah (vh , Eh (ψ)) + ah (φh , ψ)

− ah (Ih (φ), ψ) + (uf − uhf , ψ) + (q − vh , ψ) + ah (Ih (φ), ψ − Eh (ψ))

(4.11)

− (u − ud , ψ − Eh (ψ)) + ah (Ih (φ) − φ, Eh (ψ)) − ah (vh , ψ − Eh (ψ)).

By taking vh = Ih (q) in (4.11) then using the Poincar´e-Friedrich’s type inequality for piecewise H 2 functions [10] and (4.9), we find ckqh − Ih (q)k2h + kqh − Ih (q)k2 ≤αa(q, Eh (ψ)) − αah (Ih (q), Eh (ψ))

+ (uf − uhf , ψ) + (q − Ih (q), ψ) + ah (Ih (φ), ψ − Eh (ψ))

− (u − ud , ψ − Eh (ψ)) + ah (Ih (φ) − φ, Eh (ψ))

− αah (Ih (q), ψ − Eh (ψ)) + (uhf − uf , wh ) − (q − Ih (q), wh )

C 0 IP METHOD FOR DIRICHLET CONTROL PROBLEMS

This implies that

11

+ (1/δ1 )kwh k2h + ah (φ − Ih (φ), wh ) + δ1 kIh (q) − qh k2 .

c1 kqh − Ih (q)k2h + c2 kqh − Ih (q)k2 ≤αa(q, Eh (ψ)) − αah (Ih (q), Eh (ψ))

+ (uf − uhf , ψ) + (q − Ih (q), ψ) + ah (Ih (φ), ψ − Eh (ψ)) − (u − ud , ψ − Eh (ψ)) + ah (Ih (φ) − φ, Eh (ψ))

(4.12)

− αah (Ih (q), ψ − Eh (ψ)) + (uhf − uf , wh ) − (q − Ih (q), wh )

+ (1/δ1 )kwh k2h + ah (φ − Ih (φ), wh ).

Now we will estimate the right hand side of (4.12). First consider the term αa(q, Eh (ψ)) − αah (Ih (q), Eh (ψ)) and find αa(q, Eh (ψ)) − αah (Ih (q), Eh (ψ)) ≤ kq − Ih (q)kh kEh (ψ)kH 2 (Ω) ≤ Ckq − Ih (q)kh kψkh

≤ (1/δ2 )kq − Ih (q)k2h + δ2 kψk2h .

(4.13)

Next we consider ah (Ih (φ), ψ − Eh (ψ)) − (u − ud , ψ − Eh (ψ)) and αah (Ih (q), ψ − Eh (ψ)). Using the same techniques as in the [29] and taking the enriching operator as in [7] we obtain, X h2T ku − ud kT )kψkh ah (Ih (φ), ψ − Eh (ψ)) − (u − ud , ψ − Eh (ψ)) ≤C(kφ − Ih (φ)kh + T ∈Th

≤(1/δ3 )(kφ − Ih (φ)kh + (4.14)

+

and (4.15)

δ3 kψk2h ,

X

T ∈Th

h2T ku − ud kT )2

ah (Ih (q), ψ − Eh (ψ)) ≤ Ckq − Ih (q)kh ||ψ||h

≤ δ4 kψk2h + (1/δ4 )kq − Ih (q)k2h .

Using Young’s inequality for the other terms of the right hand side of (4.12) and using (4.6) we obtain, kIh (q) − qh k2h + kIh (q) − qh k2 ≤C(kq − Ih (q)k2h + kq − Ih (q)k2 + kφ − Ih (φ)k2Qh + kuf − uhf k2 X h4T ||u − ud ||2T ). + (4.16) T ∈Th

Combining all the estimates above, we complete the proof.



