RECOVERY-BASED ERROR ESTIMATORS FOR INTERFACE PROBLEMS: MIXED AND NONCONFORMING FINITE ELEMENTS

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1 SIAM J. NUMER. ANAL. Vol. 48 No. 1 pp c 2010 Socity for Industrial Applid Mathmatics RECOVERY-BASED ERROR ESTIMATORS FOR INTERFACE PROBLEMS: MIXED AND NONCONFORMING FINITE ELEMENTS ZHIQIANG CAI AND SHUN ZHANG Abstract. In [Z. Cai S. Zhang SIAM J. Numr. Anal pp ] w introducd analyzd a rcovry-basd a postriori rror stimator for conforming linar finit lmnt approximation to intrfac problms. It was shown thortically that th stimator is robust with rspct to th siz of jumps providd that th distribution of cofficints is locally monoton. Numrical xampls showd that this condition is unncssary. This papr xtnds th ida in [Z. Cai S. Zhang SIAM J. Numr. Anal pp ] to mixd nonconforming finit lmnt mthods for dvloping analyzing robust stimators. Numrical rsults on tst problms ar also prsntd. Morovr an a priori rror stimat is obtaind whn th undrlying problm has low rgularity. Ky words. problms a postriori rror stimator mixd lmnt nonconforming lmnt intrfac AMS subjct classifications. 65N30 65N15 DOI / Introduction. Th rcovry-basd a postriori rror stimators hav bn xtnsivly studid for conforming finit lmnts by many rsarchrs du to thir many appaling proprtis: simplicity asymptotic xactnss univrsality. For th mixd nonconforming finit lmnt mthods Carstnsn Bartls in [11] introducd analyzd rcovry-basd rror stimators. Thir stimators for both th mixd th nonconforming lmnts ar basd on th rcovry of th gradint in H 1 Ω 2. Ths stimators work wll for th Poisson quation vn though th gradint of th xact solution blongs only to Hdiv Hcurl for nonconvx polygonal domains. For othr typs of stimators on th mixd th nonconforming mthods s [ ] rfrncs thrin. As dmonstratd numrically in [23 9] thortically in [9] for conforming finit lmnt approximations to th intrfac problm with larg jumps xisting stimators of th rcovry typ ovrrfin rgions whr thr ar no rrors hnc fail to rduc th global rror. This is also tru for th rcovry-basd stimators in [11] for th mixd nonconforming finit lmnt mthods s Figurs Th rason for th ovrrfinmnts is that th rcovrd gradint is continuous but th tru gradint is discontinuous. To ovrcom this structural difficulty on oftn applis th mthod on ach subdomain sparatly. For rasons why this local approach is not favorabl s dtaild discussions in [23]. Mor importantly th local approach fails whn triangulations do not align with intrfacs which occurs whn intrfacs ar curvs/surfacs or hav unknown locations. In [9] w introducd analyzd a global approach for th conforming linar finit lmnt approximation by rcovring th flux in th Hdiv conforming finit lmnt spacs. Th rsulting Rcivd by th ditors April ; accptd for publication in rvisd form Novmbr ; publishd lctronically April This work was supportd in part by th National Scinc Foundation undr grants DMS DMS Dpartmnt of Mathmatics Purdu Univrsity 150 N. Univrsity Strt Wst Lafaytt IN zcai@math.purdu.du zhang@math.purdu.du. 30

