Institut für Mathematik

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1 U n i v r s i t ä t A u g s b u r g Institut für Mathmatik Ditrich Brass, Ronald H.W. Hopp, and Joachim Schöbrl A Postriori Estimators for Obstacl Problms by th Hyprcircl Mthod Prprint Nr. 02/ Januar 2008 Institut für Mathmatik, Univrsitätsstraß, D Augsburg

2 Imprssum: Hrausgbr: Institut für Mathmatik Univrsität Augsburg Augsburg ViSdP: Ronald H.W. Hopp Institut für Mathmatik Univrsität Augsburg Augsburg Prprint: Sämtlich Rcht vrblibn dn Autorn c 2008

3 A postriori stimators for obstacl problms by th hyprcircl mthod Ditrich Brass Ronald H.W. Hopp Joachim Schöbrl January 17, 2008 Abstract A postriori rror stimats for th obstacl problm ar stablishd in th framwork of th hyprcircl mthod. To this nd, w provid a gnral thorm of Pragr Syng typ. Thr is now no gnric constant in th main trm of th stimat. Morovr, th rol of dg trms is lucidatd, and th analysis also applis to othr typs of a postriori rror stimators for obstacl problms. 1 Introduction Elliptic obstacl problms oftn lad to th minimization of a quadratic functional J on a subspac V H 1 (Ω) subjct to th constraint J(v) := 1 2 a(v, v) (f, v) 0 (1.1) v(x) ψ(x) for x Ω a.. (1.2) Hr, f L 2 (Ω) and ψ C( Ω). Whn th problm is solvd by th finit lmnt mthod, th constraint (1.2) is oftn rplacd by pointwis inqualitis v h (x i ) ψ(x i ) (1.3) for all nodal points x i of th grid. This discrtization is natural, but it implis som xtra trms whn a postriori rror stimats ar computd; s,.g., [2, 4, 5, 9, 14]. Th rason is that Lagrang multiplirs for th constraints (1.3) ar point functionals. An xtnsion of th functional to H 1 (Ω) without a violation of th complmntarity condition cannot b guarantd. Th complication is lss svr whn an a postriori rror stimat is dtrmind by th hyprcircl mthod [11] that was mad popular,.g., by [10]. Th procdur known from linar thory can b adaptd for th obstacl problm. W not that no xtra trm occurs Institut of Mathmatics, Ruhr-Univrsity of Bochum, D Bochum, Grmany Dpartmnt of Mathmatics, Univrsity of Houston, Houston, TX , U.S.A., and Institut of Mathmatics, Univrsity of Augsburg, D Augsburg, Grmany Dpt. of Math. and Cntr for Comput. Engrg. Scinc, RWTH Aachn, D Aachn, Grmany 1

4 whnvr th activ point st has som rgularity, i.., if it is th closur of its intrior. Othrwis, a gnralization of th Pragr Syng thorm for th obstacl problm also yilds an additional rror trm as w find in wll-known stimators. Hr, th rsulting xtra trm ntrs into an xact xprssion for th rror. It is thrfor clar that it dos not spoil th fficincy of th rror stimat. W not that Wiss and Wohlmuth [15] obsrvd a similar phnomnon whn thy considrd inquality constraints on th boundary of th domain Ω. On th othr hand, Rpin [12] considrd th hyprcircl mthod without th rgularity assumption of th activ st, and his rsult is closr to th classical stimats than to our rsult with th hyprcircl mthod. A patch-orintd construction following Brass and Schöbrl [7] turns out to b appropriat hr, sinc th Lagrang multiplirs for th discrtizd obstacl problm ar associatd with th finit lmnt quations on patchs. Although thr is a strong rlation to th dtrmination of stimators by local Numann problms (s,.g., [1]), th lattr is focusd on lmnt-orintd constructions, and th considrations on patchs only occur in auxiliary stps. Oftn th dg trms ar dominating in a postriori stimats for linar lliptic problms. Thos trms, howvr, may ovrstimat th rror whn obstacl problms ar considrd. W can liminat this ffct in crtain cass by rlaxing th rgularity rquirmnt for th hyprcircl mthod. Th papr is organizd as follows. Sction 2 will provid a gnral thorm of Pragr Syng typ. Th prrquisits from finit lmnt thory ar prsntd in Sction 3. Th construction of th a postriori rror stimat in Sction 4 will b organizd such that no xtra trm ariss if possibl. Th fficincy will b tratd in Sctions 5 and 6. In two appndics, a discussion of th rol of dg trms in a postriori stimats lucidats th situation not only for th hyprcircl mthod. 2 A thorm of Pragr Syng typ For convninc, w rstrict ourslvs to th Poisson quation with homognous Dirichlt boundary conditions, i.., a(u, v) := u(x) v(x) dx and V = H 1 0 (Ω). Th nrgy J is to b minimizd on th convx st Ω K := {v V v ψ a.. in Ω}. Th solution u is known to b charactrizd by a(u, v u) (f, v u) 0, v K. (2.1) Th Lagrang multiplir λ is dfind by λ, v := a(u, v) (f, v) 0, v V. It follows from (2.1) that for all w V + λ, u ψ = 0 and λ, w 0, (2.2) 2

