A priori error estimates for finite element discretizations of a shape optimization problem

DOI:10.1051/m2an/2013086 www.esaim-m2an.org

A PRIORI ERROR ESTIMATES FOR FINITE ELEMENT DISCRETIZATIONS OF A SHAPE OPTIMIZATION PROBLEM

Bernhard Kiniger

and Boris Vexler

Abstract. In this paper we consider a model shape optimization problem. The state variable solves an elliptic equation on a domain with one part of the boundary described as the graph of a control function. We prove higher regularity of the control and develop a priori error analysis for the finite element discretization of the shape optimization problem under consideration. The derived a priori error estimates are illustrated on two numerical examples.

Mathematics Subject Classification. 49Q10, 49M25, 65M15, 65M60.

Received November 14, 2011. Revised March 5, 2013.

Published online October 7, 2013.

Introduction

In this paper we consider the following shape optimization problem governed by a linear elliptic equation:

minJ(q, u) =1

2u−u^q_d²_L2(Ωq)+α

2q²_L2((0,1)),

subject to −Δu+u=f^q inΩ_q,

u= 0 onΓ_q=∂Ω_q,

where the domain Ω_q is basically the unit square with one “moving” side given as the graph of the control function q, see Figure 1. The data functions u^q_d and f^q are restrictions of functions u_d and f defined on a sufficiently large (holding-all) domain ˆΩ. The precise formulation of the problem including a functional analytic setting is presented in Section1.

Similar shape optimization problems, where the unknown part of the boundary is parameterized as a graph of a function, are considered in various publications, see,e.g., [13,14,19,25]. The problem formulation in these publications involves a bound on an appropriate norm of q. Our formulation utilizes a Tikhonov-type term q²_L2((0,1)) instead.

The main contribution of this paper is ana priorierror analysis for a finite element discretization of the shape optimization problem under consideration. The control variable is discretized by Hermite finite elements allowing

Keywords and phrases.Shape optimization, existence and convergence of approximate solutions, error estimates, finite elements.

1 Lehrstuhl f¨ur Optimale Steuerung, Technische Universit¨at M¨unchen, Fakult¨at f¨ur Mathematik, Boltzmannstraße 3, 85748 Garching b. M¨unchen, Germany.kiniger@ma.tum.de; vexler@ma.tum.de

Article published by EDP Sciences c EDP Sciences, SMAI 2013

Γ

(0, 1)

Ω

(1, 1)

(0, 0) q(x) (1, 0)

Figure 1. DomainΩ_q.

for conforming discretization of the control space H²((0,1))∩H¹₀((0,1)). The corresponding discretization pa- rameter is calledσ. The state variable is discretized using usual (bi)linear finite elements with the discretization parameterh. The main result (see Thm. 3.2) is the estimate

q¯−q¯_σ,hH2((0,1))≤c(σ²+h²),

where ¯qis a local solution of the optimization problem fulfilling a second order sufficient optimality condition and (¯q_σ,h) ((σ, h)→0) is a sequence of local solutions of the discretized problem converging to ¯q. The existence of such a sequence is also shown.

There are some published results on discretization of shape optimizations problems. In [13,14] and in [8,9]

convergence of discrete solutions to a continuous one is investigated. To our best knowledge only the paper [11]

provides convergence results including the rate of convergence for a discretized shape optimization problem using a fast wavelet boundary element method for the state equation, where the discretization of the state is treated as a consistency error and is not restricted to boundary integral equation methods.

The paper is organized as follows: In the next section we discuss a precise formulation of the shape optimization problem under consideration, show the existence of at least one globally optimal solution applying the techniques similar to [13], reformulate the problem using a transformation to a reference domainΩ₀and present optimality conditions. Section 2 is devoted to the discretization. The control variable is discretized by Hermite finite elements of third order on (0,1), leading to the semidiscrete problem. Then the state variable is discretized by (bi)linear finite elements on the reference domain Ω₀ resulting in a full discretization of the problem. We show that both the semidiscrete and the discrete problems possess solutions and discuss their convergence for (σ, h) →0. In Section 3 we prove that any optimal solution ¯q possesses higher regularity,i.e. q¯∈ H⁴((0,1)).