2 RECOVERY ERROR ESTIMATORS FOR INTERFACE PROBLEMS 31 stimator is thn fr of ovrrfinmnts satisfis th fficincy rliability bounds with constants indpndnt of th siz of jumps. Th purpos of this papr is to xtnd th ida in [9] to mixd nonconforming finit lmnt approximations. To do so w nd to dtrmin what quantitis ar to b rcovrd which finit lmnt spacs ar to b usd. Th guidlin for such choics is basd on our viw that a rcovry-basd stimator is a masurmnt of th violation of finit lmnt approximations on physical continuitis. Thrfor th quantitis to b rcovrd ar thos whos finit lmnt approximations do not prsrv th physical continuity. Th intrfac problms in 2.1 hav two physical continuitis: th solution u th normal componnt of th flux σ = k u. Mathmatically this mans 1.1 u H 1 Ω σ Hdiv L 2 Ω 2. For th mixd mthod th continuity of th solution is violatd whil that of th flux is prsrvd. To masur such a violation w rcovr th gradint of th solution. To choos propr finit lmnt spacs w notic that th first proprty in 1.1 implis 1.2 u Hcurl. Physically th tangntial componnts of vctor filds in Hcurl ar continuous. Thrfor th quantity to b rcovrd is th gradint th propr finit lmnt spac is th Hcurl conforming finit lmnt spac. This choic accommodats discontinuity of th normal componnt of th gradint hnc liminats ovrrfinmnts. For nonconforming finit lmnt approximations sinc both continuitis ar violatd w rcovr both th flux th gradint in th Hdiv Hcurl conforming finit lmnt spacs rspctivly through wightd L 2 projctions. Th stimator is thn th avrag of two masurmnts: th wightd L 2 norms of diffrncs btwn th dirct th rcovrd approximations of th flux th gradint. Estimators introducd in this papr ar analyzd by stablishing th rliability fficincy bounds ar supportd by numrical rsults. In particular w prov thortically that th stimators ar robust in th sns that th rliability fficincy constants ar indpndnt of th siz of jumps providd that th distribution of cofficints is locally monoton. W also show numrically that thr is no ovrrfinmnts along intrfacs for a bnchmark tst problm whos cofficints ar not locally monoton. Rsults in this papr may b xtndd to thr dimnsions in a straightforward mannr. It is important to point out that rsarch on robust stimators for intrfac problms is limitd. For th conforming finit lmnt mthod robust a postriori rror stimators hav bn studid by Brnardi Vrfürth [6] Ptzoldt [24] for th rsidual-basd stimator by Luc Wohlmuth [18] for th quilibratd stimator by us [9] for th rcovry-basd stimator. For th nonconforming lmnts Ainsworth [1] studid a robust quilibratd stimator. For th mixd mthod s a rcnt work by Ainsworth [2]. This papr is organizd as follows. Sction 2 introducs intrfac problms variational formulations. Various finit lmnt spacs both mixd nonconforming finit lmnt approximations ar dscribd in sction 3. Rcovry procdurs a postriori rror stimators ar dfind in sctions 4 5 rspctivly. W stablish th fficincy rliability bounds of stimators introducd in this papr

3 32 ZHIQIANG CAI AND SHUN ZHANG in sction 6. Finally w prsnt numrical rsults for a bnchmark tst problm in sction Function spacs prliminaris. Lt Ω b a boundd polygonal domain in R 2 with boundary Ω = Γ D Γ N Γ D Γ N =. For a subdomain G Ω w us th stard notations dfinitions for th Sobolv spacs H s G H s Gfors 0. Th stard associatd innr products ar dnotd by sg s G thir rspctiv norms ar dnotd by sg s G. W omit th subscript G or G if G = Ω from th innr product norm dsignation whn thr is no risk of confusion. W shall us th following subspacs of H 1 Ω: H 1 D Ω := {v H1 Ω : v =0onΓ D } H 1 N Ω := {v H1 Ω : v =0onΓ N }. In two dimnsions for τ =τ 1 τ 2 t dfin th divrgnc curl oprators by τ := τ 1 x 1 τ 2 x 2 τ := τ 2 x 1 τ 1 x 2 rspctivly. For a scalar-valud function v dfin th oprator by v = Q v = v x 2 W shall us th following Hilbrt spacs: t v with Q = x Hdiv; Ω = {τ L 2 Ω 2 : τ L 2 Ω} Hcurl; Ω = {τ L 2 Ω 2 : τ L 2 Ω} quippd with th norms τ Hdiv; Ω = τ 2 0Ω τ Ω rspctivly. Dnot thir subspacs by τ Hcurl; Ω = τ 2 0Ω τ Ω H N div; Ω= {τ Hdiv; Ω : τ n ΓN =0} H D curl; Ω= {τ Hcurl; Ω : τ t ΓD =0} whr n =n 1 n 2 t t =t 1 t 2 t = Q n =n 2 n 1 t ar th unit vctors outward normal to clockwis tangnt to th boundary Ω rspctivly..

