Convergence of a finite element approximation to a state constrained elliptic control problem

Transcription

1 Als Manuskript gedruckt Technische Universität Dresden Herausgeber: Der Rektor Convergence of a finite element approximation to a state constrained elliptic control problem Klaus Deckelnick & Michael Hinze MATH-NM--6 January 6

2 Contents Introduction Finite element discretization 3 3 Error analysis 6 Numerical examples 5 Appendix

3 Convergence of a finite element approximation to a state constrained elliptic control problem Klaus Deckelnick Institut für Analysis und Numerik, Otto von Guericke Universität Magdeburg, D 396 Magdeburg, Germany Klaus.Deckelnick@mathematik.uni-magdeburg.de Michael Hinze Institut für Numerische Mathematik, Technische Universität Dresden D 69 Dresden, Germany, hinze@math.tu-dresden.de Abstract: We consider an elliptic optimal control problem with pointwise state constraints. The cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of linear finite elements and enforcing the state constraints in the nodes of the triangulation. The corresponding minima are shown to converge in L to the exact control as the discretization parameter tends to zero. Furthermore, error bounds for control and state are obtained both in two and three space dimensions. Finally, we present numerical examples which confirm our analytical findings. Mathematics Subject Classification ): 9J, 9K, 35B37 Keywords: Elliptic optimal control problem, state constraints, error estimates Introduction The aim of this paper is to analyze a finite element discretization of a control problem with pointwise state constraints. Let R d d =, 3) be a bounded, convex domain with a smooth boundary. For a given function u L ) we denote by y = Gu) the solution of the Neumann problem y + y = u in, ν y = on. Here ν denotes the outward pointing unit normal to. It is well known that y H ) and.) y H C u L. We now consider the following control problem min Ju) = y y + α u u.) u L ) subject to y = Gu) and yx) bx) in. Here, α > and y, u H ) as well as b W, ) are given functions. We denote by M ) the space of Radon measures which is defined as the dual space of C ) and endowed with the norm µ M ) = fdµ. sup f C ), f

4 The analysis of.) is well understood and sketched in [, Section 6..] for the problem under consideration. Since the state constraints form a convex set and the cost functional is quadratic it is not difficult to establish the existence of a unique solution u L ) to this problem. Moreover, from [3, Theorem 5.] we infer compare also [, Theorem ]) Theorem.. A function u L ) is a solution of.) if and only if there exist µ M ) and p L ) such that with y = Gu) there holds p v + v ).3) = y y )v + vdµ v H ) with ν v = on.) p + αu u ) = a.e. in.5) µ, yx) bx) a.e. in and b y)dµ =. The study of.) is complicated by the presence of the measure µ on the right hand side of.3). As a consequence, the solution p of this problem is no longer in H ) but only in W,s ) for all s < d. This fact also accounts for the form of the weak formulation d.3). The aim of the present paper is to develop a finite element approximation of problem.). The underlying idea consists in approximating the cost functional J by a sequence of functionals J h where h is a mesh parameter related to a sequence of triangulations. The definition of J h involves the approximation of the state equation by linear finite elements and enforces constraints on the state in the nodes of the triangulation. We shall prove that the minima of J h converge in L to the minimum of J as h and that the states convergence strongly in H as well as uniformly and derive corresponding error bounds. To the authors knowledge only few attempts have been made to develop a finite element analysis for state constrained elliptic control problems. In [] Casas proves convergence of finite element approximations to optimal control problems for semi-linear elliptic equations with finitely many state constraints. Casas and Mateos extend these results in [5] to a less regular setting for the states and prove convergence of finite element approximations to semi-linear distributed and boundary control problems. Let us comment on further approaches that tackle optimization problems for pdes with state constraints. A Lavrentiev-type regularization of problem.) is investigated in []. In this approach the state constraint y b in.) is replaced by the mixed constraint ɛu + y b, with ɛ > denoting a regularization parameter. It turns out that the associated Lagrange multiplier µ ɛ belongs to L ). The resulting optimization problems are solved either by interior-point methods or primal-dual active set strategies, compare []. The development of numerical approaches to tackle.) is ongoing. An excellent overview can be found in [8, 9], where also further references are given. The paper is organized as follows: in we describe our discretization and establish convergence of controls and states to their continuous counterparts for two- and three dimensional domains. An error analysis is carried out in 3. We obtain Oh ɛ ), if d = u u h L, y y h H = Oh ɛ ), if d = 3 ɛ > arbitrary) where u h and y h are the discrete control and state respectively. Roughly speaking, the idea is to insert the discrete solution into the continuous functional and vice