5 if w st V + := {v V v(x) 0 a..}. Morovr, w hav J(v) J(u) (2.3) = 1 2 v u λ, v u v V, and ach trm on th right-hand sid of (2.3) is nonngativ, if v K. Oftn it is mor natural to giv th rror in trms of th diffrnc J(v) J(u) and not by th nrgy norm. For instanc, a convrgnc analysis of an adaptiv P1 conforming finit lmnt approximation of (1.1), (1.2) in th sns of a guarantd rduction of th objctiv functional J has bn providd in [5]; cf. also [13] for an approach using quadratic programming tchniqus. Th dual problm is th maximization of th Trfftz functional J (τ) := 1 2 τ 2 0 (div τ + f, ψ) 0 on th dual convx con F := {τ H(div) div τ + f 0 a..}. It is known that thr is no duality gap, i.., J(u) = J ( u), and J ( u) J (τ) (2.4) = 1 2 u τ 2 (div τ + f, u ψ) 0 for all τ F. Thorm of Pragr and Syng typ for obstacl problms. Lt v K H 1 0 (Ω) and τ F H(div). Thn 2[J(v) J(u)] + 2[J ( u) J (τ)] (2.5) = [ u v λ, v u ] + [ u τ (div τ + f, ψ u) 0 ] = v τ (div τ + f, ψ v) 0. Furthrmor, if v and τ satisfy th complmntarity condition thn (div τ + f, v ψ) 0 = 0, (2.6) u v u τ 2 0 (2.7) 2[J(v) J(u)] + u τ 2 0 v τ 2 0. Rmark. W not that all innr products in (2.5) ar nonngativ. 3

6 Proof: Sinc th boundary trms vanish whn partial intgration is applid, w hav ( u τ, v u) 0 = ( u, v u) 0 + (div τ, v u) 0 = ( u, v u) 0 (f, v u) 0 + (div τ + f, v u) 0 = λ, v u + (div τ + f, v u) 0. Now, th binomial formula is applid to th sum [ v u] + [ u τ] to obtain v τ 2 0 = v u u τ λ, v u + 2(div τ + f, v u) 0 = 2[J(v) J(u)] + 2[J ( u) J (τ)] + (div τ + f, v ψ) 0. This provs (2.5). Th inquality (2.7) follows from (2.3) and (2.4), and th proof is complt. Obviously, th last trm in (2.5) corrsponds to th xtra trm in classical stimators. It will b avoidd whnvr possibl. Rmark 2.1 W mphasiz that th assumption τ H(div) may b droppd in Pragr and Syng s thorm, if w st div τ, w = (τ, w) for w H 1. In particular, if τ blongs to th brokn H(div) spac introducd in Sction 4, w hav div τ, w (2.8) = (div τ, w) 0,T [τ n]wds. T Th bnfit of this obsrvation will b lucidatd in Appndix B with a on-dimnsional xampl. Th additional frdom will b usd on dgs in th contact zon (also in highr dimnsions), in particular, if th obstacl is spcifid by a non-affin function. 3 Th Lagrang multiplir for th finit lmnt solution Th discrtization of th obstacl problm mans that th linar spac is rplacd by a finit lmnt spac V h, which will b hr th spac of linar lmnts on a triangulation T h of Ω R 2. As usual, Ω is assumd to b a polygonal domain, and th obstacl is givn by a picwis linar function ψ V h. Th corrsponding Lagrang multiplir λ h is dfind by λ h, w = a(u h, w) (f, w) 0 (3.1) 4

7 for w V h. Sinc th right-hand sid is dfind for all w V, w obtain an xtnsion of λ h to V by (3.1). Partial intgration yilds th rprsntation λ h, w (3.2) = T (f, w) 0,T + ([ u h n ], w) 0,. It shows that this xtnsion of th Lagrang multiplir contains also th information on th rsidus outsid th coincidnc st. From a computational point of viw, it is givn by th nonngativ rsidus of th finit lmnt quations in th contact zon. Lt φ i V h b th nodal basis function associatd with th nodal point x i. Thn is th rsidu in th finit lmnt inqualitis and λ h,i := λ h, φ i 0 for all i, (3.3) λ h,i = 0, if u h (x i ) ψ(x i ) > 0. (3.4) Thrfor, th discrt complmntarity condition λ h, u h ψ = 0 holds and λ h, w h = i λ h,i w h (x i ), w h V h. (3.5) Th support of φ i is th patch ω i := { } T Th x i T. Th coincidnc st (activ st) A h := {x Ω u h (x) = ψ(x)} is calld rgular, if it is th closur of its intrior. This mans that ach nodal point x i A h lis on th boundary of a triangl T which is containd in th coincidnc st A h. 4 Equilibration Th main task in th dtrmination of th a postriori rror stimat is th construction of a function σ that satisfis div σ f and morovr th complmntarity condition (2.6) whnvr possibl. (Th original rquirmnt σ H(div) will b rlaxd.) Th procdur is calld quilibration. Following [7] w construct an appropriat τ in th brokn Raviart Thomas spac Mor prcisly, th rsulting σ will satisfy RT 1 :={τ L 2 (Ω) τ(x) = a T + b T x with a T R 2, b T R in ach T }. div σ + f 0, [σ n] 0, (4.1) 5

