c 2009 Society for Industrial and Applied Mathematics

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1 SIAM J. NUMER. ANAL. Vol. 47 No. 3 pp c 2009 Socity for Industrial and Applid Mathmatics RECOVERY-BASED ERROR ESTIMATOR FOR INTERFACE PROBLEMS: CONFORMING LINEAR ELEMENTS ZHIQIANG CAI AND SHUN ZHANG Abstract. This papr studis a nw rcovry-basd a postriori rror stimator for th conforming linar finit lmnt approximation to lliptic intrfac problms. Instad of rcovring th gradint in th continuous finit lmnt spac th flux is rcovrd through a wightd L 2 projction onto H(div conforming finit lmnt spacs. Th rsulting rror stimator is analyzd by stablishing th rliability and fficincy bounds and is supportd by numrical rsults. This papr also proposs an adaptiv finit lmnt mthod basd on ithr th rcovry-basd stimators or th dg stimator through local msh rfinmnt and stablishs its convrgnc. In particular it is shown that th rliability and fficincy constants as wll as th convrgnc rat of th adaptiv mthod ar indpndnt of th siz of jumps. y words. a postriori rror stimator adaptiv mthod intrfac problms finit lmnt AMS subjct classifications. 65N30 65N15 DOI / Introduction. Th a postriori rror stimators of th rcovry typ hav bn xtnsivly studid by many rsarchrs.g. [ ] du to thir many appaling proprtis: simplicity univrsality and asymptotic xactnss. (Th univrsality is in th sns that thr is no nd for th undrlying rsidual or boundary valu problm. Lt u U b th currnt approximation thn th rcovrybasd stimator is dfind as th L 2 norm of th diffrnc btwn th dirct and postprocssd approximations of th gradint ( u U and G( u U : (1.1 η G = G( u U u U and η G = ( T η 2 G 1 2. Thr ar many postprocssing rcovry tchniqus (s survy articl [33] by Zhang and rfrncs thrin. A simpl on is th projction of th dirct approximation onto vctor-valud continuous finit lmnt spac with rspct to ithr a discrt or th L 2 innr product. Th popular Zinkiwicz Zhu (ZZ stimator [35 36] can b viwd as on basd on th discrt L 2 projction (s [28]. For stimators basd on th L 2 projction s.g. [10] and rfrncs thrin. By mploying svral multigrid smoothing aftr this L 2 projction Bank and Xu wr abl to prov that th rsulting stimator is asymptotically xact in [3] on irrgular mshs by stablishing a suprconvrgnc rsult of th rcovrd gradint. S also [ ] for asymptotically xact stimators basd on diffrnt rcovry tchniqus on irrgular mshs. For highr-ordr lmnts Bank Xu and Zhng [4] and Naga and Zhang [21] studid rcovry-basd stimators and stablishd thir asymptotic xactnss assuming that th solution of th undrlying problm is sufficint smooth. A summary Rcivd by th ditors March ; accptd for publication (in rvisd form March ; publishd lctronically Jun This work was supportd in part by th National Scinc Foundation undr grants DMS and DMS Dpartmnt of Mathmatics Purdu Univrsity Wst Lafaytt IN (zcai@math. purdu.du zhang@math.purdu.du. 2132

2 RECOVERY-BASED ESTIMATOR FOR INTERFACE PROBLEMS 2133 and bibliographical rmarks on th rcovry-basd rror stimators may b found in [2 10]. Estimators of th rcovry typ possss a numbr of attractiv faturs that hav ld to thir popularity. In particular thir as of implmntation gnrality and ability to produc quit accurat stimators hav ld to thir widsprad adoption spcially in th nginring community. Howvr for applications with nonsmooth solutions lik lliptic intrfac problms it is wll known [3] that thy ovrrfin rgions whr thr ar no rror and hnc thy fail to rduc th global rror. This is shown by Ovall in [23 24] through svral intrsting and ralistic xampls. To ovrcom this difficulty on oftn applis th mthod on ach subdomain sparatly. For rasons why this local approach is not favorabl s dtaild discussions in [24]. Mor importantly th local approach fails whn triangulations do not align with intrfacs which occurs whn intrfacs ar curvs/surfacs or hav unknown locations. On of th purposs of this papr is to dvlop a nw rcovry-basd rror stimator that compltly rsolv this difficulty for th intrfac problm using a global approach. To do so w notic that th normal componnt of th gradint and th tangntial componnts of th flux ar discontinuous across intrfacs. Hnc rcovring ithr th gradint or th flux in a continuous finit lmnt spac mans using a continuous function to approximat a discontinuous function. Th rror of such an approximation could b arbitrarily larg at whr th approximatd function is discontinuous. This is th sol rason for th failur of th xisting rcovry-basd stimators. In ordr to ovrcom this difficulty w rcovr th flux in th H(div conforming finit lmnt spacs such as thos of Raviart Thomas (RT or Brzzi Douglas Marini (BDM [8]. Mor spcifically w introduc two flux rcovry procdurs basd on th global (wightd L 2 projction onto and on th local avraging in th RT or BDM spacs. Th rsulting rcovry-basd (implicit and xplicit stimators ar thn analyzd by stablishing th rliability and fficincy bounds whr th fficincy bound for th implicit stimator is global. Both th rliability and fficincy constants ar provd to b indpndnt of th siz of jumps and hnc th stimators introducd in this papr ar robust with rspct to th diffusion cofficints. Numrically for a bnchmark tst problm w show that our stimators do not ovrrfin rgions along intrfacs and that thy ar not subjct to th constraint on th monotonicity of th diffusion cofficint in our thory. Morovr w show numrically that th implicit stimator introducd in this papr is accurat and robust with rspct to triangulations that do not align with intrfacs. Hnc it has a potntial to trat problms with intrfacs bing curvs/surfacs or having unknown locations. Th othr purpos of th papr is to study adaptiv finit lmnt mthod through local msh rfinmnt basd on ithr th rcovry-basd stimators or th dg stimator. Rcntly thr has bn intnsiv study on th convrgnc of th adaptiv mthod for conforming finit lmnts (s.g. [ ] and also for nonconforming and mixd lmnts. Dörflr in [16] provd th first convrgnc rsult for th Possion quation undr th assumption that th initial msh is fin nough. His adaptiv mthod is basd on th dg stimator. For th adaptiv mthod using th rsidual-basd stimator this fin initial msh rquirmnt was rmovd by Morin Nochtto and Sibrt in [20] by introducing a marking stratgy basd on th data oscillation. Howvr to guarant that th data oscillation monotonically dcrass th rfind lmnt has to contain a nod of th finr msh in its intrior. Vry rcntly th convrgnc of th standard adaptiv mthod basd on th xplicit rsidual-basd stimator is stablishd by Cascon t al. [13] without th marking stratgy on th data oscillation and hnc without th intrior nod rquirmnt. In this papr w