5 versa. An important tool in the analysis is the use of L error estimates for finite element approximations of the Neumann problem developed in [3]. The need for uniform estimates is due to the presence of the measure µ in.3). Finite element discretization Let T h be a triangulation of with maximum mesh size h := max T Th diamt ) and vertices x,..., x m. We suppose that is the union of the elements of T h so that element edges lying on the boundary are curved. In addition, we assume that the triangulation is quasi-uniform in the sense that there exists a constant κ > independent of h) such that each T T h is contained in a ball of radius κ h and contains a ball of radius κh. Let us define the space of linear finite elements, X h := {v h C ) v h is a linear polynomial on each T T h }. In what follows it is convenient to introduce a discrete approximation of the solution operator G. For a given function v L ) we denote by z h = G h v) X h the solution of the discrete Neumann problem ) zh v h + z h v h = vv h for all v h X h. It is well known that for all v L ).).) Gv) G h v) Ch v, Gv) G h v) L Ch d v. Here, denotes the L norm. We propose the following approximation of the control problem.):.3) min J h u) := y h P h y + α u P h u u L ) subject to y h = G h u) and y h x j ) bx j ) for j =,..., m. Here, P h denotes the L projection, i.e..) P h z v h = It is well known that z v h v h X h..5) z P h z Ch z H z H ). Problem.3) represents a convex infinite-dimensional optimization problem of similar structure as problem.), but with only finitely many equality and inequality constraints which form a convex admissible set. Again we can apply [3, Theorem 5.] which together with [, Corollary ] yields compare also the analysis of problem P) in []) 3

6 Lemma.. Problem.3) has a unique solution u h L ). There exist µ,..., µ m R and p h X h such that with y h = G h u h ) and µ h = m j= µ jδ xj we have ).6) ph v h + p h v h = y h P h y )v h + v h dµ h for all v h X h,.7) p h + αu h P h u ) = in, ).8) µ j, y h x j ) bx j ), j =,..., m and Ih b y h dµh =. Here, δ x denotes the Dirac measure concentrated at x and I h is the usual Lagrange interpolation operator. Remark.. From.7) we deduce that in problem.3) it is sufficient to minimize over controls u X h instead of u L ) in order to obtain the same unique solution u h. For the resulting finite dimensional optimization problem the result of Lemma. then follows from e.g. [, Theorem.]. We have the following convergence result. Theorem.3. Let u h L ) be the optimal solution of.3) with corresponding state y h X h and adjoint variables p h X h and µ h M ). Then, as h we have u h u in L ), y h y in H ) and in C ), where u is the solution of.) with corresponding state y. Proof. Let b := min x bx). Since b = G h b) and b bx j ) for j =,..., m we have y h P h y + α u h P h u = J h u h ) J h b) Cy, u, b). This implies that there exists a constant C which is independent of h such that.9) y h, u h, p h C for all < h. Note that the bound on p h follows from.7). In order to estimate µ h we use v h in.6) and obtain for every f C ), f fdµ h by.9). This yields m µ j fx j ) j= m µ j = j= dµ h =.) µ h M ) C for all < h. ph + P h y y h ) C In view of.9),.) there exists a sequence h and û, ˆp L ) as well as ˆµ M ) such that.) u h û, p h ˆp in L ), and µ h ˆµ in M ).