8 whr f is th L 2 projction of f in th spac of picwis constant functions. This mans that w sparat th data oscillation from th main trm of th stimat; cf. [9]. As usually, w considr th associatd rror trm as a trm of highr ordr. ch f f 0 In contrast to th tratmnt of linar lliptic problms, inqualitis ar admittd in (4.1). In addition, th complmntarity conditions ( div σ f, u h ψ) 0,T = 0, ([σ n], u h ψ) 0, = 0 (4.2) will b satisfid at last outsid a nighborhood of th coincidnc st, sinc th trms on th lft-hand sid of (4.2) ntr into th rror bound. W will silntly adopt this point and rpat it only whn ncssary. W rcall that th finit lmnt functions in th Raviart Thomas spac RT := RT 1 H(div), ar spcifid by thir normal componnts on th dgs of th grid. Similarly th functions in th brokn Raviart Thomas spac RT 1 ar givn, if th normal componnts ar known on both sids of th dgs. Obviously, u h is a brokn Raviart Thomas function with zro divrgnc in ach triangl. Th rquird σ will b obtaind by a corrction that liminats th jumps of th normal componnts on th dgs. Spcifically, th corrction shall satisfy th following proprtis: σ := τ u h (4.3) [ σ n ] [ ] u h n on ach dg, div σ f on ach triangl T. (4.4) Th dsird σ, in turn, will b computd as a sum of local corrctions with support in th patchs ω i, σ = σ ωi with supp σ ωi = ω i, (4.5) i and [ [σ ωi n] 1 uh ] 2 n, ωi, div σ ωi f T,i, T ω i, σ ωi n = 0 on ω i \ Ω, (4.6) whr f T,i := (1/ T ) fφ i dx. T Sinc ach intrior dg of th triangulation T h blongs to two patchs and i T fφ idx = T f 1 dx = T f, th proprtis (4.6) imply (4.4). 6

9 Lmma 4.1 Lt x i Ω \ Ω b a nod of th triangulation, and lt φ i V h b th nodal basis function with φ i (x i ) = 1 and φ i (x) = 0 for x Ω \ ω i. Thn 1 [ ] uh ds 2 ω i n = fφ i dx + λ h,i. (4.7) T ω i T Proof: Sinc u h is th finit lmnt solution in V h, w obtain from (3.1) with w = φ i : u h φ i dx = ω i fφ i dx + λ h,i. ω i (4.8) W rcall that u h / n is constant and φ i is linar on ach dg. Partial intgration of th lft-hand sid of (4.8) yilds u h φ i dx = u h ω i T ω T n φ idx (4.9) i = [ ] uh φ i ds ω i n = 1 [ ] uh ds. 2 n ω i Now, th assrtion of th lmma follows from (4.8) and (4.9). Nxt, w considr a patch around a nod x i. Th dsird function σ ωi will b spcifid by th intgral fluxs σ ω i T n ds on th two sids of ach dg T ω i. Th boundary condition (4.6) 3 will always silntly b assumd to hold. First, w dscrib a chap construction and distinguish four cass. In all cass Algorithm 4.2 will b applid. It follows th procdur known from linar thory; s [3, p.184]. Hr th input contains som xtra paramtrs Λ T,i and Λ,i to cop with xcss sourcs and sinks. In particular, th paramtrs Λ T,i vanish in Cas 1 blow as in th linar cas. Th paramtrs Λ,i ar st to zro in all four cass and will b activatd only latr. Th notation for th algorithm is spcifid in Fig. 1. Latr, w will prsnt a vrsion with an optimization procss in ordr to improv th fficincy of th stimator. 7

10 σ i,j,r σ i,j,l σ i,j+1,r σ i,j+1,l Figur 1: Fluxs in a patch around a vrtx x i. σ i,j,r and σ i,j,l ar th normal componnts of th fluxs that lav th triangl T j on th right and lft sid, rspctivly. Th triangls ar numratd countr clockwis, and j = T j T j+1 (with indics modulo th numbr of triangls) Algorithm 4.2 St σ i,1,r = 0; for j = 1, 2,..., until an ntir circuit around x i is compltd (or an dg on Ω is mt) { } fix σ i,j,l such that σ ωi nds = j fφ i dx σ ωi nds + Λ Tj,i; T j j 1 fix σ i,j+1,r such that [σ ωi n] = 1 2 [ u h n] + Λ j,i on j ; Th fluxs dfin a prliminary σ with support ω i. Add a constant α to all σ i,j,l and σ i,j,r for which σ 0 is minimal. Th two ruls within th bracs car that (4.6) 1 and (4.6) 2 hold. Sinc an additiv constant α dos not chang (4.6), it is fixd in th last stp for minimizing th L 2 -norm. Rmark 4.3 Thr is an asy intrprtation. By Gauss law th normal componnts of th fluxs on th thr dgs of a triangl dtrmin th magnitud of th sourc or sink in a triangl. Similarly, thr is a sourc or sink btwn th two sids of an dg that is givn by th jump of th flux on th dg and its lngth. Lmma 4.1 and (4.7) assrt that th sum of all of thm in a patch vanishs. 8

