Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains

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1 Finit lmnt approximation of Diriclt boundary control for lliptic PDEs on two- and tr-dimnsional curvd domains Klaus Dcklnick, Andras Güntr & Mical Hinz Abstract: W considr t variational discrtization of lliptic Diriclt optimal control problms wit constraints on t control. T undrlying stat quation, wic is considrd on smoot twoand tr-dimnsional domains, is discrtizd by linar finit lmnts taking into account domain approximation. T control variabl is not discrtizd. W obtain optimal rror bounds for t optimal control in two and tr spac dimnsions and prov a suprconvrgnc rsult in two dimnsions providd tat t undrlying ms satisfis som additional condition. W confirm our analytical findings by numrical xprimnts. Matmatics Subjct Classification 2000: Kywords: Elliptic optimal control problm, boundary control, control constraints, rror stimats 1 Introduction Diriclt boundary control plays an important rol in many practical applications suc as activ boundary control of flows. If on is intrstd in control by blowing and suction on parts of t boundary only, boundary controls wit low rgularity sould b admissibl wic vn may dvlop jump discontinuitis. In modl basd optimization wit boundary controls t flow oftn is modld wit t lp of t Navir-Stoks quations wos classical variational formulation dos not allow for Diriclt boundary data wit jump discontinuitis, s [6, 9], so tat t concpt of vry wak solutions [11] as to b applid instad, s [2] for a mor dtaild discussion. Morovr, pointwis bounds on t control actions av to b considrd in practic. In t prsnt work w considr as modl problm Diriclt boundary control of an lliptic quation wit L 2 -boundary controls subjct to pointwis bounds on t controls. T stat quation is posd on a boundd, smoot domain in R d, d = 2,3. Our aim is to dvlop and analyz a finit lmnt concpt wic is tailord to t numrical tratmnt of pointwis bounds, and at t sam tim is abl to cop wit t low rgularity of t control and t stat. To tis purpos w propos an approximation of t stat quation using picwis linar, continuous finit lmnts taking into account domain approximation. T controls ar not discrtizd xplicitly, but implicitly variationally troug t optimality conditions associatd wit t discrt optimal control problm. Our main rsult, s Torm 4.1, is an Institut für Analysis und Numrik, Otto von Gurick Univrsität Magdburg, Univrsitätsplatz 2, Magdburg, Grmany. Bric Optimirung und Approximation, Univrsität Hamburg, Bundsstraß 55, Hamburg, Grmany. Bric Optimirung und Approximation, Univrsität Hamburg, Bundsstraß 55, Hamburg, Grmany. 1

2 O log bound for t L 2 rror of optimal control and stat. In two spac dimnsions and undr additional conditions on t undrlying ms w ar abl driv t improvd rror bound O 3 2, wic rflcts a suprconvrgnc ffct. Tr ar only fw contributions to Diriclt boundary control rportd in t litratur. Casas and Raymond in [5] considr smilinar lliptic Diriclt boundary control problms wit pointwis bounds on two-dimnsional convx polygonal domains. Dnoting by u t optimal control ty ar abl to prov t optimal rsult u u 0, C 1 1/p. Hr, u dnots t optimal discrt boundary control wic ty find in t spac of picwis linar, continuous finit lmnts on, and p 2 dpnds on t smallst angl of t boundary polygon. For control functions of t form Bq := n q i f i i=1 wit givn f i H 5/2 and box-constraind q R n, Vxlr in [14] provids a finit lmnt analysis for two-dimnsional boundd polygonal domains and provs q q C 2. In a rcnt papr [12] May, Rannacr and Vxlr considr Diriclt boundary control witout control constraints on two-dimnsional convx polygonal domains, wr ty prsnt optimal rror stimats for t stat and t adjoint stat. Important ingrdints ar duality tcniqus and an optimal rror stimat in H 1/2 for t control. Our papr is organizd as follows. In t nxt sction w prsnt t matmatical stting and formulat t optimal control problm. In Sction 3 w xamin t finit lmnt discrtization of t stat quation taking into account t approximation of t domain. In Sction 4 w introduc t discrt control problm and prov an optimal rror stimat for t discrt controls. Sction 5 dals wit suprconvrgnc proprtis of boundary controls inducd by finit lmnt partitions wit crtain rgularity proprtis. In Sction 6 w finally prsnt numrical rsults wic confirm our analytical findings. For a domain or yprsurfac Q and s 0,1 p w dnot by W s,p Q t usual Sobolv spac and by s,p,q its norm. If p = 2 w writ W s,2 Q = H s Q wit norm s,q. 2 Matmatical stting Lt R d d = 2,3 b a boundd domain wit a smoot boundary := and considr t diffrntial oprator d Ay := xj aij y xi + a0 y, i,j=1 wr for simplicity t cofficints a ij and a 0 ar assumd to b smoot functions on. In wat follows w assum tat a ij = a ji, a 0 0 in and tat tr xists c 0 > 0 suc tat d a ij xξ i ξ j c 0 ξ 2 for all ξ R d and all x. i,j=1 2