7 Since G is compact as an operator from L ) into C ) we have, after passing to a further subsequence if necessary,.) Gu h ) Gû) in C ) and hence y h Gû) L G h u h ) Gu h ) L + Gu h ) Gû) L Ch d uh + Gu h ) Gû) L so that y h Gû) =: ŷ in C ) as h by.9) and.). A similar argument shows that y h ŷ in H ). Let us now pass to the limit in.6).8). To begin, let v H ) with ν v = on and denote by R h v the Ritz projection of v. Recalling.),.6) and the fact that R h v v in C ) we obtain ˆp v + v ) ) p h v + v = ph v + p h v ) = ph R h v + p h R h v ) = y h P h y )R h v + R h vdµ h ŷ y )v + vdˆµ. Using.) we may pass to the limit in.7) and deduce ˆp+αû u ) = a.e. in. Clearly, ˆµ ; since y h I h b in and y h ŷ in C ) we have ŷ b in. Furthermore, recalling that I hb y h )dµ h = we obtain in the limit b ŷ)dˆµ =. Lemma. now implies that û is a solution of.); as the solution of this problem is unique we must have u = û and hence y = ŷ and the whole sequence is convergent. Let us finally prove that u h u in L ). To begin, note that by.) G h u γh d ) = Gh u) Gu) + Gu) γh d Ch d u + b γh d b in, provided that γ is large enough. Evaluating the above inequality at the nodes x,..., x m we see that G h u γh d ) is admissible for the discrete problem and hence J h u h ) J h u γh d ) or α u h P h u α d u γh Ph u + G hu) γh d Ph y y h P h y. Since y h y, G h u) Gu) = y in L ) we infer that lim sup u h P h u u u lim inf u h P h u, h h where the second inequality is a consequence of the weak convergence u h P h u u u. Thus, u h P h u u u which implies u h P h u u u in L and hence u h u in L. 5

8 3 Error analysis Let us now turn to the error analysis and start with a couple of auxiliary results. Lemma 3.. Suppose that u, u h L ) are the optimal solutions of.) and.3) respectively with corresponding states y H ), y h X h. Let v L ) and z = Gv), z h = G h v). Then Ju) + z y + α 3.3) v u + b z)dµ = Jv) J h u h ) + z h y h + α ) 3.) v u h + Ih b z h dµh = J h v) Proof. An elementary calculation using.3) shows Jv) Ju) = z y + α v u + z y)y y ) + α u u )v u) = z y + α v u + p z y) + z y) ) z y)dµ + α u u )v u). Since z = Gv), y = Gu) we have p z y) + z y) ) = pv u), so that.) and.5) finally imply Jv) Ju) = z y + α v u + b z)dµ. The second claim follows in a similar way. Remark 3.. Note that in the above z = Gv), z h = G h v) do not necessarily have to be admissible for the minimization problems. The next lemma examines in more detail the approximation of J by J h. Lemma 3.3. Suppose that v W,s ) for some d d+ s. Then Jv) J h v) Ch + d d s u H v W,s + v + y H + u H ). Proof. Let z = Gv), z h = G h v). Then Jv) J h v) = z y z h P h y ) + α v u v P h u ). 6

9 Using.),.5),.) and.) we obtain z y z h P h y ) = z y z h + P h y )z y + z h P h y ) = z zh )z y + z h P h y ) y P h y )z y P h z y ) ) C z z h z + z h + y ) ) + Ch y H z H + y H Ch v + y H ). For the second term we obtain in a similar way v u v P h u ) = u P h u )w = u P h u )w P h w), where w = u + P h u v and where we have used.). Applying Lemma 5. from the Appendix we infer v u v P h u ) Ch + d d s u H w W,s This proves the lemma. Lemma 3.. Suppose that v W,s ) for some < s < Ch + d d s u H u H + v W,s). d. Then d Gv) G h v) L Ch 3 d s log h v W,s. Proof. Let z = Gv), z h = G h v). Elliptic regularity theory implies that z W 3,s ) from which we infer that z W,q ) with q = ds using a well known embedding theorem. d s Furthermore, we have 3.5) z W,q c z W 3,s c v W,s. Using Theorem. and the following Remark in [3] we have 3.6) z z h L c log h inf χ X h z χ L, which, combined with a well known interpolation estimate, yields z z h L ch d q log h z W,q ch 3 d s log h v W,s in view 3.5) and the relation between s and q. Our next aim is to derive a uniform bound on u h W,s for s < d d. Lemma 3.5. Let < s < d. Then there exists a constant c, which is independent of h, d such that u h W,s c for all < h. 7