Theorem 4.2. For the optimal state u and the adjoint state φ, the following optimal order estimate holds: k|u − uh k|h ≤ Chmin(γ,1) (kf |k + kqkH 2+γ1 (Ω) + kφkH 2+γ2 (Ω) ), k|φ − φh k|h ≤ Chmin(γ,1) (kf k + kqkH 2+γ1 (Ω) + kφkH 2+γ2 (Ω) ),

12

S. CHOWDHURY AND T. GUDI

where γ1 , γ2 and γ are as in Theorem 4.1 . Proof. We note that u = uf + q, uh = uhf + qh . Using this decomposition, Theorem 4.1 and the Poincar´e-Friedrichs inequality for piece-wise H 2 functions [10], we obtain the error estimate for the optimal state. Let Ph (φ) ∈ Vh denotes the C 0 IP approximation of φ [5]. Then it satisfies the following equation: (4.17)

ah (Ph (φ), vh ) = (u − ud , vh )

∀vh ∈ Vh .

By subtracting (3.10) from (4.17), we find (4.18)

ah (Ph (φ) − φh , vh ) = (u − uh , vh )

∀vh ∈ Vh .

Now taking vh = Ph (φ) − φh in (4.18) and using the estimate for the optimal state, we find

(4.19)

kPh (φ) − φh kQh ≤ Ck|u − uh k|h .

Finally, using the equivalence of norms k · kQh and k · kh on Qh and the equivalence of k · kh and k| · k|h on Vh (by Poincar´e-Friedrichs inequality for piece-wise H 2 functions, [10]), we obtain the estimates.  5. L2 -Norm Estimate In this section, we derive the L2 norm error estimate for the optimal control. This estimate is derived under the assumption that the optimal control q ∈ H 2+β (Ω) with β > 1/2. This regularity assumption holds true if all the interior angles of Ω are less than 120o. We refer the readers to [23, (2.4)] and [7, pg.no.2107] for the proof. With the help of this assumption, we derive some regularity for the optimal control q and a relation between the optimal control q and the adjoint state φ. Using this relation and by introducing an auxiliary dual control problem, we derive the L2 norm estimate for the optimal control. We denote the solution of this dual problem by r. Since all the interior angles of Ω are less than 120o, we have r ∈ H 2+β (Ω) for some β > 1/2. We derive a sub optimal rate of convergence of the discrete optimal control to the optimal control with the help of Aubin-Nitche duality argument, which is discussed in detail in Theorem 5.4. ∞ ¯ ¯ Since C ∞ (Ω)×C (Ω) is dense in H(div, Ω) space with respect to the natural norm defined on H(div, Ω) [52], we obtain Z Z ∂∆φ ∇(∆)φ.∇ψ + ∆2 φψ = h , ψi− 1 , 1 ,Ω ∀ ψ ∈ H 1 (Ω). (5.1) 2 2 ∂n Ω Ω By taking test functions in (2.4) from D(Ω), we obtain the following equation in the distributional sense:

(5.2)

∆2 φ = u − ud in Ω.

Combining (5.1) and (5.2), we obtain Z Z ∂∆φ (5.3) ∇(∆)φ.∇ψ + (u − ud )ψ = h , ψi− 1 , 1 ,Ω 2 2 ∂n Ω Ω

∀ψ ∈ H 1 (Ω).

C 0 IP METHOD FOR DIRICHLET CONTROL PROBLEMS

If ψ ∈ Q in (5.3), then

Z

13

∂∆φ , ψi− 1 , 1 ,Ω . 2 2 ∂n Ω We need the following auxiliary result in our error analysis: a(φ, ψ) −

(5.4)

(u − ud )ψ = −h

Lemma 5.1. Consider the variational problem, 1 (∇w, ∇p) = − h∂(∆φ)/∂n, pi ∀p ∈ H 1 (Ω). α Up to adding a constant there exists a unique solution w ∈ H 1(Ω) to the above variational formulation. Proof. The proof follows from the observation that the H 1 (Ω)-semi norm defines a norm on the quotient space H 1 (Ω)/R.  With the help of the assumptions made in the beginning of this section, we obtain the following regularity property for the optimal control q. This helps us to obtain a relation between the optimal control q and the adjoint state φ which is used to deduce the L2 norm error estimate for the optimal control. Lemma 5.2. For the optimal control q, we have ∆q ∈ H 1 (Ω) and ∇(∆q) is in H(div, Ω). Proof. We take test functions from D(Ω) in (2.5). Using (5.2) together with the definition of distributional derivative, we obtain ∆2 q = 0 in Ω.