3 Givn f L 2,u L 2 w considr t boundary valu problm Ay = f in, y = u on. 2.1 It is wll known tat 2.1 as a uniqu solution y H 1 2 wic w dnot by y = Gu. Not tat y solvs t problm in t sns tat yaφ = fφ u νa φ φ H 2 H0 1, 2.2 wr νa φ = d a ij φ xj ν i and ν is t outr unit normal to. i,j=1 In ordr to dfin an approximation of 2.1 w also introduc t bilinar form a : H0 1 H0 1 R associatd wit t diffrntial oprator A as d ay,z := aij y xi z xj + a 0 yz. i,j=1 Nxt, lt α > 0 and y 0 W 1, r, r > d b givn. W tn considr t Diriclt boundary control problm min Ju = 1 y y α u 2 u U ad subjct to y = Gu, wr U ad = {u L 2 a u b a.. on } and a,b R,a < b. Existnc of a uniqu solution u U ad of 2.3 follows by standard argumnts. Tis solution is caractrizd by t variational inquality y y 0 z y + α uv u 0 v U ad 2.4 wr z = Gv. Lt us introduc t adjoint stat p H 2 H0 1 as t solution of t following boundary valu problm: Ap = y y 0 in, p = 0 on. 2.5 It is not difficult to s tat t optimal control u is givn by u = P [a,b] 1 α ν A p a.. on 2.6 wr P [a,b] dnots t pointwis projction onto t intrval [a,b]. Lmma 2.1. Lt u U ad b t solution of 2.3 wit corrsponding stat y and adjoint stat p. Tn u H 1, y H 3 2, p W 3,r for som d < r r. Proof. Sinc p H 2 w av νa p H 1 2 and nc u H 1 2 in viw of 2.6 cf. [5], p wic in turn yilds y H 1. Elliptic rgularity implis tat p H 3 and tn νa p H 3 2. Trfor u H 1 and y H 3 2. Using an mbdding torm, t abov rgularity of νa p also implis tat u W 1 1 r,r for som r > d. Hnc, y W 1,r and sinc y 0 W 1, r w finally obtain p W 3,r for som d < r r again by lliptic rgularity. 3

4 3 Finit lmnt discrtization Lt T b a triangulation of a polygonal domain approximating. W assum tat all vrtics on =: also li on and tat at most on fac of a simplx T T blongs to. Furtrmor, w suppos tat t triangulation is quasi-uniform in t sns tat tr xists a constant κ > 0 indpndnt of suc tat ac T T is containd in a ball of radius κ 1 and contains a ball of radius κ, wr := max T T diamt is t maximum ms siz. For vry T T tr xists an invrtibl affin mapping F T : R d R d, F T ˆx = A T ˆx + b T, wic maps t standard d simplx ˆT onto T. Bsids t triangulation T wic will b usd to dfin t discrt problm and to carry out t practical calculations w also introduc an xact triangulation T of. T xistnc of suc a triangulation for sufficintly small is sown in [3]. In ssnc, for vry T T tr is a mapping Φ T C 3 ˆT, R d suc tat F T := F T + Φ T maps ˆT onto a curvd d simplx T and = T T. T Furtrmor, t mapping G wic is locally dfind by G T := F T F 1 T is a omomorpism btwn and. T construction in [3] also implis tat Φ T = 0 if T as at most on vrtx on so tat G id on all simplics wic ar disjoint from. Furtrmor, w av t stimats sup DG T Ix C, G 3,,T C, T T x T sup D F T ˆx C A T, ˆx ˆT sup D x T c 1 dt A T dt D F T ˆx c 2 dt A T, 1 F T x C A 1 T, ˆx ˆT. T T 3.1 Lt us nxt dfin t spac of linar finit lmnts, X := {φ C 0 φ T P 1 T,T T } as wll as X 0 := X H0 1. Lt γx b t rstriction to of functions in X and dnot by P : L 2 γx t L 2 projction, i.. for v L 2 w av vχ = P v χ χ γx. 3.2 Lt us introduc an approximation to t solution oprator G as follows. For a givn function u L 2 w dnot by y = G u X t uniqu solution of a y,φ = f φ, φ X 0, y = P u on, 3.3 wr a y,φ = d i,j=1 and a,ij = a ij G,a,0 = a 0 G and f = f G. a,ij y,xi φ,xj + a,0 y φ In ordr to dal wit t problm tat t solutions of 2.1 and 3.3 ar dfind on diffrnt domains w assign to ac φ X a function φ : R by φ := φ G 1 and lt X := { φ φ X } as wll as γ X = { φ φ X }. 4