This result is essential for derivinga priori error estimates. Due to the fact that the considered optimization problem is not convex in general, we have to deal with second order sufficient optimality conditions and adapt the technique from [6]. We first provide error estimates for the discretization error between a solution of the continuous problem and the corresponding solution of the semidiscrete one and then for the error between a semidiscrete and a fully discrete solution. In Section4 we present numerical examples illustrating our results.

Throughout the paper,c andc_i shall always denote generic constants which are – if not stated otherwise – independent ofσandh, but may depend onα, and have different values on different appearances. With L^p(Ω) and W^k,p(Ω) we will denote the Lebesgue and the Sobolev spaces on the domainΩ, respectively. Forp= 2 we use the standard notation H^k(Ω) = W^k,2(Ω). For an arbitrary Hilbert spaceX we will denote its scalar product with (·,·)_X and the corresponding norm with ·_X, whereas forX = L²(Ω₀) we omit the subindex. The set of allm-times continuously differentiable functions whosemth derivative is H¨older-continuous with index αwill be denoted by C^m,α(Ω). For functionsqdefined on an intervalI⊂R,qandqshall denote the first and second order weak derivative, respectively.

1. Optimization problem

1.1. Problem formulation and existence result

In this section we first describe the shape optimization problem under consideration. The control variableq from the control spaceQ= H²(I)∩H¹₀(I) withI= (0,1) characterizes the domainΩ_q through:

Ω_q={(x, y)∈R²|x∈I, y∈(q(x),1)}.

To exclude a possible degeneracy of the domainΩ_q we fix 0< ε <1 and define the set

Q¯^ad={q∈Q| q(x)≤1−εfor allx∈I}. (1.1) For each q ∈ Q¯^ad the domain Ω_q is a Lipschitz domain, which allows for the definition of the state variable u∈H¹₀(Ω_q) being the weak solution of the state equation

−Δu+u=f^q inΩ_q,

u= 0 onΓ_q=∂Ω_q. (1.2)

The shape optimization problem is then given as:

MinimizeJ(q, u) =1

2u−u^q_d²_L2(Ωq)+α

2 q²_L2(I), q∈Q¯^ad, u∈H¹₀(Ω_q), (1.3) subject to (1.2).

We define the solution operator ˜S, which assigns to eachq∈Q¯^adthe unique solution ˜u(q) = ˜S(q) of (1.2).

This allows to introduce the reduced cost functionalj: ¯Q^ad→Rby j(q) =J(q,S(q)).˜

In order to prove the existence of an optimal solution to (1.3), we need bounds in the full H²(I)-norm.

Lemma 1.1. OnQ, the H²(I)-seminorm is equivalent to the fullH²(I)-norm.

Proof. Letq∈Qbe arbitrary. AsQ⊂H¹₀(I) we know that there existsc_p>0 such thatq_L2(I)≤c_p·q_L2(I). Furthermore,

q²_L2(I)= ₁

0

q(x)²dx=− ₁

0

q(x)q(x) dx≤ q_L2(I)· q_L2(I)≤c_p· q_L2(I)· q_L2(I), which leads to

q_L2(I)≤c_p· q_L2(I).

As a result,

q²_H2(I)=q²_L2(I)+q²_L2(I)+q²_L2(I)≤(c⁴_p+c²_p+ 1)· q²_L2(I). In order to bound Q^ad in H²(I), we set q₀ = 0∈ Q¯^ad. A necessary condition for q ∈Q¯^ad to be a solution to (1.3) is

j(q)≤j(q₀),

which is

1

2S(q)−u^q_d²_L2(Ωq)+α

2q²_L2(I)≤j(q₀).

This implies

q²_L2(I)≤ 2 α

j(q₀)−1

2S(q)−u^q_d²_L2(Ωq)

≤ 2 αj(q₀).

Together with Lemma 1.1 this shows that there exists a constant ˜C = ˜C(α) >0, such that the search for a minima can be restricted to the set

Q^ad={q∈Q¯^ad| q_H2(I)≤C˜}, (1.4) which is also bounded in C¹(I).

Throughout we assume for the data

u^q_d=u_d|_Ωq, f^q =f|_Ωq, withu_d, f ∈C^2,1( ˆΩ),

where ˆΩis the all-holding domain withΩ_q ⊂Ωˆ for allq∈Q^ad. We will therefore just writef and u_d instead off^q andu^q_d.