(5.5) Now we will show that

a(q, p) = (∆q, ∆p),

a(φ, p) = (∆φ, ∆p) ∀p ∈ Q.

¯ such that φn → q in H 5/2 (Ω). Since q ∈ H 5/2 (Ω), there exists a sequence {φn } ⊂ C ∞ (Ω) Then applying Green’s formula we deduce Z 2 2 (∆φn , ∆p) − (D φn , D p) = − ∂(∇φn )/∂n · ∇p ∀p ∈ Q. ∂Ω

From this equation we find 2

2

(∆φn , ∆p) − (D φn , D p) = −

Z

[∂ 2 φn /∂n2 ∂p/∂n + ∂ 2 φn /∂n∂t∂p/∂t] ∂Ω

∀p ∈ Q.

Since φn → q in H 5/2 (Ω), by trace theory ∂φn /∂n → ∂q/∂n in H 1 (Γi ) on each line segment Γi . Hence ∂ 2 φn /∂t∂n → ∂ 2 q/∂t∂n in L2 (Γi ) for each line segment Γi (1 ≤ i ≤ m). The left hand side of the above equation is continuous in φn and after taking the limit we find m X (∆q, ∆p) − (D 2 q, D 2p) = − h∂ 2 q/∂n∂t, ∂p/∂tiΓi ∀p ∈ Q. i=1

14

S. CHOWDHURY AND T. GUDI

Further, observe that ∂ 2 q/∂n∂t = 0 on ∂Ω. Therefore a(q, p) = (∆q, ∆p). From the above equation and (2.5) we get 1 (5.6) (∆q, ∆p) = − h∂(∆φ)/∂n, pi−1/2,1/2,∂Ω ∀p ∈ Q. α From Lemma 5.1, we know that there exists a unique weak solution w ∈ H 1 (Ω) up to an additive constant of the following variational problem: 1 h∂(∆φ)/∂n, pi−1/2,1/2,∂Ω ∀p ∈ H 1 (Ω). α Now if p ∈ Q in the above equation, we have 1 (5.7) (w, ∆p) = − h∂(∆φ)/∂n, pi−1/2,1/2,∂Ω . α Subtracting (5.6) from (5.7), we obtain (∇w, ∇p) =

(w − ∆q, ∆p) = 0

∀p ∈ Q.

˜ 2 (Ω), where Hence w − ∆q belongs to the orthogonal complement of L Z ˜ 2 (Ω) = {ξ ∈ L2 (Ω) : (L ξ = 0}. Ω

Therefore, ∆q = w + constant. Hence ∇(∆q) ∈ H(div, Ω).



∞ ¯ ¯ Using Lemma 5.2, we note that ∇(∆q) ∈ H(div, Ω). Density of C ∞ (Ω)×C (Ω) in H(div, Ω) implies that (∆2 q, p) + (∇(∆q), ∇p) = h∂(∆q)/∂n, pi−1/2,1/2,∂Ω , and since ∆2 q = 0 in Ω, we have

(∇(∆q), ∇p) = h∂(∆q)/∂n, pi−1/2,1/2,∂Ω .

