Priority Program 1253

Transcription

1 Deutsce Forscungsgemeinscaft Priority Program 53 Optimization wit Partial Differential Equations K. Deckelnick, A. Günter and M. Hinze Finite element approximation of elliptic control problems wit constraints on te gradient April 7 Preprint-Number SPP53-8- ttp://

2

3 Finite element approximation of elliptic control problems wit constraints on te gradient Klaus Deckelnick, Andreas Günter & Micael Hinze Abstract: We consider an elliptic optimal control problem wit control constraints and pointwise bounds on te gradient of te state. We present a tailored finite element approximation to tis optimal control problem, were te cost functional is approximated by a sequence of functionals wic are obtained by discretizing te state equation wit te elp of te lowest order Raviart omas mixed finite element. Pointwise bounds on te gradient variable are enforced in te elements of te triangulation. Controls are not discretized. Error bounds for control and state are obtained in two and tree space dimensions. A numerical example confirms our analytical findings. Matematics Subject Classification (): 49J, 49K, 35B37 Keywords: Elliptic optimal control problem, state constraints, error estimates Introduction In steel and glass production cooling of melts forms a critical process. In order to accelerate te production process it is igly desirable to speed up te cooling processes wile avoiding damage of te products caused by large material stresses. In model based optimization, cooling processes frequently are described by systems of partial differential equations involving te temperature as a system variable, so tat large (Von Mises) stresses in te optimization process can be avoided by imposing pointwise bounds on te gradient of te temperature. o solve tese kinds of optimization problems numerically it is necessary to use derivative based optimization metods wic make use of adjoint variables. is fact ten necessitates te development of tailored discrete concepts wic take into account te low regularity of adjoint variables and multipliers involved in te optimality conditions of te underlying optimization problem. In te present work we consider a model problem wic involves te optimal control of a linear elliptic pde in te presence of pointwise bounds on te controls and on te gradient of te state. Our aim is to develop and to analyze a finite element concept wic is tailored to te numerical treatment of pointwise bounds, and at te same time is able to cope wit te low regularity of multipliers. o tis purpose we propose an approximation of te state equation using te lowest order Raviart omas mixed finite element, wile controls are not Institut für Analysis und Numerik, Otto von Guericke Universität Magdeburg, Universitätsplatz, 396 Magdeburg, Germany Scwerpunkt Optimierung und Approximation, Universität Hamburg, Bundesstraße 55, 46 Hamburg, Germany. Scwerpunkt Optimierung und Approximation, Universität Hamburg, Bundesstraße 55, 46 Hamburg, Germany.

4 discretized explicitely, but implicitly troug te optimality conditions associated wit te discrete approximation to te optimal control problem. Our main result reads u u + y y C log, and is proved in eorem 4.. Here, y,u and y,u denote te unique solutions of te optimal control problems (.4) and (3.6), respectively. Let us briefly comment on related literature. In [3] Casas and Fernandez investigate optimal control of semilinear elliptic pdes wit pointwise constraints on te gradient of te state. ey provide a complete analysis including results on te structure and on te regularity of multipliers. Numerical analysis for general semilinear elliptic control problems involving finitely many state constraints is provided by Casas and Mateos in [4]. A finite element analysis for elliptic optimal control problems wit pointwise bounds on te state is presented by te first and tird autor in [5], and is extended to te case of general constraints on te control and pointwise constraints on te state in [6]. Meyer in [] presents a finite element analysis for elliptic optimal control problems in te presence of pointwise bounds on te control and state, were e investigates piecewise constant approximations of te control. o te best of te autors knowledge tis is te first contribution to finite element analysis for elliptic control problems wit pointwise bounds on te gradient of te state. Matematical setting Let Ê d (d =,3) be a bounded domain wit a smoot boundary and consider te differential operator d ( ) Ay := xj aij y xi + a y, i,j= were for simplicity te coefficients a ij and a are assumed to be smoot functions on. In wat follows we assume tat a ij = a ji, a in and tat tere exists c > suc tat d a ij (x)ξ i ξ j c ξ for all ξ Ê d and all x. i,j= From te above assumptions we infer tat for a given u L r () ( < r < ) te elliptic boundary value problem Ay = u in y = on as a unique solution y W,r () W,r (). Furtermore, (.) y W,r C u L r, (.) were L r and W k,r denote te usual Lebesgue and Sobolev norms. By tracing te dependence on r in te above inequality it is possible to prove tat were C is independent of r. y W,r Cr u L, (.3)