3 2134 ZHIQIANG CAI AND SHUN ZHANG introduc an additional marking stratgy basd on th wightd lmnt rsidual and stablish th convrgnc of th rsulting adaptiv mthod basd on ithr th xplicit rcovry-basd stimator introducd in thispaprorthdgstimatorin[12]. Assumptions on th initial msh and on th intrior nod ar not rquird. Furthrmor th rat of convrgnc is indpndnt of th siz of jumps. This papr is organizd as follows. Th intrfac problm and its conforming linar finit lmnt approximation ar dscribd in sction 2. Th rcovry procdur and th rsulting rcovry-basd a postriori rror stimator ar introducd in sction 3. Thortical analysis and numrical xprimnts ar prsntd in sctions 4 and 5 rspctivly. Finally th adaptiv finit lmnt mthod is dscribd and analyzd in sction Problm and finit lmnt approximation. Considr th following intrfac problm (2.1 (α(x u = f in Ω with boundary conditions (2.2 u = g on Γ D and n (α u =0 onγ N whr th symbols and stand for th divrgnc and gradint oprators rspctivly f and g ar givn scalar-valud functions Ω is a boundd polygonal domain in R d (d = 2 or 3 with boundary Ω = Γ D Γ N and Γ D Γ N = n =(n 1... n d is th outward unit vctor normal to th boundary and α(x is positiv and picwis constant on polygonal subdomains of Ω with possibl larg jumps across subdomain boundaris (intrfacs: α(x =α i > 0 in Ω i for i =1... n. Hr {Ω i } n i=1 is a partition of th domain Ω with Ω i bing an opn polygonal domain. For simplicity w considr only homognous Numann boundary conditions and picwis linar data g. Also w assum that Γ D is not mpty (i.. ms (Γ D 0. W us th standard notations and dfinitions for th Sobolv spacs H s (Ω d and H s ( Ω d for s 0. Th standard associatd innr products ar dnotd by ( sω and ( s Ω and thir rspctiv norms ar dnotd by sω and s Ω. (W supprss th suprscript d bcaus th dpndnc on dimnsion will b clar by contxt. W also omit th subscript Ω from th innr product and norm dsignation whn thr is no risk of confusion. For s =0H s (Ω d coincids with L 2 (Ω d.inthis cas th innr product and norm will b dnotd by and ( rspctivly. W will also us th nrgy norm dnotd by v = v Ω = α 1/2 v. 0Ω Lt H 1 gd(ω := { v H 1 (Ω : v = g on Γ D }. Th corrsponding variational form of systm (2.1 is to find u HgD 1 (Ω such that (2.3 a(u v =f(v v H 1 0D (Ω

4 RECOVERY-BASED ESTIMATOR FOR INTERFACE PROBLEMS 2135 whr th bilinar and linar forms ar dfind by a(u v =(α(x u v and f(v =(f v rspctivly. For simplicity of prsntation considr only triangular and ttrahdra lmnts in th rspctiv two and thr dimnsions. Lt T = {} b a finit lmnt partition of th domain Ω. Assum that th triangulation T is rgular (s [15]; i.. for all T thr xists a positiv constant κ such that h κρ whr h dnots th diamtr of th lmnt and ρ th diamtr of th largst circl that may b inscribd in. Not that th assumption of th rgularity dos not xclud highly locally rfind mshs. Furthrmor assum that intrfacs F = { Ω i Ω j i j =1... n} do not cut through any lmnt T. (This assumption is ndd for analysis and for xplicit stimators but not for implicit stimators introducd in this papr. Lt P k ( b th spac of polynomials of dgr k on lmnt. Dnot th continuous picwis linar finit lmnt spac associatd with th triangulation T by U = { v H 1 (Ω : v P 1 ( T }. Lt U g = {v U: v = g on Γ D } thn th finit lmnt approximation of (2.3 is to find u U U g such that (2.4 a(u U v=f(v v U 0. Dfin α min = min 1 i n α i and α max = max 1 i n α i. It is wll known (s.g. [6] that if th solution u is in H s (Ω 1 s 2 thn th following a priori rror stimat holds: (2.5 u u U Ω C ( n 1/2 h s 1 α 1/2 u 2 s 1 Ω i i=1 with h s 1 α u ( 1/2 s 1 Ωi = h 2(s 1 α u s 1 2. T Ω i Hr and thraftr w us C with or without subscripts in this papr to dnot a gnric positiv constant possibly diffrnt at diffrnt occurrncs that is indpndnt of th msh paramtr h and th ratio α max /α min but may dpnd on th domain Ω.

5 2136 ZHIQIANG CAI AND SHUN ZHANG 3. Flux rcovry and rror stimator. Th flux dfind by (3.1 σ = α(x u in Ω is an important physical quantity which is oftn th primary concrn in practic. For th intrfac problm in (2.1 with f L 2 (Ω it is asy to s that th normal componnt of th flux is continuous but its tangntial componnt is discontinuous across th intrfacs. This typ of vctor-valud functions may b prcisly charactrizd by th following spac: H(div; Ω = { τ L 2 (Ω d : τ L 2 (Ω } L 2 (Ω d which is a Hilbrt spac undr th norm Dnot its subspac by τ H(div; Ω = ( τ 2 0Ω + τ 2 0Ω 1 2. Σ=H N (div; Ω = {τ H(div; Ω : n τ =0onΓ N }. In (3.1 dividing by α(x multiplying a tst function τ and intgrating ovr th domain Ω giv th following variational problm: find σ Σ such that (3.2 b(σ τ =u(τ τ Σ whr bilinar form b( and linar form u( ar dfind by b(σ τ = ( α 1 σ τ (σ τ Σ Σ and u(τ = ( u τ τ Σ rspctivly Flux rcovry. Th rcovry procdur introducd in this papr is basd on th conforming finit lmnt approximation to th variational problm in (3.2. Thr ar svral familis of th H(div; Ω confirming finit lmnt spacs (s.g. [8 26]. W considr only RT and BDM lmnts for simplicity. Dnot th local lowst-ordr RT and BDM spacs on lmnt T by RT 0 ( =P 0 ( d + x P 0 ( and BDM 1 ( =P 1 ( d rspctivly whr x =(x 1... x d. Thn th standard H(div; Ω conforming RT and BDM spacs ar dfind by and RT 0 = {τ Σ:τ RT 0 ( T} BDM 1 = {τ Σ:τ BDM 1 ( T} rspctivly. For convninc dnot RT 0 and BDM 1 by V. It is wll known (s [8] that V has th following approximation proprty: (3.3 inf τ RT 0 σ τ H(div; Ω C ( h σ 2 1Ω + h σ 2 1Ω for σ H 1 (Ω m m Σ with σ H 1 (Ω m and (3.4 inf τ BDM 1 σ τ 0Ω C h l σ lω for σ H l (Ω m m Σandl [1 2]. 1/2

