Priority Program 1253

Transcription

1 Deutsche Forschungsgemeinschaft Priority Program 153 Optimization with Partial Differential Equations T. Apel, J. Pfefferer and A. Rösch Finite Element Error Estimates for Neumann Boundary Control Problems on raded Meshes April 010 Preprint-Number SPP

2

3 Finite element error estimates for Neumann boundary control problems on graded meshes Thomas Apel, Johannes Pfefferer, Arnd Rösch April 0, 010 Abstract A specific elliptic linear-quadratic optimal control problem with Neumann boundary control is investigated. The control has to fulfil inequality constraints. The domain is assumed to be polygonal with reentrant corners. The asymptotic behaviour of two approaches to compute the optimal control is discussed. In the first the piecewise constant approximations of the optimal control are improved by a postprocessing step. In the second the control is not discretized; instead the first order optimality condition is used to determine an approximation of the optimal control. Although the quality of both approximations is in general affected by corner singularities a convergence order of 3/ can be proven provided that the mesh is sufficiently graded. Keywords: linear-quadratic elliptic optimal control problem, boundary control, a-priori error estimates, mesh grading, postprocessing, variational discretization 1 Introduction In this paper, we consider the discretization of the optimal control problem J(ū) = min u U ad J(u), (1.1) J(u) := F (Su, u), F (y, u) := 1 y y d L (Ω) + ν u L (Γ), Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, Neubiberg, ermany; Thomas.Apel@unibw.de Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, Neubiberg, ermany; Johannes.Pfefferer@unibw.de Universität Duisburg-Essen, Forsthausweg, Duisburg, ermany; Arnd.Roesch@uni-due.de 1

4 where the associated state y = Su to the control u is the weak solution of the state equation y + y = 0 in Ω, n y = u on Γ j, j = 1,..., m, (1.) and the control variable is constrained by a u b for a.a. x Γ. The domain Ω is a bounded, two dimensional, polygonal domain with boundary Γ and m corner points x (j), j = 1,..., m, counting counterclockwise. In particular, Γ j denotes the side on the boundary Γ which connects the corners x (j) and x (j+1) except that x (1) is the intersection of Γ m and Γ 1. The set of all corner points is denoted by S. We further distinguish between the sets of convex corners C e and concave corners C a = S\C e. The regularization parameter ν R + and the bounds a, b R are fixed numbers. The function y d W 0, (Ω) denotes the desired state. Furthermore we define the set of admissible controls by U ad := {u L (Γ) : a u b a.e. on Γ}. In that what follows we discuss the full discretization and the variational discretization of the optimal control problem. Both approaches have in common that the state and the adjoint state are discretized by a finite element method. The difference occurs when discretizing the control. The full discretization approach uses piecewise constant approximations for the control together with an additional postprocessing step. This concept was established by Meyer and Rösch in [7] for distributed control. In that case they proved a convergence order of in the L (Ω)-norm for the optimal solution. In [5] it was applied to the Neumann boundary control problem which is also considered in this paper. Mateos and Rösch proved an order of convergence of 1/p for linear finite elements on convex domains where p depends on the regularity of the solution of the optimal control problem. For nonconvex domains they proved a convergence rate of 1/ + π/ω, where ω denotes the largest inner angle in the domain. They also used higher order finite elements for the approximation of the state and the adjoint state and obtained better results. The second discretization concept was introduced by Hinze in [0] for distributed control problems. Instead of discretizing the control, the first order optimality condition is used within this approach to compute an approximation of the control. In the case of distributed control the convergence order is also in the L (Ω)-norm. Using this approach Neumann boundary control problems were primarily discussed in [11], where the focus was on convex domains, semilinear partial differential equations and not necessarily quadratic functionals. The variational approach for problem (1.1) on quasiuniform meshes was considered in [5, 1]. Casas and Mateos [11] and Hinze and Matthes [1] proved that the error in the control is bounded by ch 3/ on convex domains when using this discretization concept. Mateos and Rösch [5] obtained for the variational approach the order 1/p on convex domains and 1/ + π/ω on nonconvex domains as for the

5 fully discrete approach. But the results of Mateos and Rösch on nonconvex domains for both, the full and the variational discretization approach, have in common that the order of convergence is lower than 3/ and decreases further in case that the largest inner angle in the domain increases. In the sequel we will analyze the asymptotic behaviour of the discrete optimal control problems and prove that the error in the control is bounded by ch 3/ for both approaches if one uses appropriately graded meshes in the neighborhood of reentrant corners. Let us state some further results from the literature before we go into detail. Results on discretizations of optimal control problems by piecewise constant functions were already presented by Falk [17], eveci [18] and Malanowski [4]. Recently, Arada, Casas and Tröltzsch [7], and Casas, Mateos and Tröltzsch [1] stated some new results using piecewise constant controls. Piecewise linear controls were considered in Casas [10], Casas and Mateos [11], Casas and Tröltzsch [13], Meyer and Rösch [6] and Rösch [30, 31]. In all those papers it was proven that the error in the control in the L -norm is bounded by ch or ch 3/ on quasi-uniform meshes provided that the solution is smooth enough. In Apel, Rösch, Winkler [], Apel, Winkler [6], Apel, Sirch, Winkler [5] and Apel, Rösch, Sirch [1] mesh grading and discretization techniques were successfully applied to improve the order of convergence not only in concave D and 3D domains. The paper is organized as follows: In Section we recall regularity results in suitable spaces for the elliptic boundary value problem. We distinguish weighted spaces with homogeneous and nonhomogeneous norms. In Section 3 the discretization concept is presented and the convergence results for the fully discrete approach are formulated. Section 4 contains results from the finite element theory and from numerical integration in order to prove the supercloseness and superconvergence properties of the fully discrete approach in Section 5. Section 6 is devoted the approach of variational discretizations. The paper ends with results from a numerical experiment using the postprocessing approach and some concluding remarks. We will denote by c a generic constant which may take different values at each occurrence, but is always independent of the discretization. Regularity results in suitable spaces and first order optimality conditions The regularity of the solution of the elliptic boundary value problem can be described by using the classical Sobolev spaces W s,p (Ω), compare e.g. [19, 15]. But in our context it is more convenient to use weighted Sobolev spaces. For this, we assume that the underlying domain coincides with an infinite angle in the neighborhood N j of each corner. This is satisfied for polygonal domains. Let K j R be such an infinite angle with vertex at the point x (j) and aperture ω j (0, π), i.e. K j := {x = (x 1, x ) R : 0 < r j <, 0 < ϕ j < ω j }, where r j, ϕ j denote the polar coordinates located at the point x (j). In particular, the distance r j to the corner is defined by r j := x x (j). The sides of K j (excluding the 3