5 Next, we formulate te control problem to be considered. Let α > and y L () be given and consider min J(u) = y y + α u u K (.4) were y solves (.) and y C. Here, K = {u L () a u b a.e. in }, C = {z C ( ) d z(x) δ,x }, were a < b, δ > are constants and denotes te Euclidian norm in Ê d. Note tat, since K L r () for r > d we ave y W,r () and ence y C ( ) d by a well known embedding result. Finally we suppose tat te following Slater condition olds: û K ŷ(x) < δ, x were ŷ solves (.) wit u = û. (.5) Since û is feasible for (.4) we deduce from eorem 3 in [3], tat te above control problem as a unique solution u K. In order to formulate te optimality conditions we introduce M( ) as te space of regular Borel measures, te dual space of C ( ). e norm on M( ) is given by µ M( ) = sup fdµ. f C ( ), f eorem.. An element u K is a solution of (.4) if and only if tere exist µ M( ) d and p L t () (t < d d ) suc tat paz = (y y )z + z dµ z W,t () W,t () (.6) (p + αu)(ũ u) ũ K (.7) Here, y is te solution of (.) and t + t =. (z y) dµ z C. (.8) Remark.. Lemma in [3] sows tat te vector valued measure µ appearing in eorem. can be written in te form µ = y µ, δ were µ M( ) is a nonnegative measure tat is concentrated in te set {x y(x) = δ}. Our aim is to develop and analyze a finite element approximation of problem (.4). We start by approximating te cost functional J by a sequence of functionals J were is a mes parameter related to a sequence of triangulations. Since p as very little regularity we propose to use a mixed finite element metod based on te Raviart omas element of lowest order. It is a specialty of our approac tat it avoids explicit discretization of te controls. is procedure is motivated by te fact tat te structure of te discrete analogue to (.7) already induces a discrete structure on te control troug te discretization of te adjoint state p, compare Remark

6 3 Finite element discretization It is well known tat (.) can be written in mixed formulation. o tis purpose we introduce H(div,) := {w L () d divw L ()} and denote v = A y, were A(x) = ( a ij (x) ) d i,j=. en (y,v) satisfies A v w + y divw = w H(div, ) (3.) z divv a y z + uz = z L (). (3.) In wat follows it will be convenient to write (y,v) = G(u) for te solution of (3.), (3.). Next, let be a triangulation of wit maximum mes size := max diam(). We suppose tat is te union of te elements of ; boundary elements are allowed to ave one curved face. In addition, we assume tat te triangulation is quasi-uniform in te sense tat tere exists a constant κ > (independent of ) suc tat eac is contained in a ball of radius κ and contains a ball of radius κ. As already mentioned above we use a mixed finite element metod based on te lowest order Raviart omas element. Let V := R (, ) := {w H(div,) w R () for all }, were R () = {w : Ê d w(x) = a + βx,a Ê d,β Ê}. Furtermore, let Y := {z L () z is constant on eac }. e variational formulation (3.), (3.) gives rise to te following discrete approximation of G. For a given function u L () let (y,v ) = G (u) Y V be te solution of A v w + y divw = w V (3.3) z divv a y z + uz = z Y. (3.4) It is well known ([]) tat te difference between (y,v) = G(u) and (y,v ) = G (u) can be estimated as follows: y y + v v C ( y H + A y H ) C y H C u (3.5) by (.). In wat follows it will be crucial to control te error between v and v in L (). Lemma 3.. Let (y,v) = G(u) and (y,v ) = G (u). en y y L + v v L C log u L. Proof. see [9], Corollary 5.5, were te result is proved for te model problem a ij = δ ij and a =, but it can be extended to te general case using tecniques developed in [8]. Remark 3.. More recently, localized pointwise error estimates for general second order elliptic equations on smoot domains were proved in [7]. Next define C := {c : Ê d c is constant and c δ, }. 4

7 We approximate (.4) by te following control problem depending on te mes parameter : min J (u) := y y + α u u K ( (3.6) subject to (y,v ) = G (u) and A v C. ) Here, =. We note tat te control is not discretized in (3.6). is problem represents a convex infinite dimensional optimization problem of similar structure as problem (.4), but wit only finitely many constraints on te state. Lemma 3.3. ere exists > suc tat problem (3.6) as a unique solution u K for <. Furtermore, tere are µ Ê d, and (p,χ ) Y V suc tat wit (y,v ) = G (u ) we ave A χ w + p divw + µ A w = w V (3.7) z divχ a p z + (y y )z = z Y. (3.8) (p + αu )(ũ u ) ũ K (3.9) ( µ c A ) v c C. (3.) Proof. We first prove tat û from (.5) is feasible for (3.6). Let (ŷ,ˆv) = G(û) and (ŷ, ˆv ) = G (û). For we deduce wit te elp of Lemma 3. and (.5) A ˆv A (ˆv ˆv) + A ˆv C ˆv ˆv L + max ŷ(x) (3.) x C log + max ŷ(x) ( ǫ)δ, x ( ) for some ǫ > and <, so tat A ˆv C. e result now follows from [3, eorem 7] wit te coices U = L (), K U and C Z := Ê N Ê d, were N is te number of triangles in. ( Remark 3.4. We deduce from (3.9) tat u = P K p α ), were PK denotes te ortogonal projection in L ( () onto K. e structure of K ten yields u (x) = P [a,b] p (x)) α for x, were P [a,b] denotes te pointwise projection onto te interval [a,b]. Hence, te discrete solution is also a piecewise constant function. Similarly to Remark. we ave Lemma 3.5. e multiplier (µ ) satisfies µ = µ δ A v,. 5