5 Using t transformation rul, t fact tat ỹ = y G 1 DG 1 and 3.1 w obtain aỹ, φ a y,φ C ỹ 1,A φ 1,A y,φ X, 3.4 wr A = {x distx, < γ} and γ is cosn so larg tat T T A. Nxt, by adapting t mtods dvlopd in [13] it is possibl to sow tat tr xists an intrpolation oprator Π : L 1 X suc tat for φ W l,p 1 l 2 if p = 1, 1 p < l 2 otrwis φ Π φ m,p, C l m φ l,p,, 0 m min1,l. 3.5 In addition it is possibl to construct Π in suc way tat Π φ = 0 on providd tat φ = 0 on. If φ C 0 tn w can also dfin t usual Lagrang intrpolation oprator Ĩ : C 0 X via Ĩφ = [I φ G ] G 1 wr I is t Lagrang intrpolation oprator corrsponding to X. Abbrviating g := G w dfin for v L 2 t projction P v := [P v g ] g 1 γ X. In viw of Lmma 3.1 in [10] w av v = v g 1 d wr d = dtdg 1 DG T G 1 ν. 3.6 Applying 3.6 to 3.2 w s tat P is caractrizd by t rlation v χ d = P v χ d χ γ X. 3.7 Furtrmor on can sow tat v P v 0, C s v s,, v H s, 0 s An important ingrdint of our analysis will b an L 2 rror stimat for t approximation givn by 3.3, in particular for low rgularity of t boundary valus. A corrsponding rsult in t cas of A = and a polygonal domain can b found in [2]. Lmma 3.1. Suppos tat f L 2,u H s 0 s 1 and tat y H s+1 2, y X ar t solutions of 2.1 and 3.3 wit u = u g rspctivly. Tn tr xists 0 > 0 suc tat for 0 < 0 y ỹ 0, C s+1 2 u s, + f 0,. 3.9 Proof. In viw of t linarity of A it is sufficint to considr t problms wr itr f 0 or u 0. Lt us first assum tat f 0 and tak s = 1. W dnot by y H 3 2 t solution of Clarly, ay,φ = 0 φ H 1 0, y = P u on y s+ 1 2, C P u s,, 0 s Lt us coos φ = Π [y ỹ ] = Π y ỹ. Not tat φ X 0 sinc y = ỹ on. T llipticity of A and t fact tat a 0 0 imply togtr wit 3.10 and 3.3 c 0 y ỹ 2 ay ỹ,y ỹ = ay ỹ,y Π y + ay ỹ, Π y ỹ = ay ỹ,y Π y + [a y, Π y G y aỹ, Π y ỹ ] I + II. 5

6 In viw of 3.4, 3.5 and Poincaré s inquality w av II C ỹ 1, Π y ỹ 1, C y 1, + y ỹ 1, y Π y 1, + y ỹ 1, ǫ + C y ỹ 2 0, + C ǫ 3 2 y 2 3 2,. Estimating I wit t lp of 3.5 and Young s inquality w obtain for 0 < 0, 0 sufficintly small y ỹ 1, C y 3, In ordr to stimat t L 2 norm of y ỹ w mploy t usual duality argumnt, namly dnot by ψ H 2 t solution of Aψ = y ỹ in, ψ = 0 on Tn, 2.2 and intgration by parts imply tat y ỹ 2 = y ỹ Aψ = aỹ,ψ u P u νa ψ = I + II. Obsrving tat ψ,ĩψ H 1 0,I ψ G X 0 w infr from 3.3 and 3.10 I = ay ỹ,ψ Ĩψ + [ a ỹ, I ψ G + a y,i ψ G ] C 3 2 y 3 2, ψ 2, + C ỹ 1,A Ĩψ 1,A by 3.12, 3.4 and an intrpolation stimat. Nxt, using t continuous mbddings H 1 2 L 3, H 1 L 6 as wll as 3.12 w obtain ỹ 1,A y 1,A + y ỹ 1,A C A 1 6 y 1,3,A + C y 3 2, C1 6 y 3 2,, Ĩψ 1,A ψ 1,A + ψ Ĩψ 1,A C A 1 3 ψ 1,6,A + C ψ 2, C 1 3 ψ 2,. Tus, I C 3 2 y 3 2, ψ 2, C 3 2 P u 1, ψ 2, 3.14 in viw of For II w obtain wit t lp of 3.7 II = u P u νa ψ d + u P u νa ψd = u P u νa ψ P νa ψd + u P u νa ψd 1 and nc using 3.8 and 3.1 II C 3 2 u 1, νa ψ 1 2, + C2 u 1, νa ψ 0, C 3 2 u 1, ψ 2,. Combining tis bound wit 3.14, t stability of P in H 1 and a standard lliptic rgularity rsult w dduc tat y y 0, C 3 2 u 1, Lt us nxt look at t cas s = 0 and dfin ψ H 2 H0 1 again via As abov w obtain y ỹ 2 = I + II. 6