Applying integration by parts formula, we find ∂∆q a(q, p) = −h (5.8) , pi− 1 , 1 ,∂Ω ∀p ∈ Q. 2 2 ∂n Thus from (5.8) and (2.5), we get that ∂∆φ ∂∆q h , pi− 1 , 1 ,∂Ω = αh , pi− 1 , 1 ,∂Ω ∀p ∈ Q. 2 2 2 2 ∂n ∂n Now we will prove the following Lemma which proves a relation between the optimal control and the adjoint state. 1

Lemma 5.3. If p ∈ H 2 (∂Ω) then, ∂∆q ∂∆φ (5.9) αh , pi−1/2,1/2,∂Ω = h , pi−1/2,1/2,∂Ω . ∂n ∂n

C 0 IP METHOD FOR DIRICHLET CONTROL PROBLEMS

15

Proof. Let p ∈ H 1/2 (∂Ω). From [15, Proposition 3.32] we know that if ∂Ω is Lipschitz continuous and X = {u|∂Ω : u ∈ C ∞ (R2 )}, then X is dense in H 1/2 (∂Ω). Therefore there exists a sequence {ψn } ⊂ C ∞ (∂Ω) such that ψn → p in H 1/2 (∂Ω). Now let un be the weak solution of the following PDE: ∆2 un = 0 in Ω, un = ψn on ∂Ω, ∂un /∂n = 0 on ∂Ω. Then clearly un ∈ Q and un |∂Ω = ψn . Now

|h∂(∆φ)/∂n − α∂(∆q)/∂n, pi−1/2,1/2,∂Ω | = |h∂(∆φ)/∂n − α∂(∆q)/∂n, p − ψn i−1/2,1/2,∂Ω |

≤ k∂(∆φ)/∂n − α∂(∆q)/∂nkH −1/2 (∂Ω) kp − ψn kH 1/2 (∂Ω) ≤ ǫ, ∀ǫ > 0. 

Hence, we obtain (5.9).

Theorem 5.4. For the optimal control q, there holds kq − qh k ≤ Ch2β (kqkH 2+β (Ω) + kf k + kφkH 3 (Ω) ). Proof. We derive the estimate by using the Aubin-Nitsche duality argument using the following auxiliary optimal control problem: Find r ∈ Q such that (5.10)

j(r) = min j(p), p∈Q

where j(p) is defined by, 1 α j(p) = kup − (q − qh )k2 + |p|2H 2 (Ω) 2 2 up = w + p and w ∈ V is given by the following equation:

∀ p ∈ Q,

a(w, v) = −a(p, v) ∀ v ∈ V.

The theory of PDE constrained optimal control problems provides us that the above optimal control problem attains a unique solution. We denote it by r. For a detailed discussion on this topic we refer to [53, 38]. The solution r satisfies the following optimality condition: αa(r, p) + (ur , up ) = (q − qh , up) ∀p ∈ Q.

(5.11)

From (5.11), we find that (5.12) where ξ ∈

(5.13)

H02 (Ω)

αa(r, p) + (ur , p) − a(ξ, p) = (q − qh , p) ∀p ∈ Q,

in (5.12) satisfies the following equation:

a(ξ, v) = (ur − (q − qh ), v) ∀v ∈ H02 (Ω).

As Ω is convex, from the elliptic regularity theory for the clamped plate problems we know that ξ ∈ H 3 (Ω) ∩ H02(Ω). From (5.13), we obtain (5.14)

∆2 ξ = ur − (q − qh ),

16

S. CHOWDHURY AND T. GUDI

in the sense of distributions. Using the density of D(Ω) in L2 (Ω) we obtain ∆2 ξ = ur − (q − qh ) in Ω,

(5.15)

ξ = 0;

∂ξ ∂n

= 0 on ∂Ω.

Now we note that as Ω is convex, ξ ∈ H 3 (Ω) ∩ H02 (Ω). Therefore together with (5.15) ∈ H −1/2 (∂Ω). The density of we observe that ∇(∆ξ) ∈ H(div, Ω). This implies ∂(∆ξ) ∂n ¯ × C ∞ (Ω) ¯ in H(div, Ω) and since ∇(∆ξ) ∈ H(div, Ω), we deduce C ∞ (Ω) Z Z ∇(∆ξ)∇p + ∆2 ξp = h∂(∆ξ)/∂n, pi−1/2,1/2,∂Ω ∀p ∈ H 1 (Ω). Ω