6 RECOVERY-BASED ESTIMATOR FOR INTERFACE PROBLEMS Implicit approximation. Lt ū U Ub an approximation of th xact solution u HgD 1 (Ω of (2.3 thn w rcovr th flux by solving th following problm: find σ V V such that (3.5 b(σ V τ =ū U (τ τ V with ū U (τ = ( ū U τ. Thorm 3.1. Lt u and σ V b th solutions of (2.3 and (3.5 rspctivly. Thn thr xists a positiv constant C indpndnt of th ratio α max /α min such that th following a priori rror bound (3.6 α 1/2 (σ σ V 0Ω C ( inf τ V α 1/2 (σ τ holds. Proof. Dnot th tru rrors of th solution and th flux by = u ū U and E = σ σ V + u ū U Ω 0Ω rspctivly. Diffrnc btwn (3.2 and (3.5 givs th following rror quation: b(e τ =u(τ ū U (τ τ V whr (3.7 u(τ ū U (τ = ( τ. Using th abov rror quation and th Cauchy Schwarz inquality yilds α 1/2 E 2 = b(e E =b(e σ τ +b(e τ σ V 0Ω α 1/2 α E 1/2 (σ τ + u(τ σ V ū U (τ σ V 0Ω 0Ω τ V. By th Cauchy Schwarz and triangl inqualitis w hav u(τ σ V ū U (τ σ V = ( τ σ V α 1/2 α 1/2 (τ σ V 0Ω 0Ω ( α 1/2 Ω (τ σ + α 1/2 E. 0Ω 0Ω Now th rror bound in (3.6 follows from th abov inqualitis and th ɛ inquality (ab 1 2ɛ a2 + ɛ 2 b Explicit approximation. In this subsction w dscrib an xplicit approximation of th flux basd on th RT 0. Dnot th st of all dgs/facs of th triangulation by E := E Ω E D E N whr E Ω is th st of all intrior lmnt dgs/facs and E D and E N ar th st of boundary dgs/facs blonging to th rspctiv Γ D and Γ N. For ach E dnot a unit vctor normal to by n. Whn E D E N assum that n is th

7 2138 ZHIQIANG CAI AND SHUN ZHANG unit outward normal vctor. Th nodal basis function φ of RT 0 corrsponding to E Ω E D is charactrizd by (3.8 φ n = δ E whr δ is th ronckr dlta. For ach intrior dg/fac E Ω lt + and b th two lmnts sharing th common dg/fac such that th unit outward normal vctor of + coincids with n.lta ± b th vrtics of ± opposit to. Thn th nodal basis function of RT 0 corrsponding to E Ω has of th form ( x a + d + for x + (3.9 φ (x := d (x a for x 0 lswhr whr and ± ar th d 1anddmasur of and ± rspctivly. boundary dg/fac E D th corrsponding nodal basis function is ( x a + (3.10 φ (x := d + for x + 0 lswhr. For Lt τ = α(x ū U dfin its approximation ˆσ RT 0 (ū U inrt 0 by (3.11 ˆσ RT 0 (ū U = ˆσ φ (x E Ω E D whr ˆσ is th normal componnt of ˆσ RT 0 on E Ω E D dfind by ( ( γ τ n (3.12 ˆσ := + +(1 γ τ n for E Ω τ n for E D for som constant γ [0 1]. W choos (3.13 γ = α + α + α to nsur that th fficincy constant is indpndnt of th ratio α max /α min (s Thorm 4.3. Rmark 3.1. If th normal componnt of τ is continuously across dg/fac thn th normal componnt of its approximation dfind in (3.12 quals to th normal componnt of τ ; i.. ˆσ = τ n Error stimators. Lt σ V b th solution of problm (3.5 for any lmnt T dfin th following local a postriori rror indicator by (3.14 η V = α 1/2 σ V + α 1/2 0 ū U.

8 RECOVERY-BASED ESTIMATOR FOR INTERFACE PROBLEMS 2139 Thn th corrsponding global a postriori rror stimator is ( 1/2 (3.15 η V = (η V 2 = α 1/2 σ V + α 1/2 0Ω ū U. T It is asy to s that (3.16 η V =minα 1/2 τ + α 1/2 0Ω ū U. τ V This stimator rquirs numrical solution of a systm of linar quations with a mass matrix. Such a systm can b solvd vry fficintly by svral swps of th Jacobi itration or bttr by prconditiond conjugat gradint mthod with th Jacobi prconditionr. Not that this stimator dos not rquir th alignmnt btwn triangulations and th intrfacs of th undrlying problm. Hnc it can b applid to problms with intrfacs bing curvs/surfacs or having unknown locations. Nxt basd on th xplicit approximation in (3.11 w dfin xplicit local a postriori rror indicator by (3.17 ˆη RT0 = α 1/2 ˆσ RT0 + α 1/2 ū U 0 for any T and xplicit global a postriori rror stimator by ( 1/2 (3.18 ˆη RT0 = (ˆη RT0 2 = α 1/2 ˆσ RT0 + α 1/2 0Ω ū U. T This xplicit stimator is similar to that introducd by Luc and Wolhmuth [19] but thy diffr in th rcovry procdur. Th lattr is mor complicatd xpnsiv and probably accurat than th formr. Nvrthlss both th stimators ar subjct to th alignmnt assumption btwn triangulations and intrfacs. W study this xplicit stimator bcaus it is usd in our analysis and it is probably appaling to th nginring community. 4. Rliability and fficincy bounds. In this sction w stablish rliability and fficincy bounds for both implicit and xplicit stimators undr th assumption that triangulations align with intrfacs Clémnt-typ intrpolation. Clémnt-typ intrpolation oprators hav bn intnsivly studid in th litratur (s.g. [6 25] and thy ar oftn usd for stablishing th rliability bound of a postriori rror stimators. In this sction w follow [6] to dfin a wightd Clémnt-typ intrpolation oprator and to stat its approximation and stability proprtis. To this nd dnot by N and N th sts of all vrtics of th triangulation T and of lmnt T rspctivly. For any z Ndnotbyφ z th nodal basis function lt ω z = suppt (φ z and dnot by ˆω z th union of lmnts in ω z whr th cofficint α achivs th maximum for ω z. For a givn function v dfin its wightd avrag ovr ˆω z by (4.1 vdx= ˆω z ˆω z vφ z dx ˆω z φ z dx.

9 2140 ZHIQIANG CAI AND SHUN ZHANG Now following [6] dfin th intrpolation oprator I : L 2 (Ω U 0 by (4.2 Iv = z N(π z vφ z (x whr th nodal valu at z is dfind by { ˆω (Iv(z =π z v = z vdx z N\Γ D 0 z N Γ D. Not that th wightd avrag in (4.1 implis (4.3 (1 φ z (v π z vˆωz = φ z (v π z v dx =0 z N\Γ D ˆω z which will b usd in th subsqunt sction in ordr to handl a trm involving th right-hand sid function f. In this and nxt subsctions assum that th Hypothsis 2.7 in [6] holds. That is assum that for any two diffrnt subdomains Ω i and Ω j which shar at last on point thr is a connctd path passing from Ω i to Ω j through adjacnt subdomains such that th diffusion cofficint α is monoton along this path. This assumption is waknd to th quasi-monotonicity in [25]. Lmma 4.1. For any T z N andv H0D 1 (Ω thr xists a positiv constant C indpndnt of th ratio α max /α min such that (4.4 (v π z vφ z 0 Ch α 1/2 and that v Δ (4.5 (v π z v φ z 0 Cα 1/2 v Δ whr Δ is th union of all lmnts that shar at last on vrtx with. Proof. Using th following inqualitis φ z Ch 1 and φ z 0 Ch (d 2/2 (4.4 and (4.5 can b provd in a similar fashion as that in [6]. Lmma 4.2. For any T and v H0D 1 (Ω th following stimats hold: (4.6 v Iv 0 Ch α 1/2 and v Δ (4.7 (v Iv 0 Cα 1/2 v Δ. Proof. Using th idntity z N φ z (x =1in whav (4.8 v Iv 0 = (v π z vφ z 0 (v π z vφ z 0 z N z N and (4.9 (v Iv = (v π z vφ z = z N φ z v + (v π z v φ z. z N z N