Forq∈Q^ad andv∈H¹₀(Ω_q) we define an extension ˜v∈H¹₀( ˆΩ) by

˜ v(x) =

v(x) ifx∈Ω_q, 0 else.

Ifv_n∈H¹₀(Ω_qn) forq_n ∈Q^ad, then the sequence (v_n)_n∈Nis said to (weakly) converge to an elementv∈H¹₀(Ω_q) if the extended functions ˜v_n (weakly) converge to ˜v in H¹₀( ˆΩ).

The following proposition states the continuity of the solution operator ˜S, see [13] for the proof.

Proposition 1.2. Let q_n, q∈Q^ad,q_n→q inL^∞(I)andu_n = ˜S(q_n). Then there existsu˜∈H¹₀( ˆΩ)with

˜

u_n→u˜ inH¹₀( ˆΩ) for n→ ∞, such that u= ˜u|_Ωq holds withu= ˜S(q).

A direct consequence is the following existence theorem.

Theorem 1.3. Problem (1.3)has a global solution.

Proof. We haveQ^ad=∅ andJ(q, u)≥0. So there exists a minimizing sequence

q_n, u_n= ˜S(q_n)

n∈Nwith j= inf

q∈Q^adj(q) = lim

n→∞J(q_n, u_n).

The boundedness ofQ^adin H²(I) implies the existence of a subsequence (q_nk)_k∈N⊂(q_n)_n∈Nand ¯q∈Q^adwith q_nk q¯ in H²(I),

q_nk→q¯ in C⁰( ¯I) fork→ ∞,

due to the compact embedding of H²(I) into C⁰( ¯I). With Proposition1.2it follows that there exists ¯u∈H¹₀(Ω_¯q) with ˜u_nk →u˜¯in H¹₀( ˆΩ) and ¯u= ˜S(¯q). Now let

J₁: (q, u)→ u−u_d²_L2(Ωq)=u˜−˜u_d²_L2( ˆΩ), J₂:q→ q²_L2(I).

The functional J₂ is continuous and convex in H²(I) and therefore weakly lower semi-continuous. By the definition ofj it follows that

k→∞lim J₁(q_nk, u_nk) =J₁(¯q,u),¯ lim inf

k→∞ J₂(q_nk)≥J₂(¯q).

By adding these two inequalities we get

J(¯q,u)¯ ≤lim inf

k→∞ J(q_nk, u_nk) =j,

and conclude thatJ(¯q,u) =¯ j. Hence, (¯q,u) is a solution of the initial problem (1.3).¯ Remark 1.4. Although the state equation (1.2) is linear, the solution operator ˜S is nonlinear and one cannot expect the reduced cost functionaljto be convex. Therefore uniqueness of an optimal solution cannot be shown in general.

1.2. Optimality conditions

In order to find the optimal solution ¯qand to formulate optimality conditions, one usually needs some sort of derivative of the reduced cost functionalj. In general, the derivativej(q)(δq) can be representedviaa domain integral. This is done later, see (3.8). Given sufficient regularity ofq, it can also be representedviaa boundary integral over the moving part of the boundary. This general principle is known as the Hadamard Formula and can be found in [26], Theorem 2.27. Ifqis sufficiently smooth (q∈C²(I)), then it can be shown thatjis Fr´echet differentiable with

j(q)(δq) =

Γq

1

2(u−u_d)²+∂_nu ∂_nz

V_q,δq, ndΓ_q(x) +α ₁

0

q δqdx. (1.5)

Within (1.5), the so-called adjoint variablez∈H¹₀(Ω_q) is the weak solution of −Δz+z=u−u_d inΩ_q,

z= 0 onΓ_q, (1.6)

and the vector fieldV_q,δq, describing a transformation fromΩ_q toΩ_q+δq, is given by V_q,δq(x, y) =

0 (1−y)_1−q(x)^δq(x)

.

For an overview on how to derive such a representation we refer to [15,26]. Within Subsection3.1 we will first present the domain integral representation of j(q)(δq) which holds for all q ∈ Q^ad. Second, we will use that representation to show that the optimal solution ¯qis sufficiently regular, such that (1.5) actually holds in the optimal solution ¯q.