10 Proof. In view of.7) we have u h W,s α p h W,s + P h u H α p h W,s + c, so that it is sufficient to bound p h W,s. Let s be such that + = and suppose that φ L s ). Let us denote by ψ W,s ) s s the unique solution of the Neumann problem ψ + ψ = φ in ν ψ = on. Integration by parts and.6) yield p h φ = ph ψ + p h ψ ) = ph R h ψ + p h R h ψ ) 3.7) = y h P h y )R h ψ + R h ψdµ h, where R h ψ is the Ritz projection of ψ. Arguing similarly as in Theorem of [] one shows that there exists a unique solution p h W,s ) of the problem 3.8) p h v + v ) = y h P h y )v + vdµ h v H ) with ν v = on. Furthermore, there exists a constant c = cs) > such that 3.9) p h W,s c y h P h y + µ h M ) ) c uniformly in h in view of.9) and.). If we use v = ψ in 3.8) and combine it with 3.7) we obtain p h p h )φ = y h P h y )ψ R h ψ) + ψ R h ψ)dµ h ch ψ H yh + P h y ) + ψ R h ψ L µ h M ) ch ψ H + ch d s log h ψ W,s ch d s log h φ L s. Note that we have again applied 3.6) in order to control ψ R h ψ L. Since φ L s ) is arbitrary we infer p h p h L s ch d s log h. Interpolation and inverse estimates then give by 3.9) and since d s p h L s c p h L s + ch d s log h c = d d s) >. s d Let us finally turn to an error estimate for the optimal controls and the optimal states. 8

11 Theorem 3.6. Let u and u h be the solutions of.) and.3) respectively. For every ɛ > there exists C ɛ > such that u u h + y y h H C ɛ h d ɛ. Proof. Let us define ỹ h := Gu h ) H ) and ỹ h := G h u) X h. Then Lemma 3. implies Ju) + ỹ h y + α u h u + b ỹ h )dµ = Ju h ) J h u h ) + ỹ h y h + α ) u u h + Ih b ỹ h dµh = J h u). Since u = u p W,s d ) for all s < d we obtain with the help of Lemma 3.3 α d+ d ỹ h y + ỹ h y h + α u h u ) 3.) = Ju h ) Ju) + J h u) J h u h ) b ỹ h )dµ Ih b ỹ h dµh ) ) Ch + d d s u H u W,s + u h W,s + u + u h + y H + u H + ỹ h b)dµ + ỹ h I h b)dµ h. Let us first consider the last two integrals. We have for x ỹ h x) bx) = ỹ h x) y h x)) + y h x) I h b)x)) + I h b)x) bx)) Gu h ) G h u h ) L + I h b b L, since y h x j ) bx j ), j =,..., m implies that y h I h b in. If we combine Lemma 3. with Lemma 3.5 we infer ỹ h b)dµ ch 3 d s log h uh W,s + Ch b W, ch 3 d s log h. Similarly we have from.5) ỹ h x) I h b)x) = ỹ h x) yx)) + yx) bx)) + bx) I h b)x)) G h u) Gu) L + b I h b L, so that.) and Lemma 3. give yh I h b ) dµ h ch 3 d s log h u W,s + Ch b W, ch 3 d s log h. Inserting these estimates into 3.) and applying again Lemma 3.5 we derive If we now choose s sufficiently close to u u h + y y h ch 3 d s log h. d d we obtain u u h + y y h C ɛ h d ɛ. 9