6 vertices) are denoted by Γ 1 j (ϕ j = ω j ) and Γ j (ϕ j = 0). Moreover, let ξ j, j = 1,..., m, be infinitely differentiable cut-off functions defined in the closure of Ω, which are equal to one in a neighborhood of the corner x (j) and equal to zero in Ω\N j. Moreover, we set ξ 0 = 1 m j=1 ξ j and = ( 1,..., m ), a real-valued vector. To ensure ξ 0 0 the sets K j should not overlap. In the sequel we distinguish between two types of weighted Sobolev spaces V k, (Ω) := {v D (Ω) : v V k, < } (Ω) and W k, (Ω) := {v D (Ω) : v W k, < } (Ω) with k N 0 and R m, equipped with the norms and v V k, (Ω) := ξ 0v W k, (Ω) + v W k, (Ω) := ξ 0v W k, (Ω) + m j=1 m j=1 ξ j v V k, j (K j ) ξ j v W k, j (K j ), where the Sobolev spaces W k,p (Ω)(= H k (Ω) for p = ) are defined as usual. By using standard multi-index notation the weighted norms for each angle K j are defined by v V k, (K j ) := j K j α k r ( j k+ α ) j D α v dx 1 and v W k, (K j ) := j K j α k r j j D α v dx 1. The corresponding trace spaces are V k 1/ (Γ) and W k 1/ (Γ) for k > 0. The norm in W k 1/ (Γ) is defined by v k 1/ W (Γ) := inf{ u k, W k, : u W (Ω) (Ω), u Γ\S = v}. Recall that the set S is the set of all corner points. We could analogously define the norm in V k 1/ (Γ), but it will be more suitable to have a norm which is given by integral expressions. Hence we define an equivalent norm in the trace space of V k, (Ω) by v V k 1/ m (Γ) := m ξ 0 v W k 1/ (Γ j ) + j=1 j=1 i=1 ξ j v V k 1/ j (Γ i j ), 4

7 with v V k 1/ j (Γ i j ) := Γ i j + α =k 1 α k 1 Γ i j r ( j k+1/+ α ) j D α v ds x Γ i j rj (x) j (D α v)(x) r j (y) j (D α v)(y) x y ds x ds y compare [3, Remark 1.1 together with Theorem 1.1]. At first glance both norms only differ in the exponent of the weighting function. But a closer examination reveals, that the space W 1, (Ω) does not belong automatically to the scale of the first kind of weighted spaces. The space W 1, (Ω) V 1, 0 (Ω) only contains functions which vanish at the corner. But the solution of the Neumann problem does not satisfy this condition in general. By contrast the second kind of weighted spaces contains the classical Sobolev spaces for = 0. Thus we will use the spaces W k, (Ω) to describe the regularity of the Neumann problem. But as we will see in the sequel the space V k, (Ω) and its trace space are more convenient for the numerical analysis of problem (1.1) having regard to the following lemmas. Lemma.1. For (0, 1) m one has W (Ω) = V (Ω) ξ 1Π 0 ξ m Π 0, 1 where ξ j Π 0 is the set of all functions, which have the form ξ j p and p is a constant. In particular, for any v W (Ω) one can write v = v s + m j=1 ξ jv j with v s V (Ω) and v j = v(x (j) ), j = 1,..., m. Moreover, the norm equivalence is valid. m v W (Ω) v s V (Ω) + v(x (j) ) j=1, Proof. One can set n =, l = and s = 1 in Theorem of [3] to get the result for domains with one corner. The extension to general polygonal domains is obvious by writing y = m j=0 ξ jy, see also [3, p. 73]. Lemma.. Let 0 < j, j+1 < 1. Then W 3/ (Γ j ) = V 3/ (Γ j ) ξ j Π 0 ξ j+1 Π 0, i.e. for any v W 3/ (Γ j ) one can write v = v s + ξ j v j + ξ j+1 v j+1 with v s V 3/ (Γ j ) and v i = v(x (i) ), i = j, j + 1. Furthermore, the norm equivalence holds true. v W 3/ (Γ j ) v s V 3/ (Γ j ) + v(x(j) ) + v(x (j+1) ) 5

8 Proof. This can be deduced from Theorem 7.1. and Theorem in [3] by setting n =, l = and s = 1. Remark.3. Lemma.1 and. remain valid for j = 0, j = 1,..., m, if the function v is equal to zero in the neighborhood N j of the corner x (j). Next we state a lemma concerning the solvability and the regularity of the elliptic boundary value problem (1.). Lemma.4. Let λ j = π/ω j and 1 > j > 1 λ j, j = 1,..., m. Moreover, let f 1/ (Ω) and g W (Γ). Then the problem W 0, y + y = f in Ω, n y = g on Γ j, j = 1,..., m, (.1) has a unique weak solution and the a priori estimate ( y W (Ω) c f W 0, (Ω) + g W 1/ is valid with a constant c independent of y. Proof. The technique is adapted from the proof of Theorem 3.. in [4], where the special case g = 0 for Ω R 3 is considered. We get from the Lax-Milgram Theorem the unique solvability of the elliptic boundary value problem in H 1 (Ω). Thus we can conclude for j > 1 λ j the a priori estimate ( ) y W (K j) c f W 0, (K j) + g W 1/ (Γ 1 j ) + g W 1/ (Γ j ) and the existence of a unique solution y W (K j) by setting k = 0 in Theorem 3.. of [34]. The Neumann problem (.1) can be locally transformed near each corner x (j) into a boundary value problem in the infinite angle K j. Thus we obtain the desired result by using the partition of unity method and the local results in K j. We set λ = (λ 1,..., λ m ) a real-valued vector. In the sequel all inequalities containing vectorial parameters must be understood component-by-component. For ω j < π we obtain λ j > 1 and we use the freedom to choose j = 0 in the following. This means weights are active in concave corners only. Moreover, we will always choose j < 1. This is always possible for concave corners since λ j > 0 (even > 1/). Now we apply the results given so far to the optimal control problem. The exposition follows [5]. The solution operator S : L (Γ) L (Ω) which associates a state y = Su to a control u is given via (1.). We also introduce the problem (Γ) ) p + p = z in Ω, n p = 0 on Γ, (.) 6

