u = f in Ω, u = q on Γ. (1.2)

Transcription

1 ERROR ANALYSIS FOR A FINITE ELEMENT APPROXIMATION OF ELLIPTIC DIRICHLET BOUNDARY CONTROL PROBLEMS S. MAY, R. RANNACHER, AND B. VEXLER Abstract. We consider the Galerkin finite element approximation of an elliptic Dirichlet boundary control model problem governed by the Laplacian operator. The functional theoretical setting of this problem uses L 2 controls and a very weak formulation of the state equation. However, the corresponding finite element approximation uses standard continuous trial and test functions. For this approximation, we derive a priori error estimates of optimal order, which are confirmed by numerical experiments. The proofs employ duality arguments and known results from the L p error analysis for the finite element Dirichlet and Neumann projection. Key words. Dirichlet boundary control, finite elements, a priori error estimates. AMS subject classifications. 65K10, 65N30, 65N21, 49M25, 49K Introduction and statement of results. We consider the following elliptic Dirichlet boundary control problem posed on a convex polygonal domain Ω R 2 with boundary Γ = Ω : min J(u, q) := 1 2 u u d 2 L 2 (Ω) + α 2 q 2 L 2 (Γ) (1.1) with respect to {u, q}, under the constraint u = f in Ω, u = q on Γ. (1.2) Here, u d and f are sufficiently smooth prescribed functions, while α > 0 is a regularization parameter. For simplicity, we assume that at least u d, f L 2 (Ω). The natural functional analytic setting of this problem, which is also most convenient for numerical approximation, uses Q := L 2 (Γ) as control space. This prohibits the choice of the associated state space to be H 1 (Ω) as the trace operator γ : H 1 (Ω) L 2 (Γ) is not surjective. To overcome this dilemma, we use a very weak formulation of the state equation (1.2) allowing for solutions u L 2 (Ω) (see Grisvard [9], [10], and Berggren [3]): For given q L 2 (Γ) find u L 2 (Ω) such that (u, ϕ) + q, n ϕ = (f, ϕ) ϕ H 1 0 (Ω) H 2 (Ω). (1.3) Here, (, ) = (, ) L2 (Ω) is the L 2 inner product on the domain Ω and, =, L2 (Γ) that on its boundary Γ. The corresponding norms are = L2 (Ω) and = L2 (Γ), respectively. There are alternative variational formulations of Dirichlet boundary optimal control problems of the type (1.1), (1.2). For a brief survey, we refer to Kunisch/Vexler [12]. However, the formulation considered here appears to be the most attractive one from the computational point of view. The finite element discretization of this optimization problem uses a standard weak formulation of the state equation which is possible due to higher regularity of Institute of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 294, D Heidelberg, Germany. Centre for Mathematical Sciences, Technische Universität München, Bolzmannstraße 3, D Garching b. München, Germany This work has been supported by the German Research Foundation (DFG) within the Priority Programme 1253 Optimization with Partial Differential Equations. 1

2 2 S. MAY, R. RANNACHER, AND B. VEXLER the actual solution pair {û, ˆq}. For this approximation the estimate ˆq ˆq h + û û h = O(h 1 1/p ) (1.4) has been given in Casas/Raymond [5] for a problem with additional inequality constraints for the control q, which was expected to be only suboptimal for the state. The contribution of this paper consists in the improved L 2 -error estimate û û h = O(h 3/2 1/p ). (1.5) and in optimal-order error estimates with respect to weaker norms of the form ˆq ˆq h H 1/2 (Γ) = O(h3/2 1/p 1/r ), û û h H 1 (Ω) = O(h 2 1/p 1/r ). (1.6) Here, the values of p [2, p d ) and r [2, p Ω ) depend on the regularity of the data and the limited elliptic regularity on domains with corners, respectively. Further, for the associated adjoint state the error estimate ẑ ẑ h = O(h 2 1/p 1/r ) (1.7) is obtained. In view of the results of the computational experiments presented at the end of this paper, these estimates for the primal state and the control seem to be order-optimal. For maximum regularity, e.g., for smooth data on a rectangular domain, almost the order O(h 2 ) of convergence is obtained which is best possible for linear or bilinear finite elements. The proofs employ duality arguments based on the KKT system (Karush-Kuhn-Tucker system) associated to the optimization problem (1.1), (1.2) and uses various results from the finite element error analysis in L p for p 2. The contents of this paper is organized as follows. Section 2 contains the variational formulation of the Dirichlet boundary control problem and its Galerkin finite element approximation. This includes the derivation of the continuous and discrete KKT systems representing the first order necessary optimality conditions which form the basis of the error analysis. In Section 3 several auxiliary results on elliptic regularity and finite element approximation are provided. These are used in Section 4 to prove some suboptimal-order error estimates, followed by the final optimal-order ones in Section 5. The last Section 6 contains the results of some test calculations made to check the theoretical predictions. 2. The Dirichlet boundary control problem The state equation in very weak form. For later use, we provide some notation and results from the theory of Sobolev spaces and elliptic boundary value problems. The standard Sobolev spaces on Ω and Γ will be denoted by H s (Ω), H0(Ω) s, W s,p (Ω), and H s (Γ), respectively, for s R + and 1 p (see Adams/Fournier [1]). For functions v H 1 (Ω) the strong traces v Γ L 2 (Γ) exist and form the natural trace space H 1/2 (Γ) equipped with the generic norm v H 1/2 (Γ) := inf { ψ H1 (Ω), ψ H 1 (Ω), ψ Γ = v }. Accordingly, we have the following trace inequality for functions v H 1 (Ω) : v H 1/2 (Γ) c v H1 (Ω). (2.1)

3 FINITE ELEMENTS FOR ELLIPTIC BOUNDARY CONTROL 3 For right hand side f L 2 (Ω) and boundary function q H 1/2 (Γ) the boundary value problem (1.2) has a standard weak solution u H 1 (Ω), which is determined by u Γ = q and ( u, ϕ) = (f, ϕ) ϕ H 1 0 (Ω). (2.2) For q = 0, this weak solution is in H 2 (Ω) as Ω is a convex polygonal domain. Furthermore, it obeys the a priori bound u H 2 (Ω) c f. (2.3) Let H 1/2 (Γ) denote the dual space of H 1/2 (Γ) equipped with the natural norm v H 1/2 (Γ) := sup χ H 1/2 (Γ) v, χ χ H 1/2 (Γ) For functions v H 2 (Ω) the gradient v has a trace v Γ H 1/2 (Γ) 2. On a domain with smooth boundary Γ the outer normal unit vector n is continuous and, thus, the normal derivative n v = n v H 1/2 (Γ) is well defined for functions v H 2 (Ω) and satisfies n v H 1/2 (Γ) c v H2 (Ω), v H 2 (Ω). (2.4) This estimate does not make sense if Γ is polygonal, i.e. only Lipschitz continuous. But for v H 2 (Ω), we still have n v Γi H 1/2 (Γ i ) on each of the straight components Γ i, i = 1,..., m, of Γ. Accordingly, we introduce the space H 1/2 (Γ) := {q L 2 (Γ), q H 1/2 (Γ i ), i = 1,..., m}. Further, on L 2 (Γ), we define the dual norm v H 1/2 (Γ) := sup ψ H 1 0 (Ω) H2 (Ω) v, n ψ ψ H2 (Ω). sup χ H 1/2 (Γ) v, χ χ H1/2 (Γ) and denote by H 1/2 (Γ) the completion of L 2 (Γ) with respect to this norm. In the case of a smooth boundary Γ, the mapping n : H0 1 (Ω) H 2 (Ω) H 1/2 (Γ) is onto and we therefore have H 1/2 (Γ) = H 1/2 (Γ). The following lemma states the well-posedness of the general boundary value problem (1.3) in the very weak form. As a special case it also guarantees the existence of the very weak harmonic extension of general boundary data q H 1/2 (Γ). Lemma 2.1. For any given q H 1/2 (Γ) the state equation in its very weak form (1.3) possesses a unique solution u = u(q) L 2 (Ω). There holds the a priori estimate u c q H 1/2 (Γ) + c f H 2 (Ω), (2.5) where H 2 (Ω) denotes the dual space of H 1 0 (Ω) H 2 (Ω). Proof. First, suppose that q H 1/2 (Γ) and f H 1 (Ω). Then, there exists a unique weak solution u = u(q) H 1 (Ω) of the boundary value problem (2.2). By integration by parts, we find that this solution fulfills (u, ϕ) + q, n ϕ = (f, ϕ) ϕ H 1 0 (Ω) H 2 (Ω). To prove the a priori estimate, we use a duality argument. Let w H 1 0 (Ω) be the solution of the auxiliary problem w = u in Ω, w Γ = 0.