11 Cas 1. x i Ω \ Ω and u h (x i ) > ψ(x i ). Hr (4.7) holds with λ h,i = 0, and w apply Algorithm 4.2 with Λ T,i = 0 for all T. Cas 2. x i Ω \ Ω, u h (x i ) = ψ(x i ), and u h = ψ holds at last in on triangl T ω i. Lt m = m i b th numbr of triangls in th patch on which u h = ψ holds. By assumption, m 1. W st { 1 m Λ T,i := λ h,i, if u h (x) = ψ(x), x T, (4.10) 0, othrwis. Th algorithm now yilds a corrction that satisfis (4.6) 1, (4.6) 2 and th complmntarity rlation T ω i (div σ ωi, u h ψ) 0 = 0. Cas 3. x i Ω \ Ω, u h (x i ) = ψ(x i ), and u h (x) ψ(x) holds for at last on point in ach triangl T ω i. Lt m b th numbr of triangls in ω i. St Λ T,i = 1 m λ h,i. Th algorithm yilds a corrction that satisfis (4.6) 1, (4.6) 2, but thr will now b a nonzro contribution of th complmntarity trm to th rror stimat. Cas 4. x i Ω. Th dgs and triangls in ω i ar numratd such that th algorithm starts at an dg on Ω and stops at th othr dg on th boundary. Sinc th circuit is incomplt, no condition has to b satisfid, and w can prform th algorithm with Λ T,i = 0 for all T ω i. By construction, (4.6) is guarantd in all four cass. Th cas 1 corrsponds to th construction for linar lliptic quations [3, p. 184], and it is optimal in th framwork of local procdurs. In th cass 2 and 3 th fficincy of th rsult can b improvd. Instad of fixing th variabls Λ T,i a priori, thy ar dtrmind by a small quadratic program. In ordr to hav a unifid dscription and to avoid th distinction of th cass, th optimization is gnrally includd: η P S,i := (4.11) min{ τ (div τ + fφ i, ψ u h ) 0 + 2([(τ + u h ) n], u h ψ) 0, } with τ dtrmind by Algorithm 4.2 xcutd with paramtrs subjct to th constraints Λ T,i + Λ,i = λ h,i, T ω i ω i Λ T,i 0 for all T ω i, Λ,i 0 for all ω i. Th optimization problm is solvabl, sinc a fasibl solution xists. This follows from th procdur with a priori fixd paramtrs. If th nod in th intrior of th patch dos not 9

12 blong to th coincidnc st, thn λ h,i = 0, all slack variabls vanish, and th optimization is trivial. W hav not usd th short notation with (2.8) in ordr to s th jumps mor clarly. Aftr summing th corrctions on all th patchs w obtain th final stimat. Thorm 4.4 Lt ach σ ωi b dtrmind as dscribd abov and σ by (4.5). Thn w hav th a postriori rror stimat u u h 2 0 J(u h ) J(u) σ ch 2 f f (div σ + f, ψ u h ) 0 + 2( [ σ n ], u h ψ) 0,. Hr, th trm in th third lin gts nonzro contributions only via Stp 3, and in ach triangl T, div σ f = 1 Λ T,i. T 5 Rlation to rsidual stimators By Thorm 4.4, th hyprcircl mthod rsults in a rliabl stimator. For studying its fficincy w will compar th stimator with th classical ons for th obstacl problm. In particular, w focus on rsidual stimators. Th rlation to th tru rror of th finit lmnt solution will b invstigatd in th nxt sction. Th optimization problm (4.11) on a patch ω i will b modifid to achiv a simplr, but quivalnt on. For simplicity w drop th indx i whnvr thr is no dangr of confusion. (Thr is,.g., th xcption f T,i.) First, (4.11) is rwrittn, { } η P S = min τ div τ + fφ i, ψ u h, (5.1) subjct to div τ f T,i, τ RT 1 (ω), i [τ n] (1/2)[ u h n], (5.2) τ n = 0 on ω. For τ RT 1 (ω)/kr(div RT 1 ), a scaling argumnt shows that whr c 1 τ 2 0 div τ 2 1,h c 1 1 τ 2 0, div τ 2 1,h := T ω h 2 T div τ 2 0,T + ω h [τ n] 2 0,. 10

13 Th jumps of τ on th dgs ar now also considrd as (distributional) parts of div τ in th spirit of Rmarks 2.1 and 4.3. Thrfor, w dfin s = div τ by stting s T := (div τ) T, s := [τ n]. In particular, s is givn by 2m ral numbrs if ω consist of m triangls and τ RT 1 (ω). Hnc, subjct to c 1 η P S η s := (5.3) { min s 2 1,h + 2(s + fφ i, ψ u h ) 0,ω + 2(s [ u h n], u h ψ) 0, }, s T f T,i, s (1/2)[ u h n], (5.4) s, 1 = 0. Hr, th total divrgnc on th patch is dfind by s, 1 := T ω T s T + ω s. Equation (5.4) 3 was hiddn in (5.1) by th condition τ n = 0 on ω. Th limination of th condition (5.4) 3 will mak th construction simplr. Lmma 5.1 Assum that u h = ψ holds in at last on triangl of ω, and lt ω consist of m triangls. Lt subjct to η s,2 := min{ s 2 1,h + 2(s + fφ i, ψ u h ) 0,ω +2(s [ u h n], u h ψ) 0, }, (5.5) s T f T,i, s (1/2)[ u h n] b th rror stimator without th constraint (5.4) 3. Thn with th constant c 2 dpnding only on th shap paramtr. η s η s,2 (1 + 2m) 2 c 2 η s (5.6) Proof: Th two trms in (5.5) 1 ar nonngativ. Thr is a constant c 2 that dpnds only on th shap paramtr such that c 2 h 2 T T c 1 2 h2 T. 11