4 RECOVERY ERROR ESTIMATORS FOR INTERFACE PROBLEMS Intrfac problms variational forms. Considr th following intrfac problm: 2.1 { kx u =f in Ω u =0 onγ D n k u =0 onγ N whr f is a givn scalar-valud function in L 2 Ω kx is positiv picwis constant on polygonal subdomains of Ω with possibl larg jumps across subdomain boundaris intrfacs: kx =k i > 0inΩ 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 boundary conditions. Also w assum that Γ D is not mpty i.. ms Γ D 0. Dnot bilinar linar forms rspctivly by au v =kx u v fv =f v. Thn th variational form of problm 2.1 is to find u HD 1 Ω such that 2.2 au v =fv v H 1 D Ω. Dfin th flux by σ = kx u in Ω thn th mixd variational formulation is to find σ u H N div; Ω L 2 Ω such that { k 1 σ τ τu=0 τ H N div; Ω 2.3 σ v = f v v L 2 Ω. 3. Finit lmnt approximations Finit lmnt spacs. For simplicity of prsntation considr only triangular lmnts. Lt T = {K} b a finit lmnt partition of th domain Ω dnot by h K th diamtr of lmnt K. Assum that th triangulation T is rgular [13] that intrfacs F = { Ω i Ω j i j =1... n} do not cut through any lmnt K T. Dnot th st of all dgs of th triangulation by E := E Ω E D E N whr E Ω is th st of all intrior lmnt dgs E D E N ar th sts of all boundary dgs blonging to th rspctiv Γ D Γ N.Forach Ednotbym h th midpoint th lngth of th dg rspctivly. For ach K TltP b th spac of polynomials of dgr k. Dnot th conforming nonconforming linar finit lmnt spacs [13 15] associatd with th triangulation T by U = {v H 1 Ω : v K P 1K K T} U nc = {v L 2 Ω : v K P 1K K T v is continuous at m E Ω} rspctivly thir rspctiv subspacs by U D = {v U: v =0onΓ D } UD nc = {v Unc : vm =0 E D }. Th Hdiv; Ω conforming Raviart Thomas RT Brzzi Douglas Marini BDM spacs [7] of th lowst ordr ar dfind by RT 0 = {τ H Ndiv; Ω : τ K RT 0K K T}with RT 0K =P 0K 2 x 1x 2 P 0K

5 34 ZHIQIANG CAI AND SHUN ZHANG BDM 1 = {τ H Ndiv; Ω : τ K BDM 1K K T} with BDM 1K =P 1K 2 rspctivly. Th Hcurl ; Ω conforming first scond typs of Ndlc spacs of th lowst ordr [21 22] ar dfind by ND 1 ={τ H Dcurl; Ω : τ K ND 1K K T } with ND 1K=P 0K 2 x 2 x 1P 0K ND 2 = {τ H D curl; Ω : τ K ND 2 K K T} with ND 2 K =P 1 K 2 rspctivly. For convninc dnot RT 0 /BDM 1 by V N ND 1 /ND 2 by W D. Also lt P 0 = {v L 2 Ω : v K P 0 K K T}. Finally w dfin th discrt gradint divrgnc curl oprators as follows: h v K := v K h τ K := τ K h τ K := τ K for all K T rspctivly Finit lmnt approximations. Th mixd finit lmnt mthod is to find σ m u m V N P 0 such that 3.1 { k 1 σ m τ τu m =0 τ V N σ m v=f v v P 0. Lt σ uσ m u m b th solutions of rspctivly dnot th tru rror of th mixd finit lmnt approximation by 3.2 E m m =σ σ m u u m. Thn th diffrnc btwn givs th following rror quations: 3.3 { k 1 E m τ τ m =0 τ V N E m v=0 v P 0. A stard argumnt givs th following a priori rror stimat. Thorm 3.1. Assum that th solution σu of problm 2.3 blongs to H s Ω H 1s Ω with 0 s 1. Thn w hav th following a priori rror bound: 3.4 k 1/2 E m 0Ω C h s k 1/2 u sω with h s k 1/2 u sω = K T h2s K k1/2 u sk 2 1/2. Hr hraftr in this papr w us C with or without subscripts to dnot a gnric positiv constant possibly diffrnt at diffrnt occurrncs that is indpndnt of th msh paramtr h K th ratio k max /k min but may dpnd on th domain Ω. Th nonconforming finit lmnt mthod is to find u nc UD nc such that 3.5 k h u nc h v=fv v U nc D.