4 4 S. MAY, R. RANNACHER, AND B. VEXLER By elliptic regularity, we have w H 2 (Ω) and conclude u 2 = (u, w) = (f, w) q, n w. Hence, using the dual norms defined above gives us u 2 c { f H 2 (Ω) + q H 1/2 (Γ)} w H 2 (Ω). Since w H 2 (Ω) c u, the a priori estimate (2.5) follows. Now, since the subspaces H 1 (Ω) H 2 (Ω) and H 1/2 (Γ) H 1/2 (Γ) are dense, the existence of a solution to the very weak variational problem for given data f H 2 (Ω) and q H 1/2 (Γ) follows by a standard continuation argument. The a priori bound (2.5) carries over to these solutions by continuity and therefore also implies uniqueness. We will call a function v L 2 (Ω) very weakly harmonic if it satisfies (v, ϕ) q, n ϕ = 0 ϕ H 1 0 (Ω) H 2 (Ω), (2.6) with some function q H 1/2 (Γ). Then, the function q is the very weak trace of v on Γ. For this, almost by definition, we have the following trace estimate: q H 1/2 (Γ) = sup ψ H 1 0 (Ω) H2 (Ω) (v, ψ) c v. (2.7) ψ H 2 (Ω) We also call Bq := v the harmonic extension of the boundary data q H 1/2 (Γ) to Ω. For q H 1/2 (Γ), there holds Bq H 1 (Ω) and ( Bq, ϕ) = 0 ϕ H 1 0 (Ω), Bq Γ = q. (2.8) In the course of the further analysis, we will frequently use certain Sobolev trace inequalities and a priori bounds for the harmonic extension. From Grisvard [9, Theorem f.] and the literature cited therein, we have the following result. Lemma 2.2. Let Ω R d be a bounded domain with Lipschitz boundary Γ. For 1 < p < and s < 1 + 1/p, such that s 1/p is not an integer, the trace operator is continuously defined from W s,p (Ω) onto W s 1/p,p (Γ), with a right continuous inverse, and there holds v W s 1/p,p (Γ) c v W s,p (Ω). (2.9) Particularly, the harmonic extension is continuously defined from W s 1/p,p (Γ) into W s,p (Ω) and satisfies Bq W s,p (Ω) c q W s 1/p,p (Γ). (2.10) We note that in [9] the result of Lemma 2.2 is stated for 1/p < s < 1 + 1/p, and only the existence of a linear extension operator satisfying (2.10) is claimed. Here, we assume that the estimate (2.9) holds for the slightly larger parameter range 0 < s < 1 + 1/p and that the estimate (2.10) actually holds for the harmonic extension. The following lemmas collect some further properties of the trace operator and the harmonic extension which are not directly contained in Lemma 2.2. Lemma 2.3. Let Ω R 2 be a bounded convex polygonal domain with boundary Γ.

5 FINITE ELEMENTS FOR ELLIPTIC BOUNDARY CONTROL 5 (i) For any fixed p > 2, the trace operator is continuously defined from W 1/2,p (Ω) into L p (Γ) and there holds v Lp (Γ) c v W 1/2,p (Ω). (2.11) (ii) For v H0 1 (Ω) W 2,p (Ω), 2 < p < p Ω, with p Ω defined in (2.32) below, the normal derivative n v exists in W 1 1/p,p (Γ) and there holds n v W 1 1/p,p (Γ) c v W 2,p (Ω). (2.12) In the exceptional case p = 2, the normal derivative n v exists in H s (Γ) for 0 s < 1/2 and there holds n v H s (Γ) c v H 2 (Ω). (2.13) (iii) For q H 1 1/p (Γ), 2 p <, the normal derivative n Bq H 1/p (Γ) exists and there holds n Bq H 1/p (Γ) c q H 1 1/p (Γ). (2.14) In all cases the constants c may depend on p or s, respectively. Proof. (i) For the estimate (2.11), we refer to Scott [17]. (ii) The estimate (2.12) can be derived by using the estimate (2.11) for the partial derivatives i v and observing that n v(x) 0 as x approaches a corner point of Γ. For details see Jakovlev [11] where also the estimate (2.13) can be found. (iii) For q H 1 1/p (Γ) the results of Lemma 2.2, for s = 3/2 1/p, imply that Bq H 3/2 1/p (Ω) and thus Bq H 1/2 1/p (Ω) 2. Then, again using Lemma 2.2, we obtain n Bq H 1/p (Γ) and the asserted estimate (2.14). This completes the proof. Lemma 2.4. For functions v W 1,p (Ω), 1 < p <, there holds v L p (Γ) ε v L p (Ω) + cp 1 p 1 ε 1 p 1 v L p (Ω), 0 < ε 1. (2.15) Proof. For completeness, we supply the simple proof (see also Grisvard [9, Theorem ]. Let p = p(p 1) 1. By the usual trace inequality for W 1,1 functions, there holds v p L p (Γ) c vp L 1 (Ω) + c v p L 1 (Ω). Then, using Hölder s and Young s inequality gives us v p L p (Γ) c v p L p (Ω) + cp v L p (Ω) v p 1 L p (Ω) = c v p L p (Ω) + ( )( ε v Lp (Ω) cε 1 p v p 1 ) L p (Ω) This implies the asserted estimate. c v p L p (Ω) + εp v p L p (Ω) + cp p p ε p v p L p (Ω).

6 6 S. MAY, R. RANNACHER, AND B. VEXLER 2.2. The optimization problem. For deriving existence results for the optimization problem considered, we define the solution operator S : L 2 (Γ) L 2 (Ω) by Sq = u(q) = u and (u, ϕ) + q, n ϕ = (f, ϕ) ϕ H 1 0 (Ω) H 2 (Ω). This operator is affine linear and continuous due to the a priori estimate (2.5). Then, the optimization problem can be rephrased in the following compact form: j(q) := J(Sq, q) min for q L 2 (Γ). (2.16) Lemma 2.5. The optimization problem (1.1) together with the very weak formulation (1.3) of the state equation possesses a uniquely determined solution {û, ˆq} L 2 (Ω) L 2 (Γ). This solution satisfies the necessary and in this case also the sufficient optimality condition j (ˆq), χ = 0 χ L 2 (Γ), (2.17) with the derivative j (ˆq) : L 2 (Γ) L 2 (Γ) L 2 (Γ). Proof. (i) The proof of existence is by the direct method of variational calculus. The reduced functional j( ) is bounded from below on L 2 (Γ) and strictly convex since the solution operator S is affine linear. Therefore, it is weakly lower semicontinuous in L 2 (Γ). Hence, there exists a minimizing sequence (q k ) k N, inf q L2 (Γ) j(q) = lim k j(q k ), which is bounded in L 2 (Γ) due to the coercivity of j( ) on L 2 (Γ). For any of its weak accumulation points ˆq there holds j(ˆq) lim k j(q k ). Such an accumulation point is a unique global minimum of the reduced functional. (ii) To prove the necessary optimality condition, we note that 1( ) j(ˆq + εχ) j(ˆq) 0, ε for any χ L 2 (Γ). Then, letting ε 0 yields j (ˆq), χ 0 for all χ L 2 (Γ), which implies (2.17). As the reduced cost functional is strictly convex, the condition (2.17) is not only necessary but also sufficient. Lemma 2.6. The directional derivative of j( ) at some point q L 2 (Γ) is given by j (q)(χ) = α q, χ n z, χ (2.18) for χ L 2 (Γ), where z = z(q) H 1 0 (Ω) H 2 (Ω) is the solution of the associated adjoint problem (ψ, z) = (Sq u d, ψ) ψ L 2 (Ω). (2.19) Proof. We introduce the Lagrangian functional L: L 2 (Ω) L 2 (Γ) {H 1 0 (Ω) H 2 (Ω)} R by L(u, q, z) := J(u, q) + (f, z) + (u, z) q, n z. Then, with u(q) = Sq, there holds j(q) = J(u(q), q) = L(u(q), q, z(q)), and for χ L 2 (Γ) : j (q)(χ) = L u(u(q), q, z(q))(u q(q)(χ)) + L q(u(q), q, z(q))(χ) + L z(u(q), q, z(q))(z q(q)(χ)).