10 RECOVERY-BASED ESTIMATOR FOR INTERFACE PROBLEMS 2141 Equation (4.6 is a dirct consqunc of (4.8 th triangl inquality and (4.4. Equation (4.7 follows from th triangl inquality th fact that φ z 1 and (4.5 that (v Iv 0 φ z v 0 + (v π z v φ z 0 z N z N d v 0 + C This complts th proof of th lmma Rliability. Lt H f = α 1 h2 f ω z and z N (F Γ D ( Ĥ f = α 1/2 hf = z N α 1/2 v Δ Cα 1/2 v Δ. z N \(F Γ D T ω z α 1 h2 α 1 h2 f 2 0 1/2. ω f fdx z Rmark 4.1. Th scond trm in H f is a highr-ordr trm for f L 2 (Ω and so is th first trm for f L p (Ω with p>2(s[12]. Lmma 4.3. For any v H0D 1 (Ω thr xists a positiv constant C indpndnt of th ratio α max /α min such that (4.10 (f v Iv CH f v and that (4.11 (f v Iv C Ĥf v. Proof. Equation (4.3 and th fact that ˆω z = ω z for z N\(F Γ D giv (1 φ z (v π z v ωz =0 z N\(F Γ D which togthr with z N φ z(x =1inΩgivs (f v Iv= z N(f (v π z vφ z ωz = = = z N (F Γ D z N (F Γ D + z N \(F Γ D z N (F Γ D + (f (v π z vφ z ωz + (f (v π z vφ z ωz z N \(F Γ D ( f fdx(v π z vφ z ω z z N \(F Γ D ω z ω z (f (v π z vφ z ω z ( f fdx(v π z vφ z ω z (f (v π z vφ z ωz Now (4.10 is a dirct consqunc of th Cauchy Schwarz inquality and ( /2

11 2142 ZHIQIANG CAI AND SHUN ZHANG Equation (4.11 follows from th idntity (f v Iv= (f (v π z vφ z z N ω z th Cauchy Schwarz inquality and (4.4. Thorm 4.1. Assum that ū U = u U is th solution of (2.4. Thn th stimator dfind in (3.15 satisfis th following global rliability bound: η V (4.12 C (η V + H f and th following bound: ( C r (η 2 V + Ĥ2 f whr C and C r ar constants indpndnt of th ratio α max /α min. Proof. It follows from th orthogonality proprty of th finit lmnt solution intgration by parts (2.1 and th Cauchy Schwarz inquality that 2 = a( I=(α (u u U ( I =(α u + σ V ( I (σ V + α u U ( I (f σ V I+η V I which combining with th fact that (α(x u U =0 T (4.7 (4.10 and th Cauchy Schwarz inquality implis 2 (f I T ( (σ V + α u U I + Cη V C (H f + η V + ( T ( T h 2 h 2 α I 2 0 1/2. Using th invrs inquality and (4.6 w thn hav 1/2 (α 1/2 σ V + α 1/2 2 u U 0 2 C (H f + η V + Cη V = C (H f + η V which lads to (4.12. Using (4.11 instad of (4.10 on can prov th validity of (4.13 in th sam way. Thorm 4.2. Undr th sam assumption of Thorm 4.1 th xplicit stimator ˆη RT 0 dfind in (3.18 satisfis th following global rliability bound: (4.14 C (ˆη RT 0 + H f and th following bound: ( C r (ˆη 2 RT 0 + Ĥ2 f whr C and C r ar constants indpndnt of th ratio α max /α min.

12 RECOVERY-BASED ESTIMATOR FOR INTERFACE PROBLEMS 2143 Proof. Th rliability bounds in (4.14 and (4.15 ar an immdiat consqunc of th rspctiv (4.12 and (4.13 and th fact that ˆη RT0 α(x min 1/2 τ + α(x 1/2 0Ω u U = η RT0. τ RT 0 This complts th proof of th thorm Efficincy. For any E Ω and any vctor-valud function ρ that is picwis constant with rspct to th triangulation T dnot th jump of th normal componnt of ρ across = + by For any E\E Ω st J (ρ =[ρ n ]=(ρ + ρ n. J (ρ =0. For any Ednotbyω th union of all lmnts that shar dg/fac. Dfin a modification of th dg rror stimator as follows: ( 1/2 ( (4.16 η E := η 2 with η = E α + 1/2 2h J (α u + α U ds 2 whr h is th diamtr of dg/fac. Without assumptions on th distribution of th cofficint α it was provd by Ptzoldt (s quation (5.7 in [25] that thr xists a constant C>0indpndnt of α max /α min h andh such that ( (4.17 η 2 C 2 ω + h2 f α + + α f 2 0 T ω whr f is th avrag of f ovr : f = 1 fdx. Lmma 4.4. For any lmnt T th constant vctor τ on has th following rprsntation in RT 0 : (4.18 τ = τ φ (x whr τ =(τ n is th normal componnt of τ on dg. Proof. Sinc constant vctor on blongs to RT 0 ( τ can b writtn as τ = τ φ (x. Now using (3.8 yilds τ =(τ n and hnc th lmma. Thorm 4.3. Thr xists a constant C>0indpndnt of α max /α min such that ( (4.19 ˆη 2 C RT 0 2 ω + h2 T f ft 2 α 0T T T T ω

13 2144 ZHIQIANG CAI AND SHUN ZHANG whr ω is th union of lmnts sharing a common dg/fac with and that (4.20 η BDM1 η RT0 ˆη RT0 C Ω + C ( T h 2 α f f 2 0 1/2. Proof. For any lmnt T and for any dg/fac without loss of gnrality assum that n is th outward unit vctor normal to. Dnotby th adjacnt lmnt with common dg/fac. Lt τ = α u U thn for any x (3.11 (3.12 (3.13 and (4.18 giv ˆσ RT 0 τ = (ˆσ τ φ (x = \ Ω = \ Ω (1 γ (τ τ φ (x (1 γ J (τ φ (x = \ Ω α α + α J (τ φ (x. Sinc J (τ is constant in and φ (x 2 0 C it thn follows from th triangl inquality that ˆη 2 α = 1/2 (ˆσ RT 0 RT 0 τ 2 C 1 ( 0 α + 2 J (τ φ (x 2 0 α (4.21 C C \ Ω \ Ω \ Ω 1 α + α J (τ 2 φ (x 2 0 η 2 which togthr with (4.17 implis (4.19. Th first two inqualitis in th global fficincy bound (4.20 follow from th fact that RT 0 BDM 1 and (3.16 rspctivly. To prov th third on in (4.20 summing up (4.19 ovr all T givs ˆη 2 RT 0 = α 1/2 ˆσ RT0 + α 1/2 2 u C ( U 2 ω 0 + T T ( C 2 h 2 + C f f 2 α 0 T T T ω which in turn implis (4.20. This complts th proof th thorm. h2 T f ft 2 α 0T T 5. Numrical xprimnts. In this sction w rport som numrical rsults for an intrfac problm with intrscting intrfacs usd by many authors.g. [ ] which is considrd as a bnchmark problm. Lt Ω = ( and u(r θ =r β μ(θ