6 RECOVERY ERROR ESTIMATORS FOR INTERFACE PROBLEMS 35 Lt u u nc b th solutions of rspctivly dnot th tru rror of th nonconforming finit lmnt approximation by nc = u u nc. Sinc U D UD nc w hav th following rror quation: 3.6 k h nc v =k u h u nc v =0 v U D. For K T any v H 1s K 0 <s 1 dnot by { h s K k 1/2 v sk B sk h K v= s 1/2 1] h s K k1/2 v sk h K k 1/2 K f 0K s 0 1/2]. Lmma 3.2. For any w UD nc any K Tlt w K b th man valu of w K ovr dg K. Assum that th solution u of problm 2.2 blongs to H 1s Ω with s 0 1] thn 3.7 n k uw w K ds CB skh K u w 0K. K For v H 1s K not that intgral n k v wds is th stard intgration in L 2 ifs > 1/2. Whn s 0 1/2] it should b viwd as th duality pairing n k vw whrn k v H ɛ1/2 w H 1/2ɛ for any positiv ɛ<s. Proof. Th dfinition of w K implis 3.8 n k uw w K ds = n k u ζ w w K ds for any constant ζ.whns 1/2 1] 3.7 may b provd by a stard argumnt s.g. [8]. Whn s 0 1/2] using 3.8 th dfinition of th dual norm th approximation proprty w hav n k uw w K ds n k u ζ 1/2ɛ w w K 1/2ɛ Ch ɛ n k u ζ 1/2ɛ w 0K. Lt ζ b th man valu of k u ovr K. Choosing ζ = ζ n using 2.1 th fact [5] that φ n 1/2ɛ C φ ɛk h 1ɛ K Δφ 0K for any φ H 1ɛ K with Δφ L 2 K forany0<ɛ<1/2 w hav n k u ζ 1/2ɛ = n k u ζ 1/2ɛ C k u ζ ɛk h 1ɛ K f 0K C h sɛ K k u sk h 1ɛ K f 0K. Combining th abov two inqualitis yilds 3.7 hnc th proof of th lmma. Thorm 3.3. Assum that th solution u of problm 2.2 blongs to H 1s Ω with 0 <s 1. Thn w hav th following a priori rror bound: k 1/2 h u u nc 0Ω C 1/2 BsKh 2 K u. K T

7 36 ZHIQIANG CAI AND SHUN ZHANG Proof. Ltu c b th orthogonal projction of u onto U D with rspct to th innr product k thnthfactthatu D UD nc implis inf v U nc D k 1/2 h u v 0Ω k 1/2 h u u c 0Ω C K T h 2s K k1/2 u 2 sk 1/2. Th scond inquality abov may b provd similarly as that of Proposition 2.4 in [6]. For any w UD ncanyk Tany K ltk b th lmnt sharing th common dg thn th continuity of w at th midpoint of implis that w K = w K. For E D wm = 0 implis that w K = 0. Thn it follows from intgration by parts 2.1 th continuity of n k u across dg that k u h wf w = wds= K T Kn k u n k uw w K ds K T K which togthr with th triangl inquality Lmma 3.2 giv k u h w f w C 1/2 BsKh 2 K u k 1/2 h w 0Ω. K T Now Thorm 3.3 is a dirct consqunc of Strang s lmma s.g. [13]. 4. Gradint /or flux rcovry Implicit approximation. For th mixd finit lmnt approximation σ m u m th continuity of th solution hnc th continuity of th tangntial componnt of th gradint ar violatd whil that of th flux is prsrvd. This suggsts to rcovr th gradint in th Hcurl conforming finit lmnt spac ND 2. Sinc u = k 1 σ it is rcovrd by finding ρ m ND 2 such that 4.1 k ρ m τ =σ m τ τ ND 2. It may b provd asily that ρ m is a good approximation to u up to th accuracy of th mixd finit lmnt approximation. Thorm / 4.1 s [8]. Thr xists a constant C>0indpndnt of th ratio k max kmin such that 4.2 k 1/2 u ρ m 0Ω C inf k 1/2 u τ 0Ω k 1/2 σ σ m 0Ω. τ ND 2 For th nonconforming finit lmnt approximation u nc both th continuitis of th tangntial componnt of th gradint th normal componnt of th flux ar violatd. Hnc w rcovr both th gradint flux as follows: finding ρ nc W D such that 4.3 kρ nc τ =k h u nc τ τ W D finding σ nc V N such that 4.4 k 1 σ nc τ = h u nc τ τ V N.