7 Using 3.14 togtr wit an invrs inquality w av I C 3 2 P u 1, ψ 2, C 1 2 P u 0, ψ 2,. Rturning to 3.15 w infr for t scond trm II C u 0, + P 1 u 0, 2 νa ψ 1 2, + 1 ν A ψ 0, C 2 u 0, ψ 2,. Combining t abov two bounds w dduc tat y ỹ 0, C 1 2 u 0, T cas 0 < s < 1 tn follows from 3.16 and 3.17 by intrpolation. If u 0,f L 2 w can procd in a similar way as abov, starting wit a bound of t form y ỹ 1, C f 0, followd by a duality argumnt to giv y ỹ 0, C 3 2 f 0,. Sinc our primary intrst lis on t boundary valus w lav t dtails to t radr. Our nxt aim is to bound t discrt solution corrsponding to f 0 in trms of u 0,. In ordr to formulat t rsult w introduc t distanc function d x := distx,. It follows from [7], 14.6 tat tr xists δ > 0 suc tat d C 3 δ, wr r := {x d x < r} for r > 0. Coos a function η C 3 suc tat 0 η 1, ηx = 1,x δ 2. Tn, ρx := ηxd x + 1 ηx δ 2, x blongs to C 3 and ηx = 0,x \ 2δ 3 satisfis Furtrmor, lt and ρx = d x, x δ, ρx δ 2 2, x \ δ ωx := ρx +, x. Lmma 3.2. Lt u L 2 and suppos tat z X is t solution of a z,φ = 0 φ X 0, z = P u g on Tn z 2 + ω z 2 C u 2 0, Proof. Lt y b again t solution of Sinc P u g = P u g and P 2 = P, Lmma 3.1 for s = 0 implis tat Combining tis stimat wit 3.11 w dduc y z 0, C P u 0, C u 0, z 0, y 0, + C u 0, C u 0, On t otr and, an invrs stimat, 3.5, 3.11 and 3.21 yild z 0, z Π y 0, + Π y 0, C 1 z Π y 0, + C y 1, 3.23 C 1 z y 0, + C y 1, C 1 2 u 0, + C P u 1 2, C 1 2 u 0,. 7

8 It rmains to bound ρ z 2. T llipticity of A and t fact tat a 0 0 imply c 0 ρ z 2 d i,j=1 ρa ij z,xi z,xj a z,ρ z 1 2 d i,j=1 a ij ρ xi z 2 x j I + II. Sinc ρx = d x = 0,x, w av tat φ := I ρ G z X0. Hnc, 3.19 and 3.4 yild I = a z,ρ z Ĩρ z + [a z,ĩρ z a z,i ρ G z ] 3.24 C z 1, ρ z Ĩρ z 1, + C z 1,A Ĩρ z 1,A. For fixd T T w av obsrving 3.1 togtr wit t fact tat z P 1 T ρ z Ĩρ z 1, T 3.25 C ρ G z I ρ G z 1,T C D 2 [ρ G z ] 0,T C z D 2 ρ G 0,T + ρ G z 0,T + z ρ G 0,T C z 1,T C z 1, T, wr dnots t dyadic product of two vctors. In particular Ĩ ρ z 1, T ρ z Ĩ ρ z 1, T + ρ z 1, T C z 1, T Insrting 3.25 and 3.26 into 3.24 w dduc wit t lp of 3.22 and 3.23 Finally, intgration by parts and 3.22 imply II = 1 2 d i,j=1 I C z 2 1, C u 2 0, aij,xj ρ xi + a ij ρ xi x j z d i,j=1 νa ρ z 2 C z 2 0, + z 2 0, C u 2 0, + P u 2 0, C u 2 0,. Combining tis stimat wit 3.27 complts t proof. 4 Error analysis for t control problm W approximat 2.3 using t variational discrtization from [8]. Tis lads to t following control problm dpnding on : min J u = 1 y y,0 2 + α u 2 u U,ad subjct to y = G u, wr U,ad = {u L 2 a u b a.. on } and y,0 = y 0 G. It is not difficult to s tat 4.1 as a uniqu solution u U,ad and tat tis solution is caractrizd by t variational inquality y y,0 z y + α u v u 0 v U,ad

9 Hr z = G v X. It is asy to sow tat compar 2.6 u = P [a,b] 1 α ν A p, wr p X 0 and ν A p γx ar dfind by a φ,p = y y,0 φ and φ X 0 ν A p w = a w,p y y,0 w w X. 4.3 Torm 4.1. Lt u and u b t solutions of 2.3 and 4.1 wit corrsponding stats y and y rspctivly. Tn for all 0 < 0. Hr, ũ = u g 1. u ũ 0, + y ỹ 0, C log Proof. Using v = ũ U ad in 2.4 and v = u g U,ad in 4.2 w obtain y y 0 y y + α uũ u y y,0 z y + α u u g u wr y = Gũ and z = G u g. Transforming 4.5 to and rspctivly w obtain ỹ y 0 z ỹ dtdg 1 + α ũ u ũ d 0 or quivalntly ỹ y 0 z ỹ + α wr, using 3.1 togtr wit t fact tat ỹ 0,, ũ 0, C, ũ u ũ + δ δ C z ỹ 0, + u ũ 0, C y ỹ 0, + y z 0, + u ũ 0, ǫ y ỹ 2 0, + u ũ 2 0, + Cǫ 2 + C y z 2 0,. 4.7 Combining 4.4, 4.6 and 4.7 w dduc α u ũ 2 0, y y 0 y y + ỹ y 0 z ỹ + δ = y ỹ 2 + y ỹ y z y y 0 y y z ỹ + δ 1 2 y ỹ 2 0, y y 0 y y z ỹ +ǫ y ỹ 2 0, + u ũ 2 0, + Cǫ 2 + C ǫ y z 2 0, and nc aftr coosing ǫ > 0 small noug and rcalling Lmma 3.1 α 2 u ũ 2 0, y ỹ 2 0, C2 y y 0 y y z ỹ