7 FINITE ELEMENTS FOR ELLIPTIC BOUNDARY CONTROL 7 Since by construction L u(u(q), q, z(q))( ) = 0 and L z(u(q), q, z(q))( ) = 0, we obtain j (q)(χ) = L q(u(q), q, z(q))(χ) = α χ, q χ, n z, which proves the asserted representation. As a consequence of the foregoing lemmas, we have the following result. Lemma 2.7. The solution of the optimization problem (1.1), (1.3) is characterized by the Euler-Lagrange principle stating that the pair {û, ˆq} L 2 (Ω) L 2 (Γ) is a solution if and only if there exists an adjoint state ẑ H 1 0 (Ω) H 2 (Ω), such that the triplet {û, ˆq, ẑ} solves the optimality system (Karush-Kuhn-Tucker system) (û, ϕ) + ˆq, n ϕ = (f, ϕ) ϕ H 1 0 (Ω) H 2 (Ω), (2.20) α ˆq, χ n ẑ, χ = 0 χ L 2 (Γ), (2.21) (ψ, ẑ) (û, ψ) = (u d, ψ) ψ L 2 (Ω). (2.22) Proof. Let {û, ˆq} L 2 (Ω) L 2 (Γ) be a solution of the optimization problem. Then, by definition there hold (2.20) and (2.22). The necessary condition (2.17) and the representation (2.18) of j ( ) imply (2.21). In turn, for each solution {û, ˆq, ẑ} L 2 (Ω) L 2 (Γ) {H 1 0 (Ω) H 2 (Ω)} of the KKT system, the necessary (and sufficient) optimality condition (2.17) is satisfied implying that ˆq is a minimum Galerkin finite element approximation. For the approximation of the optimization problem (1.1), (1.3), we consider a finite element method. Let V h H 1 (Ω) be finite element subspaces defined on meshes T h = {K} consisting of triangles or quadrilaterals and satisfying the usual conditions of shape and size regularity. These meshes are characterized by the mesh-width parameter h (0, 1] where h max{diam(k), K T h }. Further, we set V h,0 := V h H0 1 (Ω) and let Vh be the trace space corresponding to V h. For simplicity, we only consider lowest-order finite elements, i.e., piecewise linear or bilinear trial and test functions. The discrete solutions {û h, ˆq h } V h Vh are determined by the discrete optimization problems under the constraints J(û h, ˆq h ) = min J(u h, q h ), (2.23) u h V h,q h Vh ( u h, ϕ h ) = (f, ϕ h ) ϕ h V h,0, u h Γ = q h. (2.24) By analogous arguments as used for the continuous optimization problem (1.1), (1.3), we see that their discrete counterparts (2.23), (2.24) possess uniquely determined solutions. These solutions may be computed by applying the gradient or the Newton method for the discrete reduced functional j h (q h ) := J(S h q h, q h ), which requires the evaluation of first and second directional derivatives of j h ( ). Here, S h : Vh V h denotes the discrete solution operator defined by S h q h = u h (q h ) = u h and (2.24). This equation may be rewritten in the form ( v h, ϕ h ) + ( B h q h, ϕ h ) = (f, ϕ h ) ϕ h V h,0, (2.25) for the function v h := u h B h q h V h,0, where B h : V h V h is an arbitrary extension operator of discrete boundary data to all of Ω.

8 8 S. MAY, R. RANNACHER, AND B. VEXLER Lemma 2.8. With the foregoing notation the first directional derivative of j h ( ) at some point q h Vh is given by j h(q h )(χ h ) = α χ h, q h + (B h χ h, u h u d ) ( B h χ h, z h ), (2.26) for χ h V h, where u h = S h q h and z h = z h (q h ) V h,0 is the solution of the discrete adjoint problem ( ψ h, z h ) = (u h u d, ψ h ) ψ h V h,0. (2.27) Proof. The argument is analogous to that used in Lemma 2.6 on the continuous level. For the discrete Lagrangian functional L h : V h,0 V h V h,0 R, defined by L h (v h, q h, z h ) := J(v h + B h q h, q h ) + (f, z h ) ( v h, z h ) ( B h q h, z h ), we have j h (q h ) = J(v h + B h q h, q h ) = L h (v h, q h, z h (q h )). Further, j h(q h )(χ h ) = L h,v(v h, q h, z h (q h ))(v h,q(χ h )) + L h,q(v h, q h, z h (q h ))(χ h ) + L h,z(v h, q h, z h (q h ))(z h,q(q h )(χ h )) = L h,q(v h, q h, z h (q h ))(χ h ) = (v h + B h q h u d, B h χ h ) + α χ h, q h ( B h χ h, z h ) = (u h u d, B h χ h ) + α χ h, q h ( B h χ h, z h ), which proves the asserted representation. As a consequence of the foregoing lemma, we obtain the following result which is analogous to the corresponding one on the continuous level, Lemma 2.7: Lemma 2.9. The solution of the discrete optimization problem (2.23), (2.24) is characterized by the Euler-Lagrange principle stating that the pair {û h, ˆq h } V h V h is a solution if and only if there exists an adjoint state ẑ h V h,0, such that the triplet {û h, ˆq h, ẑ h } solves the discrete KKT system û h Γ = ˆq h, ( û h, ϕ h ) = (f, ϕ h ) ϕ h V h,0, (2.28) α ˆq h, χ h + (û h, B h χ h ) ( B h χ h, ẑ h ) = (u d, B h χ h ) χ h V h, (2.29) ( ψ h, ẑ h ) (û h, ψ h ) = (u d, ψ h ) ψ h V h,0. (2.30) The solution {û h, ˆq h, ẑ h } is independent of the particular choice of the extension operator B h. The numerical results for this approximation presented in Section 6 suggest the following partially optimal rates of convergence under generic assumptions on the regularity of the solution: h 1 1/r ˆq ˆq h + h 1/2 1/r û û h + ẑ ẑ h = O(h 2 1/p 1/r ). (2.31) Here, 2 r < p Ω and 2 p < p d p, p := min{p d, p Ω } are essentially determined by the regularity ẑ H0 1 (Ω) W 2,p (Ω), depending on the maximum interior angle ω max of the polygonal domain Ω like p Ω = 2ω max 2ω max π, (2.32)

9 FINITE ELEMENTS FOR ELLIPTIC BOUNDARY CONTROL 9 including the special case p Ω = for ω = π 2, and the regularity of the data u d L pd (Ω), p d > 2. These convergence rates turn out to be better for weaker error measures, e.g., for the mean values: ˆq ˆq h, 1 + (û û h, 1) = O(h 2 1/p 1/r ). (2.33) It is the main goal of the following analysis to provide theoretical support for these practically observed convergence rates. This will also cover the case of solutions with reduced regularity induced by irregular data. Remark 1. The assumption that the domain Ω is polygonal and convex is made for simplifying the arguments in the proofs. A curved boundary complicates the numerical approximation while reentrant corners reduce the solution s regularity. On a general polygonal (or piecewise smoothly bounded) domain, the optimization problem is well-posed if in the state equation (1.3) the test space D( ) := {ϕ H0 1 (Ω), ϕ L 2 (Ω)} is used The KKT systems of the optimization problems. The error analysis of the finite element approximation (2.23), (2.24) of the optimization problem (1.1), (1.3) is based on their equivalent formulation in terms of the corresponding KKT systems. We begin by recasting the KKT system (2.20), (2.21), (2.22) in a form which can be approximated by a standard finite element method using only continuous trial and test functions. This is possible, since the solution of the very weak optimality system (2.20), (2.21), (2.22) turns out to be more regular than required for its definition. For later use, we determine its degree of regularity, which is guaranteed in general on a convex polygonal domain. Actually, the regularity of the solution pair {û, ˆq} is essentially determined by that of the adjoint state ẑ. Lemma Suppose that f L 2 (Ω) and u d L pd (Ω), p d > 2. Let p Ω 2 be defined by (2.32) and p := min{p d, p Ω }. Then, the solution {û, ˆq} L 2 (Ω) L 2 (Γ) of the optimization problem (1.1), (1.3) and the associated adjoint state ẑ H 1 0 (Ω) H 2 (Ω) determined by (2.19) have the additional regularity properties {û, ˆq} H 3/2 1/p (Ω) H 1 1/p (Γ), ẑ W 2,p (Ω), 2 p < p. (2.34) Proof. By Lemma 2.7, the triplet {û, ˆq, ẑ} satisfies the equations (2.20), (2.21), (2.22). Since ˆq L 2 (Γ), according to Berggren [3], we have û H s (Ω) for 0 s < 1 2 and therefore û Lp (Ω) for some p > 2. Let this p be chosen such that p < p. This in turn implies that ẑ W 2,p (Ω) for this p > 2 (see Grisvard [9]). Hence, n ẑ W 1 1/p,p (Γ) by Lemma 2.3(ii). Then, from (2.21), we infer that ˆq W 1 1/p,p (Γ) H 1 1/p (Γ) H 1/2 (Γ). Using this in (2.20) yields û H 1 (Ω), i.e., û is the usual weak H 1 solution of the boundary value problem (1.2). By elliptic regularity theory, this implies ẑ W 2,p (Ω) for 2 p < p. Finally, in view of Lemma 2.2, by elliptic regularity theory, û Γ = ˆq W 1 1/p,p (Γ) H 1 1/p (Γ) implies that û H 3/2 1/p (Ω), which completes the proof. Remark 2. The right hand side in the equation for ẑ is û u d. Hence, under the mere assumption that u d L (Ω), in general, we cannot expect ẑ W 2, (Ω) or higher regularity, even on a rectangle. This restricts all our results to the case 2 p < with constants blowing up as p. However, in the special case ω max = π 2, we have (2.34) with p =, provided that u d C γ ( Ω) for some γ > 0 and u d (x i ) = 0 in all cornerpoints x i of Ω. This follows from Grisvard [9] due to the