9 and the according solution operator P : L (Ω) H 1 (Ω) which associates a function p = P z with a function z via (.). The adjoint operator of S is S : L (Ω) L (Γ), where we have S z = (P z) Γ = p Γ [5]. Applying the reen formula shows that (Su, z) L (Ω) = (u, S z) L (Γ) u L (Γ), z L (Ω). We can also relate an adjoint state p u := P (Su y d ) to every control u, which is the unique solution of p u + p u = y y d in Ω, n p u = 0 on Γ. (.3) Note that S (Su y d ) = (P (Su y d )) Γ = p u Γ. Before finishing this section with the theorem about the solvability and regularity of the optimal control problem, we define the projection Π [a,b] f(x) := max(a, min(b, f(x))). (.4) Theorem.5. The optimal control problem (1.1) has a unique solution ū. The corresponding state ȳ = Sū and adjoint state p = pū are given by (1.) and (.3). The variational inequality ( p + νū, u ū) L (Γ) 0 u U ad (.5) is necessary and sufficient for the optimality of ū. This condition can be expressed equivalently by ū = Π [a,b] ( 1 ) ν p Γ. (.6) Moreover, there holds for > 1 λ, u L (Γ) and y d W 0, (Ω) that ȳ W (Ω) C( Ω), p W (Ω) C( Ω), p Γ W 3/ (Γ) C(Γ) and ū C(Γ) m j=1 H1 (Γ j ) and there exists a constant c which depends only on the data of the problem such that ū L (Γ) + m j=1 ū H 1 (Γ j ) + ȳ W (Ω) + p W (Ω) + p W 3/ c. (.7) (Γ) Proof. First we recall some results from the literature which hold for linear quadratic and strictly convex optimal control problems. The unique solvability of the optimal control problem, the necessity and sufficiency of the variational inequality for an optimal control and the equivalence between the variational inequality and the projection formula have been discussed in several papers, see e.g. [1, 11, 5]. We have y H 3/ (Ω) for any function u L (Γ), see Theorem.1. in [1]. Since the space H 3/ (Ω) is embedded in the space W 0, (Ω) (definitely for 0) we get with Lemma.4 that p W (Ω) for > 1 λ. This implies that the restriction of p on every side Γ j, j = 1,..., m, belongs to H 1 (Γ j ) W 3/ (Γ j ) and the restriction on the complete boundary belongs to H 1 ɛ (Γ), ɛ > 0 arbitrary, due to the trace theorem [33, Theorem 8.7]. Since ū is determined by the projection formula we get that ū belongs to the space m j=1 H1 (Γ j ). 7

10 The space H 1 (Γ j ) is embedded in the space W 1/ (Γ j ) since 0. Thus we can conclude ȳ W (Ω) by means of Lemma.4. Moreover we have p, ū C(Γ) due to the embedding H 1 ɛ (Γ) C(Γ), ɛ < 1, and ȳ, p C( Ω) due to the embedding H 3/ (Ω) C( Ω). 3 Discretization and fully discrete approach We discretize the optimal control problem by a finite element method. To state the discretization we introduce a family of graded triangulations {T h } of Ω, which is admissible in the sense of Ciarlet [14] with h being the the global mesh parameter. Note that there is a segmentation h of the boundary given by the triangulation T h in a natural way. Furthermore we denote by µ j (0, 1], j = 1,..., m, the grading parameters which are summarized in the vector µ. With r T,j := inf (x1,x ) T x x (j) and r,j := inf (x1,x ) x x (j) we denote the distance of the triangle T and of the edge to the corner x (j), respectively. For edges with T there holds r T,j r,j. We assume that the element size h T := diamt satisfies c 1 h 1/µ j h T c h 1/µ j for r T,j = 0, c 1 hr 1 µ j T,j h T c hr 1 µ j T,j for 0 < r T,j R j, (3.1) c 1 h h T c h for r T,j > R j for j = 1,..., m and some constants R j. We emphasize that the number of elements of such a discretization is of order h, see e.g. [3]. Moreover, we introduce the finite dimensional spaces V h := {y h C( Ω) : y h T P 1 T T h }, U h := {u h L (Γ) : u h P 0 h }, U ad h := U h U ad, where P k, k = 0, 1 is the space of polynomials of degree less than or equal to k. Now we can state the discrete version of the state equation. Find for each u L (Γ) the unique element y h = S h u V h satisfying a(y h, v h ) = (u, v h ) L (Γ) v h V h, where S h : L (Γ) L (Ω) is the discrete solution operator, a : H 1 (Ω) H 1 (Ω) R is the bilinear form a(y, v) = y v + yv dx. Ω The finite dimensional approximation of the optimal control problem reads as follows J h (ū h ) = min u h U ad h J h (u h ), (3.) J h (u h ) := 1 S hu h y d L (Ω) + ν u h L (Γ). 8

11 To be coherent with the definitions of the continuous optimal control problem, we also introduce the discrete solution operator P h : L (Ω) H 1 (Ω) defined by p h = P h z with p h V h such that a(v h, p h ) = (z, v h ) L (Ω) v h V h, The adjoint of the discrete solution operator is equal to the discretized version of the adjoint solution operator. Thus we can write S h : L (Ω) L (Γ) with S h z = (P hz) Γ. The discrete adjoint state is the unique element p h (u) V h such that Note that S h (S hu y d ) = p h (u) Γ and a(v h, p h (u)) = (S h u y d, v h ) L (Ω) v h V h. (S h u, z) L (Ω) = (u, S h z) L (Γ) z L (Ω), u L (Γ). Remark 3.1. The discrete optimal control problem (3.) admits a unique solution ū h. The optimal discrete state S h ū h is denoted by ȳ h and the optimal discrete adjoint state p h is given by p h (ū h ). The first order optimality condition reads as follows ( p h + νū h, u h ū h ) L (Γ) 0 u h U h ad. (3.3) For the rest of the paper we make the following assumption to state the superconvergence properties of the fully discrete approach. Since the optimal control ū is given by the projection formula (.6) we can easily classify the edges h in two sets K 1 an K. Assumption 3.. Let K 1 and K be defined as follows K 1 :=, K := h :ū/ W 3/ 3(1 µ)/ () h :ū W 3/ 3(1 µ)/ (). We assume with a constant c greater than zero. meas(k 1 ) ch Note that this assumption is satisfied in many practical cases, see Section 4 in [5] for details. Now we state the main results for the fully discrete approach. Theorem 3.3. Assume that Assumption 3. is valid. If the grading parameters satisfy µ < λ then the estimates hold true. ȳ ȳ h L (Ω) ch 3/, (3.4) p p h L (Γ) ch 3/ (3.5) 9