12 Finally, in order to obtain the error bound for y in H we note that ) y yh ) v h + y y h )v h = u u h )v h for all v h X h, from which one derives the desired estimate using standard finite element techniques and the bound on u u h. In general we only expect weak convergence of µ h to µ. Nevertheless we have the following partial result. Corollary 3.7. Let K be compact with K suppµ =. For every ɛ > there exists a constant C ɛ such that µ h K) C ɛ h d ɛ. Proof. By Lemma 5. in the Appendix there exists a nonnegative function φ C ) which satisfies φ on K, φ = on suppµ, ν φ = on. Since µ h we obtain from.6) µ h K) φ dµ h = φ R h φ)dµ h + R h φ dµ h = φ R h φ)dµ h + ph R h φ + p h R h φ ) y h P h y )R h φ = φ R h φ)dµ h + ph φ + p h φ ) y h P h y )R h φ = φ R h φ)dµ h + p h φ + φ) y h P h y )R h φ, where R h is again the Ritz projection. On the other hand,.3) and the fact that φ = on suppµ imply y y )φ p φ + φ) =. Combining this relation with the first estimate we derive µ h K) φ R h φ)dµ h + p h p) φ + φ) + + y y h y + P h y )φ y h P h y )φ R h φ) φ R h φ L µ h M ) + p p h φ H + y h + P h y ) φ R h φ + y y h + y P h y ) φ C φ R h φ L + C ɛ h d ɛ C ɛ h d ɛ in view of.),.7) and Theorem 3.6. Remark 3.8. We mention here a second approach that differs from the one discussed above in the way in which the inequality constraints are realized. Denote by D,..., D m the cells of the dual mesh. Each cell D i is associated with a vertex x i of T h and we have = m i= D i, intd i ) intd j ) =, i j.

13 In.3), we now impose the constraints 3.) y h I h b) for j =,..., m D j on the discrete solution y h = G h u). Here, we have abbreviated D j f = D j D j f. The measure µ h that appears in Lemma. now has the form µ h = m j= µ j D j dx, and the pointwise constraints in.8) are replaced by those of 3.). The error analysis for the resulting numerical method can be carried out in the same way as shown above with the exception of Theorem 3.6, where the bounds on ỹ b and ỹ h I h b require a different argument. In this case, additional terms of the form f f L D j ) D j have to be estimated. Since these will in general only be of order Oh), this analysis would only give u u h, y y h H = O h). The numerical test example in suggests that at least u u h = Oh), but we are presently unable to prove such an estimate. Numerical examples Example.. The following test problem is taken - in a slightly modified form - from [], Example 6.. Let := B ), α >, y x) := + π π x + π log x, u x) := + απ x log x απ and bx) := x +. We consider the cost functional Ju) := y y + α u u, where y = Gu). By checking the optimality conditions of first order one verifies that u is the unique solution of.) with corresponding state y and adjoint states px) = π x π log x and µ = δ. The finite element counterparts of y, u, p and µ are denoted by y h, u h, p h and µ h. For an error functional Eh) we define the experimental order of convergence as EOC = ln Eh ) ln Eh ) ln h ln h. To investigate EOCs for our model problem we choose a sequence of uniform partitions of containing five refinement levels, starting with eight triangles forming a uniform octagon as initial triangulation of the unit disc. The corresponding grid sizes are h i = i for i =,..., 5. As error functionals we take Eh) = u, y) u h, y h ) and Eh) = u, y) u h, y h ) H and note, that the error p p h is related to u u h via.7). We solve problems.3) using the QUADPROG routine of the MATLAB OPTIMIZATION TOOLBOX. The required finite

14 element matrices for the discrete state and adjoint systems are generated with the help of the MATLAB PDE TOOLBOX. Furthermore, for discontinuous functions f we use the quadrature rule fx)dx f ) x st ) T, T T h where x st ) denotes the barycenter of T. In all computations we set α =. In Table, we present EOCs for problem.3) case S = D) and the approach sketched in Remark 3.8 case S = M). As one can see, the error u u h behaves in the case S = D as predicted by Theorem 3.6, whereas the errors y y h and y y h H show a better convergence behaviour. On the finest level we have u u h =.3733, y y h =.386 and y y h H = Furthermore, all coefficients of µ h are equal to zero, except the one in front of δ whose value is The errors u u h, y y h and y y h H in the case S = M show a better EOC than in the case S = D. This can be explained by the fact that the exact solutions y and u are very smooth, and that the relaxed form of the state constraints introduce a smearing effect on the numerical solutions at the origin. On the finest level we have u u h =.98, y y h =.656 and y y h H = Furthermore, the coefficient of µ h corresponding to the patch containing the origin has the value Figures and present the numerical solutions y h and u h for h = 5 in the case S = D and S = M, respectively. We note that using equal scales on all axes would give completely flat graphs in all four figures. S=D) S=M) S=D) S=M) S=D) S=M) Level u u h u u h y y h y y h y y h H y y h H Table : Experimental order of convergence Example.. The second test problem is taken from [], Example. It reads min Ju) = y y + u u u L ) Here, denotes the unit square, bx) = subject to y = Gu) and yx) bx) in. x +, x <,, x, y x) = x, x <, x =, 3 x >,