8 RECOVERY ERROR ESTIMATORS FOR INTERFACE PROBLEMS 37 Thorm / 4.2 s [8]. Thr xists a constant C>0indpndnt of th ratio k max kmin 4.5 k 1/2 u ρ nc 0Ω C inf k 1/2 u τ 0Ω k 1/2 u h u nc 0Ω τ W D that 4.6 k 1/2 σ σ nc 0Ω C inf k 1/2 σ τ 0Ω k 1/2 u h u nc 0Ω τ V N Explicit approximations. Lt δ δ ij dnot th Kronckr dlta. Nodal basis functions of RT 0 BDM 1 ND 1 ND 2 corrsponding to dg E ar charactrizd as follows: 1 For RT 0 φ is uniquly dtrmind by φ n ds = δ E. 2 For BDM 1 φ i i =1 2 ar uniquly dtrmind by s j1 φ i n ds = δ δ ij E for j =1 2 whr s is a local coordinat on ranging from 1 to1. Noticthatφ 1 of BDM 1 φ of RT 0 ar th sam. Sinc φ 1 n =1/ whavth following orthogonality proprty: φ 1 n φ 2 n ds = 1 φ 2 n ds =0. 3 For ND 1 ψ is uniquly dtrmind by ψ t ds = δ E. 4 For ND 2 ψ i i =1 2 ar uniquly dtrmind by s j1 ψ i t ds = δ δ ij E for j =1 2. Similarly ψ 1 of ND 2 ψ of ND 1 coincid satisfy th following orthogonality: 4.7 ψ 1 t ψ 2 t ds =0. Lmma 4.1. Evry constant vctor τ on K T has th following rprsntations: τ = τ φ τ = τ n ds τ = τ ψ τ = τ t ds K K rspctivly in RT 0 ND 1. Evry linar vctor τ on K has th following rprsntations: τ = τ 1 φ 1 τ 2 φ 2 with τ i = s i1 τ n ds K

9 38 ZHIQIANG CAI AND SHUN ZHANG τ = τ 1 ψ 1 τ 2 ψ 2 with τ i = s i1 τ t ds K rspctivly in BDM 1 ND 2. Proof. Sinc RT 0 K ND 1 K bothbdm 1 K ND 2 K contain th rspctiv constant linar vctors th lmma is thn a dirct consqunc of th charactristic quations of nodal basis functions. Lmma 4.2. For a linar function v dfind on dg lt{ψ i } 2 i=1 b th ND 2 basis functions thn 4.8 v = v 1 ψ 1 t v 2 ψ 2 t with v i = s i1 vds 4.9 v 2 ds = v 2 1 ψ 1 2 ds v 2 2 ψ 2 2 ds. Proof. 4.8 follows from th fact that span{ψ i t } 2 i=1 =span{1 s}. 4.9 is a consqunc of Now w ar rady to introduc xplicit approximations to th flux th gradint. For ach Ednotbyn a unit vctor normal to. Whn E D E N assum that n is th unit outward normal vctor. For ach intrior dg E Ω lt K K b th two lmnts sharing th common dg such that th unit outward normal vctor of K coincids with n. For σ m V N ltτ 1 = k 1 σ m. Sinc τ 1 is a linar vctor-valud function on ach K its approximation ˆρ m ND 2 may b dfind by 4.10 ˆρ m σ m = E ˆρ1 ψ 1 ˆρ 2 ψ 2 whr ˆρ 1 ˆρ 2 ar th zro- on-momnts of th tangntial componnt of ˆρ m on th dg E Ω E D rspctivly dfind by 4.11 { γ1 ˆρ i := si1 τ 1 K t ds 1γ 1 si1 τ 1 K t ds for E Ω si1 τ 1 t ds for E D for som paramtr γ 1 [0 1]. Lt τ 2 = k h u nc.sincτ 2 is picwis constant thn it suffics to approximat it using RT 0. Dfin th xplicit approximation ˆσ nc u nc inrt 0 =span{φ : E} by 4.12 ˆσ nc u nc = ˆσ φ E whr ˆσ is th normal componnt of ˆσ nc u nc onthdg E dfind by { γ ˆσ := τ n 2 K ds 1γ 2 τ 2 n K ds for E Ω τ 2 n ds for E D