14 Thrfor, w considr th minimization of th quivalnt xprssion ( s ) 2 T ω( T s T ) 2 + ω + 2(s + fφ i, ψ u h ) 0,ω + 2(s [ u h n], u h ψ) 0,. Lt s b th minimizr of th problm (5.5). W construct a fasibl candidat s that satisfis th avraging constraint (5.4) 3, and th functional will incras only by th givn m-dpndnt factor. Cas (a): Assum that s, 1 > 0. Lt T b a triangl with u h = ψ on T. W st s := s and rdfin it on th spcial triangl s T := s T T 1 s, 1 without changing th othr valus. A straight forward calculation shows that s s 2 2m s 2. Hnc, s (1 + 2m) s 2 holds for th modifid ( 1, h)-norm. Obviously, s is fasibl du to th ngativ corrction, and th scond trm on th right-hand sid of (5.5) dos not chang. Th assrtion holds in this cas. Cas (b). Assum that s, 1 < 0. St ŝ T := f T,i, s := ŝ := s. Sinc thr xists a fasibl solution of th minimization problm, it follows that ŝ, 1 0. For T ω, th convx combination s T := s T + s, 1 ŝ, 1 s, 1 T 1 (ŝ T s T ) yilds a fasibl solution. Sinc w hav distributd th man valu on on or mor triangls, th 1,h norm is not mor incrasd than in cas 1. Th scond trm in (5.5) was diminishd or unchangd by th choic abov, and th proof of th nontrivial part is complt. Th inquality η s η s,2 is obvious. Sinc th quadratic trms in (5.5) ar diagonal, th variabls ar now sparatd in th problm, and th minimizr is asily dtrmind. Adding now th labl of th patch, w hav s T,i = R T,i := min{f T,i, h 2 u h ψ}, (5.7) s,i = R,i := min{ 1 [ uh ], h 1 u h ψ}. 2 n Hr and in th squl, an ovrlind quantity rfrs to th man valu on th subst undr considration. W associat to th choic (5.7) rsidual typ rror stimators. Thr ar lmnt trms (ara-basd trms) η T,i (5.8) := h 2 T ( f T,i, min{ f T,i, h 2 u h ψ}) 0,T { h 2 = T f T,i 2 0,T, if f T,i h 2 T u h ψ, ( f T,i, u h ψ) 0,T, othrwis 12

15 and, with th abbrviation λ := [ u h n ], th dg trms η,i :=h (λ, min{λ, h 1 u h ψ }) 0, { h λ = 2 0,, if h λ u h ψ (λ, u h ψ) 0,, othrwis. (5.9) Th quantitis abov rflct th fact that a continuous transition btwn th coincidnc st and th points in its nighborhood is rasonabl. Thorm 5.2 Th Pragr-Syng rror stimator is quivalnt to th rsidual rror stimator, i.., η P S,i η T,i + η,i. T ω i ω i Proof: W prsnt th proof for th lmnt trms, th dg trms can b tratd in th sam way. Morovr, w rcall th quivalnc of η P S,i with th variants in Lmma 5.1. Cas 1. f T,i h 2 T u h ψ. Thn w hav s T = f T,i and th contribution of th lmnt T to η s in (5.3) is η s,t = h 2 T f T,i 2 0,T + 2 ( f T,i + f T,i, ψ u h ) 0,T = h 2 T f T,i 2 0,T. Cas 2. f T,i > h 2 T u h ψ. Thn w hav s T = h 2 T u h ψ. Not that th two contributions ar nonngativ. Hnc, w may multiply by a factor of two for liminating inconvnint trms η s,t =h 2 T u h ψ 2 0,T + + 2( h 2 u h ψ f T,i, u h ψ) 0,T 2h 2 T u h ψ 2 0,T + + 2( h 2 u h ψ f T,i, u h ψ) 0,T = 2 ( f T,i, u h ψ) 0,T. Similarly, by taking half of th scond trm w obtain η s,t ( f T,i, u h ψ) 0,T. 6 Efficincy W procd with th analysis of th fficincy and focus our attntion on th hyprcircl mthod. Howvr, th rsults will b of intrst for rsidual-typ stimators as wll. First, w s that solving local Dirichlt problms is fficint. Lmma 6.1 Assum that v i H0 1 (ω i ) and v i ψ u h. Thn {J(u h ) J(u h + v i )} i { } 3 J(u h ) J(u). 13

16 Proof: Lt ach v i satisfy th assumption of th lmma, and lt m b th maximal numbr of ovrlapping patchs. Obviously, m 3 holds in 2-spac. Th lmnt w := 1 m vi is in th convx st, and thus From Young s inquality it follows that J(u h ) J(u h + w) J(u h ) J(u) (6.1) w 2 a = 1 m Th diffrncs of th nrgis valuats to vi 2 a 1 m vi 2 a. J(u h ) J(u h + v) (6.2) = 1 2 a(u h, u h ) (f, u h ) a(u h, u h ) a(u h, v) 1 2 a(v, v) + (f, u h + v) 0 = a(u h, v) + (f, v) 0 1 a(v, v) 2 = λ h, v 1 a(v, v). 2 By applying (6.2) to v = w and v = v i and rcalling (6.2) w obtain {J(u h ) J(u h + v i )} i = i { λ h, v i 12 v i 2 } λ h, mw m 1 2 w 2 = m {J(u h ) J(u h + w)} m {J(u h ) J(u)}. Following Lmma 6.1 w will construct a corrction v ψ u h such that th improvmnt (6.2) dominats th rsidual rror stimator. This shows its fficincy. Som lmntary proprtis of th lmnt bubbl functions b T and th dg bubbl functions b ar rquird. Thy ar dfind in trms of th barycntric coordinats b T := λ 1 λ 2 λ 3, b := λ 1 λ 2. Lmma 6.2 (1) Lt g b a linar function that is non-ngativ on a triangl T and ḡ b its man-valu on T. Thn 12 ḡ b T (x) g(x) for all x T. (6.3) (2) Thr is a constant c 1 (c 12) such that bt c b 2 h 2 (6.4) 14