15 Figure : Numerically computed state y h left) and control u h right) for h = 5 in the case S = D Figure : Numerically computed state y h left) and control u h right) for h = 5 in the case S = M. and u x) = 5 x, x <, 9 x. The exact solution is given by y and u in. The corresponding Lagrange multiplier p H ) is given by px) = x, x <,, x. The multiplier µ has the form.) fdµ = fds + fdx, f C ). {x = } {x > } In our numerical computations we use uniform grids generated with the POIMESH function of the MATLAB PDE TOOLBOX. Integrals containing y, u are numerically evalu- 3

16 ated by substituting y, u by their piecewise linear, continuous finite element interpolations I h y, I h u. The grid size of a grid containing l horizontal and l vertical lines is given by h l =. Fig. 3 presents the numerical results for a grid with h = in the case S=D). l+ 36 The corresponding values of µ h on the same grid are presented in Fig.. They reflect the fact that the measure consists of a lower dimensional part which is concentrated on the line {x x = } and a regular part with a density χ {x > }. We again note that using equal scales on all axes would give completely flat graphs for y h as well as for u h. We compute EOCs for the two different sequences of grid-sizes s o = {h, h 3,..., h 9 } and s e = {h, h,..., h 8 }. We note that the grids corresponding to s o contain the line x =. Table presents EOCs for s o, and Table 3 presents EOCs for s e. For the sequence s o we observe super-convergence in the case S=D), although the discontinuous function y for the quadrature is replaced by its piecewise linear, continuous finite element interpolant I h y. Let us note that further numerical experiments show that the use of the quadrature rule.) for integrals containing the function y decreases the EOC for u u h to 3, whereas EOCs remain close to for the other two errors y y h and y y h H. For this sequence also the case S=M) behaves twice as good as expected by our arguments in Remark 3.8. For the sequence s e the error u u h in the case S=D) approximately behaves as predicted by our theory, in the case S=M) it behaves as for the sequence s o. The errors y y h and y y h H behave that well, since the the exact solutions y and u are very smooth. For h 9 we have in the case S=D) u u h =.38, y y h =.333 and y y h H =.555, and in the case S=M) u u h =.77577, y y h =.585 and y y h H = We observe that the errors in the case S = M are two magnitudes larger than in the case S=D). This can be explained by the fact that an Ansatz for the multiplier µ with a linear combination of Dirac measures is better suited to approximate measures concentrated on singular sets than a piecewise constant Ansatz as in the case S=M). Finally, Table presents µ i and µ i for s o in the case S=D). As one can see µ i tends x i {x =/} x i {x >/} x i {x =/} to, the length of {x = /}, and µ i tends to /, the area of {x > /}. These x i {x >/} numerical findings indicate that µ h = m µ i δ xi well approximates µ, since dµ h = m µ i, i= and that µ h also well resolves the structure of µ, see.). For all numerical computations of this example we have µ i = for x i {x < /}. 5 Appendix Lemma 5.. Let d d+ s and v W,s ). Then Proof. The assertion is clear if s = d d+ us write v P h v = v P h v Ch + d d s v W,s. or if s = so that we may assume d d+ v P h v sd d+s s v P h v d s) s i= < s <. Let