Ω

Thus by using integration by parts and (5.12), we find that r satisfies

∂(∆ξ) , pi−1/2,1/2,∂Ω ∀p ∈ Q. ∂n Since all the interior angles of Ω are less than 120o , we deduce that r ∈ H 2+β (Ω), where β > 1/2 depends on the interior angle at the corner points [7, pg.no. 2107]. By taking the test functions p ∈ D(Ω) in (5.11), we derive αa(r, p) = −h

∆2 r = 0,

in the sense of distributions. Consequently by using density of D(Ω) in L2 (Ω), we conclude that ∆2 r = 0 in Ω.

(5.16)

Since r ∈ H 2+β (Ω) for β > 1/2, we obtain by using the similar arguments used for proving Lemma 5.2 that ∇(∆r) ∈ H(div, Ω). Therefore ∂(∆r) ∈ H −1/2 (∂Ω). This implies that ∂n

∂(∆r) ∂(∆ξ) , pi−1/2,1/2,∂Ω = h , pi−1/2,1/2,∂Ω ∀p ∈ Q. ∂n ∂n Now by similar arguments used for proving Lemma 5.2, we deduce that ∂(∆r) ∂(∆ξ) αh (5.17) , pi−1/2,1/2,∂Ω = h , pi−1/2,1/2,∂Ω ∀p ∈ H 1/2 (∂Ω). ∂n ∂n Since ∇(∆ξ) ∈ H(div, Ω), by standard density argument [52] the following integration by parts formula holds true: Z Z ∇(∆ξ)∇p + ∆2 ξp = h∂(∆ξ)/∂n, pi−1/2,1/2,∂Ω ∀p ∈ H 1 (Ω). αh

Ω

Ω

For the L2 norm error estimate, we will use the test function space as hqi + Qh . 1+s i We note R that if v ∈ H (Ω), with s > 1/2 and ∇v ∈ H(div, Ω) then for e ∈ Eh , there holds e [[∂v/∂n]]ψ = 0 ∀ψ ∈ L2 (e). Using this fact in the above equation and after taking p = ph ∈ hqi + Qh , we find by triangle wise integration by parts that Z ∂(∆ξ) , ph i = (ur − (q − qh ))ph . ah (ξ, ph ) + h ∂n Ω

C 0 IP METHOD FOR DIRICHLET CONTROL PROBLEMS

Hence from (5.17), we get ∂(∆r) ah (ξ, ph ) + hα , ph i − ∂n

Z

Ω

u r ph = −

Z

Ω

17

(q − qh )ph .

∂ Also since hα ∂(∆r) , ph i = h ∂n (∆r), uhph i, we find ∂n Z Z ∂ h ah (ξ, ph ) − ur (ph − uph ) − ur uhph + αh (∆r), uhph i (5.18) ∂n Ω Ω Z Z = − (q − qh )(ph − uhph ) − (q − qh )uhph . Ω

Ω

1+s

As before if ∇ψ ∈ H(div, Ω), ψ ∈ H (Ω) for s > 1/2 and v ∈ H 1 (Ω), then for e ∈ Ehi we R have e [[∂ψ/∂n]]v = 0. Therefore, we have h∂(∆r)/∂n, uhph i = −ah (r, uhph ) and Z Z h (5.19) ah (ξ, ph ) − ur (ph − uph ) − ur uhph − αah (r, uhph ) Ω Z ZΩ = − (q − qh )(ph − uhph ) − (q − qh )uhph , Ω

and hence

ah (ξ, ph −

uhph )

Ω

Z

(ur − (q − qh ))(ph − uhph ) Ω Z Z h h + ah (ξ, uph ) − ur uph − αah (r, ph ) = − (q − qh )uhph .

−

Ω

As ξ ∈ H

3

(Ω) ∩ H02 (Ω)

Ω

satisfies the discrete formulation (similar to the proof in [5]), we have Z Z h h αah (r, ph ) + ur uph = ah (ξ, uph ) + (q − qh )uhph .