17 and b T = 1 T. 60 Proof: (1) Lt α i dnot th non-ngativ valu of g at th vrtx i. W writ g(x) = 3 i=1 α iλ i and not that ḡ = (1/3) 3 i=1 α i. If th indics i, j, k ar in cyclic ordr, w obtain g(x) = = 4 3 α i λ i i=1 3 α i λ i (4λ j λ k ) i=1 3 α i b T = 12ḡ b T. i=1 (2) Th stimat (6.4) follows by standard scaling argumnts. Th last quation is obtaind by simpl computation of th intgral. Nxt w rfr to th lowr bounds of th rror that rsult from th local Dirichlt problms on lmnts or dgs and thir nighborhood E D,T := sup J(u h ) J(u h + v) v H 0 1(T ) v ψ u h E D, := sup J(u h ) J(u h + v) v H 0 1(ω ) v ψ u h In particular, Lmma 6.1 yilds E D,T + T { } E D, c J(u h ) J(u). Thorm 6.3 Thr xists a constant c such that th ara portion of th stimator η P S satisfis η T c E D,T. Proof: Givn T ω i, lt v := cb T max{h 2 T f T,i, ψ u h } H 1 0 (T ), whr c is th constant in Lmma 6.2. By dfinition, v cb T ψ u h 12b T ψ u h ψ u h. Hnc, u h + v ψ. Sinc th support of v is containd in th lmnt T, it follows from (3.1) that E D,T J(u h ) J(u h + v) = 1 2 v 2 0 λ h, v = 1 2 v 2 0 ( f T, v). 15

18 W distinguish two cass. Cas 1. h 2 T f T < ψ u h. Thn v = cb T ψ u h and f T is ngativ. Cas 2. h 2 T f T ψ u h. Thn v = cb T h 2 T f T,i and E D,T 1 2 c2 b T 2 ψ u h 2 b T cf T ψ u h 1 2 c b T ψ u h 2 c b T f T ψ u h c 120 T f T u h ψ. E D,T 1 2 c2 b T 2 h 2 T λ T 2 + b T ch 2 T ft c b T h 2 T ft 2 = c 120 h2 T f T 2 0,T. In both cass, th local improvmnt E D,T dominats a multipl of th rsidual rror stimator η T. Thorm 6.4 Thr xists a constant c such that th dg portion of th stimator η P S satisfis with th fficincy masur χ dfind as and η c χ E D,. χ = 1 unlss [ ] uh < 0 and h 2 T f T < u h ψ (6.5) n { χ = max 1, max min T ω Th proof is postpond to Appndix C. { hft [ uh n ], h2 f T }}. (6.6) u h ψ Th thorms show that th lmnt trms of th stimator du to Pragr and Syng ar fficint, but that w hav a wakr rsult for th ingrdints of th dgs. W may summariz th rsults as J(u h ) J(u) c 1 σ 2 0 c 2 h 2 T f f 2 0 T [ ] uh c 3 h n,t T f T, 16

19 whr th prim at th last sum indicats that it runs ovr thos pairs with T, [ u h ] n < 0, and h 2 T f T < u h ψ. An xampl in Appndix A shows th loss of fficincy with xactly th factor (6.6), sinc th xtra trms ar much largr than th tru rror. Th discussion of th xampl also lucidats that thr is an inhrnt handicap with obstacl problms. Fortunatly, this is no drawback in actual computations, if local rfinmnts tak car of xtra trms. Ths xtra trms can b data oscillations on patchs [2] or associatd with th rror in th Lagrang multiplirs [14]. In ordr to achiv an rror rduction, th rfinmnt has to b organizd in such a way that not only th stimator but also th xtra trms ar rducd within th adaptiv cycl; s,.g., [5] and [6]. On th othr hand, w dmonstrat in Appndix B that th stimator du to Pragr Syng dals vry wll with othr phnomna of non-affin obstacls if on-sidd jumps ar admittd with th quilibratd fluxs. A A Countrxampl Th handicap of a postriori rror stimats for obstacl problms and thir fficincy is lucidatd by a on-dimnsional xampl. Th obstacl will b vn affin linar. Lt b d > 0, and considr th variational problm in H 1 ( 1, +1): v (x) 2 dx + b v(x)dx min! 1 (A.1) with th constraint ψ = 0 and th boundary conditions v( 1) = 0, v(1) = d. A boundary point z of th contact zon is givn by z z2 = d/b, i.., z d/b. Th solution of th variational problm is 0, 1 x z, 1 u(x) = 2 b(z + x)2, z x 0, 1 2 bz2 + bz x, 0 x 1; s Figur 2. Th corrsponding finit lmnt solution with on nod at 0 is { 0, 1 x 0, u h (x) = x, 0 x 1. A straight-forward calculation yilds u u h 2 1 = z 1 b 2 (x + z) 2 dx + (d bz) 2 dx = 1 3 b2 z 3 + ( 1 2 bz2 ) 2 d 3 /b. Sinc th jump of u h quals d, th rror bound is d2. Hnc, th quotint of th rror stimat and th tru rror is b/d f T /λ, and th formula (6.6) for th fficincy masur is sharp (modulo a constant). Not that th sam dg trm as in th stimat du 0 17