10 10 S. MAY, R. RANNACHER, AND B. VEXLER fact that û(x i ) = ˆq(x i ) = α 1 n ẑ(x i ) = 0, cf. the argument in the proof of Lemma 2.3 (ii), and û C γ ( Ω) by an embedding theorem. Next, we rewrite equation (2.21) using equation (2.22) to obtain α ˆq, χ n ẑ, χ = α ˆq, χ ( ẑ, Bχ) ( ẑ, Bχ) = α ˆq, χ + (û u d, Bχ), where Bχ H 1 (Ω) is the harmonic extension of the boundary function χ H 1/2 (Γ) defined by (2.8). Hence, in view of Lemma 2.10, the solution {û, ˆq, ẑ} H 3/2 1/p (Ω) W 1 1/p,p (Γ) [H 1 0 (Ω) W 2,p (Ω)] of (2.20), (2.21), (2.22) also satisfies the following set of equations: û Γ = ˆq, ( û, ϕ) = (f, ϕ) ϕ H 1 0 (Ω), (2.35) α ˆq, χ + (û, Bχ) = (u d, Bχ) χ H 1/2 (Γ), (2.36) ( ψ, ẑ) (û, ψ) = (u d, ψ) ψ H 1 0 (Ω). (2.37) In order to remove the nonhomogeneous boundary condition, we introduce the function ˆv := û Bˆq H 1 0 (Ω). Then, the triplet {ˆv, ˆq, ẑ} H 1 0 (Ω) H 1/2 (Γ) H 1 0 (Ω) satisfies the system ( ˆv, ϕ) = (f, ϕ) ϕ H 1 0 (Ω), (2.38) α ˆq, χ + (ˆv + Bˆq, Bχ) = (u d, Bχ) χ H 1/2 (Γ), (2.39) ( ψ, ẑ) (ˆv + Bˆq, ψ) = (u d, ψ) ψ H 1 0 (Ω). (2.40) The corresponding finite element approximation {û h, ˆq h, ẑ h } V h V h V h,0 is characterized by the discrete KKT system (see Lemma 2.7) û h Γ = ˆq h, ( û h, ϕ h ) = (f, ϕ h ) ϕ h V h,0, (2.41) α ˆq h, χ h + (û h, B h χ h ) ( ẑ h, B h χ h ) = (u d, B h χ h ) χ h V h, (2.42) ( ψ h, ẑ h ) (û h, ψ h ) = (u d, ψ h ) ψ h V h,0, (2.43) or incorporating the nonhomogeneous boundary condition for û h into the variational formulation: ( ˆv h, ϕ h ) + ( B hˆq h, ϕ h ) = (f, ϕ h ) ϕ h V h,0, (2.44) α ˆq h, χ h + (ˆv h + B hˆq h, B h χ h ) ( ẑ h, B h χ h ) = (u d, B h χ h ) χ h V h, (2.45) ( ψ h, ẑ h ) (ˆv h + B hˆq h, ψ h ) = (u d, ψ h ) ψ h V h,0, (2.46) where ˆv h := û h B hˆq h V h,0. The solution of this system is independent of the particular choice of the extension operator B h : Vh V h. From now on, we choose B h to be the discrete harmonic extension defined by ( B h q h, ϕ h ) = 0 ϕ h V h,0, B h q h Γ = q h. (2.47) Then, the system (2.44), (2.45), (2.46) reduces to ( ˆv h, ϕ h ) = (f, ϕ h ) ϕ h V h,0, (2.48) α ˆq h, χ h + (ˆv h + B hˆq h, B h χ h ) = (u d, B h χ h ) χ h V h, (2.49) ( ψ h, ẑ h ) (ˆv h + B hˆq h, ψ h ) = (u d, ψ h ) ψ h V h,0. (2.50)

11 FINITE ELEMENTS FOR ELLIPTIC BOUNDARY CONTROL 11 Writing the equations for {ˆv, ˆq, ẑ} for discrete test functions and subtracting the corresponding discrete equations (2.48), (2.49), (2.50) yields the following equations for the errors e v := ˆv ˆv h, e q := ˆq ˆq h, and e z := ẑ ẑ h : ( e v, ϕ h ) = 0 ϕ h V h,0, (2.51) α e q, χ h + (ˆv + Bˆq u d, Bχ h ) (ˆv h + B hˆq h u d, B h χ h ) = 0 χ h V h, (2.52) ( e z, ψ h ) (e v + Bˆq B hˆq h, ψ h ) = 0 ψ h V h,0. (2.53) Remark 3. Notice that, since B h B, the system (2.48), (2.49), (2.50) is not the Galerkin approximation of (2.38), (2.39), (2.40). In this situation the general paradigm that Galerkin discretization and optimization (i.e. forming the necessary optimality condition) commute does not hold. This essentially complicates the error analysis as several additional terms need to estimated, which originate from the lacking Galerkin orthogonality of the approximation. Remark 4. The choice of B : H 1/2 (Γ) H 1 (Ω) and B h : Vh V h as the harmonic, respectively, discrete harmonic extension operators is for convenience of the argument used in the following error analysis. Actually, the continuous as well as the discrete optimal solutions are independent of this particular choice. For practical computations B h is usually chosen to satisfy B h q h (a i ) = 0 in all interior nodal points a i. 3. Auxiliary estimates Auxiliary error and stability estimates. Next, we collect some known results on the approximation behavior of finite element methods. (I) We will use the following inverse estimate for finite element functions χ h V h (notice that Γ is one-dimensional): χ h H r (Γ) ch s r χ h H s (Γ), (3.1) for 0 s r 1. This can be proven by combining estimates in Ciarlet [7] and Brenner/Scott [4] with standard results from operator interpolation theory. (II) Let I h : C( Ω) V h and I h : C(Γ) Vh denote the natural nodal interpolation operators which satisfy (see Ciarlet [7] and Brenner/Scott [4]) (v I h v) Lp (Ω) ch 2 v Lp (Ω), (3.2) for v W 2,p (Ω), 1 p. The interpolation operator I h is local, but only defined for continuous functions. A locally defined quasi-interpolation operator Ĩh : L 2 (Ω) V h has been devised by Scott/Zhang [18]. This operator preserves (polynomial) boundary conditions and satisfies Ĩh Vh = id. For this, there holds the same estimate (3.2) as for the standard nodal interpolation I h and in addition the estimates for v H s (Ω). v Ĩhv + h (v Ĩhv) ch s v H s (Ω), 1 s 2, (3.3) (III) The L 2 projection P h : L2 (Γ) V h is defined by q P h q, χ h = 0 χ h V h.