(5.20)

Ω

Ω

By taking ph = q − qh in (5.20), we find Z Z h h αah (r, q − qh ) + ur uq−qh = ah (ξ, uq−qh ) + (q − qh )uhq−qh , Ω

and

ah (ξ, uhq−qh )

Ω

Z

(q − qh )(uhq − uq + uq − uhqh ) Ω Z = αah (r, q − qh ) + ur (uhq − uq + uq − uhqh ).

+

Ω

Thus, 2

Z

(q − qh )(uhq − uq ) − ah (ξ, uhq−qh ) Ω Z Z h + αah (r, q − qh ) + ur (uq − uq ) + ur (uq − uhqh )

kq − qh k = −

Ω

Ω

18

S. CHOWDHURY AND T. GUDI

=−

Z

Ω

(q − qh )(uhq − uq ) − ah (ξ, uhq−qh )

Z

+ αah (r − rh , q − qh ) + αah (rh , q − qh ) + ur (uhq − uq ) Ω Z + ur (uq − uhqh ) ZΩ = − (q − qh )(uhq − uq ) − ah (ξ, uhq−qh ) Ω Z + αah (r − rh , q − qh ) + ah (φ, rh ) − (u − ud )rh − ah (φh , rh ) Z Z ZΩ + (uh − ud )rh + ur (uhq − uq ) + ur (uq − uhqh )) Ω Ω ZΩ = − (q − qh )(uhq − uq ) − ah (ξ, uhq−qh ) Ω Z + αah (r − rh , q − qh ) + ah (φ − φh , rh ) − (u − ud )rh Ω Z Z Z + (uh − ud )rh + ur (uhq − uq ) + ur (uq − uhqh ) Ω Ω ZΩ = − (q − qh )(uhq − uq ) − ah (ξ, uhq−qh ) Ω

+ αah (r − rh , q − qh ) + ah (φ − φh , rh − r) + ah (φ − φh , r) Z Z Z h − (u − uh )rh + ur (uq − uq ) + ur (uq − uhqh ) Ω Ω ZΩ = − (q − qh )(uhq − uq ) − ah (ξ, uhq−qh ) Ω

(5.21)

+ αah (r − rh , q − qh ) + ah (φ − φh , rh − r) + ah (φ − φh , r) Z Z Z h h − (uf − uf )rh − (uq − uqh )(r − rh ) + ur (uhq − uq ). Ω

Ω

Ω

Now we will estimate each term one by one on the right hand side of (5.21). Consider Z − (q − qh )(uhq − uq ) Ω

i.e. we aim to estimate kuhq − uq k. We will use duality argument to get the estimate kuhq

(uhq − uq , w) − uq k = sup . kwk w∈L2 (Ω),w6=0

C 0 IP METHOD FOR DIRICHLET CONTROL PROBLEMS

19

To this end, consider ∆2 φw = w in Ω, ∂φw φw = 0, = 0 on ∂Ω. ∂n Then (uhq − uq , w) = ah (φw , uhq − uq ) = ah (φw − Ph (w), uhq − uq )

≤ Ckφw − Ph (φw )kQh kuhq − uq kQh + Chkwk kuhq − uq kQh .

But

kuhq − uq kQh ≤ kuhq − uhqh kQh + kuhqh − uq kQh .

(5.22)

It can be easily checked that

kuhq−qh kQh ≤Ckq − qh kQh

≤C(kq − Ih (q)kQh + kIh (q) − qh kQh ) ≤C(kq − Ih (q)kQh + kIh (q) − qh kh )

≤C(kq − Ih (q)kQh + kIh (q) − qkh + kq − qh kh ),

(5.23)

and then by using the Lagrange interpolation of q say Ih (q) and from (5.22)-(5.23), we find (5.24) Hence

kuhq − uq kQh ≤ C(kq − Ih (q)kQh + kIh (q) − qkh + kq − qh kh ). kuhq − uq k ≤ Ch1+β (kqkH 2+β (Ω) + kf k + kφkH 3 (Ω) ).