10 Using 2.5, 2.2, intgration by parts, t dfinition of P and t fact tat a z y,φ = 0 for φ X 0 w obtain y y 0 y y z ỹ = y y Ap z ỹ Ap = u ũ νa p ap, z ỹ + P u ũ νa p = ap Ĩp, z ỹ u ũ P u ũ νa p +[a I p G,z y aĩp, z ỹ ] I + II + III. 4.9 T first intgral is stimatd wit t lp of an intrpolation inquality and Lmma 3.2: I ω 1 p Ĩp ω z ỹ 21 2 C p 2,, ω u ũ 0, C p 3,r, ω u ũ 0,. In viw of 3.18 and t coara formula w av so tat ω 1 1 δ2 d + + \ δ2 2 δ C δ 2 0 {d =τ} 1 dadτ + C C log τ + I ǫ u ũ 2 0, + C ǫ 2 log Nxt, II = II 1 + II 2 wr II 1 = u ũ P u ũ νa p d II 2 = u ũ P u ũ νa pd 1. W infr from 3.7 and 3.8 tat II 1 = u ũ νa p P νa p d On t otr and, 3.1 implis C 3 2 νa p 3 2, u ũ 0, ǫ u ũ 2 0, + C ǫ 3. II 2 C u ũ 0, νa p 0, ǫ u ũ 2 0, + C ǫ 2 so tat in conclusion Finally, rcalling 3.4 w av II ǫ u ũ 2 0, + C ǫ III C Ĩp 1,A z ỹ 1, C A 1 2 p 1,,A z ỹ 1, C p 1,, u ũ 0, ǫ u ũ 2 0, + C ǫ in viw of Lmma 3.2. Insrting 4.10, 4.11 and 4.12 into 4.8 and coosing ǫ small noug yilds t rsult. 10

11 5 Suprconvrgnc In t following sction w dmonstrat tat it is possibl to improv t ordr of convrgnc undr additional conditions on t undrlying ms. W assum from now on tat d = 2 making us of t tory dvlopd in [1], wr t following dfinition can b found: Dfinition 5.1. T triangulation T is calld O 2σ irrgular if t following olds: a T st of intrior dgs of T can b dcomposd into two disjoint sts E 1 and E 2 wit t following proprtis: For ac E 1 lt T,T T wit T T =. Tn in t paralllogram formd by T T t lngts of any two opposit dgs only diffr by O 2. E 2 T + T = O 2σ. b T st of boundary vrtics P can b dcomposd into two disjoint sts P 1 and P 2 wit t following proprtis: For ac vrtx x P 1 dnot by T, T t two boundary dgs saring x and lt t, t b t unit tangnts. Also dnot by,f,g and,f,g t dgs obtaind by making a clockwis travrsal of T, T rspctivly. Tn t t = O, = O 2, f f = O 2, g g = O 2. P 2 C wr C is indpndnt of. T following rsult is ssntially provd in [1], Lmma 2.5 for functions f blonging to W 3,. Sinc w would lik to us a corrsponding stimat for t solution of t adjoint problm wic only blongs to W 3,r for som r > 2 w rquir a suitabl modification allowing a boundary trm of t discrt tst function φ. Lmma 5.2. Suppos tat t triangulation T is O 2σ irrgular and lt f W 3,r for som r > 2. Tn f I f φ C f 3,r, 1+min1,σ φ 1, φ 0, φ X. Proof. Lmma 2.3 in [1] givs f I f φ = = T T T T T T f I f φ 2 f q α t 2 + β 2 f φ t n t T T T λ =3, µ =1 γ T,λµ λ f µ φ I 1 + I Hr, q is t quadratic function vanising at t ndpoints of and bing qual to 1 4 at t midpoint. Furtrmor, n is t unit normal to pointing away from T wil t dnots t unit tangnt wit t tangnts on T bing orintd countrclockwis. T numbrs α,β and functions γ T,λµ dpnd on t gomtry of T and tir prcis form can b found in [1]. For our purposs it is sufficint to not tat t conditions in Dfinition 5.1 imply α, β, γ T,λµ C 2, E 1 E 2, 5.2 α α, β β C3, T T = E 1, 5.3 α α, β β C 3,,, = {x},x P