Throughout we make the following assumption.

Assumption 1.5. We assume that for the optimal solution ¯q under consideration the restriction ¯q ≤ 1−ε from (1.1) is not active,i.e. there existsδ >0 with ¯q(x)≤1−ε−δfor allx∈I.

1.3. Transformation of the problem

In what follows we provide an equivalent formulation of (1.3) by the transformation onto the reference domain Ω₀= (0,1)². To this end we introduce

V_q(x, y) =

0 (1−y)q(x)

, (1.7)

a velocity field, and let

T_q(x, y) = (Id+V_q)(x, y) =

x y+ (1−y)q(x)

be a transformation with Ω_q =T_q(Ω₀). The following functions derived from this transformation will be used in the sequel:

DT_q(x, y) =

1 0 (1−y)q(x) 1−q(x)

, (1.8)

γ_q(x, y) = det (DT_q(x, y)) = 1−q(x), (1.9)

A_q(x, y) = γ_q·DT_q⁻¹·DT_q^−T

(x, y) =

1−q(x) −(1−y)q(x)

−(1−y)q(x) ^(1−y)_1−q(x)^2q(x)2+1

, (1.10)

A_q,δq(x, y) = d

dtA_q+t·δq(x, y) t=0

=

−δq(x) −(1−y)δq(x)

−(1−y)δq(x) δq(x)+2(1−y)²q(x)δq(x)(1−q(x))+(1−y)²q(x)²δq(x) (1−q(x))²

, (1.11)

A_{q,δq,τ q}(x, y) = d

dtA_{q+t·τ q,δq}(x, y) t=0

. (1.12)

With these explicit definitions at hand, one can easily derive some stability results which follow by a direct calculation and the boundedness ofQ^adin H²(I), as well as the fact thatu_d, f ∈C^2,1( ˆΩ).

Lemma 1.6. Forq, p∈Q^adwe have

• γq−γ_pL∞(Ω0)≤c· q−p_H2(I),

• Tq−T_pL∞(Ω0)≤c· q−p_H2(I),

• T_q⁻¹−T_p⁻¹

L^∞(Ω0)≤c· q−p_H2(I),

• Vq−V_pL∞(Ω0)≤c· q−p_H2(I),

• f◦T_q−f◦T_pL∞(Ω0)≤c· q−p_H2(I),

• ∇f ◦T_q− ∇f ◦T_pL∞(Ω0)≤c· q−p_H2(I),

• ∇²f◦T_q− ∇²f◦T_p

L^∞(Ω0)≤c· q−p_H2(I),

• ud◦T_q−u_d◦T_pL∞(Ω0)≤c· q−p_H2(I),

• ∇ud◦T_q− ∇ud◦T_pL∞(Ω0)≤c· q−p_H2(I),

• ∇²u_d◦T_q− ∇²u_d◦T_p

L^∞(Ω0)≤c· q−p_H2(I),

• div(Vq)_L∞(Ω0)≤c· q_H1(I).

Furthermore, the expressionsγ_q,T_q,T_q⁻¹,V_q,f◦T_q,∇f◦T_q.∇²f◦T_q,u_d◦T_q,∇u_d◦T_q,∇²u_d◦T_q anddiv(V_q) are bounded in L^∞(Ω₀).

Lemma 1.7. Forq, p∈Q^ad,δq∈Q, it holds that:

• Aq_L∞(Ω0)≤c,

• A_q−A_pL∞(Ω0)≤c· q−p_H2(I),

• A_q,δq

L^∞(Ω0)≤c· δq_H2(I),

• A_q,δq,δq

L^∞(Ω0)≤c· δq²_H2(I).