10 RECOVERY ERROR ESTIMATORS FOR INTERFACE PROBLEMS 39 for som constant γ 2 [0 1]. Lt τ 3 = h u nc. Sinc τ 3 is picwis constant w approximat it in ND 1 =span{ψ : E}by 4.14 ˆρ nc u nc = ˆρ ψ E whr ˆρ is th tangntial componnt of ˆρ nc u nc onthdg E dfind by γ 3 τ 3 K t ds 1γ 3 τ 3 K t ds for E Ω 4.15 ˆρ := τ 3 t ds for E N for som constant γ 3 [0 1]. To nsur th fficincy bound indpndnt of th siz of jumps w choos 4.16 γ 2 = γ 3 =. 5. A postriori rror stimators Error stimators for mixd lmnts. Basd on th solution of 4.1 on th xplicit approximation in 4.10 w dfin a postriori rror stimators as follows: 5.1 η mk = k 1/2 ρ m k 1 σ m 0K K T η m = k 1/2 ρ m k 1 σ m 0Ω 5.2 ˆη mk = k 1/2 ˆρ m k 1 σ m 0K K T ˆη m = k 1/2 ˆρ m k 1 σ m 0Ω rspctivly. Ths stimators ssntially masur th violation of th continuity of th tangntial drivativs of th tru solution on th dgs by th mixd lmnts. It is obvious that 5.3 η m = min k 1/2 τ k 1 σ m 0Ω ˆη m. τ ND Error stimators for nonconforming lmnts. For vry lmnt K T lt η nc1k = k 1/2 σ nc k 1/2 u nc 0K η nc2k = k 1/2 ρ nc u nc 0K ˆη nc1k = k 1/2 ˆσ ncrt0 k 1/2 u nc 0K ˆη nc2k = k 1/2 ˆρ ncn u nc 0K whr σ nc ρ nc ar th solutions of rspctivly ˆσ ncrt0 ˆρ ncn ar th xplicit approximations dfind in th rspctiv Th sums of thir squars ar dnotd by η nc1 = k 1/2 σ nc k 1/2 u nc 0Ω η nc2 = k 1/2 ρ nc u nc 0Ω ˆη nc1 = k 1/2 ˆσ ncrt0 k 1/2 u nc 0Ω ˆη nc2 = k 1/2 ˆρ ncn u nc 0Ω. It is obvious that 5.4 η nc1 = min k 1/2 τ k 1/2 h u nc 0Ω τ V N η nc2 = min k 1/2 τ h u nc 0Ω. τ W D