20 u u h -1 -z 0 1 Figur 2: Exact and finit lmnt solution of th problm in Appndix A. u τ 1 u h ψ Figur 3: Exact and finit lmnt solution of th problm in Appndix B and th quilibratd flux. to Pragr Syng is ncountrd in th typical classical stimators [2, 4, 5, 9, 14]. It is no drawback in actual computations; cf. Sction 6. On purpos, w hav chosn an xampl with an affin obstacl. W gt a similar xampl with zro data oscillation if non-affin obstacls ar chosn. If w xtnd th load in (A.1) to th complt domain and st ψ(x) = x on [0, 1], thn th finit lmnt solution is th sam. A symmtry argumnt shows that th xact solution changs so littl, that th fficincy problm is th sam. Obviously th kink of th obstacl implis th dtrioration hr. B Effcts of dg trms with inqualitis Anothr on-dimnsional xampl shows that th hyprcircl mthod can cop with nonaffin obstacls bttr than som wll-known stimators. W gain appropriat flxibility by admitting quilibratd fluxs τ H(div) as statd in Rmark 2.1. This is positiv in contrast to th xampl in th prcding appndix, but th situation is diffrnt, sinc th jump thr has th opposit sign. Th discussion of th xampl may b of intrst indpndntly of th hyprcircl mthod. Thrfor, som argumnts of Sction 2 ar rpatd. Lt 0 < b 1, and considr th variational problm in H 1 0 ( 1, +1): with th constraint v (x) 2 dx 2b +1 1 ψ(x) = 1 2 x and homognous Dirichlt boundary conditions. v(x)dx min! Th solution is obviously u(x) = 1 x + b x (1 x ); s Figur 3. Th finit lmnt solution with linar lmnts and on nod at th midpoint of th intrval is u h (x) = 1 x. 18

21 Lt τ b a picwis polynomial with a possibl jump at x = 0. W hav u h u u τ 2 0 u h τ 2 0 (B.1) if (u τ, u h u ) 0 0. W start as in th prof of (2.7), but procd in th spirit of Rmark 2.1. In this xampl, w hav f = 2b and (u τ, u h u ) 0 (B.2) = (u, u h u ) 0 (f, u h u) τ(u h u )dx + (f, u h u) 0 = λ, u h u (τ + 2b)(u h u)dx [τ(0+) τ(0 )](u u h )(0). From th charactrization of th xact solution w know that th first trm in (B.2) is nonngativ. Th scond trm vanishs, if w hav pointwis τ = b. Sinc x = 0 blongs to th activ point st, w hav (u u h )(0) 0, and th last trm is nonngativ whnvr th jump of τ is nonpositiv. Thrfor, th appropriat quilibration lads to τ(x) = { ρ bx, x < 0, ρ bx, x 0 with an arbitrary ρ 0. Th rsulting stimator u h τ 0 attains its minimum for ρ = 1 b/2. Hr u h τ 0 = b/ 6 and th stimator quals u h u 0. This provs th fficincy. Error stimators which contain jump trms of u h or of (u h ψ) cannot b fficint for small valus of th paramtr b. C Proof of Thorm 6.4 Lmma C.1 (1) Lt g b a linar function that is non-ngativ on a triangl T and ḡ b its man-valu on th dg T. Thn (2) Thr is a constant c 1 (c 2) such that b c b 2 h. (3) Lt 0 < z < 1 and 2 ḡ b (x) g(x) for all x T. (C.1) (C.2) b (z) = max{0, (λ 1 λ 3 /z)(λ 2 λ 3 /z)} (C.3) b a bubbl function whos support is rducd to a strip of with 2zh. Thn b (z) 2 0,T h 2 z, b (z) 2 0,T z 1, b (z) 2 0, h. 19

22 Proof: (1) Lt α i dnot th non-ngativ valu of g at th vrtx i. W writ g(x) = 3 i=1 α iλ i and not that ḡ = (1/2) 2 i=1 α i. W obtain g(x) = 3 α i λ i i=1 2 α i λ i i=1 2 α i λ 1 λ 2 = 2ḡ b. i=1 (2) Th stimat (C.2) follows by standard scaling argumnts. Morovr, w hav b = (1/6). (3) Th stimats follow by standard scaling argumnts. Proof of th thorm. W rcall λ = [ u h ] n. Givn ωi, lt v := cb max{ h λ, ψ u h } H 1 0 (ω ), whr c is th constant in Lmma C.1. By dfinition, v cb ψ u h 2b ψ u h ψ u h. Hnc, u h + v ψ. From (3.1) w obtain for v H 1 0 (ω ): W distinguish two cass. E D, J(u h ) J(u h + v) = 1 2 v 2 0 λ h, v (C.4) = 1 2 v 2 0 ( f, v) ω (λ, v). Cas 1. h λ u h ψ. Thn λ is positiv and v = cb u h ψ is ngativ. W hav Now (C.4) yilds η = (λ, u h ψ ) 0, u h ψ 2. E D, 1 2 c2 b 2 ψ u h 2 + fv b cλ ψ u h ω 1 2 c b h 1 ψ u h 2 + fv ω + c b λ ψ u h c 12 λ u h ψ + fv ω c η + f T v. T ω T (C.5) 20