14 RECOVERY-BASED ESTIMATOR FOR INTERFACE PROBLEMS 2145 in th polar coordinats at th origin with cos((π/2 σβ cos((θ π/2+ρβ if 0 θ π/2 cos(ρβ cos((θ π + σβ if π/2 θ π μ(θ = cos(σβ cos((θ π ρβ if π θ 3π/2 cos((π/2 ρβ cos((θ 3π/2 σβ if 3π/2 θ 2π whr σ and ρ ar numbrs. Th function u(r θ satisfis th intrfac quation in (2.1 with Γ N = f =0and { R in (0 1 2 ( α(x = 1 in Ω\ ([0 1] 2 [ 1 0] 2. Th numbrs β R σ andρ satisfy som nonlinar rlations (.g. [20 14]. xampl whn β =0.1 thn For R ρ = π/4 and σ Not that whn β =0.1 this is a difficult problm for computation. Rmark 5.1. This problm dos not satisfy Hypothsis 2.7 in [6] and th distribution of its cofficints is not quasi monoton. Starting with a coars triangulation T 0 a squnc of mshs is gnratd by using standard adaptiv mshing algorithm that adopts th Dörflr s bulk marking stratgy i.. Marking Stratgy E dscribd in sction 7.1 of [16] with θ E =0.2. Th choic of θ E =0.2 is not critical but rcommndd in [14] for bttr prformanc. Markd triangls ar rfind rgularly by dividing ach into four congrunt triangls. Additionally irrgularly rfind triangls ar ndd in ordr to mak th triangulation admissibl. For mor dtails on adaptiv msh rfinmnt algorithms s.g. [11 7]. Not that th solution u(r θ isonlyinh 1+β ɛ (Ω for any ɛ>0 and hnc it is vry singular for small β at th origin. This suggsts that rfinmnt is cntrd around th origin. Th tru rror can b computd by rr := α 1/2 (u u U 2 = α(n u(u 2u U ds + α 1/2 2 u. U 0Ω Ω 0 T Sinc th tru solution u is vry smooth nar th boundary th intgrations on th boundary can b computd vry accuratly. Th rlativ rror stimator will b calculatd as th ratio of th stimator and α 1/2 u 0Ω : η ff-indx := α 1/2 (u u U 0 which is th so-calld ffctivity indx. W will us th following stopping critria: rl-rr := α 1/2 / (u u U α 1/2 u tol. 0Ω 0Ω Dnot by k th numbr of lvls of rfinmnt and by n th numbr of vrtics of triangulation. Numrical xprimnts hr will also involv th following rror stimators: (1 ZZ gradint rcovry-basd stimator [35]: (5.1 η ZZf = G( u U u U 0

15 2146 ZHIQIANG CAI AND SHUN ZHANG Tabl 5.1 Comparison of stimators for rlativ rror lss than 0.5. k n Err Rl-rr η Eff-indx η ZZg η ZZf η C η BV η RT η BDM whr G( u U U d and its nodal valu at vrtx z N is dfind by (G( u U z = 1 u ω z U dx. ω z (2 ZZ flux rcovry-basd rror stimator [35]: (5.2 η ZZf = α 1/2 (G( α u U +α u U 0 whr G( α u U U d and its nodal valu at vrtx z N is dfind by (G( α u U z = 1 α u ω z U dx. ω z (3 Carstnsn flux rcovry-basd rror stimator [9]: α (5.3 η C =min 1/2 (α u U + τ. τ U d 0Ω (4 Brnardi-Vrfürth (BV rror stimator [6] (an improvd xplicit rsidual basd stimator for diffusion problms: (5.4 η BV := ( 1 2 1/2 h α 1/2 [α u U ] n h 2 α 1 f + (α u U 0 2 whr α =max ω (α. Sinc f = 0 in this xampl th BV stimator may also b viwd as an dg stimator. In th first st of numrical xprimnts triangulations align with intrfacs of th problm. In particular w start with th coarsst triangulation T 0 obtaind from halving 16 congrunt squars by conncting th bottom lft and uppr right cornrs. W rport numrical rsults with th stopping critria tol = 0.5 tol = 0.15 and tol = 0.1. For tol = 0.5 Tabl 5.1 shows that th η ZZg stimator nds about six tims as many grid points as th rst stimators. Comparing Figurs 5.1 and 5.2 it is clar that th η ZZg stimator introducs unncssary rfinmnts along th intrfacs. For tol = 0.15 th η ZZg stimator will gnrat too many grid points for computrs to handl. Numrical rsults for th rst stimators ar rportd in Tabl 5.2. Figurs 5.3 and 5.4 show that both th stimators η ZZf and η C ovrrfin rgions along th intrfacs. This is bcaus th tangntial componnt of th flux is discontinuous but th rcovrd flux has continuous tangntial componnt.

16 RECOVERY-BASED ESTIMATOR FOR INTERFACE PROBLEMS 2147 Fig Msh gnratd by η ZZg. Fig Msh gnratd by η RT. Tabl 5.2 Comparison of stimators for rlativ rror lss than k n Err Rl-rr η Eff-indx η ZZf η C η BV η RT η BDM Fig Msh gnratd by η ZZf. Fig Msh gnratd by η C. Tabl 5.3 Comparison of stimators for rlativ rror lss than 0.1. k n Err Rl-rr η Eff-indx η BV η RT η BDM For tol = 0.1 both th stimators η ZZf and η C fail. Numrical rsults for th rst stimators ar rportd in Tabl 5.3 and Figurs Mshs gnratd by η BV η RT andη BDM ar similar. By inspcting th ffctivity indx both η RT and η BDM ar mor accurat than η BV and thy ar possibly asymptotically xact.

17 2148 ZHIQIANG CAI AND SHUN ZHANG η BV / A 1/2 u 0 and A 1/2 ( u u h 0 / A 1/2 u η / A 1/2 u BV 0 A 1/2 ( u u / A 1/2 u h rfrnc lin with slop 1/ numbr of nods Fig Msh gnratd by η BV. Fig Error vrsus stimator η BV. η rt / A 1/2 u 0 and A 1/2 ( u u h 0 / A 1/2 u η / A 1/2 u rt 0 A 1/2 ( u u / A 1/2 u h rfrnc lin with slop 1/ numbr of nods Fig Msh gnratd by η RT. Fig Error vrsus stimator η RT. η bdm / A 1/2 u 0 and A 1/2 ( u u h 0 / A 1/2 u η / A 1/2 u bdm 0 A 1/2 ( u u / A 1/2 u h rfrnc lin with slop 1/ numbr of nods Fig Msh gnratd by η BDM. Fig Error vrsus stimator η BDM. Th BV stimator η BV is subjct to th assumption that th intrfacs do not cut through any lmnt of triangulations and so is th analysis prsntd in sction 4 for th stimators introducd in this papr. Howvr it is asy to s that th stimator η V dfind in (3.15 is fr of this assumption. In practic it is important to considr th cas that triangulations do not align with intrfacs bcaus this happns whn intrfacs ar curvs/surfacs or thir locations ar unknown. Hnc th purpos of