Using transformation of integrals we obtain the following equivalence of the weak formulations onΩ_qandΩ₀: Lemma 1.8. Let q∈Q^ad,u∈H¹₀(Ω_q)andu^q=u◦T_q ∈H¹₀(Ω₀). Then the following variational formulations are equivalent

Ωq

(∇u∇z+uz) dx=

Ωq

f zdx ∀z∈H¹₀(Ω_q), (1.13)

Ω0

(∇u^qA_q∇z^q+u^qz^qγ_q) dx=

Ω0

(f◦T_q)z^qγ_qdx ∀z^q ∈H¹₀(Ω₀). (1.14) The advantage of (1.14) over (1.13) is its independence of the domain, instead the coefficients of the matrix A_q are changeable. To shorten notation, we will also from now on make use of the following forms

a(q)(u, z) =

Ω0

(∇uAq∇z+uzγ_q) dx, (1.15)

l(q)(z) =

Ω0

(f◦T_q)zγ_qdx. (1.16)

Then (1.14) can be written as

a(q) (u^q, z^q) =l(q) (z^q) ∀z^q ∈H¹₀(Ω₀).

Lemma 1.8 motivates the introduction of another solution operator S which assigns to each control q the

“transported” solution,i.e.letS(q) = ˜S(q)◦T_q∈H¹₀(Ω₀).

In the following lemmata we summarize some properties of the bilinear forma(q)(·,·) and the linear functional l(q)(·), which are direct consequences of the definitions, Lemma1.6 and Lemma1.7.

Lemma 1.9. The bilinear form a(q)(·,·) is continuous and coercive in H¹₀(Ω₀), i.e. there exist c₁, c₂ > 0, independent ofq, such that for all q∈Q^ad andu, z∈H¹₀(Ω₀)it holds that

|a(q)(u, z)| ≤c₁· ∇u_L2(Ω0)· ∇z_L2(Ω0), a(q)(u, u)≥c₂· ∇u²_L2(Ω0).

Furthermore, there existsc₃>0, independent of q, such that foru∈W^1,p(Ω₀), p∈[1,∞] andz∈W^1,p(Ω₀) with p⁻¹+p⁻¹= 1the following H¨older-like inequality holds

|a(q)(u, z)| ≤c₃· u_W1,p(Ω0)· z_W1,p(Ω0). Lemma 1.10. Forq, p∈Q^ad andu, z∈H¹(Ω₀)it holds that

|a(q)(u, z)−a(p)(u, z)| ≤c· q−p_H2(I)· u_H1(Ω0)· z_H1(Ω0). Lemma 1.11. Forq, p∈Q^ad andz∈H¹(Ω₀)it holds that

|l(q)(z)−l(p)(z)| ≤c· q−p_H2(I)· z_L2(Ω0).

In the following sections we will make heavy use of various regularity results for elliptic partial differential equations. Here we state them all at once.

Theorem 1.12. Let Ω ⊂ R² be bounded with Lipschitz boundary Γ, f ∈ H⁻¹(Ω), and let the matrix A be symmetric and uniformly elliptic with coefficients a_i,j ∈ L^∞(Ω₀). Furthermore, let u ∈ H¹₀(Ω) be the weak

solution of −div (A∇u) =f in Ω,

u= 0 onΓ.

1. If1/2≥t > s >0and the coefficients ofAbelong toC^0,t(Ω), then for allf ∈H^−1+s(Ω)it holdsu∈H^1+s₀ (Ω) and there existsc_s>0with u_H1+s(Ω)≤c_s· f_H−1+s(Ω).

2. If the coefficients of A are Lipschitz and there exists ε > 0 with f ∈ H^−1/2+ε(Ω), then u ∈ H^3/2(Ω). In addition, iff ∈L²(Ω), then there exists c >0 withu_H3/2(Ω)≤c· f_L2(Ω).

3. If Ωis polygonal and convex, the coefficients of A are Lipschitz andf ∈L²(Ω), then u∈H²(Ω) and there existsc >0, depending only on the diameter ofΩ, withu_H2(Ω)≤c· f_L2(Ω).

4. IfΩis polygonal and convex, and the coefficients ofAare Lipschitz then there existsp_Ω >2such that for all p < p_Ω andf ∈L^p(Ω)it holds thatu∈W^2,p(Ω)and there existsc >0, depending on the Lipschitz-constant ofA, such that u_W2,p(Ω)≤c· f_Lp(Ω).

Proof. Part (1) can be found in [22], (2) can be found in [16,17,24], part (3) is proven in [12,18] and the proof

for the last part (4) can be found in [12].