(5.25)

Now to estimate the second term of the right hand side of (5.21) i.e. ah (ξ, uhq−qh ), consider (5.26)

ah (ξ, uhq−qh ) = ah (ξ − ξh , uhq−qh ) ≤ Ckξ − ξh kQh kuhq−qh kQh .

Using (5.18) and (5.21), we find the required estimate for ah (ξ, uhq−qh ) i.e., (5.27)

ah (ξ, uhq−qh ) ≤ Ch2 kξkH 3(Ω) (kqkH 2+β (Ω) + kf k + kφkH 3 (Ω) ).

Now by substituting p = r in (5.11) and using the fact that ur = r, we get that krkΩ ≤ kq−qh kΩ . Note that the elliptic regularity estimate for (5.8) gives that kξkH 3 (Ω) ≤ Ckq−qh kΩ . Hence from (5.22), we find (5.28)

ah (ξ, uhq−qh ) ≤ Ch1+β kq − qh kΩ (kqkH 2+β (Ω) + kf || + kφkH 3 (Ω) ).

Now, we will show that First we show that

krkH 2+β (Ω) ≤ Ckq − qh kΩ . |r|H 2+β (Ω) ≤ Ckq − qh kΩ .

20

S. CHOWDHURY AND T. GUDI

To this end, we will use the similar techniques as in [7]. Let z = αr − ξ. Then z satisfies the following variational problem: a(z, v) = a(αr − ξ, v) ∀v ∈ Q. Using the integration by parts on the right hand side of the above equation, we obtain a(z, v) = −(∆2 ξ, v) ∀v ∈ Q. Then by the elliptic regularity theory for the biharmonic equation with Cahn Hilliard boundary condition [4], we obtain z ∈ H 2+β (Ω). Further |z|H 2+β (Ω) ≤ k∆2 ξk

≤ kur − (q − qh )k

≤ kr − (q − qh )k ≤ Ckq − qh k.

Taking p = r in (5.11), we obtain |r|H 2 (Ω) + krk ≤ Ckq − qh k. Then by using the integration by parts, we get |r|H 1(Ω) ≤ C(|r|H 2 (Ω) + krk) ≤ Ckq − qh k. Hence from the above estimates, we obtain krkH 2+β (Ω) ≤ Ckq − qh k. To estimate the third term of the right hand side of (5.21), note that ah (r − rh , q − qh ) ≤ Ckr − rh kQh k|q − qh k|Qh

≤ Ch2β (kqkH 2+β (Ω) + kf k + kφkH 3 (Ω) )krkH 2+β (Ω) ≤ Ch2β (kqkH 2+β (Ω) + kf k + kφkH 3 (Ω) )kq − qh k

≤ Ch2β (kqkH 2+β (Ω) + kf k + kφkH 3 (Ω) )kq − qh k.

(5.29)

Using Theorem 4.2 and the arguments in the proof of (5.29), we find ah (φ − φh , rh − r) ≤ kφ − φh kQh kr − rh kQh

≤ Chβ (kφ − Ih (φ)kQh + kIh (φ) − φh kQh )kq − qh k ≤ Chβ (kφ − Ih (φ)kQh + kIh (φ) − φh kh )kq − qh k

(5.30)

≤ Chβ (kφ − Ih (φ)kQh + kIh (φ) − φkh + kφ − φh kh )kq − qh k ≤ Ch1+β (kqkH 2+β (Ω) + kf k + kφkH 3 (Ω) )kq − qh k.

Note that (5.31)

ah (φ − φh , r) = 0.

C 0 IP METHOD FOR DIRICHLET CONTROL PROBLEMS

21

The rest of the terms of (5.21) can be easily estimated and those estimates are given as follows: Z (5.32) − (uf − uhf )rh ≤ Ckuf − uhf k krkH 2+β (Ω) ≤ Ch2 kf k kq − qh k. Ω

(by Aubin − Nitche duality argument)

(5.33)

−

Z

Ω

(uq − uhqh )(r − rh ) ≤ Ch2+2β (kqkH 2+β (Ω) + kf k + kφkH 3 (Ω) )kq − qh k.