12 In viw of 5.2 w av Nxt, w writ as in [1] wr I 2 C 2 f 3, φ 1,. 5.5 I 1 = I 11 + I 12 + I 13, I 1j = { q α α f 2 E t 2 + β β 2 f } φ t n t, j = 1,2, j I 13 = { 2 f q α t 2 + β 2 f } φ t n t. Arguing as in [1] w av I 11 + I 12 C σ f 2,, φ 1,. 5.6 In ordr to trat I 13 w procd in a sligtly diffrnt mannr compard to [1]. Lt us st 2 f B f := α t 2 + β 2 f t n, as wll as B f := 1 B f. Tn w can writ I 13 = q B f φ t = q B f B f φ t + q B f φ t. A Poincaré typ inquality along wit a scaling argumnt yilds for g H 1, q T g 1 g 0,q, C 1+1 q 2 q g 0, q,t, T,1 + 1 > q 2 q Applying tis stimat wit q = q = 2 and using 5.2 as wll as an invrs inquality w dduc q B f B f φ t C B f B f 0, φ 0, C 2 f 3, φ 0,. For t scond trm w writ as in [1] q B f φ = B f φ q = B f φ t t t 6 = 1 B f 6 B f φ x + 1 B f 6 B f φ x, x P 1 wr and ar t dgs saring x. Using 5.4 as wll as t t = O for = {x} w av for x P 1 B f B f B f B fx + B f B fx + B fx B fx x P 2 C B f B f 0,, + B f B f 0,, + C 3 D 2 fx C 3 2 r f 3,r,T T 12

13 by 5.7 wit q =, q = r. On t otr and w av for x T φ x φ, C 1 2 φ 0, + C 1 2 r φ 0,r,T. Tus, B f B f φ x x P 1 f 3,r,T T 1 2 φ 0, r φ 0,r,T x P f r r 3,r,T φ , 1 1 r T T x P +C 4 2 r r r φ r r 0,r,T T T C 3 2 r C r T T f r 3,r,T C 2 1 r f 3,r, φ 0, + C 2 f 3,r, φ 0,, sinc x P 1 C 1 and r < 2. Furtrmor, rcalling tat P 2 C, B f B f φ x C 2 D 2 f 0,, φ 0,, C 3 2 f 3,r, φ 0,. x P 2 In conclusion, I 13 C 3 2 f 3,r, φ 0, + C 2 f 3,r, φ 0,. 5.8 Combining 5.8 wit 5.6 and 5.5 w finis t proof of t lmma. Rmark 5.3. Lmma 5.2 continus to old if t triangulation T is picwis O 2σ irrgular, tat is, if can b writtn as t union of a boundd numbr of polygonal subdomains ac of wic is O 2σ irrgular cf. Torm 4.4 in [1]. In ordr to simplify t subsqunt analysis w assum from now on tat R 2 is convx and tat A =. As a consqunc, and y = G u is dfind by y φ = fφ, φ X 0, y = P u on, 5.9 wr P is again givn by 3.2. W xtnd a function φ X to as follows: if is t subst of \ boundd by t boundary dg T and t curvd sgmnt ẽ, tn φ is givn by t linar xtnsion of φ from T. Furtrmor, lt g : b dfind by g x := x + δxν x, x, wr ν is t constant normal to on and δx is cosn in suc a way tat g x. Not tat g is bijctiv for small. Givn u H s,0 s 1, it follows from Torm 1 in [4] tat y ỹ 0, C 2 f 0, + s+1 2 u s,, y = Gu,y = G u g W ar now in position to prov t main rsult of tis sction. 13

14 Torm 5.4. Suppos tat t triangulation T is picwis O 2 irrgular. Lt u and u b t solutions of 2.3 and 4.1 wit y 0, = y 0. Tn for all 0 < 0. Hr, ũ = u g 1 u ũ 0, + y ỹ 0, C 3 2 and y,y ar t corrsponding stats rspctivly. Proof. As in t proof of Torm 4.1 lt y = Gũ,z = G u g. W again av ỹ y 0 z ỹ + α ũ u ũ + δ wr now δ = ỹ y 0 z ỹ + α ũ u ũ d 1. \ Sinc d 1 C 2 in our stting w obtain δ y 0 0,\ + ỹ 0,\ z ỹ 0,\ + C 2 u ũ 0, Using Lmma 2 in [4] w infr tat y 0 0,\ C y 0 0, + 2 y 0 1, C. On t otr and it follows from 2.10 in [4] tat for φ X φ 0,\ C φ 0, + 2 φ 1, C φ 0, + 2 φ 1, Combining t bounds y 1, C 1 2 u 0, + f 0, C 1 2, z y 1, C 1 2 u g u 0, wit 5.13 w dduc from 5.12 δ C 2 u g u 0, + u ũ 0, C 2 u ũ 0, Tus, w dduc from 4.4, 5.11 and 5.14 similarly as in t proof of Torm 4.1 α u ũ 2 0, 1 2 y ỹ 2 0, y y 0 y y z ỹ +ǫ y ỹ 2 0, + u ũ 2 0, + Cǫ 4 + C y z 2 0, and nc aftr coosing ǫ sufficintly small and applying 5.10 wit s = 1 α 2 u ũ 2 0, y ỹ 2 0, C 3 y y 0 y y z ỹ Using 2.5 for our cas A = as wll as intgration by parts w av y y 0 y y z ỹ 5.16 = y y p + z y p + z ỹ p \ = u ũ ν p z y p + P u g u ν p + z ỹ p S 1 + S 2 + S 3 + S 4. \ 14