11 40 ZHIQIANG CAI AND SHUN ZHANG Now a postriori rror stimators for nonconforming lmnts ar dfind as follows: 5.5 η 2 nck = c 2 η 2 nc1k 1 c 2 η 2 nc2k K T η 2 nc = c 2 η 2 nc1 1 c 2 η 2 nc2 5.6 ˆη 2 nck = c 2ˆη 2 nc1k 1 c 2 ˆη 2 nc2k K T ˆη 2 nc = c 2ˆη 2 nc1 1 c 2 ˆη 2 nc2 whr 0 <c<1is a paramtr to b chosn.g. c 2 =1/2. Ths stimators ssntially masur th violations of th continuity of both th normal componnts of th flux th tangntial drivativs of th tru solution on th dgs by th nonconforming lmnts. Lt u nc ũ nc b th solutions of 3.5 with th right-h sids f f h rspctivly whr f h is picwis constant with f h K = 1 K fdxfor all K T. K It is asy to s that 5.7 k 1/2 h u nc ũ nc 0Ω C k 1/2 hf f h 0Ω. Lt σ m RT 0 b th mixd finit lmnt approximation lt x K b th cntr of inrtia of K. By th wll-known fact [19] that σ m k h ũ nc K = 1 2 f h K x x K w hav σ m k h u nc K = 1 2 f h K x x K k h u nc k h ũ nc K. Hnc to avoid th flux rcovry w may rplac η nc1k by η ncfk = 1 2 k1/2 f h x x K 0K to obtain th following stimators: 5.8 η 2 nck = η 2 ncfk η 2 nc2k K T η 2 nc = c 2 η 2 ncf 1 c 2 η 2 nc2 5.9 η 2 nck = η 2 ncfk ˆη 2 nc2k K T η 2 nc = c 2 η 2 ncf 1 c 2 ˆη 2 nc2 whr η ncf = 1 2 K T k1/2 f h x x K 2 0K 1/2. Now it follows from th triangl inquality that 5.10 η nc ˆη nc η nc η nc C k 1/2 hff h 0Ω η nc η nc C k 1/2 hff h 0Ω. 6. Rliability fficincy bounds. This sction analyzs th stimators introducd in th prvious sction by stablishing th rliability fficincy bounds with constants indpndnt of th siz of jumps. To this nd assum that Hypothsis 2.7 in [6] holds. That is for any two diffrnt subdomains Ω i 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 kx is monoton along this path. This assumption is waknd to th quasi-monotonicity in [24]. It is a stard tchniqu s.g. [4 14] to analyz stimators for th mixd nonconforming lmnts by using th Hlmholtz dcomposition s.g. [15]. Lmma 6.1 Hlmholtz dcomposition. For a vctor-valud function τ L 2 Ω 2 thrxistα HD 1 Ω β H1 N Ω such that 6.1 τ = kx α β k 1 τ τ =k α αk 1 β β.

12 RECOVERY ERROR ESTIMATORS FOR INTERFACE PROBLEMS 41 Not that for all α H 1 D Ω all β H1 N Ω 6.2 α β=0. It is also common to us Clémnt-typ intrpolation oprators s.g. [6 24] for stablishing th rliability bound of a postriori rror stimators. Following [6] on can dfin th intrpolation oprator J : L 2 Ω U D s [9] for mor dtails so that 6.3 v Jv 0K h K v Jv 0K Ch K k 1/2 K k1/2 v 0ΔK v H 1 D Ω whr Δ K is th union of all lmnts that shar at last on vrtx with K that 6.4 f v Jv CH f k 1/2 v 0Ω v H 1 D Ω whr H f is highr ordr for f L p Ω with p>2. Lt U N = {v U: v =0onΓ N }. Similarly on can dfin a robust intrpolation J : L 2 Ω U N so that 6.5 vj v 0K h K vj v 0K Ch K k 1/2 K k1/2 v 0ΔK v H 1 Ω that 6.6 f v J v CG f k 1/2 v 0Ω v H 1 Ω whr G f is highr ordr for f L p Ω with p> Rliability on mixd lmnts. Thorm 6.2. Th stimator η m dfind in 5.1 satisfis th following global rliability bound: 6.7 k 1/2 E m 0Ω C η m k 1/2 hf f h 0Ω G h k 1 σ m. If V N = RT 0 thn 6.8 k 1/2 E m 0Ω C η m k 1/2 hf f h 0Ω. Proof. LtE m = σ σ m = k u σ m H N div; Ω; by Lmma thr xist α m HD 1 Ω β m HN 1 Ω such that 6.9 E m = k α m β m k 1/2 E m 2 0Ω = k 1/2 α m 2 0Ω k 1/2 β m 2 0Ω. Th uppr bound of th first trm in 6.9 follows asily from 6.2 intgration by parts E m n =0onΓ N α m =0onΓ D th first quation in 3.3 th Cauchy Schwarz inquality that k 1/2 α m 2 0Ω =E m α m = E m α m = E m Q h α m α m which implis =f f h Q h α m α m C k 1/2 hf f h 0Ω k 1/2 α m 0Ω 6.10 k 1/2 α m 2 0Ω C k1/2 hf f h 2 0Ω.