18 RECOVERY-BASED ESTIMATOR FOR INTERFACE PROBLEMS 2149 Fig An initial msh: Two horizontal lins ar y =0.5 and y = 0.5. Fig An initial msh: Two horizontal lins ar y = and y = 0.5. Tabl 5.4 Estimator η RT with th initial msh in Figur k n Err Rl-rr η Eff-indx η RT η rt / A 1/2 u 0 and A 1/2 ( u u h 0 / A 1/2 u η / A 1/2 u rt 0 A 1/2 ( u u / A 1/2 u h rfrnc lin with slop 1/ numbr of nods Fig Msh gnratd by η RT with th initial msh Figur Fig Error vrsus stimator η RT with th initial msh Figur th scond st of numrical xprimnts is to tst our stimator η RT for initial mshs not aligning with th intrfacs. Considr two initial mshs dpictd in Figurs 5.11 and 5.12 whr two horizontal lins y = 0.5 and y = 0.5 in Figur 5.11 and y = and y = 0.5 in Figur 5.12 do not coincid with th intrfac y =0. For th initial msh in Figur 5.11 svral stps of rfinmnts gnrat a triangulation that aligns with th intrfac y = 0. Hnc numrical rsults dpictd in Tabl 5.4 and Figurs 5.13 and 5.14 ar in a good agrmnt with thos rportd prviously. For th initial msh in Figur 5.12 w choos th horizontal lin y = so that rfinmnts nvr gnrat a triangulation that aligns with th intrfac y =0. Numrical rsults for this tst ar rportd in Tabl 5.5 and Figurs 5.15 and As xpctd (s Figur 5.15 th msh is rfind along th intrfac y = 0 du to th nonalignmnt of th mshs and th intrfac.

19 2150 ZHIQIANG CAI AND SHUN ZHANG Tabl 5.5 Estimator η RT with th initial msh in Figur k n Err Rl-rr η Eff-indx η RT η rt / A 1/2 u 0 and A 1/2 ( u u h 0 / A 1/2 u η / A 1/2 u rt 0 A 1/2 ( u u / A 1/2 u h rfrnc lin with slop 1/ numbr of nods Fig Msh gnratd by η RT with initial msh Figur Fig Error vrsus stimator η RT with initial msh Figur Adaptiv mthod. This sction proposs an adaptiv finit lmnt mthod and analyzs its convrgnc. Quantitis w usd for marking lmnts for rfinmnt ar diffrnt from thos in [16 20] and hnc convrgnc of our adaptiv algorithm is not subjct to constraints on ithr th sufficintly small initial msh [16] or th intrior nods in th rfind lmnts [20] Adaptiv algorithm. Givn an initial triangulation T 0 a squnc of nstd conforming triangulations T k is gnratd through th following loop: Solv Estimat Mark Rfin. Th Solv stp solvs (2.4 in th finit lmnt spac corrsponding to th triangulation T k for th discrt solution u k U g (k whr U g (k is th finit lmnt spac dfind on T k accordingly. Hr and thraftr w shall xplicitly xprss th dpndnc of a quantity on k by ithr th subscript lik u k or th variabl lik U g (k. Th Estimat stp computs som quantitis andthmark stp is to mark lmnts whr thos quantitis ar larg for rfinmnt. Th choic of th quantitis usd for marking lmnts is crucial for convrgnc analysis of th corrsponding adaptiv algorithms. For xampl Dörflr in [16] uss only local indicators (dgbasd and convrgnc of his algorithm is subjct to th sufficintly small initial msh; Morin Nochtto and Sibrt in [20] us both local indicators (rsidual-basd and oscillations and hnc thir algorithm is no longr subjct to th initial msh constraint but is undr th intrior nod assumption. In this papr w propos to us both local indicators η (k (dg- and rcovry-basd and wightd lmnt rsiduals α 1/2 h(kf 0 = α 1/2 h (k f 0 T k. Thn th corrsponding marking stratgis ar as follows:

20 RECOVERY-BASED ESTIMATOR FOR INTERFACE PROBLEMS 2151 Marking Stratgy E: Giving a paramtr 0 <θ E < 1 construct a minimal subst ˆT k of T k such that (6.1 η 2 (k θ2 Eη 2 (k ˆT k whr η 2 (k = T k η 2 (k; Marking Stratgy R: Giving a paramtr 0 <θ 0 < 1 and th subst ˆT k T k producd by th Marking Stratgy E nlarg ˆT k to a minimal st (dnotd again by ˆT k such that (6.2 α 1/2 h(kf 2 0 θ2 0 α 1/2 h(kf 2. 0Ω ˆT k Finally th Rfin stp is to rfin th lmnts in ˆT k obtaind in Marking Stratgy R to gnrat a nw triangulation T k+1 such that U 0 (k U 0 (k +1 and that ach of its dgs/facs contains a nod of th finr msh T k+1 in thir intrior. Not that som lmnts in T k \ ˆT k adjacnt to lmnts in ˆT k ar also rfind to avoid hanging nods. Not also that nw intrior nods in th lmnts in ˆT k ar not rquird. In summary th adaptiv finit lmnt algorithm may b dfind as follows. Adaptiv algorithm. For a givn initial msh T 0 choos paramtrs θ E θ 0 (0 1. For k = prform (1 u k = Solv (T k fgα(x. (2 {η (k α 1/2 h(kf 0 } Tk = Estimat (T k u k fgα(x. (3 ˆT k = Mark (θ E θ 0 ; T k {η (k α 1/2 h(kf } Tk. (4 T k+1 = Rfin (T k ˆT k Convrgnc analysis. Th analysis prsntd hr is similar to that of [20]. To stablish th convrgnc of th adaptiv mthod w start with th following assumptions on a postriori rror stimators. Assumption R. Assum that thr xists a positiv constant C r such that ( (6.3 u u k 2 Ω C r η 2 (k+ α 1/2 h(kf 2 0Ω for k = Assumption E. Assum that thr xists a positiv constant C l such that (6.4 C l θeη 2 2 (k u k+1 u k 2 Ω + α 1/2 h(kf for k = Th Assumption R is similar to but wakr than th global rliability bound. This bound is stablishd for th stimators introducd in this papr in sction 4.2 and for th dg stimator dfind in (4.16 (s th proof of Thorm 5.3 in [25] with C r indpndnt of th siz of jumps. It also holds for th ZZ stimator but th constant C r dpnds on th siz of jumps (s.g. [28]. Th Assumption E will b vrifid in th nxt sction for various stimators. Lmma 6.1. Undr th Assumptions R and E w hav (6.5 u k+1 u k 2 Ω δ 1 u u k 2 Ω (1 + δ 0 α 1/2 h(kf 2 0Ω whr δ 0 = C l θ 2 E and δ 1 = δ 0 /C r ( Ω