12 The proof is postponed to section 5. Following [7, 5], we construct a control in a postprocessing step and compute a control ũ h by a projection ( of the discrete adjoint state p h to the set of admissible controls U ad, i.e. ũ h := Π [a,b] 1 ν p h). Thus this projection is piecewise linear and continuous, but in general ũ h / U h. However we can prove the following superconvergence property. Theorem 3.4. Assume that Assumption 3. is fulfilled. For a family of meshes with grading parameters µ < λ, the estimate is valid. ū ũ h L (Γ) ch 3/ (3.6) Proof. The projection operator Π [a,b] is a bounded continuous operator from L (Γ) to L (Γ). Thus we obtain ν ū ũ h L (Γ) = ν Π [a,b] ( 1ν ) ( p Π [a,b] 1 h) ν p where we used (3.5) in the last step. c p p h L (Γ) ch 3/, 4 Results from the finite element theory L (Γ) In this section we collect some results from the finite element theory for elliptic equations and from numerical integration to prove the supercloseness and superconvergence results in the next section. Lemma 4.1. Let u L (Γ), and z W 0, (Ω) with > 1 λ. The discretization error for the state and the adjoint state can be estimated by Su S h u L (Ω) + h Su S h u W 1, (Ω) ch, (4.1) P z P h z L (Ω) + h P z P h z W 1, (Ω) ch, (4.) provided that the mesh grading parameters fulfil µ < λ. Proof. To prove (4.1) we use standard techniques for estimates on finite element errors. The result for the adjoint state holds analogously. Due to the continuity and the coercivity of the bilinear form a we can conclude by means of Cea s Lemma y y h W 1, (Ω) c inf v h V h y v h W 1, (Ω). (4.3) Since y C( Ω) the linear Lagrange interpolant I h y is well defined and belongs to V h. We have y y h W 1, (Ω) c y I h y W 1, (Ω). 10

13 To derive an estimate for the interpolation error we write according to Lemma.1 y = m ξ j y j = ξ 0 y + j=0 j C e ξ j y + j C a ξ j y s + j C a ξ j y j with ξ j y s V (Ω), y j Π 0, j C a, and ξ j being the cut-off functions defined in Section. Hence we can estimate y y h W 1, (Ω) c ξ 0 y I h (ξ 0 y) W 1, (Ω) + ξ j y I h (ξ j y) W 1, (Ω) j C e + ξ j y s I h (ξ j y s ) W 1, (Ω) + ξ j y j I h (ξ j y j ) W 1, (Ω). j C a j C a Since ξ 0 y, ξ j y for j C e and ξ j y j for j C a belong to W (Ω) we can use standard interpolation error estimates, see e.g. [9], for the first term and for the terms in the first and third sum. The function ξ j y s belongs to V j (K j ) for j C a. Thus we can use interpolation error estimates in weighted Sobolev spaces with homogeneous norm in angles to estimate the terms in the second sum, see e.g. [8, 9, 8]. Collecting everything we arrive at y y h W 1, (Ω) ch ξ 0 y W (Ω) + ( ξ j y W (Ω) + j C e j C a ch ξ 0 y W (Ω) + ξ j y W (Ω) + j C e j C a ξ j y s V (K j ) + ξ jy j W (Ω) j ( ) ξ j y s V (K j ) + c j y j, j where we used y j Π 0, µ < λ and that c j := ξ j W (Ω) is constant independent of the mesh parameter h. With Lemma.1 and Theorem.5 we get y y h W 1, (Ω) ch y W ch. (Ω) Finally we can use the Nitsche method to double the order of convergence in the L -norm, see e.g. [8]. ) Lemma 4.. The norms of the discrete solution operators S h and Sh several norms by a constant c always independent of h. We have are bounded in S h u L (Ω) c u L (Γ), S h z H 1/ (Γ) c z L (Ω), S h z L (Γ) c z L (Ω). 11

14 Proof. Due to the variational formulation of the elliptic boundary value problem we have S h u H 1 (Ω) = (u, S hu) L (Γ) u L (Γ) S h u L (Γ) c u L (Γ) S h u H 1 (Ω), where we used the trace theorem [9, Theorem 1.6.6] in the last step. With the inclusion H 1 (Ω) L (Ω) we get the first inequality. Analogously we get P h z H 1 (Ω) c z L (Ω). (4.4) To prove the second inequality we apply the trace theorem [33, Theorem 8.7]. We have that S h z H 1/ (Γ) c P hz H 1 (Ω) c z L (Ω), (4.5) where we inserted (4.4). For the proof of the third inequality we begin with introducing intermediate functions S h z L (Γ) S h z I hs z L (Γ) + I h S z L (Γ). For the first term we get with Lemma 3 of [] S h z I hs z L (Γ) S h z I hs z L (Ω) c z L (Ω). The second term can be estimated by I h S z L (Γ) S z L (Ω) c S z H 1+ɛ (Ω) c S z W 1 ɛ (Ω) c z L (Ω), where we used the boundedness of I h, usual embedding theorems, the trace theorem [33, Theorem 8.7], Theorem 1.3 in [3] and Lemma.4 with 0 < ɛ < λ. Lemma 4.3. Let T be an arbitrary element in h and Ĝ the reference element. Denote F : ˆx Ĝ F (ˆx) = Bˆx + b the affine linear change of variables such that F Ĝ = and denote ˆv = v F with v Hs (), s [0, ]. Then there exists a constant c such that ˆv H s (Ĝ) c 1/ h s T v H s (). (4.6) Proof. One can deduce the inequality for integer s from Theorem 15.1 in [14]. For realvalued s we refer to Example 3 in [16]. Notice, that B ch T and det(b) h T. Next we define a projection operator from the space of the continuous functions into the space of piecewise constant functions on the boundary. Furthermore we state some properties of this operator which will be used in the error analysis in the sequel. Definition 4.4. Let h be the triangulation of the boundary of Ω which is induced by the triangulation T h. The projection operator R h is the 0-interpolator onto U h. We define R h : C(Γ) U h f R h f with where S is the midpoint of the edge. (R h f)(x) := f(s ) if x, 1