23 Cas 2. h λ < u h ψ. In this cas w hav η = h 2 λ 2 and considr a tst function with th modifid bubbl function v = αb (z) h λ. Th paramtrs α > 0 and z < 1 will b fixd latr. Sinc b (z) abov. By th invrs inqualitis in Lmma C.1 and (C.4) w gt E D, 1 2 v 2 0 λ h, v b, w gt u h + v ψ as Now w choos α = c 2z 2c 1 = 1 2 α2 h 2 λ 2 b (z) 2 0 +αhλ 2 b (z) + fv ω α 2 h 2 c 1 λ 2 z +c 2 αh 2 λ 2 + fv. ω to absorb th first trm by th scond on and obtain E D, c 3 zη + T ω T f T v. (C.6) Th intraction of th dg bubbls with th lmnt bubbls is givn by th last trms in (C.5) and (C.6). Th trms will b absorbd by th obsrvation with th xcption spcifid in Thorm 6.2. E D,T E D, if T (C.7) By dfinition, th tst function v has th opposit sign as λ. Thrfor, w can drop th trm if λ f T < 0. Othrwis w distinguish thr cass. In all of thm η T = h 2 T f T 2 0,T, and w st z = 1. Cas a) h λ u h ψ and f T > 0. A standard scaling argumnt yilds h 1 b 0,T c. Morovr, η = hλ u h ψ ( u h ψ ) 2, and f T v hf T 0,T h 1 v 0,T T = hf T 0,T h 1 b 0,T c ψ u h cη 1/2 T η1/2. 21

24 Cas b) 0 h λ < u h ψ and f T > 0. A similar scaling argumnt and η = h 2 λ 2 yilds f T v hf T 0,T h 1 v 0,T T = hf T 0,T h 1 b (z) 0,T α hλ αη 1/2 T η1/2. Cas c) λ < 0 and u h ψ h 2 T f T < 0. Hr η = h 2 λ 2 and T f T v can b boundd as in Cas b). In any of th thr cass, by Young s inquality it follows that E D, c η c c η c η 1/2 T η1/2 T ω T ω η T. (C.8) Finally, combining Thorm 6.3 and (C.7) w absorb th last trm in (C.5) and (C.6) to obtain η ce D,. As a prcaution w rcall that th gnric constant c can attain diffrnt valus at diffrnt placs. In th cas that was xcludd, i.., λ < 0 and h 2 T f T < u h ψ w obtain only a wakr bound of h 2 T f T in trms of η T, h 2 f T 2 h2 f T u h ψ η T. Th factor on th right-hand sid lads to th last trm in th fficincy masur (6.6). It guarants fficincy if w ar far away from th obstacl. W altrnativly rstart with (C.6) and insrt v and choos α = c 2z 2c 1 : E D, c 3 zη αh λ (f T, b (z) ) 0,T T ω c 3 zη c 4 z 2 η 1/2 W procd with a Young inquality to gt and st z = T ω h 2 f T. E D, c 5 zη c 6 z 3 h 4 f T 2 hλ 2h 2 f T c5 /c 6 to absorb th scond trm by th first on to gt th bound E D, c λ η = c η1/2 hf T h 2 η. f T This stimat is advantagous if th stimator is larg compard to th load. 22

25 Rfrncs [1] M. Ainsworth and T.J. Odn, A Postriori Error Estimation in Finit Elmnt Analysis. Wily, Chichstr [2] S. Bartls and C. Carstnsn. Avraging tchniqus yild rliabl a postriori finit lmnt rror control for obstacl problms. Numr. Math. 99, (2004). [3] D. Brass, Finit Elmnts: Thory, Fast Solvrs and Applications in Solid Mchanics. 3rd dition. Cambridg Univrsity Prss [4] D. Brass. A postriori rror stimators for obstacl problms anothr look. Numr. Math. 101, (2005). [5] D. Brass, C. Carstnsn and R.H.W. Hopp, Convrgnc analysis of a conforming adaptiv finit lmnt mthod for an obstacl problm. Numr. Math. 107, (2007). [6] D. Brass, C. Carstnsn and R.H.W. Hopp, Error rduction in adaptiv finit lmnt approximation of lliptic obstacl problms. (in prparation). [7] D. Brass and J. Schöbrl, Equilibratd rsidual rror stimator for Maxwll s quations. Math. Comp. (to appar). [8] R. Luc and B. Wohlmuth, A local a postriori rror stimator basd on quilibratd fluxs. SIAM J. Numr. Anal. 42, (2004). [9] P. Morin, R.H. Nochtto and K.G. Sibrt. Data oscillation and convrgnc of adaptiv FEM. SIAM J. Numr. Anal. 38, (2000). [10] P. Nittaanmäki and S. Rpin, Rliabl Mthods for Computr Simulation. Error Control and A Postriori Estimats. Elsvir. Amstrdam [11] W. Pragr and J.L. Syng, Approximations in lasticity basd on th concpt of function spacs. Quart. Appl. Math. 5, (1947). [12] S. I. Rpin. Estimats of dviations from xact solutions of lliptic variational inqualitis. J. Math. Scincs 115, (2003). [13] K. Sibrt and A. Vsr, A unilatrally constraind quadratic minimization with adaptiv finit lmnts. SIAM J. Optimization 18, (2007). [14] A. Vsr. Efficint and rliabl a postriori rror stimators for lliptic obstacl problms. SIAM J. Numr. Anal. 39, (2001). [15] A. Wiss and B. Wohlmuth, A postriori rror stimator and rror control for contact problms. IANS rport 12/2007, Univrsity of Stuttgart. Institut of Mathmatics, Ruhr-Univrsity of Bochum, D Bochum, Grmany, Dpartmnt of Mathmatics, Univrsity of Houston, Houston, TX , U.S.A. and Institut of Mathmatics, Univrsity of Augsburg, D Augsburg, Grmany Dpt. of Math. and Cntr for Comput. Engrg. Scinc, RWTH Aachn, D Aachn, Grmany 23