15 Taking into account 5.9 and t fact tat I p X 0 w infr wit t lp of Lmma 5.2 S 2 = z y p I p C p 3,r, 2 z y 1, z y 0, C 3 2 u g u 0, C 3 2 u ũ 0,. Sinc p H 3 w dduc similarly as abov tat S 4 C 2 u ũ 0,. Nxt, rcalling t rlation [ P v] g = P v g as wll as 3.6 w av S 3 = P u ũ [ p ν ] g 1 d = P u ũ ν p d + P u ũ [ p ν ] g 1 p ν d. In ordr to dal wit t scond trm w lt x = g x. Sinc p = 0 on w av tat p = ν p ν on. Hnc [ p ν ]g 1 x p ν x = px ν x pg x νg x = px pg x ν x + ν pg xνg x ν x νg x As a consqunc, = px pg x ν x 1 2 νpg x νg x ν x 2. [ p ν ] g 1 p ν C2 on sinc g x x C 2, νg x ν x C. Finally, w may writ S 1 + S 3 = u ũ P u ũ ν pd + r = u ũ ν p P ν p d + r wr r C 2 u ũ 0,. Now, 3.8 implis tat S 1 + S 3 C 3 2 ν p 3 2, u ũ 0, + r C 3 2 u ũ 0,. Rturning to 5.16 w finally obtain y y 0 y y z y C 3 2 u ũ 0, and t rsult follows aftr insrting tis stimat into Numrical xampls For our numrical xprimnts w considr t variational discrtization 4.1 of problm 2.3 wit t unit circl = B 1 0 as domain and A = as diffrntial oprator. W st α = 1, a = 0 and b = 1. For t numrical solution of t optimal control problm 4.1 w apply t fixpoint itration v U,ad givn v + := P [a,b] 1 α ν A p v 15

16 v := v +. Hr, for givn v U,ad t function ν A p v is dfind by 4.3 wit y = G v. W not tat t variational discrt solution u may admit activ sts wos boundaris do not coincid wit finit lmnt nods, compar Fig. 1a, wr t boundary control u bold is dpictd on a coars ms togtr wit function 1 α ν A p u dottd. W considr two xampls and invstigat t rror functionals E 0 u = u ũ 0,, E 0 y = y y 0,, E 1 y = y y 1,, E 0 p = p p 0,, E 1 p = p p 1,, bot on a squnc of arbitrary mss and on a squnc of congruntly rfind, picwis O 2 irrgular mss. Fig. 1b sows an arbitrary ms wil Fig. 1c dpicts a grid of t typ wic w us to numrically confirm our suprconvrgnc rsult of Torm 5.4. Rmark 6.1. T triangulation in Fig. 1c is picwis O 2 irrgular, but only O irrgular. It is automatically constructd by congrunt rfinmnt from t initial grid formd by t 8 bold sctor boardrs togtr wit t corrsponding sctor scants. Hr w not tat nw boundary nods ar projctd onto t unit circl. T rsulting triangulation in ac of t 8 sctors tn is O 2 irrgular. Picwis O 2 irrgular mss ar oftn gnratd automatically by congrunt rfinmnt, say from an initial grid T 0 containing finitly many triangls T combind wit projcting boundary nods onto smoot domain boundaris. Evry sub-triangulation obtaind in tis way from som T T 0 tn is O 2 irrgular. Tis in viw of Torm 5.4 xplains wy in practic on oftn obsrvs bttr rats of convrgnc tan xpctd from t gnral tory, compar t discussion in [1]. Tabls 1 and 2 summariz t ms-proprtis in trms of t numbr of triangls nt, t numbr of nods np and t ms paramtr. For an rror functional w dfin t xprimntal ordr of convrgnc by EOC = log E 1 log E 2 log 1 log 2. Finally for an arbitrary function g : B 1 0 R w abbrviat ĝr,φ := gr cos φ,r sin φ, wr r,φ 0,1] [0,2π. a Variational discrtization. b Arbitrary ms i = 2. c Suprconvrgnc ms j = 4. Figur 1: Variational discrtization and considrd triangulations. 16