15 Lemma 4.5. Let E h h be a triangulation of Γ j with grading parameters µ j and µ j+1. Then for any element E h the following estimates hold true ch 3/ 1/ f 3/ V 3(1 µ j )/ () for f ξ j g, g V 3/ 3(1 µ j )/ (Γ j ) ch 3/ 1/ f 3/ (f R h f)ds x V 3(1 µ j+1 )/ () for f ξ j+1 g, g V 3/ 3(1 µ j+1 )/ (Γ1 j+1 ) ch 3/ 1/ f W 3/ () for f W 3/ (Γ j ) ch 1/ f H 1 () for f H 1 (Γ j ). Proof. We prove the first inequality and deduce the second one analogously. The basic idea is that the formula is a exact numerical scheme for polynomials of order one. We distinguish between edges with r,j > 0 and r,j = 0. For r,j > 0 we get (f R h f)ds x c ( ˆf ˆR h ˆf)dsˆx Ĝ = c ( ˆf ˆp ˆR h ( ˆf ˆp))dsˆx Ĝ ( ) c ( ˆf ˆp)dsˆx + ( ˆR h ( ˆf ˆp))dsˆx Ĝ Ĝ c ( ˆf ˆp L 1 (Ĝ) + ˆf ) ˆp L (Ĝ) c ˆf ˆp H 1 (Ĝ), where ˆp is a arbitrary polynomial of order one. Next we use Theorem 6.1 in [16] to arrive at ˆf ˆp H 1 (Ĝ) c ˆf ˆp H 3/ (Ĝ) c ˆf H 3/ (Ĝ). (4.7) Using (4.6) we can continue with (f R h f)ds x c 1/ h 3/ T f H 3/ (). (4.8) Inserting the definition of h T for r T,j > 0 we get with α := 3(1 µ j )/ (f R h f)ds x c 1/ h 3/ rt,j f α H 3/ () ( c 1/ h 3/ r j (x) α D x f(x) r j (y) α D y f(y) r j (x) α r j (y) α + x y ds x D y f(y) ds y x y ds x ds y ) 1/. 13

16 A Taylor series expansion of r j (y) α yields with θ (0, 1) ( (f R h f)ds x c 1/ h 3/ r j (x) α D x f(x) r j (y) α D y f(y) x y ds x ds y ) 1/ + r j ((1 θ)x + θy) (α 1) ds x D y f(y) ds y. For every with r,j > 0 there holds min x r j (x) max x r j (x). Thus we get ( (f R h f)ds x c 1/ h 3/ r j (x) α D x f(x) r j (y) α D y f(y) x y ds x ds y ) 1/ + r j (y) (α 1) D y f(y) ds y ( c 1/ h 3/ r j (x) α D x f(x) r j (y) α D y f(y) x y ds x ds y ) 1/ + r j (y) (α 3/+1) D y f(y) ds y = c 1/ h 3/ f 3/ V (), 3(1 µ j )/ where we used h T and h T cr T,j which is satisfied for r T,j > 0. For the case that r,j = 0 we can proceed by using that R h is a bounded operator from L () to L () with norm 1 and V 3/ 3(1 µ j )/ () W 3µ j/ () L () for all µ j in the possible range (1/3, 1). This leads to the estimate (f R h f)ds x c f L () c ˆf L (Ĝ) c ˆf 3/ V c 1/ h 3/ 3(1 µ j)/ T f V 3/ 3(1 µ j )/ (). 3(1 µ j )/ (Ĝ) This can analogously be proven to (4.6) together with ˆr γ j h γ T rγ j for r T,j = 0. The first inequality is proven by observing that h T ch 1/µ j for r T,j = 0. One can directly deduce the third inequality from (4.8), since f belongs to W 3/ (Γ j ) and h T ch. The inequality for functions in H 1 (Γ j ) is similar, only ˆf ˆp H 1 (Ĝ) c ˆf H 1 (Ĝ) is used instead of (4.7). In the following we introduce the L -projection in the space of piecewise constant functions on the boundary and state some properties of this projection operator. Definition 4.6. Let h be the triangulation of the boundary. The L -projection of a function f L (Γ) is the piecewise constant function that fulfills Q h f 1 f(x)ds x on any element h. 14

17 Lemma 4.7. For any element h and any function f H s (), s [0, 1], the estimate f Q h f L () ch s T f H s () is valid. Proof. First we observe that Q h p = p for any p P 0 (). Thus we can write for f H s () f Q h f L () = f p Q h (f p) L () c f p L () c 1/ ˆf ˆp L (Ĝ) c 1/ ˆf ˆp H s (Ĝ) c 1/ ˆf H s (Ĝ) ch s T f H s () where we used Theorems 3. and 6.1 of [16] and (4.6). Corollary 4.8. For any element h and any functions f H s (), s = [0, 1], and v H 1 () the estimate is valid. (f Q h f, v) L () ch s+1 T f H s () v H 1 () Proof. Due to the definition of Q h we have the orthogonality (f Q h f, p) L () = 0 for all p P 0 (). Thus we get (f Q h f, v) L () = (f Q h f, v Q h v) L () f Q h f L () v Q h v L () ch s+1 T f H s () v H 1 (), where we used the Cauchy-Schwarz inequality and Lemma 4.7. Lemma 4.9. With mesh parameters µ < λ and under Assumption 3. the estimate holds. S h ū S h R h ū L (Ω) ch 3/ (4.9) Proof. Let z h = S h ū S h R h ū and v h = S h z h. Due to the relation between S h and S h and by inserting Q h ū we can write S h ū S h R h ū L (Ω) = (z h, z h ) L (Ω) = (ū R hū, S h z h) L (Γ) = (ū R h ū, v h ) L (Γ) = (ū Q h ū, v h ) L (Γ) + (Q h ū R h ū, v h ) L (Γ). (4.10) We estimate both terms in (4.10) separately. For the first term we get (ū Q h ū, v h ) L (Γ) = (ū Q h ū, v h ) L () c h 3/ T ū H 1 () v h H 1/ () h h ch 3/ m j=1 ū H 1 (Γ j ) v h H 1/ (Γ j ) ch3/ z h L (Ω), 15