8 Proof. Fix. e assertion is clear if µ =. Suppose tat µ and define c : Ê d by c := A v,, δ µ µ, =. Clearly, c C so tat (3.) implies µ (δ µ µ A ) v, and terefore δ µ µ A v δ µ. Hence we obtain µ µ = δ A v and te lemma is proved. As a consequence of Lemma 3.5 we immediately infer tat µ = µ δ A v,. (3.) We now use (3.) in order to derive an important a priori estimate. Lemma 3.6. Let u K be te optimal solution of (3.6) wit corresponding state (y,v ) Y V and adjoint variables (p,χ ) Y V, µ,. en y, µ C for all <. Proof. Combining (3.) wit (3.) we deduce µ A (v ˆv ) δ µ ( ǫ)δ µ = ǫδ µ. Coosing w = v ˆv in (3.7) and using te symmetry of A as well as te definition of G we ence obtain ǫδ µ µ A (v ˆv ) = A χ (v ˆv ) p div(v ˆv ) = (y ŷ )divχ a (y ŷ )p + (u û)p. If we use z = y ŷ in (3.8) and ũ = û in (3.9) we finally deduce ǫδ µ (y y )(y ŷ ) + α u (û u ) y α u + C ( y + ŷ + û) and te result follows. Remark 3.7. For te measure µ M( ) d defined by f dµ := µ fdx, f C ( ) d, it follows immediately tat µ M( ) d C, <. 6

9 4 Error analysis eorem 4.. Let u and u be te solutions of (.4) and (3.6) wit corresponding states y and y respectively. en for all <. u u + y y C log Proof. Inserting ũ = u into (.7) and ũ = u into (3.9) we derive α u u p(u u) + p (u u ) I + II. (4.3) In order to treat te first term we introduce (y,v ) = G(u ) and note tat Lemma 3. yields v v L C log u L C log, (4.4) since u K. Recalling (.6) we ave I = p ( Ay Ay ) = (y y )(y ( y) + y y ) dµ = (y y )(y ( y) + Pδ ( y ) y ) dµ + ( y P δ ( y ) ) dµ were P δ denotes te ortogonal projection onto B δ () = {x Ê d x δ}. Note tat P δ (x) P δ ( x) x x x, x Ê d. (4.5) Since x P δ ( y (x)) C we infer from (.8) I (y y )(y y) + max y (x) P δ ( y (x)) µ M( ) d. (4.6) x Let x, say x for some. Since u is feasible for (3.6) we ave tat A v B δ () so tat (4.5) implies y (x) P δ ( y (x)) y (x) A v + P δ ( y ( (x)) P δ A ) v y (x) A v. (4.7) Using a well known interpolation estimate along wit (.3) we obtain y (x) A v y (x) y + A (v v ) C d r y W,r + C v v L Cr d r u L + C v v L 7

10 for r > d. us, we deduce after coosing r = log and recalling Lemma 3. y (x) A v C log, wic combined wit (4.6) and (4.7) yields I (y y )(y y) + C log. (4.8) Next, let us introduce (ỹ,ṽ ) := G (u) Y V. Using (3.4) and (3.7) we infer for te second term II = p div ( ) ) ṽ v + a p (ỹ y = A χ (ṽ ) v + µ A ( ) ) ṽ v + a p (ỹ y = A χ (ṽ ) ) v + a p (ỹ y + ( µ (P δ A ) ) ṽ A v + ( µ A ( ṽ P δ A ) ) ṽ. Since (P δ ( A ṽ ) ) C we deduce from (3.) tat II A χ (ṽ ) v + ) a p (ỹ y + max A ( ṽ P δ A ) ṽ µ. In order to estimate te last term we note tat y C implies tat ( y) = ( A v ) C and ence again by Lemma 3. A ( ṽ P δ A ) ṽ A (ṽ v) + ( P δ A (ṽ v) ) wic combined wit Lemma 3.6 yields II A χ (ṽ ) v + C ṽ v L C log, a p (ỹ y ) + C log. e symmetry of A, (3.3) and (3.8) ten give ) ) II (ỹ y divχ + a p (ỹ y + C log = (y y ) ( ) ỹ y + C log. (4.9) 8