17 i nt np Tabl 1: Ms paramtrs for t squnc of arbitrary mss. j nt np Tabl 2: Ms paramtrs for t squnc of picwis O 2 irrgular mss. In our first xampl w considr problm 2.3 wit continuous data f and smoot data y 0. For tis purpos w st ŷr,φ = r 3 max0,cos 3 φ ŷ 0 r,φ = 7r 2 cos 2 φ + 6r 2 6rcos φ + ŷr,φ and ˆfr,φ = 6r max0,cos φ. Tn it is asy to cck tat û1,φ = ûφ = max0,cos 3 φ solvs 2.3 and t associatd adjoint variabl is givn by ˆpr,φ = r 3 r 1cos 3 φ. In t prsnt xampl w dal wit classical solutions in t sns tat y,p C 2 and u C 2, s Figs. 2a and 2b. Tabl 3 summarizs t numrical rsults for t squnc of arbitrary mss from Tabl 1. In addition to t EOCs for two conscutiv mss also t avrag and t EOC btwn coarsst and finst grid is computd in t rows and 1 9. T EOC for E0 u bavs as prdictd by Torm 4.1, wras t L 2 -rror of t stat Ey 0 convrgs wit a rat of 1.5 fastr tan prdictd. In Tabl 4 w prsnt t numrical rsults for our squnc of O 2 irrgular mss. On clarly obsrvs t suprconvrgnc ffct for picwis O 2 irrgular grids prdictd by Torm 5.4. Again t rat of convrgnc for Eu 0 bavs as xpctd wras t EOC for Ey 0 is narly quadratic. Nxt, lt us construct an analytical solution to problm 2.3 in t sam way as in t prvious xampl but wit lss rgular data and nc lss rgular optimal control. W coos ŷr,φ = r 3 max0,cos φ, ŷ 0 r,φ = 15r 2 8rcos φ + ŷr,φ and st f := y. Tn û1,φ = ûφ = max0,cos φ solvs 2.3 and t associatd adjoint variabl is givn by ˆpr,φ = r 3 r 1cos φ. Lt us not tat f = y as to b undrstood in t distributional sns, i.. 1 f,ζ = 8rx 1,x 2 cosφx 1,x 2 ζx 1,x 2 dx 1 dx 2 x 2 2 ζ0,x 2dx 2 ζ C0, 1 wr = { x 1,x 2 x 1 > 0 }. Fig. 3a sows t optimal stat y wit t optimal boundary control u and Fig. 3b prsnts t associatd adjoint stat p. T convrgnc baviour of our rror functionals is similar to tat obsrvd in t prvious xampl. For arbitrary mss Eu 0 convrgs linarly as is sown in Tabl 5. On our squnc of picwis O 2 irrgular mss t convrgnc 17

18 rat of tis rror functional improvs to 1.5 as displayd in Tabl 6. Again in bot cass t baviour of Ey 0 is bttr tan prdictd and t convrgnc rat on our squnc of picwis O 2 irrgular mss is igr tan on t squnc of arbitrary mss. a Optimal stat y wit boundary control u. b Adjoint stat p. Figur 2: Analytical solution of Exampl 1. i Eu 0 EOC Ey 0 EOC Ey 1 EOC Ep 0 EOC Ep 1 EOC Tabl 3: Errors and EOCs for arbitrary mss of Exampl 1. Acknowldgmnts T autors acknowldg support of t DFG Priority Program 1253 troug grants DFG and DFG Rfrncs [1] Bank, R.E., Xu, J.: Asymptotically xact a postriori rror stimators, Part I: Grids wit suprconvrgnc. SIAM J. Numr. Anal. 41,

19 j Eu 0 EOC Ey 0 EOC Ey 1 EOC Ep 0 EOC Ep 1 EOC Tabl 4: Errors and EOCs for picwis O 2 irrgular mss of Exampl 1. a Optimal stat y wit boundary control u. b Adjoint stat p. Figur 3: Analytical solution of Exampl 2. [2] Brggrn, M.: Approximations of vry wak solutions to boundary valu problms. SIAM J. Numr. Anal. 42, [3] Brnardi, C.: Optimal finit lmnt intrpolation on curvd domains. SIAM J. Numr. Anal. 26, [4] Brambl, J.H., King, T.: A robust finit lmnt mtod for nonomognous Diriclt problms in domains wit curvd boundaris. Mat. Comp. 63, [5] Casas, E., Raymond, J.-P.: Error stimats for t numrical approximation of Diriclt boundary control for smilinar lliptic quations. SIAM J. Control Optim. 45, [6] Fursikov, A.V., Gunzburgr, M.D., Hou, L.S.: Boundary valu problms and optimal boundary control for t Navir Stoks systms: T two dimnsional cas. SIAM J. Control Optim. 36, [7] Gilbarg, D., Trudingr, N.S.: Elliptic partial diffrntial quations of scond ordr, 2nd d.. Springr

20 i Eu 0 EOC Ey 0 EOC Ey 1 EOC Ep 0 EOC Ep 1 EOC Tabl 5: Errors and EOCs for arbitrary mss of Exampl 2. j Eu 0 EOC Ey 0 EOC Ey 1 EOC Ep 0 EOC Ep 1 EOC Tabl 6: Errors and EOCs for picwis O 2 irrgular mss of Exampl 2. [8] Hinz, M.: A variational discrtization concpt in control constraind optimization: t linar-quadratic cas. Computational Optimization and Applications 30, [9] Hinz, M., Kunisc, K.: Scond ordr mtods for boundary control of t instationary Navir Stoks systm. Z. Angw. Mat. Mc. 84 3, [10] Ito, K., Kunisc, K., Picl, G.H.: Variational approac to sap drivativs for a class of Brnoulli problms. J. Mat. Anal. Appl. 314, [11] Lions, J.L., Magns, E.: Non-omognous boundary valu problms and applications. Springr [12] May, S., Rannacr, R., Vxlr, B.: Error analysis of a finit lmnt approximation of lliptic Diriclt boundary control problms. Lrstul für Numrisc Matmatik, Univrsität Hidlbrg, Prprint 05/2008. [13] Scott, L.R., Zang, S.: Finit lmnt intrpolation of nonsmoot functions satisfying boundary conditions. Mat. Comp. 54, [14] Vxlr, B.: Finit lmnt approximation of lliptic Diriclt optimal control problms. Numr. Funct. Anal. Optim. 28,