12 12 S. MAY, R. RANNACHER, AND B. VEXLER By standard results for finite elements there holds the error estimate (see Ciarlet [7], Brenner/Scott [4], and Casas/Raymond [6]) q P h q L p (Γ) + h s P h q W s,p (Γ) ch s q W s,p (Γ), 0 s 1, 1 < p <, (3.4) for q W s,p (Γ). (IV) For a function u H0 1 (Ω) let Rh Du V h,0 projection ( Dirichlet projection ) defined by denote the corresponding Ritz ( (u R D h u), ϕ h ) = 0 ϕ h V h,0. We recall the following standard results from the literature Lemma 3.1. On a convex polygonal domain, the Ritz projection Rh D satisfies the stability estimate V h,0 : H1 0 (Ω) R D h u Lp (Ω) c u Lp (Ω), 1 < p, (3.5) for u W 1,p (Ω). Furthermore, there holds the error estimate u R D h u Lp (Ω) + h u R D h u W 1,p (Ω) ch 2 u W 2,p (Ω), 2 p < p Ω. (3.6) Proof. The proofs can be extracted from the results in Rannacher/Scott [15]. (V) Finally, we introduce the Neumann projection Rh Nu V h H 1 (Ω) defined by of a function u ( (u R N h u), ϕ h ) + (u R N h u, ϕ h ) = 0 ϕ h V h. For this approximation, we recall the following results from the literature. Lemma 3.2. On a convex polygonal domain, the Neumann projection Rh N satisfies the error estimate u R N h u L p (Ω) + h 1/p u R N h u L p (Γ) ch 2 u W 2,p (Ω), (3.7) for q Ω < p < p Ω, where q Ω = p Ω (p Ω 1) 1. Proof. The bound for the first term in (3.7) can be obtained by combining (in a nontrivial way) results from Rannacher/Scott [15] and Scott [16], which hold for q Ω < p < p Ω. One of these results is the estimate (u R N h u) L p (Ω) ch u W 2,p (Ω). Then, the bound for the second term in (3.7) follows by using the relation (2.15) with ε := h 1 1/p, u Rh N u L p (Γ) cε (u Rh N u) L p (Ω) + cε 1/(p 1) u Rh N u L p (Ω) ch 1 1/p (u Rh N u) Lp (Ω) + ch 1/p u Rh N u Lp (Ω) ch 2 1/p u W 2,p (Ω). This completes the proof.

13 FINITE ELEMENTS FOR ELLIPTIC BOUNDARY CONTROL Properties of discrete harmonic extension. In this section, we provide some a priori bounds and error estimates for the discrete harmonic extensions. Lemma 3.3. For the discrete harmonic extension B h q h V h of the boundary data q h Vh there hold the a priori estimates B h q h c q h H (Γ), 1/2 (3.8) B h q h c q h, (3.9) B h q h L p (Ω) ch 1/p 1 q h L p (Γ), 1 < p <. (3.10) Proof. (i) The poof is by referring back to the corresponding estimates for B. For the modified interpolation operator Ĩh in (3.3) there holds (B h ĨhB)q h Γ = 0. Hence by the properties of B h and the estimate (3.3) for s = 1, we have B h q h 2 = ( B h q h, (B h ĨhB)q h ) + ( B h q h, ĨhBq h ) Then, the stability estimate (2.10) yields (3.8), B h q h ĨhBq h c B h q h Bq h H1 (Ω). B h q h c Bq h H1 (Ω) c q h H 1/2 (Γ). (ii) Next, let w H 1 0 (Ω) H 2 (Ω) be the solution of the auxiliary problem w = B h q h in Ω, w Γ = 0. Then, using several of the foregoing estimates in a standard way, we obtain B h q h 2 = (B h q h, w) = ( B h q h, w) q h, n w = ( B h q h, (w R D h w)) q h, n w B h q h (w R D h w) + q h w H2 (Ω) c { h q h H 1/2 (Γ) + q h } w H 2 (Ω) c q h B h q h. This yields (3.9). (iii) To prove (3.10), we recall the estimate v Lp (Ω) c sup w W 1,q 0 (Ω) ( v, w), (3.11) w Lq (Ω) which holds for 1 < p < and q = p(p 1) 1, particularly on convex polygonal domains. This follows from a result in Alkhutov/Kondratev [2], which states that the boundary value problem u = f in Ω, u Γ = 0, possesses for f W 1,p (Ω) a uniquely determined solution u W 1,p 0 (Ω) satisfying u W 1,p (Ω) c f W 1,p (Ω).

14 14 S. MAY, R. RANNACHER, AND B. VEXLER For given q h Vh let B h q h V h denote that extension which coincides with q h at each nodal point a i Γ, but vanishes at each interior nodal point a i Ω. Then, for v h := B h q h B h q h V h,0 there holds ( v h, ϕ h ) = ( B h q h, ϕ h ) ( B h q h, ϕ h ) = ( B h q h, ϕ h ), ϕ h V h,0. Now, let v H 1 0 (Ω) be defined by the equation ( v, ϕ) = ( B h q h, ϕ) ϕ H 1 0 (Ω). Then, v h can be viewed as the Ritz projection of v. The estimate (3.11) implies that v L p (Ω) c B h q h L p (Ω). Then, by Lemma 3.1, we obtain the estimate This implies that v h Lp (Ω) c B h q h Lp (Ω), 1 < p <. B h q h L p (Ω) v h L p (Ω) + B h q h L p (Ω) c B h q h L p (Ω). (3.12) Therefore it remains to estimate the norm B h q h L p (Ω), which is localized to a strip S h along the boundary of width h. For this purpose let T T h be a cell of the decomposition of Ω, which nontrivially intersects the boundary: ΓT := T Ω. Then, by a standard argument employing transformations to a reference unit cell, there holds B h q h p L p (T ) ch1 p q h p L p (Γ T ), and summing this over all such cells belonging to the strip S h, B h q h L p (S h ) ch 1/p 1 q h L p (Γ). This proves the asserted estimate. Lemma 3.4. For the harmonic extensions B : H 1 (Γ) H 3/2 (Ω) and B h : Vh there hold the error estimates V h (B B h )q h ch 1/2 1/p q h H (Γ), 1 1/p (3.13) (ψ, (B B h )q h ) ch 2 1/p 1/r q h H (Γ) ψ 1 1/p Lr (Ω), (3.14) for q h Vh, ψ Lr (Ω), 2 r p < p Ω. Particularly, there holds (B B h )q h ch 3/2 1/p q h H (Γ). 1 1/p (3.15) Proof. Notice that B h q h V h is just the Ritz projection of Bq h corresponding to the same boundary values q h V h, ( (B B h )q h, ϕ h ) = 0 ϕ h V h,0, (B B h )q h Γ = 0. (i) For the modified interpolation operator Ĩh B h )q h Γ = 0. Hence, defined in (3.3) there holds (ĨhB (B B h )q h 2 = ( (B B h )q h, (B ĨhB)q h ) (B B h )q h (B ĨhB)q h.

15 FINITE ELEMENTS FOR ELLIPTIC BOUNDARY CONTROL 15 The interpolation estimate (3.3) together with the stability estimate (2.10) yields which proves the estimate (3.13). (B ĨhB)q h ch 1/2 1/p Bq h H 3/2 1/p (Ω) ch 1/2 1/p q h H 1 1/p (Γ), (ii) For an arbitrary but fixed ψ L r (Ω) let w H 1 0 (Ω) be the solution of the auxiliary problem w = ψ in Ω, w Γ = 0 satisfying w W 2,r (Ω), for 2 r < p Ω, and w W 2,r (Ω) c ψ L r (Ω) (see Grisvard [9]). Then, with the Neumann projection Rh N : H1 (Ω) V h and observing (B B h )q h Γ = 0, (ψ, (B B h )q h ) = ( w, (B B h )q h ) = ( w, (B B h )q h ) = ( (w R N h w), (B B h )q h ) + ( R N h w, (B B h )q h ) =: Λ 1 + Λ 2. The two terms Λ 1 and Λ 2 are estimated separately using the properties of B and Rh N : Λ 1 = ( (w R N h w), (B B h )q h ) = ( (w R N h w), Bq h ) + (w R N h w, B h q h ) = w R N h w, n Bq h (w R N h w, Bq h ) + (w R N h w, B h q h ) = R N h w, n Bq h + (w R N h w, B h q h ). The first term on the right is estimated using the inverse estimate (3.1), the a priori estimate (2.14), and the error estimate (3.7) as follows: Rh N w, n Bq h Rh N w H (Γ) 1/p n Bq h H 1/p (Γ) ch 1/p Rh N w q h H 1 1/p (Γ) ch 1/p w Rh N w Lr (Γ) q h H 1 1/p (Γ) ch 2 1/r 1/p w W 2,r (Ω) q h H 1 1/p (Γ) ch 2 1/r 1/p ψ L r (Ω) q h H (Γ). 1 1/p For the second term, by similar but simpler arguments we have (w Rh N w, B h q h ) w Rh N w B h q h ch 2 ψ Lr (Ω) q h H (Γ). 1 1/p Thus, Λ 1 ch 2 1/r 1/p ψ L r (Ω) q h H (Γ). 1 1/p Next, we denote by a i the nodal points of the mesh T h and by ϕ i h the corresponding nodal basis functions satisfying ϕ i h L 2 (supp(ϕ i h )) c. Then, using the properties