18 where we used Corollary 4.8, the discrete Cauchy-Schwarz inequality, h T ch, Theorem.5 and Lemma 4.. For the second term in (4.10) we split the integral into two integrals over the sets K 1 and K (Q h ū R h ū, v h ) L (Γ) = (Q h ū R h ū, v h ) L (K 1 ) + (Q h ū R h ū, v h ) L (K ). (4.11) For the second term in (4.11) we begin with applying the Cauchy-Schwarz inequality and the estimate v h L (K ) v h L (Γ) = S h z h L (Γ) c z h L (Ω) to get (Q h ū R h ū, v h ) L (K ) Q h ū R h ū L (K ) v h L (K ) c Q h ū R h ū L (K ) z h L (Ω). For Q h ū R h ū L (K ) we write Q h ū R h ū L (K ) = K (Q h ū R h ū) ds x m j=1 h Γ j K (Q h ū R h ū) ds x. Using the definition of Q h and that R h ū is constant on each edge we can continue I j := h Γ j K = h Γ j K (Q h ū R h ū) ds x = h Γ j K ( ) 1 (ū R h ū) ds ξ ds x = h Γ j K ( 1 ūds ξ R h ū) ds x ( ) 1 (ū R h ū) ds ξ. (4.1) Now we consider the case that both corners x (j) and x (j+1) are concave. The case of only one concave corner or no concave corner can analogously be deduced by simple modifications. In particular, for a convex corner x (j) we don t need the following splitting, since ξ j ū belongs to H 3/ (Γ j ). For µ j, µ j+1 (1/3, 1) we have 3/(1 µ j ), 3/(1 µ j+1 ) (0, 1). Thus we get ū = ξ 0 ū + ξ j ū + ξ j+1 ū = ξ 0 ū + ξ j ū s + ξ j+1 ū s + ξ j ū j + ξ j+1 ū j+1 (4.13) on Γ j according to Lemma. with ξ 0 ū H 3/ (Γ j ), ξ j ū s V 3/ 3/(1 µ j ) (Γ j ), ξ j+1ū s V 3/ 3/(1 µ j+1 ) (Γ1 j+1 ) and ū j, ū j+1 Π 0. Inserting (4.13) in (4.1) yields I j = h Γ j K 1 j+1 + i=j j+1 (ξ 0 ū R h ξ 0 ū) ds ξ + i=j (ξ i ū i R h ξ i ū i ) ds ξ. (ξ i ū s R h ξ i ū s ) ds ξ 16

19 With Lemma 4.5 we arrive at j+1 I j c h 3 ξ 0 ū W 3/ () + h Γ j K j+1 ch 3 ξ 0 ū W 3/ (K Γ j ) + j=1 h Γ j K i=j i=j j+1 ξ i ū s 3/ V 3/(1 µ i ) () + i=j j+1 ξ i ū s 3/ V 3/(1 µ i ) (K Γ j ) + i=j ū i ξ i W 3/ () since ξ i W 3/ (K Γ j ) c independent of the mesh parameter h. Using the norm equivalence of Lemma. we get m ( ) 1 (ū R h ū) ds ξ ch 3 ū W 3/ (K ) ch3, (4.14) where we used µ j < λ j, j = 1,..., m, together with Theorem.5 in the last steps. For the estimation of the first term in (4.11) we begin with (Q h ū R h ū, v h ) L (K 1 ) v h L (Γ) Q h ū R h ū ds x = v h L (Γ) = v h L (Γ) h K 1 h K 1 h K 1 1 ū R h ūds ξ. ū i ūds ξ 1 R h ūds ξ ds x using the definition of Q h. We can continue by means of Lemma 4.5 and (3.1) ū R h ūds ξ c h 1/ ū H 1 () h h K 1 K 1 ch c h K 1 1/ ū h K 1 m h 3/ ū H 1 (Γ j ) ch 3/, j=1 H 1 () where we used the discrete Cauchy-Schwarz inequality, Assumption 3. and Theorem.5. We finally get with Lemma 4. This finishes the proof. (Q h ū R h ū, v h ) L (K 1 ) ch 3/ z h L (Ω)., 1/ 17

20 Lemma With mesh parameters µ < λ and under Assumption 3. the estimate is valid. p S h (S hr h ū y d ) L (Γ) ch 3/ (4.15) Proof. We begin by introducing intermediate functions and use the triangle inequality to obtain p S h (S hr h ū y d ) L (Γ) p S h (ȳ y d) L (Γ) + S h (ȳ y d) S h (S hū y d ) L (Γ) + S h (S hū y d ) S h (S hr h ū y d ) L (Γ). For the first term we use the trace theorem (Theorem in [9]) and Lemma 4.1 to get p S h (ȳ y d) L (Γ) c p P h (ȳ y d ) 1/ L (Ω) p P h(ȳ y d ) 1/ H 1 (Ω) (h p W (Ω))1/ (h p W (Ω))1/ ch 3/. For the second term we use Lemma 4. and again the results of Lemma 4.1 to arrive at S h (ȳ y d) S h (S hū y d ) L (Γ) c ȳ y d (S h ū y d ) L (Ω) = c ȳ S h ū L (Ω) ch ȳ W (Ω) ch. For the third term we use Lemma 4. and Lemma 4.9 to obtain S h (S hū y d ) S h (S hr h ū y d ) L (Γ) = S h (S hū S h R h ū) L (Γ) c S h ū S h R h ū L (Ω) ch 3/. 5 Supercloseness result and proof of Theorem 3.3 In this section the supercloseness result ū h R h ū L (Γ) ch 3/ and Theorem 3.3 will be proven. Before doing this we recall the following lemma which is nested in Proposition 4.5 of [5]. The proof does not require quasi uniform meshes. Lemma 5.1. The inequality is valid. ν R h ū ū h L (Γ) (R h p p h, ū h R h ū) L (Γ) (5.1) Lemma 5. (Supercloseness). Assume that Assumption 3. is fulfilled. Let ū h be the solution of the discrete optimal control problem (3.) on a family of meshes with grading parameters µ < λ. Then the estimate holds true. ū h R h ū L (Γ) ch 3/ (5.) 18