21 2152 ZHIQIANG CAI AND SHUN ZHANG Proof. Equation (6.5 is a dirct consqunc of th Assumptions R and E. Nxt w show that th wightd lmnt rsidual α 1/2 h(kf 0Ω as a function of th rfinmnt lvl k is monotonically dcrasing. Lmma 6.2. Lt 0 <γ 0 < 1 b th rduction factor of lmnt siz associatd with on rfinmnt stp. Lt Tk ˆ b a subst of T k satisfying Marking Stratgy R. If T k+1 is gnratd by th Rfin stp from T k thn th following lmnt rsidual rduction occurs: (6.6 α 1/2 h(k +1f ζ α 1/2 h(kf 0Ω 0Ω with ζ = 1 (1 γ0 2 θ2 0. Proof. Dnot th collction of lmnts in T k+1 containd in lmnts of ˆT k by T k+1 = { T k+1 ˆ ˆT } k. By th assumption of th lmma w hav h (k +1 γ 0 h ˆ(k T k+1 which combining with α = α ˆ for T k+1 implis α 1 h2 (k +1 f 2 0 γ0 2 α 1 ˆ h2ˆ(k f 2 0 ˆ T k+1 ˆ ˆT k (6.7 = γ0 2 α 1/2 h(kf 2 0Ω α 1 h2 (k f 2 0 T k \ ˆT k. Not that T k+1 \ T k+1 = { T k+1 T k \ ˆT k or ˆ T k \ ˆT } k and that for any T k+1 \ T k+1 or Hnc h (k +1=α 1/2 h (k if T k α 1/2 α 1/2 h (k +1 α 1/2 ˆ h ˆ(k if ˆ T k \ ˆT k. T k+1 \ T k+1 α 1 h2 (k +1 f 2 0 T k \ ˆT k α 1 h2 (k f 2 0 which togthr with (6.7 yilds α 1/2 h(k +1f 2 = α 1 0Ω h2 (k +1 f T k+1 T k+1 \ T k+1 α 1 h2 (k +1 f 2 0 γ0 2 α 1/2 h(kf 2 +(1 0Ω γ2 0 α 1 h2 (k f 2 0. T k \ ˆT k

22 RECOVERY-BASED ESTIMATOR FOR INTERFACE PROBLEMS 2153 Combining with th following consqunc of (6.2 α h2 (k f 2 0 (1 θ2 0 1/2 h(kf T k \ ˆT k α 1 givs th validity of (6.6. This complts th proof of th lmma. Now w ar rady to stablish th rror rduction proprty of th adaptiv finit lmnt mthod. For convninc w introduc som matrix notations. For a vctor y k =(y 1 y 2 t and a matrix B =(b ij 2 2 dnotby y k l2 = y1 2 + y2 2 and ρ(b =max λ(b th rspctiv l 2 norm and spctral radiuswhrλ(b dnots th ignvalu of B. Thorm 6.1. Lt δ =max{ 1 δ 1 ζ} with δ 1 and ζ givn in th rspctiv Lmmas 6.1 and 6.2. Undr th Assumption R and Assumption E th squnc {u k } gnratd by th adaptiv finit lmnt mthod satisfis th following rror rduction proprty: (6.8 u u k Ω C 0 δ k with C 0 = u u 0 2 Ω + α 1/2 h(0f 2 0Ω. Proof. Sinc U 0 (k U 0 (k + 1 it follows from th orthogonal proprty of th finit lmnt approximations and Lmma 6.1 that 2 0Ω u u k 2 Ω = u u k+1 2 Ω + u k+1 u k 2 Ω u u k+1 2 Ω + δ 1 u u k 2 Ω (1 + δ 0 α 1/2 h(kf which implis (6.9 u u k+1 2 Ω (1 δ 1 u u k 2 Ω +(1+δ 0 α 1/2 h(kf 2. 0Ω 2 0Ω Lt y k =( u u k Ω α 1/2 h(kf 0Ω t and ( 1 δ1 1+δ 0 B = 0 ζ thn (6.9 and Lmma 6.2 lad to Hnc y k B y k 1 = B k y 0. u u k Ω y k l2 ρ ( B k y 0 l2. Now (6.8 is an immdiat consqunc of th facts that { ρ(b k =max (1 δ 1 k/2 ζ k} = δ k and C 0 = y 0 l2. This complts th proof of th thorm.

23 2154 ZHIQIANG CAI AND SHUN ZHANG 6.3. Assumption E. In this sction w vrify th Assumption E for svral stimators. For simplicity w analyz only problm (2.1 with pur homognous Dirichlt boundary conditions i.. g =0andΓ D = Ω in (2.2. Th xtnsion to mixd boundary conditions for th stimators analyzd in sction 4 is straightforward. Lt η (k b th dg stimator dfind in (4.16; th following lmma stablishs a local uppr bound. Th proof hr is similar to thos in [16 20]. Lmma 6.3. For vry E k containing a vrtx of T k+1 as its intrior point thr xists a positiv constant C such that ( (6.10 η(k 2 C u k+1 u k 2 ω + α 1/2 h(kf 2. 0ω Proof. For any v U 0 (k + 1 th orthogonality proprty of th finit lmnt approximation u k+1 implis a(u k+1 u k v=a(u k+1 u v+a(u u k v=a(u u k v which togthr with intgration by parts (2.1 th fact that (α(x u k = 0 T k and th continuity of th normal componnt of th xact flux givs (6.11 a(u k+1 u k v=a(u u k v= fvdx+ J (α(x u k vds. E k T k Sinc α(x is a picwis constant thn J (α(x u k onach E k is a constant dnotd by j.ltψ U 0 (k + 1 b th nodal basis function associatd with on of intrior nods on thn it is asy to s that supp (ψ ω ψ =0 E k ψ 0ω Ch (k ψ 0ω C and ψ ds Ch (k whr ω = + and C is a positiv constant indpndnt of h (k. Choosing v = j ψ in (6.11 w thn hav Ch (k j 2 j 2 ψ ds = a(u k+1 u k j ψ fj ψ dx ω ( ( 1/2 C α + + α j u k+1 u k ω + α 1/2 h(kf. 0ω Hnc η (k = ( 2 j h (k 1/2 C (α + α + u k+1 u k ω + α 1/2 h(kf ω which lads to (6.10 and hnc th proof of th lmma. Lmma 6.4. For all ˆT k thr xists a positiv constant C such that ( ˆη 2 (k C RT 0 u k+1 u k 2 ω + α 1/2 (6.12 h(kf 2. 0ω Proof. Equation (6.12 is a dirct consqunc of Lmma 6.3 and th rspctiv (4.21.