Further from (5.18), we get Z ur (uhq − uq ) ≤ kuhq − uq kΩ krkH 2+β (Ω) Ω

(5.34)

≤ Ch1+β (kqkH 3 (Ω) + kf k + kφkH 3 (Ω) )kq − qh k.

Hence from (5.21), (5.25) and (5.27)-(5.34), we obtain the desired estimate.



6. Numerical Examples In this section, we illustrate the theoretical results by performing a numerical experiment. In the experiment, we test the validity of the a priori error estimates derived in the energy and L2 norms derived in Theorem 4.1, Theorem 4.2 and Theorem 5.4. To this end, we modify the model problem to construct an example with known solution by as follows: First ˜ by modify the cost functional J, denoted by J, ˜ p) = 1 ||w − ud ||2 + α |q − qd |2 2 J(w, (6.1) H (Ω) 2 2 subject to the condition that (w, p) ∈ Q × Q satisfies w = S(f, p). Then it is easy to check that the optimality conditions take the form: w = uf + q, a(w, v) = (f, v) − a(q, v) ∀v ∈ V, a(φ, v) = (u − ud , v) ∀v ∈ V,

αa(q, p) = a(φ, p) − (u − ud , p) + αa(qd , p) ∀p ∈ Q. For this example, the domain Ω is taken to be the unit square (0, 1)×(0, 1) and the parameter α = 1. We choose the data of the problem as follows. Choose the state to be u(x, y) = sin2 πx sin2 πy +cos πx cos πy, the adjoint state to be φ(x, y) = sin4 πx sin4 πy and the control to be q(x, y) = cos πx cos πy. Then compute f = ∆2 u and ud = u − ∆2 φ. The choice of φ leads to qd = q. We generate a sequence of meshes with mesh size h as shown in the Table 6.1 by uniformly refining successively each triangulation. We have used our in house MATLAB code for the computations. Figure 6.1 shows the graph of the computed (left) and the exact (right) solution.

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S. CHOWDHURY AND T. GUDI

1

1

0.5

0.5

0

0

−0.5

−0.5

−1 1

−1 1 1 0.5

0.5

1 0.5

0 0

0.5 0 0

Figure 6.1. The comparison between the computed (left) and the exact (right) solution The computed errors and orders of convergence in the energy and the L2 norms are shown in Table 6.1 and Table 6.2, respectively. The experiment clearly illustrates the expected rates of convergence. h ku − uh kh 1/4 11.7593 1/8 5.1160 1/16 2.5328 1/32 1.2612 1/64 0.6303 1/128 0.3144

order kφ − φh kh – 16.7950 1.2007 7.7830 1.0143 4.2441 1.0059 2.1249 1.0007 1.0578 1.0002 0.4852

order kq − qh k0,Ω – 7.6730 1.1096 2.4142 0.8792 1.1352 0.9981 0.5594 1.0063 0.2792 1.0032 0.1398

order – 1.6683 1.0886 1.0210 1.0025 1.0003

Table 6.1. Errors and orders of convergence in energy norm

h ku − uh k0,Ω 1/4 519.5152 1/8 0.0385 1/16 0.0115 1/32 0.0030 1/64 0.0008 1/128 0.0042

order kφ − φh k0,Ω – 0.4618 13.7209 0.0370 1.7368 0.0116 1.9247 0.0032 1.9227 0.0008 1.9542 0.0041

order kq − qh k0,Ω – 519.5482 3.6409 0.0345 1.6725 0.0105 1.8480 0.0028 1.9310 0.0007 1.9664 0.0018

order – 13.8771 1.7231 1.9021 1.9642 1.9898

Table 6.2. Errors and orders of convergence in L2 norm

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