21 Proof. We rewrite inequality (5.1): ν R h ū ū h L (Γ) (R h p p h, ū h R h ū) L (Γ) = (R h p p, ū h R h ū) L (Γ) + ( p S h (S hr h ū y d ), ū h R h ū) L (Γ) + (S h (S hr h ū y d ) p h, ū h R h ū) L (Γ). For the first term we can use that it represents a formula for the numerical integration, since ū h R h ū is piecewise constant. We obtain (R h p p, ū h R h ū) L (Γ) = (R h p(x) p(x))(ū h (x) R h ū(x))ds x h = h (ū h (S ) R h ū(s )) 1/ 1/ = h ū h R h ū L () 1/ (R h p p)ds x (R h p p)ds x c ū h R h ū L ( ) 1 (Γ) (R h p p)ds x h where we used in the last step the discrete Cauchy-Schwarz inequality. Using Lemma 4.5 as in the proof of Lemma 4.9 between (4.1) and (4.14) with λ j > λ j > µ j, j = 1,..., m, we arrive at (R h p p, ū h R h ū) L (Γ) ch 3/ ū h R h ū L (Γ) p W 3/ (Γ). For the second term we use the Cauchy-Schwarz inequality and Lemma 4.10 to get 1/, ( p S h (S hr h ū y d ), ū h R h ū) L (Γ) p S h (S hr h ū y d ) L (Γ) ū h R h ū L (Γ) ch 3/ ū h R h ū L (Γ). The third term can be omitted, since it is smaller than zero, (Sh (S hr h ū y d ) p h, ū h R h ū) L (Γ) = (Sh (S hr h ū y d ) Sh (S hū h y d ), ū h R h ū) L (Γ) = (Sh (S h(r h ū ū h )), ū h R h ū) L (Γ) = (S h (R h ū ū h ), S h (ū h R h ū)) L (Ω) = S h (R h ū ū h ) L (Ω) 0, where we used the definition of p h and Sh. Now we can state the missing proof of Theorem

22 Proof of Theorem 3.3. We introduce intermediate functions and apply the triangle inequality, use ȳ = S h ū h and obtain ȳ ȳ h L (Ω) ȳ S h ū L (Ω) + S h ū S h R h ū L (Ω) + S h R h ū S h ū h L (Ω) ȳ S h ū L (Ω) + S h ū S h R h ū L (Ω) + c R h ū ū h L (Γ). The last inequality holds due to Lemma 4.. The three terms were estimated separately in Lemma 4.1, Lemma 4.9 and in Lemma 5.. Thus we can conclude For the error in the adjoint state we get ȳ ȳ h L (Ω) ch 3/. p p h L (Γ) p S h (S hr h ū y d ) L (Γ) + S h (S hr h ū y d ) p h L (Γ). The first term was estimated in Lemma For the second holds S h (S hr h ū y d ) p h L (Γ) = S h (S hr h ū y d ) S h (S hū h y d ) L (Γ) = S h S h(r h ū ū h ) L (Γ) c R h ū ū h L (Γ) ch 3/, where we used Lemma 4. and Lemma 5. in the last two steps. 6 Variational approach In this section we consider the variational discretization concept first presented in [0] for distributed control problems. We consider as discretized optimal control problem: min J h (u) = F (S h u, u). (6.1) u U ad Problem (6.1) admits a unique solution u h. In that what follows we denote by y h = S hu h the discrete state and by p h = p h(u h ) the discrete adjoint state. The first order necessary and sufficient condition for the optimality of u h to problem (6.1) can be written as (p h + νu h, u u h ) L (Γ) 0 u U ad. (6.) Lemma 6.1. Let u h be the solution of problem (6.1) on a family of triangulations T h of Ω. Then the estimate is valid. ν ū u h L (Γ) S (Sū y d ) S h (Sū y d) L (Γ) + c Sū S h ū L (Ω) (6.3) Proof. The proof is given in Section 7 of [5]. It also holds for graded meshes. 0

23 Theorem 6.. Let u h be the solution of problem (6.1) on a family of meshes with mesh grading parameters µ < λ according to (3.1). Then the estimate holds. Proof. We get from (6.3) ν ū u h L (Γ) ch 3/ ν ū u h L (Γ) S (Sū y d ) Sh (Sū y d) L (Γ) + c Sū S h ū L (Ω) ( c p P h (Sū y d ) 1/ L (Ω) p P h(sū y d ) 1/ H 1 (Ω) + h ȳ W ( ) c (h p W (Ω))1/ (h p W (Ω))1/ + h ȳ W (Ω) ch 3/, (Ω) ) where we used the trace theorem [9, Theorem 1.6.6], the definition of S h and S h, Lemma 4.1 and Theorem.5 in the last steps. 7 Numerical example and concluding remarks In this section we present a numerical example for the fully discrete approach to compare the proven order of convergence with the experimental one. The example is taken from [5]. Let r, φ be the polar coordinates. For ω [0, π) define the circular sector S ω := {x R : (r(x), φ(x)) (0, ] [0, ω]} and the domain Ω ω := ( 1, 1) S ω with its boundary Γ ω. Consider the optimal control problem in the form y + y = 0 in Ω ω, n y = u + g a.e. on Γ ω, p + p = y y d in Ω ω, n p = g 1 a.e. on Γ ω, u = Π [a,b] ( p Γω ) on Γ ω. The data y d, g 1 and g are chosen in the following way y d = r λ cos(λφ) in Ω ω, g 1 = n y d on Γ ω, g = Π [a,b] (y d ) on Γ ω, with λ = π/ω, a = 0.5 and b = 0.5. The unique solution of this problem is given by ȳ = 0 in Ω ω, p = y d in Ω ω, ū = g on Γ ω. 1