Remark 1.13. All the statements of Theorem1.12remain true if a zero order term pops up. Let b ∈C¹(Ω) withb≥c >0 inΩ. Let nowu∈H¹₀(Ω) be the weak solution of

−div (A∇u) +bu=f inΩ, u= 0 onΓ.

Which can equivalently be rewritten as

−div (A∇u) =f −bu inΩ,

u= 0 onΓ. (1.17)

Now we can apply Theorem1.12to (1.17) and use the common H¹-regularity results to show that Theorem1.12 still holds.

Corollary 1.14. Forq∈Q^adit holds that the corresponding stateuas well as the corresponding adjoint statez, defined via (1.2)and (1.6), respectively, possess the higher regularity u, z∈H^3/2(Ω_q).

Proof. This corollary follows from Theorem1.12, part (2), together with Remark1.13.

2. Discretization

In order to solve (1.3) numerically we discretize the problem using (bi)linear finite elements. To this end, we first discretize the control q, afterwards the state u is being discretized. For the discretization of the control we split the interval (0,1) intoN pairwise disjoint nonempty subintervals,i.e. we choose 0 =x₀< x₁< . . . <

x_N−1 < x_N = 1 and setI_i = (x_i−1, x_i) for i∈ {1, . . . , N}. Furthermore, let |Ii| ≤σ for a given discretization parameterσ >0 and alli∈ {1, . . . , N}. We use Hermite elements to define the set of the discretized controls by

Q_σ=

q∈Q| q|_Ii∈ P3(I_i) ∀i∈ {1, . . . , N}

, (2.1)

Q^ad_σ =Q^ad∩Q_σ, (2.2)

where Pn(J) shall denote the set of all polynomials of degree less than or equal to n over the set J. It can be seen immediately that for q_σ ∈Q^ad_σ we have q_σ ∈ W^2,∞(I)→ C^1,1(I). As polynomials of degree 3 have 4 degrees of freedom, anyq_σ ∈Q^ad_σ is uniquely determined by the values ofq_σ(x_i) andq_σ(x_i). As Q^ad_σ ⊂H¹₀(I), the function values atx₀andx_N are set to 0.

Corollary 2.1. For q_σ ∈ Q^ad_σ we have S(q_σ) ∈ H²(Ω₀) and there exists c > 0 independent of σ with S(qσ)_H2(Ω0)≤c· f◦T_qσγ_qσL2(Ω0).

Proof. Because of q_σ ∈ C^1,1(I) the matrix A_qσ is Lipschitz, and the proof follows with Theorem 1.12,

part (3).

The problem with discretized control now reads as

Minimizej(q_σ), q_σ∈Q^ad_σ . (2.3)

Now we discretize the state u. As later on all computations are carried out on the reference domain Ω₀, we just discretize the “transported” solution u^q. Let (T_h)_h>0 be a family of partitions of Ω₀ into either triangles or quadrilaterals which should also fulfill the usual regularity assumptions like shape-regularity etc. h_K shall denote the diameter of the element K and h = max_K∈Thh_K is a discretization parameter for the state. The space of discrete test functions is now defined as

V_h=

v_h ∈H¹₀(Ω₀) v_h|_K ∈ P1(K)∀K∈ Th

for triangles, and V_h=

v_h ∈H¹₀(Ω₀) v_h|_K ∈ Q1(K)∀K∈ Th

for quadrilaterals.

Here Qn(K) shall denote the set of all polynomials overK whose exponents inxand y are less than or equal ton. We are now able to formulate the fully discretized problem

MinimizeJ q_σ, u^h◦T_q⁻¹_σ

= 1

2u^h◦T_q⁻¹_σ −u_d²

L²(Ω_qσ)+α

2 q_σ²_L2(I), q_σ ∈Q^ad_σ , u^h∈V_h, (2.4) whereq_σ andu^hfulfill the following variational equality

a(q_σ) u^h, z^h

=l(q_σ) z^h

∀z^h∈V_h, (2.5)

which is just the discrete counterpart of (1.14).

Again, for eachq∈Q^adthere exists a uniqueu^h(q)∈V_hwhich solves (2.5). Therefore we can define a discrete analogue to the operatorS and the functionalj.

Definition 2.2. Let S_h : Q^ad → V_h be the operator which assigns to each q ∈ Q^ad the unique solution u^h of (2.5), and letj_h:Q^ad→R, j_h(q) =J(q, S_h(q)◦T_q⁻¹) be the reduced discrete cost functional.