11 Inserting (4.8) and (4.9) into (4.3) we finally obtain α u u (y y ) ( y y ) + (y y ) ( ) ỹ y + C log = y y ( + (y y )(y ỹ ) + (y y )(y y ) ) + C log y y + C ( y ỹ + y y ) + C log y y + C ( u + u ) + C log in view of (3.5) and te result follows. 5 Numerical example We consider (.4) wit te coices = B () Ê, α =, K = {u L () u a.e. in }, C = {z C ( ) z(x),x } as well as y (x) := 4 + ln 4 x, x, ln ln x, < x. In order to construct a test example we allow an additional rigt and side f in te state equation and replace (.) by were y = f + u in y = on,, x, f(x) :=, < x. e optimization problem ten as te unique solution, x u(x) =, < x wit corresponding state y y. We note tat te bounds on te control are not active, so tat we obtain equality in (.7), i.e. p = u. Furtermore, te measure µ is given by µ = xl B (). For te numerical solution we use te routine fmincon contained in te Matlab optimization toolbox. e state equation was approximated wit te elp of te Matlab implementation of te lowest order Raviart omas element provided by []. For an error functional E() we define te experimental order of convergence by In able we investigate te error functionals EOC = ln E( ) ln E( ) ln ln. E u () := u u, E y () := y y, and E P y () := y yp, 9

12 u u E u () y y E y () y y P EP y () e e e e e e e e e e e e-.84 able : Errors and EOCs for te controls, te state and te piecewise linearly post processed state µ able : Beaviour of te discrete multipliers were te superscript P is assigned to te piecewise linearly post processed state associated to u. It turns out tat te controls sow te beaviour predicted by eorem 4., wereas te L Norm of te state seems to converge linearly. e post processed state sows te same order of convergence, but as a smaller error. In able we display te values of µ, were (µ ) is given by (3.). ese values are expected to converge to µ ( ) = π as. In Figs. 5 we present te numerical approximations y,y P,u,v and µ on a grid containing m = 89 gridpoints. Fig. 5 clearly sows tat te support of µ is concentrated around x =. Acknowledgements e autors acknowledge support of te DFG Priority Program 53 troug grants DFG6-38 and DFG6-38. x Figure : State: Piecewise constant (left), and error (rigt)

13 x Figure : State: Post processed (left), and error to piecewise constant approximation (rigt) Figure 3: Control: discrete solution (left), error (rigt) Figure 4: v : first component (left), second component (middle) and vector-field (rigt) Figure 5: Measure: µ (left), µ (rigt).

14 References [] Bariawati, C., Carstensen, C.: ree Matlab Implementations Of e Lowest-Order Raviart-omas MFEM Wit A Posteriori Error Control, Computational Metods in Applied Matematics 5, (5). Software download at [] Brezzi, F., Fortin, M.: Mixed and ybrid finite element metods, Springer Series in Computational Matematics, 5, Springer Verlag, New York, 99. [3] Casas, E., Fernández, L.: Optimal control of semilinear elliptic equations wit pointwise constraints on te gradient of te state, Appl. Mat. Optimization 7, (993). [4] Casas, E., Mateos, M.: Uniform convergence of te FEM. Applications to state constrained control problems. Comp. Appl. Mat. (). [5] Deckelnick, K., Hinze, M.: Convergence of a finite element approximation to a state constrained elliptic control problem, MAH-NM--6, Institut für Numerisce Matematik, U Dresden (6). [6] Deckelnick, K., Hinze, M.: A finite element approximation to elliptic control problems in te presence of control and state constraints, Hamburger Beiträge zur Angewandten Matematik, Preprint HBAM7- (7). [7] Demlow, A.: Localized pointwise error estimates for mixed finite element metods, Mat. Comp. 73, (4). [8] Gastaldi, L., Nocetto, R.H.: On L accuracy of mixed finite element metods for second order elliptic problems, Mat. Apl. Comput. 7, 3 39 (988). [9] Gastaldi, L., Nocetto, R.H.: Sarp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations. RAIRO Modél. Mat. Anal. Numér. 3, 3 8 (989). [] Meyer, C.: Error estimates for te finite element approximation of an elliptic control problem wit pointwise constraints on te state and te control, WIAS Preprint 59 (6).