16 16 S. MAY, R. RANNACHER, AND B. VEXLER of the harmonic extensions, we estimate as follows: Λ 2 = ( R N h w, (B B h )q h ) = a i Ω R N h w(a i )( ϕ i h, (B B h )q h ) = a i Γ R N h w(a i )( ϕ i h, (B B h )q h ) Rh N w(a i ) ϕ i h L2 (supp(ϕ i h )) (B B h )q h L2 (supp(ϕ i h )) a i Γ ch 1/2( h Rh N w(a i ) 2) 1/2( ) 1/2 (B B h )q h 2 L 2 (supp(ϕ i h )) a i Γ a i Γ ch 1/2 Rh N w w (B B h )q h. Then, by the estimates (3.7) of Lemma 3.2 and the already proved estimate (3.13), Λ 2 ch 1/2 h 2 1/r w W 2,r (Ω)h 1/2 1/p q h H 1 1/p (Γ) ch 2 1/r 1/p ψ Lr (Ω) q h H (Γ). 1 1/p Combining the estimates for Λ 1 and Λ 2, we obtain (ψ, (B B h )q h ) ch 2 1/p 1/r ψ L r (Ω) q h H 1 1/p (Γ), which proves the estimate (3.14). (iii) The L 2 -norm estimate (3.15) follows from (3.14) by setting ψ := (B B h )q h and r = 2. This completes the proof. 4. Basic error estimates. In the following, we will use the quantity Σ p := ˆq H 1 1/p (Γ) + f + ẑ W 2,p (Ω), which is bounded for 2 p < p, according to Lemma 2.10, with p := min{p d, p Ω }. The constants c α appearing below may blow up for p p. We begin with the estimate of the reduced state error. Lemma 4.1. For the reduced state error e v = ˆv ˆv h there holds e v ch 2 f. (4.1) Proof. The function ˆv = û Bˆq H 1 0 (Ω) is the solution of the boundary value problem ˆv = f in Ω, ˆv Γ = 0, and ˆv h V h,0 denotes its Ritz projection satisfying ( e v, ϕ h ) = 0, ϕ h V h,0. Since f L 2 (Ω), we have ˆv H 2 (Ω) and the estimate (3.6) of Lemma 3.1 yields the asserted error estimate. As starting point for the estimate of the control, we recall the perturbed Galerkin orthogonality equation (2.52), α e q, χ h + (û u d, Bχ h ) (û h u d, B h χ h ) = 0, χ h V h, (4.2)

17 FINITE ELEMENTS FOR ELLIPTIC BOUNDARY CONTROL 17 where û = ˆv + Bˆq and û h = ˆv h + B hˆq h. This can be rearranged into the form α e q, χ h = (Bˆq B hˆq h, B h χ h ) (ˆv ˆv h, B h χ h ) (û u d, (B B h )χ h ). (4.3) Theorem 4.2. For the control error e q := ˆq ˆq h and the state error e u := û û h there holds the estimate for 2 p < p, where c α 1 + α 1. Proof. (i) We begin with the relation Setting χ h := P h e q = P h ˆq ˆq h e q + e u c α h 1 1/p Σ p, (4.4) α e q 2 = α e q, ˆq P h ˆq + α e q, P h e q. in (4.3) and rearranging terms, we obtain α e q, P h e q = (Bˆq B hˆq h, B h P h e q ) (ˆv ˆv h, B h P h e q ) (û u d, (B B h )P h e q ) = (Bˆq B hˆq h, (B B h )P h ˆq) (Bˆq B hˆq h, B(P h ˆq ˆq)) (Bˆq B hˆq h, Bˆq B hˆq h ) (ˆv ˆv h, B h P h e q ) (û u d, (B B h )P h e q ). Combining this with the first equation results in α e q 2 + Bˆq B hˆq h 2 = α e q, ˆq P h ˆq + (Bˆq B hˆq h, (B B h )P h ˆq) (Bˆq B hˆq h, B(P h ˆq ˆq)) (ˆv ˆv h, B h P h e q ) (û u d, (B B h )P h e q ). The five terms on the right hand side of (4.5) will be treated separately. First term: By the error estimate (3.4) for P h, α e q, ˆq P h ˆq α e q ˆq P h ˆq cαh 1 1/p e q ˆq H 1 1/p (Γ) α 4 e q 2 + cαh 2 2/p ˆq 2 H 1 1/p (Γ). Second term: By the L 2 error estimate (3.15) and the stability estimate (3.4), (Bˆq B hˆq h, (B B h )P h ˆq) 1 4 Bˆq B hˆq h 2 + (B B h )P h ˆq Bˆq B hˆq h 2 + ch 3 2/p P h ˆq 2 H 1 1/p (Γ) 1 4 Bˆq B hˆq h 2 + ch 3 2/p ˆq 2 H 1 1/p (Γ). Third term: By the stability estimate (2.10) and the error estimate (3.4), (Bˆq B hˆq h, B(P h ˆq ˆq)) 1 4 Bˆq B hˆq h 2 + B(P h ˆq ˆq) Bˆq B hˆq h 2 + c P h ˆq ˆq Bˆq B hˆq h 2 + ch 2 2/p ˆq 2 H 1 1/p (Γ). (4.5) Fourth term: By the estimate (3.9) of Lemma 3.3, the L 2 stability of the projection Ph, and the estimate (4.1) of Lemma 4.1, (ˆv ˆv h, B h P h e q ) ˆv ˆv h B h P h e q c ˆv ˆv h P h e q c ˆv ˆv h e q c α ˆv ˆv h 2 + α 4 e q 2 c α h4 f 2 + α 4 e q 2.

18 18 S. MAY, R. RANNACHER, AND B. VEXLER Fifth term: We recall that ẑ H 1 0 (Ω) H 2 (Ω) satisfies ẑ = û u d in Ω. Hence, observing that (B B h )P h e q Γ = 0, with p = p(p 1) 1 2, and using the properties of the harmonic extension operators B and B h, we deduce that (û u d, (B B h )P h e q ) = ( ẑ, (B B h )P h e q ) = ( ẑ, (B B h )P h e q ) + n z, (B B h )P h e q = ( (ẑ I h ẑ), B h P h e q ). Further, using the interpolation error estimate (3.2), the a priori estimate (3.10) of Lemma 3.3, and the L 2 stability of the projection P h (û u d, (B B h )P h e q ) (ẑ I h ẑ) L p (Ω) B h P h e q L p (Ω) Collecting the foregoing estimates, we obtain ch ẑ W 2,p (Ω)h 1/p 1 P h e q = ch 1 1/p ẑ W 2,p (Ω) P h e q cα 1 h 2 2/p ẑ 2 W 2,p (Ω) + α 4 e q 2. α e q 2 + Bˆq B hˆq h α e q 2 + c(1 + α)h 2 2/p ˆq 2 H 1 1/p (Γ) Bˆq B hˆq h 2 and absorbing terms into the left hand side, + cα 1 h 4 f 2 + cα 1 h 2 2/p ẑ 2 W 2,p (Ω), α e q 2 ch 2 2/p ˆq 2 H 1 1/p (Γ) + cα 1{ h 4 f 2 + h 2 2/p ẑ 2 W 2,p (Ω)}. This proves the asserted estimate of e q. (ii) Further, observing that e u = e v + Bˆq B hˆq h and the estimate (4.1) of Lemma 4.1 for e v, we obtain the estimate of e u. Remark 5. The estimate of the control error provided by Theorem 4.2 is orderoptimal with respect to the regularity of the solution, but that for the state error is only suboptimal. The latter will be improved in the next section. Corollary 4.3. The discrete controls admit the uniform bound ˆq h H 1 1/p (Γ) c α Σ p, 2 p < p. (4.6) Proof. Using the foregoing results together with the estimate (3.4) for the L 2 projection Ph, we estimate as follows: ˆq h H 1 1/p (Γ) ˆq h Ph ˆq H 1 1/p (Γ) + Ph ˆq ˆq H 1 1/p (Γ) + ˆq H 1 1/p (Γ) ch 1+1/p ˆq h Ph ˆq + c ˆq H 1 1/p (Γ) ch 1+1/p{ e q + ˆq Ph ˆq } + c ˆq H 1 1/p (Γ) ch 1+1/p e q + c ˆq H 1 1/p (Γ) { c α ˆq H 1 1/p (Γ) + f + ẑ W 2,p (Ω)}. This implies the asserted estimate.