Next, one is interested whether the problems (2.3) and (2.4) also have a solution, and what happens if we take the limit for (σ, h)→0. Again, most of the proofs are similar to [13] and can easily be transformed to our problem. We will just list two of the most important theorems which answer the questions from above.

Theorem 2.3. The (partially) discretized problems (2.3)and (2.4)have a solution.

Theorem 2.4. Let q¯_σn,hn,u¯^σn,hn=S_h(¯q_σn,hn)

n∈N be a sequence of optimal pairs for (2.4)with (σ_n, h_n)→ 0 for n → ∞. Then there exists q¯ ∈ Q^ad and u¯ ∈ H¹₀(Ω_q¯) with u¯ = S(¯˜ q) and a subsequence

¯

q_σnk,hnk,u¯^σnk,hnk

k∈N⊂ q¯_σn,hn,u¯^σn,hn

n∈N with

¯

q_σnk,hnk →q¯ in H²(I),

¯

u^σnk,hnk →u¯◦T_q¯ in H¹₀(Ω₀) fork→ ∞, and the pair (¯q,u)¯ is a solution of (1.3).

Remark 2.5. Theorem2.4also holds if a sequence (¯q_σ,h,u¯_σ,h)_σ,h>0of local optimal pairs for (2.4) is considered.

In that case, the limit (¯q,u) is of course just a local solution of (1.3).¯

3. A priori error estimates

Within this section we are going to prove somea priori bounds for the error between the optimal control ¯q in the continuous case and its fully discretized counterpart ¯q_σ,h. As the optimal control may not be unique, it is not guaranteed that one can find converging sequences of globally optimal discretized solutions for each ¯q.

However, we will show that for each ¯qone can find converging subsequences of at least local optimal solutions of (2.4).

Let ¯qbe a local solution of (1.3) which satisfies Assumption1.5. The optimality conditions of first and second order, respectively, are

j(¯q)(δq) = 0 ∀δq∈Q, (3.1)

j(¯q)(δq, δq)≥0 ∀δq∈Q. (3.2)

The differentiability of the reduced cost functionalj will be shown within this section.

In order to proof our main result later on, we have to make the following assumption, which is just slightly stronger than (3.2).

Assumption 3.1. Let ¯q be a local minimum of (1.3). We assume that j(¯q)(δq, δq)>0 ∀δq∈Q\ {0}. Now we are able to state our main result:

Theorem 3.2. Let q¯be a local solution of (1.3) which fulfills the Assumptions 1.5 and3.1. Then, for (σ, h) sufficiently small, there exist c₁, c₂>0, independent of σandh, with

q¯−q¯_σ,hH2(I)≤c₁·σ²+c₂·h², where(¯q_σ,h)_σ,h>0 is a converging sequence of local optimal solutions of (2.4).

In order to prove Theorem3.2, the error will be split,

q¯−q¯_σ,hH2(I)≤ q¯−q¯_σH2(I)+q¯_σ−q¯_σ,hH2(I), (3.3) where (¯q_σ)_σ>0 and (¯q_σ,h)_σ,h>0 are sequences of local optimal solutions of (2.3) and (2.4), respectively, which converge to ¯q in H²(I). The existence of such sequences will be shown in the sequel.

The first part on the right hand side of (3.3) refers to the discretization of the control, whereas the latter part refers to the discretization of the state. These two parts will be estimated separately.

In addition, from now on let ¯qbe a fixed local solution of (1.3) which fulfills Assumption1.5 and3.1.

First of all, we have to show that the operatorS is at least twice Fr´echet differentiable. This can be done by using the Implicit Function Theorem for Banach spaces. The following version can be found in [4], Theorem 2.3.

Theorem 3.3. Let F ∈C^k(X^ad×Y, Z),k≥1, whereY andZ are Banach spaces andX^ad is an open subset of Banach space X. Suppose that F(x^∗, y^∗) = 0 and thatF_y(x^∗, y^∗)is continuously invertible. Then there exist neighbourhoods Θ of x^∗ in X,Φ of y^∗ in Y and a mapg ∈C^k(Θ, Y) such that F(x, g(y)) = 0for all x∈ Θ.