24 RECOVERY-BASED ESTIMATOR FOR INTERFACE PROBLEMS 2155 With th local discrt bounds in Lmmas 6.3 and 6.4 it is thn straightforward to show th validity of th Assumption E. Lmma 6.5. Th Assumption E is valid for th stimators η (k and ˆη RT0 (k. Proof. Sinc ω contains at most d + 2 lmnts it thn follows from (6.1 and (6.12 that θe 2 ˆη 2 RT 0 (k ( u k+1 u k 2 ω + α 1/2 h(kf 2 0ω ˆη 2 RT T ˆ k 0 (k C ˆ T k ( (d +2C u k+1 u k 2 α Ω + 1/2 h(kf 2 0Ω which provs th validity of th Assumption E with C l =(C (d +2 1 for th stimators ˆη RT0 (k. It may b provd in th sam fashion for th stimator η (k. REFERENCES [1] M. Ainsworth and A. W. Craig A postriori rror stimation in finit lmnt mthod Numr. Math. 36 (1991 pp [2] M. Ainsworth and J. T. Odn A Postriori Error Estimation in Finit Elmnt Analysis Pur Appl. Math. Wily-Intrscinc John Wily & Sons Nw York [3] R. Bank and J. Xu Asymptotically xact a postriori rror stimators Part I: Grids with supprconvrgnc SIAM J. Numr. Anal. 41 (2003 pp ; Part II: Gnral unstructurd grids SIAM J. Numr. Anal. 41 (2003 pp [4] R. Bank J. Xu and B. Zhng Suprconvrgnt drivativ rcovry for Lagrang triangular lmnts of dgr p on unstructurd grids SIAM J. Numr. Anal. 45 (2007 pp [5] I. Babuska and T. Strouboulis Th Finit Elmnt Mthod and Its Rliability Numr. Math. Sci. Comput. Oxford Univrsity Prss Oxford [6] C. Brnardi and R. Vrfürth Adaptiv finit lmnt mthods for lliptic quations with non-smooth cofficints Numr. Math. 85 (2000 pp [7] S. C. Brnnr and C. Carstnsn Finit lmnt mthods in Encyclopdia of Computational Mchanics John Wily and Sons Vol. 1: Fundamntals E. Stin R. d Borst T. Hughs ds pp [8] F. Brzzi and M. Fortin Mixd and Hybrid Finit Elmnt Mthods Springr-Vrlag Nw York [9] C. Carstnsn All first-ordr avraging tchniqu for a postriori finit lmnt rror control on unstructur grids ar fficint and rliabl Math. Comp. 73 (2003 pp [10] C. Carstnsn and S. Bartls Each avraging tchniqu yilds rliabl a postriori rror control in FEM on unstructur grids. Part I: Low ordr conforming nonconforming and mixd FEM Math. Comp. 71 (2002 pp [11] C. Carstnsn S. Bartls and R. los An xprimntal survy of a postriori Courant finit lmnt rror control for th Poisson quation Adv. Comput. Math. 15 (2001 pp [12] C. Carstnsn and R. Vrfürth Edg rsiduals dominat a postriori rror stimats for low ordr finit lmnt mthods SIAM J. Numr. Anal. 36 (1999 pp [13] J. M. Cascon C. ruzr R. H. Nochtto and. G. Sibrt Quasi-optimal Convrgnc Rat for an Adaptiv Finit Elmnt Mthod SIAM J. Numr. Anal. 46 (2008 pp [14] Z. Chn and S. Dai On th fficincy of adaptiv finit lmnt mthods for lliptic problms with discontinuous cofficints SIAM J. Sci. Comput. 24 (2002 pp [15] P. G. Ciarlt Th Finit Elmnt Mthod for Elliptic Problms North-Holland Amstrdam [16] W. Dörflr A convrgnt adaptiv algorithm for Poisson s quation SIAMJ. Numr. Anal. 33 (1996 pp [17] W. Hoffmann A. H. Schatz L. B. Wahlbin and G. Wittum Asymptotically xact a postriori stimators for th pointwis gradint rror on ach lmnt in irrgular mshs

25 2156 ZHIQIANG CAI AND SHUN ZHANG Part I: A smooth problm and globally quasi-uniform mshs Math. Comp. 70 (2001 pp [18] R. B. llogg On th Poisson quation with intrscting intrfacs Appl. Anal. 4 (1975 pp [19] R. Luc and B. I. Wohlmuth A local a postriori rror stimator basd on quilibratd fluxs SIAM J. Numr. Anal. 42 (2004 pp [20] P. Morin R. H. Nochtto and. G. Sibrt Convrgnc of adaptiv finit lmnt mthods SIAM Rv. 44 (2002 pp [21] A. Naga and Z. Zhang Th polynomial-prsrving rcovry for highr ordr finit lmnt mthods in 2D and 3D Discrt Contin. Dyn. Syst. Sr. B 5-3 (2005 pp [22] R. H. Nochtto Adaptiv finit lmnt mthods for lliptic PDE Lctur nots Cntr for Nonlinar Analysis Carngi Mllon Univrsity Pittsburgh PA [23] J. S. Ovall Two Dangrs to Avoid Whn Using Gradint Rcovry Mthods for Finit Elmnt Error Estimation and Adaptivity Tchnical rport 6 Max-Planck-Institut fur Mathmatick in dn Naturwissnschaftn Bonn Grmany [24] J. S. Ovall Fixing a Bug in Rcovry-typ A Postriori Error Estimators Tchnical rport 25 Max-Planck-Institut fur Mathmatick in dn Naturwissnschaftn Bonn Grmany [25] M. Ptzoldt A postriori rror stimators for lliptic quations with discontinuous cofficints Adv. Comput. Math. 16 (2002 pp [26] P. A. Raviart and I. M. Thomas A Mixd Finit Elmnt Mthod for Scond Ordr Elliptic Problms Lctur Nots in Math. 606 Springr-Vrlag Brlin and Nw York (1977 pp [27] A. H. Schatz and L. B.Wahlbin Asymptotically xact a postriori stimators for th pointwis gradint rror on ach lmnt in irrgularmshs Part II: Th picwis linar cas Math. Comp. 73 (2004 pp [28] R. Vrfürth A Rviw of A Postriori Error Estimation and Adaptiv Msh-Rfinmnt Tchniqus Wily-Tubnr Stuttgart Grmany [29] N. Yan and A. Zhou Gradint rcovry typ a postriori rror stimats for finit lmnt approximations on irrgular mshs Comput. Mthods Appl. Mch. Engrg. 190 (2001 pp [30] D. Yu Asymptotically xact a postriori rror stimators for lmnts of bi-vn dgr Chins J. Numr. Math. Appl. 13 (1991 pp [31] D. Yu Asymptotically xact a postriori rror stimators for lmnts of bi-odd dgr Chins J. Numr. Math. Appl. 13 (1991 pp [32] Z. Zhang A postriori rror stimats on irrgular grids basd on gradint rcovry Adv. Comput. Math. 15 (2001 pp [33] Z. Zhang Rcovry tchniqus in finit lmnt mthods in Adaptiv Computations: Thory and Algorithms Mathmatics Monogr. Sr. 6 T. Tang and J. Xu ds. Scinc Publishr Nw York 2007 pp [34] Z. Zhang and J. Z. Zhu Analysis of th suprconvrgnt patch rcovry tchniqu and a postriori rror stimator in th finit lmnt mthod. Part 1 Comput. Mthods Appl. Mch. Engrg. 123 (1995 pp ; Part 2 Comput. Mthods Appl. Mch. Engrg. 163 (1998 pp [35] O. C. Zinkiwicz and J. Z. Zhu A simpl rror stimator and adaptiv procdur for practical nginring analysis Intrnat. J. Numr. Mthods Engrg. 24 (1987 pp [36] O. C. Zinkiwicz and J. Z. Zhu Th suprconvrgnt patch rcovry and a postriori rror stimats Intrnat. J. Numr. Mthods Engrg. 33 (1992 Part 1: Th rcovry tchniqu pp ; Part 2: Error stimats and adaptivity pp