17 Figure 3: Numerically computed state y h left) and control u h right) for h = 36 S = D. in the case Figure : Numerically computed multiplier µ h for h = 36 in the case S = D. and apply Hölder s inequality with p = s sd d+s, q = s d s) s) We infer from [6] that v P h v v P h v sd d+s s L s v P hv d s) s v P h v sd d+s s L s v L ds d s L ds d s which implies + P hv L ds d s ) d s) s. v P h v L s Ch v W,s, P h v L ds d s C v L ds d s which, together with the continuous embedding W,s ) L ds d s ), gives so that the assertion follows. v P h v ch sd d+s s v W,s 5

18 S=D) S=M) S=D) S=M) S=D) S=M) Level u u h u u h y y h y y h y y h H y y h H Table : Experimental order of convergence, x = grid line S=D) S=M) S=D) S=M) S=D) S=M) Level u u h u u h y y h y y h y y h H y y h H Table 3: Experimental order of convergence, x = not a grid line Lemma 5.. Suppose that K and K are two disjoint compact subsets of. Then there exists a nonnegative function φ C ) which satisfies ν φ = on, φ on K, φ = on K. Proof. For r > let us define r := {x distx, ) < r}. In view of the smoothness of there exists δ > such that for each x δ there exists a unique point y = yx) with x = y distx, )νy) 6

19 Level µ i µ i x i {x =/} x i {x >/} Table : Approximation of the multiplier in the case S=D), x = grid line see [7],.6). Since K K = we may assume that distk, K) > δ. Let us define Γ K := {yx) x K δ }, Γ K := {yx) x K δ }. Γ K and Γ K are disjoint, compact subsets of, since distk, K) > δ and x yx) is continuous. Let φ C ) be a nonnegative function satisfying φ on Γ K, φ = on Γ K. By setting φ x) = φ yx)) we extend φ as a C function to δ. Clearly, ν φ = on. Let ψ C ) be a nonnegative cut off function with ψ = in δ Then φ := ψφ satisfies and ψ = in \ δ. ν φ = on, φ on K δ, φ = on K. Finally, choose a nonnegative function φ 3 C ) with φ 3 on K \ δ ), φ 3 x) = if distx, K \ δ )) δ 8. Then, ν φ 3 = on, φ 3 = on K and φ := φ + φ 3 has the required properties. Acknowledgments We thank Ulrich Matthes TU Dresden) for coding the numerical examples and Alan Demlow University of Freiburg) for pointing out reference [3] to us. References [] Casas, E.: L estimates for the finite element method for the Dirichlet problem with singular data, Numer. Math. 7, ). 7

20 [] Casas, E.: Control of an elliptic problem with pointwise state constraints, SIAM J. Cont. Optim., ). [3] Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints, SIAM J. Cont. Optim. 3, ). [] Casas, E.: Error Estimates for the Numerical Approximation of Semilinear Elliptic Control Problems with Finitely Many State Constraints, ESAIM, Control Optim. Calc. Var. 8, ). [5] Casas, E., Mateos, M.: Uniform convergence of the FEM. Applications to state constrained control problems. Comp. Appl. Math. ). [6] Douglas, J., Dupont, T., Wahlbin, L.: The stability in L q of the L projection into finite element function spaces, Numer. Math. 3, ). [7] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, nd ed.). Springer, 983. [8] Hintermüller, M., K., Kunisch, K.: Path following methods for a class of constrained minimization methods in function spaces, Report RICAM-7, RICAM Linz ). [9] Hintermüller, M., K., Kunisch, K.: Feasible and non-interior path following in constrained minimization with low multiplier regularity, Report, Universität Graz 5). [] Meyer, C., Prüfert, U., Tröltzsch, F.: On two numerical methods for state-constrained elliptic control problems, Technical Report 5-5, Institut für Mathematik, TU Berlin 5). [] Meyer, C., Rösch, A., Tröltzsch, F.: Optimal control problems of PDEs with regularized pointwise state constraints, Preprint, Institut für Mathematik, TU Berlin, to appear in Computational Optimization and Applications ). [] Nocedal, J., Wright, S.J.: Nonlinear optimization. Springer Series in Operations Research, Springer 999. [3] Schatz, A.H.: Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I: Global estimates, Math. Comput. 67, No.3, ). [] Tröltzsch, F.: Optimale Steuerung mit partiellen Differentialgleichungen. Vieweg, 5. 8