19 FINITE ELEMENTS FOR ELLIPTIC BOUNDARY CONTROL Improved error estimates. The orders of convergence derived in the preceding section for the state û may not be optimal as is demonstrated by the numerical results presented in Section 6, below. The key to improved error estimates for the state is the proof of higher order error estimates for the control in norms weaker than the L 2 norm. To this end, we will employ duality arguments based on the KKT system. The modified KKT system (2.38), (2.39), (2.40) can be written in compact form for the triplet ˆX := {ˆv, ˆq, ẑ} H 1 0 (Ω) H 1/2 (Γ) H0 1 (Ω) as follows: A( ˆX, Φ) = F (Φ), (5.1) for all Φ = {ϕ z, ϕ q, ϕ v } H 1 0 (Ω) H 1/2 (Γ) H 1 0 (Ω), with the bilinear form A( ˆX, Φ) := ( ˆv, ϕ v ) + α ˆq, ϕ q + (ˆv + Bˆq, Bϕ q ) + ( ẑ, ϕ z ) (ˆv + Bˆq, ϕ z ) and the right hand side F (Φ) := (f, ϕ v ) (u d, ϕ z ) + (u d, Bϕ q ). For a given linear functional J( ) let W = {w v, w q, w z } H 1 0 (Ω) H 1/2 (Γ) H 1 0 (Ω) be the solution of the dual problem A(Ψ, W ) = J(Ψ) Ψ = {ψ z, ψ q, ψ v }. (5.2) For J(Ψ) = J u (ψ v ) + J q (ψ q ) + J z (ψ z ) this is equivalent to the system ( ψ v, w v ) = J u (ψ v ) ψ v H 1 0 (Ω), (5.3) α ψ q, w q + (Bψ q, Bw q ) (Bψ q, w v ) = J q (ψ q ) ψ q H 1/2 (Γ), (5.4) ( ψ z, w z ) (ψ z, w v ) + (ψ z, Bw q ) = J z (ψ z ) ψ z H 1 0 (Ω). (5.5) In the special case J(Ψ) = J q (ψ q ), the first equation (5.3) has the unique solution w v = 0, and we obtain the triangular system α ψ q, w q + (Bψ q, Bw q ) = J q (ψ q ) ψ q H 1/2 (Γ), (5.6) ( ψ z, w z ) + (ψ z, Bw q ) = 0 ψ z H 1 0 (Ω). (5.7) By coercivity arguments it is easily seen that for J q ( ) L 2 (Γ) there is a unique solution {w q, w z } L 2 (Γ) H0 1 (Ω). Lemma 5.1. For J q (Ψ) = (Bψ q, ψ) with a fixed ψ L r (Ω), 2 r < p Ω, the solution {w q, w z } of the dual system (5.6), (5.7) has the regularity {w q, w z } H 1 1/r (Γ) W 2,r (Ω) and there holds the a priori estimate w q H 1 1/r (Γ) + w z W 2,r (Ω) cα 1 ψ Lr (Ω). (5.8) Proof. For Bw q L 2 (Ω), we infer that w z H 1 0 (Ω) H 2 (Ω). Since (5.6) also holds for ψ q L 2 (Γ), we can test with ψ q = w q, which gives us α w q + Bw q c ψ. Further, by elliptic regularity, we have w z H 2 (Ω) c ψ. Next, we employ a duality argument. Let η H 1 0 (Ω) be the weak solution of the auxiliary problem η = ψ Bw q in Ω.

20 20 S. MAY, R. RANNACHER, AND B. VEXLER Since Bw q ψ L 2 (Ω), we have η H 2 (Ω), and n η Γ H s (Γ) for 0 s < 1/2 by Lemma 2.3. Further, η H 2 (Ω) + n η H s (Γ) c ψ. Then, from α χ, w q = (Bχ, ψ Bw q ) = (Bχ, η) = ( Bχ, η) + χ, n η = χ, n η we infer that also w q = α 1 n η H s (Γ). In view of the Lemma 2.2 this implies that Bw q H s+1/2 (Ω). As s can be taken arbitrarily close to 1/2, by the Sobolev embedding theorem we conclude that Bw q L r (Ω) for r 2 as considered and Bw q Lr (Ω) c ψ. Now, this implies that η H 1 0 (Ω) W 2,r (Ω) and consequently, in virtue of Lemma 2.3, w q = α 1 n η W 1 1/r,r (Γ) H 1 1/r (Γ) and w q H 1 1/r (Γ) cα 1 ψ L r (Ω). Finally, again by elliptic regularity theory, we obtain w z H 1 0 (Ω) W 2,r (Ω) and w z W 2,r (Ω) cα 1 ψ Lr (Ω). This completes the proof. Theorem 5.2. For the control error e q := ˆq ˆq h and any ψ L r (Ω) there holds (Be q, ψ) c 2 αh 2 1/p 1/r Σ p ψ Lr (Ω), (5.9) for 2 p < p (depending on the regularity of the data and the domain) and 2 r < p Ω (depending on the regularity of the domain), where c 2 α 1 + α 2. Proof. We use the dual problem described above with ψ L p (Ω). Taking the test pair {ψ q, ψ z } = {e q, e v } in the equations (5.6) and (5.7) gives us α e q, w q + (Be q, Bw q ) = (Be q, ψ), (5.10) ( e v, w z ) + (e v, Bw q ) = 0. (5.11) Then, subtracting the L 2 projection P h wq V h in (5.10), yields (Be q, ψ) = α e q, w q P h w q + (Be q, B(w q P h w q )) + α e q, P h w q + (Be q, BP h w q ). (5.12) We recall the Galerkin orthogonality relations (2.51), (2.52), (2.53): ( e v, ϕ h ) = 0 ϕ h V h,0, (5.13) α e q, χ h + (ˆv + Bˆq u d, Bχ h ) (ˆv h + B hˆq h u d, B h χ h ) = 0 χ h V h, (5.14) ( e z, ψ h ) (e v + Bˆq B hˆq h, ψ h ) = 0 ψ h V h,0. (5.15)

21 FINITE ELEMENTS FOR ELLIPTIC BOUNDARY CONTROL 21 We also set χ h = P h wq in (5.14) and rearrange terms to obtain 0 = α e q, Ph w q + (ˆv + Bˆq u d, BPh w q ) (ˆv h + B hˆq h u d, B h Ph w q ) = α e q, Ph w q + (Be q, BPh w q ) (Be q, (B B h )Ph w q ) (Be q, B h Ph w q ) + (ˆv + Bˆq u d, BPh w q ) (ˆv h + B hˆq h u d, B h Ph w q ) = α e q, Ph w q + (Be q, BPh w q ) (Be q, (B B h )Ph w q ) (Be q, B h Ph w q ) + (ˆv + Bˆq u d, (B B h )Ph w q ) + (ˆv + Bˆq u d, B h Ph w q ) (ˆv h + B hˆq h u d, B h Ph w q ) = α e q, Ph w q + (Be q, BPh w q ) (Be q, (B B h )Ph w q ) + (e v + Bˆq B hˆq h Be q, B h Ph w q ) + (û u d, (B B h )Ph w q ) = α e q, Ph w q + (Be q, BPh w q ) (Be q, (B B h )Ph w q ) + (e v, B h Ph w q ) + ((B B h )ˆq h, B h Ph w q ) + (û u d, (B B h )Ph w q ). Combining this with (5.12) gives us (Be q, ψ) = α e q, w q P h w q + (Be q, B(w q P h w q )) + (Be q, (B B h )P h w q ) (e v, B h P h w q ) ((B B h )ˆq h, B h P h w q ) (û u d, (B B h )P h w q ). (5.16) The six terms on the right hand side of (5.16) will be treated separately with the sixth term being the most difficult one. First term: By the properties of the L 2 projection Ph for 2 p p, 2 r p Ω, α e q, w q P h w q = α ˆq P h ˆq, w q P h w q α ˆq P h ˆq w q P h w q and the error estimate (3.4), cαh 1 1/p ˆq H 1 1/p (Γ)h 1 1/r w q H 1 1/r (Γ) cαh 2 1/p 1/r Σ p w q H 1 1/r (Γ). Second term: By the stability estimate (2.10), the error estimate (3.4), and the estimate (4.4) of Theorem 4.2, for 2 p p d, 2 r p Ω, (Be q, B(w q P h w q )) Be q B(w q P h w q ) c e q w q P h w q c α h 1 1/p Σ p h 1 1/r w q H 1 1/r (Γ) = c α h 2 1/p 1/r Σ p w q H 1 1/r (Γ). Third term: By the stability estimate (2.10), the error estimate (3.15) of Lemma 3.4, the result (4.4) of Theorem 4.2, and the stability estimate (3.4), for 2 p p d, (Be q, (B B h )P h w q ) Be q (B B h )P h w q c e q h 3/2 1/r P h w q H 1 1/r (Γ) c α h 1 1/p Σ p h 3/2 1/r P h w q H 1 1/r (Γ) c α h 5/2 1/p 1/r Σ p w q H 1 1/r (Γ).