24 Figure 1: Ω 3/π with ungraded and graded mesh (µ = ) The function ȳ vanishes, but p and ū have exactly the singular behaviour discussed above. To solve the optimal control problem we have used an active set strategy. The discrete solutions of the PDEs have been computed by using a finite element method with graded meshes as discussed in Section 3, see Figure 1. In Table 1 one can find the computed errors ū ũ h L (Γ ω) and the experimental order of convergence (eoc) for ω = 3/π and different mesh grading parameters when the mesh parameter h becomes small. On uniform meshes (µ = 1) the order of convergence for this angle is 1/ + λ = 1.16, compare Theorem 6. in [5]. We can see that the convergence rate can be improved by using the described mesh grading technique. For a mesh grading parameter µ = < /3 = λ we have a convergence order of about One can also see that the mesh grading parameter µ = < 7/9 = 1/3 + λ/3 would suffice to get the proven order of convergence 3/. This weaker condition was already stated in Lemma 4.9. Tracing through the proofs of the fully discrete approach as well as of the variational approach reveals that we only used the condition µ < λ to get p p h L (Γ) ch 3/. The corresponding L (Ω)-error already has quadratic convergence on such meshes, compare Lemma 4.1. We do not need mesh grading techniques at all to prove the L (Ω)-convergence rate 3/. Thus, the finite element error estimate on the boundary seems to be not sharp. For the corresponding interpolation error one can easily show the desired estimate p I h p L (Γ) ch 3/ for µ < 1/3 + λ/3, but the transfer to the finite element error is still an open question. Numerical experiments indicate also that an approximate rate close to can be obtained for mesh grading with µ < 1/4 + λ/. This condition can be expected from the consideration of the interpolation error on the boundary. It is stronger than the condition µ < λ but weaker than the condition µ < λ/ which is expected to be necessary and sufficient for providing an L (Ω)-error estimate of order ɛ which also would yield an

25 number of dof µ = 1 µ = µ = Ω Γ value eoc value eoc value eoc e e e e e e e e e e e e e e e e e e e e e Table 1: L -error of the postprocessed control ũ h for ω = 3π/ L (Γ)-error estimate. But as in the discussion before there are missing results in the finite element approximation on the boundary for elliptic equations. Acknowledgement The authors would like to thank unar Matthies for providing the finite element library MooNMD []. They further acknowledge the support of the DF priority program 153 Optimization with partial differential equations. References [1] Th. Apel, A. Rösch, and D. Sirch. L -error estimates on graded meshes with application to optimal control. SIAM J. Control Optim., 48(3): , 009. [] Th. Apel, A. Rösch, and. Winkler. Optimal control in non-convex domains: a priori discretization error estimates. CALCOLO, 44(3): , 007. [3] Th. Apel, A. Sändig, and J. Whiteman. raded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci., 19:63 85, [4] Th. Apel and D. Sirch. L -error estimates for the Dirichlet- and Neumann problem on anisotropic finite element meshes. SPP 153, Preprint SPP , 008. [5] Th. Apel, D. Sirch, and. Winkler. Error estimates for control constrained optimal control problems: Discretization with anisotropic finite element meshes. DF Priority Program 153, Preprint SPP , 008. [6] Th. Apel and. Winkler. Optimal control under reduced regularity. Appl. Numer. Math., 59: , 009. [7] N. Arada, E. Casas, and F. Tröltzsch. Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl., 3():01 9, 00. 3

26 [8] T. Babuška, R. B. Kellogg, and Pitkäranta. Direct and inverse error estimates for finite elements with mesh refinements. Numer. Math., 33: , [9] S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods, volume 15 of Texts in Applied Mathematics. Springer, third edition, 008. [10] E. Casas. Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems. Adv. Comput. Math., 6: , 007. [11] E. Casas and M. Mateos. Error estimates for the numerical approximation of Neumann control problems. Computational Optimization and Applications, 39(3):65 95, 008. [1] E. Casas, M. Mateos, and F. Tröltzsch. Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Computational Optimization and Applications, 31():193 19, 005. [13] E. Casas and F. Tröltzsch. Error estimates for linear-quadratic elliptic control problems. Analysis and optimization of differential systems. Kluwer Academic Publisher, 003. [14] P.. Ciarlet. Basic error estimates for elliptic problems. Handbook of numerical analysis, :17 351, [15] M. Dauge. Elliptic boundary value problems on corner domains - smoothness and asymptotics of solutions, volume 1341 of Lecture Notes in Mathematics. Springer- Verlag, Berlin, Smoothness and asymptotics of solutions. [16] T. Dupont and R. Scott. Polynomial approximation of functions in Sobolev spaces. Mathematics of Computation, 34(150): , [17] M. Falk. Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl., 44:8 47, [18] T. eveci. On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO Anal. Numér., 13:313 38, [19] P. risvard. Elliptic Problems in Nonsmooth Domains. Pitman, Boston, [0] M. Hinze. A variational discretization concept in control constrained optimization: the linear-quadratic case. Computational Optimization and Applications, 30(1):45 61, 005. [1] M. Hinze and U. Matthes. A note on variational discretization of elliptic Neumann boundary control. Hamburger Beiträge zur Angewandten Mathematik, 008-1, 008. [] V. John and. Matthies. MoonNMD a program package based on mapped finite element methods. Computing and Visualization in Science, 6: ,

27 [3] V. A. Kozlov, V.. Maz ya, and J. Roßmann. Elliptic Boundary Value Problems in Domains with Point Singularities. American Mathematical Society, Providence, Rhode Island, [4] K. Malanowski. Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems. Appl. Math. Optim., 8:69 95, 198. [5] M. Mateos and A. Rösch. On saturation effects in the Neumann boundary control of elliptic optimal control problems. Computational Optimization and Applications, puplished online 009. [6] C. Meyer and A. Rösch. L -estimates for approximated optimal control problems. SIAM J.Control and Optimization, 44: , 005. [7] C. Meyer and A. Rösch. Superconvergence properties of optimal control problems. SIAM J. Control Optim., 43(3): , 005. [8] L. A. Oganesyan, L. Rukhovets, and V. Rivkind. Variational-difference methods for solving elliptic equations, volume 8 of Differential equations and their applications. Izd. Akad. Nauk Lit. SSR, Vilnius, [9]. Raugel. Résolution numéique de problèmes elliptiques dans des domaines avec coins. PhD thesis, Université Rennes (France), [30] A. Rösch. Error estimates for parabolic optimal control problems with control constraints. Z. Anal. Anwend., 3: , 004. [31] A. Rösch. Error estimates for linear-quadratic control problems with control constraints. Optim. Methods Softw., 1(1):11 134, 006. [3] J. Roßmann. ewichtete Sobolev-Slobodetskij-Räume und Anwendungen auf elliptische Randwertprobleme in ebieten mit Kanten. Habilitationsschrift, Universität Rostock, [33] J. Wloka. Partielle Differentialgleichungen. B.. Teubner, 198. [34] V. Zaionchkovskii and V. A. Solonnikov. Neumann problem for second-order elliptic equations in domains with edges on the boundary. Journal of Mathematical Sciences, 7(): ,