Furthermore,F(x, y) = 0 for (x, y)∈Θ×Φimpliesy =g(x).

Corollary 3.4. The OperatorS is at least twice continuously Fr´echet differentiable.

Proof. We set X^ad = int Q^ad

, X = Q, Y = H¹₀(Ω₀) and Z = H⁻¹(Ω₀). Furthermore, define F : Q^ad× H¹₀(Ω₀)→ H⁻¹(Ω₀), F(q, u) =a(q)(u,·)−l(q)(·). Then F is affine in uand twice continuously differentiable with respect toq, as can be shown by a direct computation using the definitions (1.8)−(1.12). The result then

follows with Theorem3.3.

We now recall the definition of the operatorS and its derivatives, which follow by a direct calculation.

1. u=S(q)∈H¹₀(Ω₀) is the solution of

a(q)(u, z) = (f◦T_q, zγ_q) ∀z∈H¹₀(Ω₀). (3.4) 2. δu=S(q)(δq)∈H¹₀(Ω₀) is the solution of

a(q)(δu, z) = (∇f◦T_q)^T ·V_δq, zγ_q

+ (f◦T_q, zdiv(V_δq))− ∇u, A_q,δq∇z

−(u, zdiv(V_δq)) ∀z∈H¹₀(Ω₀). (3.5)

3. δτ u=S(q)(δq, τ q)∈H¹₀(Ω₀) is the solution of a(q)(δτ u, z) = V_{τ q}^T · ∇²f◦T_q·V_δq, zγ_q

− ∇u, A_{q,δq,τ q}∇z + (∇f◦T_q)^T ·V_{τ q}, zdiv(V_δq)

+ (∇f◦T_q)^T ·V_δq, zdiv(V_{τ q})

− ∇τ u, A_q,δq∇z

− ∇δu, A_{q,τ q}∇z

−(τ u, zdiv(V_δq))−(δu, zdiv(V_{τ q})) ∀z∈H¹₀(Ω₀), (3.6) whereτ u=S(q)(τ q).

Remark 3.5. The representations (3.6) as well as (3.9) show that the second derivatives are symmetric with respect to the directions.

Lemma 3.6. Forq∈Q^ad,δq∈Qit holds withq-independent constants that 1. S(q)_H1(Ω0)≤c,

2. S(q)(δq)_H1(Ω0)≤c· δq_H2(I), 3. S(q)(δq, δq)_H1(Ω0)≤c· δq²_H2(I).

Proof. (1) By the uniform coercivity ofa(q)(·,·) we have c· S(q)²_H1(Ω0)≤a(q)(S(q), S(q)) =l(q) (S(q))

≤ γq_L∞(Ω0)· f◦T_qL2(Ω0)· S(q)_H1(Ω0)≤c· S(q)_H1(Ω0).

(2) and (3) are also proven using the uniform coercivity ofa(q)(·,·) as well as the Lemmata1.6 and1.7.

Lemma 3.7. Forq, p∈Q^ad,δq∈Qit holds that 1. S(q)−S(p)_H1(Ω0)≤c· q−p_H2(I),

2. S(q)(δq)−S(p)(δq)_H1(Ω0)≤c· q−p_H2(I)· δq_H2(I),

3. S(q)(δq, δq)−S(p)(δq, δq)_H1(Ω0)≤c· q−p_H2((0,1))· δq²_H2(I). Proof. (1) Let e=S(q)−S(p). With Lemma1.10we have

c· e²_H1(Ω0)≤a(q)(e, e) =a(q)(S(q), e)−a(q)(S(p), e)

≤a(q)(S(q), e)−a(p)(S(p), e) +c· S(p)_H1(Ω0)· e_H1(Ω0)· q−p_H2(I). Due to the definition ofS(q), S(p) and Lemma1.11we get

a(q)(S(q), e)−a(p)(S(p), e) =l(q)(e)−l(p)(e)≤c· q−p_H2(I)· e_H1(Ω0). AsS(p)_H1(Ω0)is bounded, the proof for this part is finished.

(2) and (3) are proven similarly to the first part, one additionally has to make use of the Lemmata1.6,1.10,

1.11and3.6.