22 22 S. MAY, R. RANNACHER, AND B. VEXLER Fourth term: By the error estimate (4.1) of Lemma 4.1, the stability estimates (3.9) of Lemma 3.3 and (3.4), (e v, B h P h w q ) e v B h P h w q ch 2 f P h w q ch 2 Σ p w q H 1 1/r (Γ). Fifth term: By the error estimate (3.14) of Lemma 3.4, the stability estimates (3.8), (3.9) of Lemma 3.3, the stability estimate (4.6) of Corollary 4.3, the stability estimate (3.4), for 2 r p Ω and 2 p p, ((B B h )ˆq h, B h P h w q ) ch 2 1/r 1/p ˆq h H 1 1/p (Γ) B h P h w q Lr (Ω) ch 2 1/r 1/p ˆq h H 1 1/p (Γ) B h P h w q H 1 (Ω) ch 2 1/r 1/p ˆq h H 1 1/p (Γ) P h w q H 1/2 (Γ) c α h 2 1/r 1/p Σ p w q H 1 1/r (Γ). Sixth term: To estimate the sixth term, we recall that ẑ H0 1 (Ω) H 2 (Ω) satisfies ẑ = û u d in Ω. Hence, we can use the error estimate (3.14) of Lemma 3.4 and the stability estimate (3.4) to obtain, for 2 r p Ω, 2 p p, (û u d, (B B h )P h w q ) ch 2 1/p 1/r ẑ W 2,p (Ω) P h w q H 1 1/r (Γ) Combining all these estimates gives us and hence in view of Lemma 5.1, ch 2 1/p 1/r Σ p w q H 1 1/r (Γ). (Be q, ψ) c α h 2 1/p 1/r Σ p w q H 1 1/r (Γ), (Be q, ψ) c 2 αh 2 1/p 1/r Σ p ψ L r (Ω). This completes the proof. Corollary 5.3. For the control error e q := ˆq ˆq h and the state error e u := û û h there holds e q H 1/2 (Γ) + e u c 2 αh 3/2 1/p Σ p, (5.17) for 2 p < p, where c 2 α 1 + α 2. Proof. (i) Observing that Be q is harmonic, by the trace extimate (2.7), we have e q H 1/2 (Γ) c Be q. Taking ψ := Be q in the estimate (5.9) of Theorem 5.2 for r = 2, we obtain which implies the first part of the assertion. (ii) From the identity Be q c 2 αh 3/2 1/p Σ p, (5.18) (e u, ψ) = (e v, ψ) + ((B B h )ˆq h, ψ) + (Be q, ψ), (5.19)

23 we conclude that FINITE ELEMENTS FOR ELLIPTIC BOUNDARY CONTROL 23 e u e v + (B B h )ˆq h + Be q. Hence, by the estimate (4.1) of Lemma 4.1, the estimate (3.15) of Lemma 3.4, and the just proven estimate (5.18), e u ch 2 f + ch 3/2 1/p ˆq h H 1 1/p (Γ) + c 2 αh 3/2 1/p Σ p. In view of the estimate (4.6), this implies the second part of the assertion. Corollary 5.4. For the primal state error e u := û û h and the adjoint state error e z := ẑ ẑ h there holds e u H 1 (Ω) + e z c 2 αh 2 1/p 1/r Σ p, (5.20) for 2 p < p and 2 r < p Ω, where c 2 α 1 + α 2. Proof. (i) We recall the identity (5.19). By Lemma 4.1, Lemma 3.4 and Theorem 5.2, we obtain (e u, ψ) c { h 2 f + h 2 1/r 1/p ˆq h H 1 1/p (Γ) + c 2 αh 2 1/p 1/r Σ p } ψ L r (Ω), for 2 p p and 2 r < p Ω. By arguments already used before, this implies e u H 1 (Ω) = sup ψ H 1 (Ω) (e u, ψ) ψ H1 (Ω) c 2 αh 2 1/p 1/r Σ p. (ii) For proving the error estimate of the adjoint state, we recall the equation (2.53) in the form ( e z, ψ h ) = (e u, ψ h ), ψ h V h,0. (5.21) The adjoint state ẑ H 1 0 (Ω) is determined by the boundary value problem ẑ = û u d in Ω, ẑ Γ = 0, and has the regularity ẑ W 2,p (Ω), with 2 p < p. Let w H 1 0 (Ω) H 2 (Ω) be the solution of the auxiliary problem w = e z in Ω, w Γ = 0, satisfying w H2 (Ω) c e z. Then, using (2.53), we conclude e z 2 = ( e z, (w R D h w)) + ( e z, R D h w) = ( (ẑ I h ẑ), (w R D h w)) + (e u, R D h w) (ẑ I h ẑ) (w R D h w) + e u H 1 (Ω) R D h w H1 (Ω) ch 2 ẑ H2 (Ω) w H2 (Ω) + c 2 αh 2 1/p 1/r Σ p w H2 (Ω) c { h 2 + c 2 αh 2 1/p 1/r} Σ p e z. This proves the asserted estimate. Corollary 5.5. For the primal state error e u := û û h e q := ˆq ˆq h there holds and the control error (e u, 1) + e q, 1 c 2 αh 2 1/p 1/r Σ p, (5.22)

24 24 S. MAY, R. RANNACHER, AND B. VEXLER for 2 p < p and 2 r < p Ω, where c 2 α 1 + α 2. Proof. In view of (e u, 1) Ω 1/2 sup (e u, ϕ) ϕ H 1 (Ω) ϕ H1 (Ω) = Ω 1/2 e u H 1 (Ω), the first part of the asserted estimate follows from Corollary 5.4. Next, we recall the Galerkin orthogonality relation (2.52) in the form α e q, χ h = (û u d, Bχ h ) + (û h u d, B h χ h ) χ h V h. Hence observing that the function χ h 1 satisfies χ h V h and Bχ h B h χ h 1, it follows that α e q, 1 = (e u, 1). This implies the second part of the asserted estimate. Remark 6. The numerical experiments shown in the next section indicate that for the adjoint state there may hold the improved error estimate e z c 2 αh 2 1/r Σ p, 2 r < p Ω. (5.23) 6. Numerical tests. In this section, we present some numerical results obtained for the optimization problem (1.1), (1.2) by the discretization described above. The purpose is to clarify the convergence rates to be expected for several configurations. For the computation the software libraries GASCOIGNE [8] and RoDoBo [14] have been used. Three different configurations have been considered in order to illustrate the sharpness of our theoretically derived error estimates: 1. Regular domain (unit square) and known analytic solution. 2. General domain with ω max = 5 6π and unknown solution. 3. Regular domain (unit square) and singular data Example on unit square with known analytic solution. The domain is the unit square Ω = (0, 1) 2 and the data is chosen as f = 4 α, u d = ( α) (x(1 x) + y(1 y)), α = 0.01, such that the optimal solution is given by ˆq = 1 α (x(1 x) + y(1 y)), û = 1 α (x(1 x) + y(1 y)) and ẑ = xy(1 x)(1 y). The obtained results are presented in Tables 6.1, 6.2 and 6.3. The test on uniform cartesian meshes shows second-order convergence for all quantities which is probably due to superapproximation effects which are not captured by our analysis. These effects disappear on irregular meshes and the observed orders of convergence agree well with the theoretically predicted rates.