13 42 ZHIQIANG CAI AND SHUN ZHANG To bound th scond trm in 6.9 k 1/2 β m 2 0Ω notic first that U N RT 0. Thn th first quation in 3.3 givs 6.11 k 1 E m v= v m =0 v U N. By 6.2 th fact that J β m U N 6.11 intgration by parts boundary conditions of ρ m u on Γ D I J β m on Γ N whav k 1/2 β m 2 0Ω =k1 E m β m =k 1 E m β m J β m = ρ m u I J β m ρm k 1 σ m I J β m = ρ m I J β m ρ m k 1 σ m I J β m which togthr with th Cauchy Schwarz inquality 6.5 implis k 1/2 β m 2 0Ω ρ m I J β m Cηm k 1/2 β m 0Ω = h ρ m k 1 σ m I J β m h k 1 σ m I J β m Cηm k 1/2 β m 0Ω. Using th Cauchy Schwartz inquality th invrs inquality w obtain that h ρ m k 1 σ m I J β m C k 1/2 h h ρ m k 1 σ m 0Ω k 1/2 h 1 I J β m 0Ω Cη m k 1/2 β m 0Ω that h k 1 σ m I J β m CG h k 1 σ m k 1/2 β m 0Ωz. Combining th abov thr inqualitis dividing th quantity k 1/2 β m 0Ω squaring on both sids giv k 1/2 β m 2 0Ω C η m G h k 1 σ m 2 which togthr with yilds 6.7. Finally if V N = RT 0 thn h k 1 σ m = 0. Hnc 6.8 is a dirct consqunc of 6.7 th fact that G 0 = 0. This complts th proof of th thorm. Corollary 6.3. Th rliability bounds in Thorm 6.2 hold for th xplicit rror stimator ˆη m. Proof. Th corollary is a dirct consqunc of Thorm Rliability on nonconforming lmnts. Thorm 6.4. Th stimator η nc dfind in 5.5 satisfis th following global rliability bound: 6.12 k 1/2 h nc 0Ω C η nc H f. Proof. Lt nc = u u nc ; by Lmma thr xist α nc HD 1 Ω β nc HN 1 Ω such that 6.13 k h nc = k α nc β nc k 1/2 h 2 0Ω = k1/2 α nc 2 0Ω k1/2 β nc 2 0Ω.

14 RECOVERY ERROR ESTIMATORS FOR INTERFACE PROBLEMS 43 By using 3.6 intgrations by parts boundary conditions on k u σ nc I Jα nc th Cauchy Schwarz inquality th fact that h k h u nc =0th invrs inquality w hav k 1/2 α nc 2 0Ω =k h nc α nc =k h nc I Jα nc =k u σ nc I Jα nc σ nc k h u nc I Jα nc f σ nc α nc Jα nc Cη nc1 k 1/2 α nc 0Ω =f α nc Jα nc h σ nc k h u nc α nc Jα nc Cη nc1 k 1/2 α nc 0Ω C η nc1 H f k 1/2 α nc 0Ω. To bound th scond trm in 6.13 first by intgration by parts th orthogonality of nonconforming lmnts i.. [u nc] ds =0forany Ewhav h u nc J β nc = K T K u nc J β nc n ds = E J β nc n [u nc] ds =0. It thn follows from intgration by parts boundary conditions on ρ nc I J β nc th fact that h k 1 h u nc = 0 th Cauchy Schwarz invrs inqualitis 6.5 that k 1/2 β nc 2 0Ω = h β nc = h u nc I J β nc =ρ nc h u nc I J β nc ρ nc I J β nc =ρ nc h u nc I J β nc n ρ nc h u nc I J β nc Cη nc2 k 1/2 β nc 0Ω. Now th rliability bound in 6.12 is a dirct consqunc of th abov two inqualitis. This complts th proof of th thorm. Corollary 6.5. Lt η dnot th stimators η nc ˆη nc or η nc thn w hav th following rliability bound: k 1/2 h nc 0Ω C η H f k 1/2 hf f h 0Ω. Proof. Th corollary is a dirct consqunc of Thorm Efficincy. To stablish th fficincy bounds w mak us of th known rsult on th dg rror stimators in [11]. To this nd for any E Ω any vctorvalud function τ that is picwis linar with rspct to th triangulation T dnot th jump of th normal tangntial componnts of τ across = K K by J n τ =[τ n ]=τ K τ K n J t τ =[τ t ]=τ K τ K t rspctivly. For any E\E Ω wst J n τ =0 J t τ =0.

15 44 ZHIQIANG CAI AND SHUN ZHANG Efficincy on mixd lmnts. W dfin an dg rror stimator for mixd lmnts as follows: /2 η me := ηm 2 with η m = E kk 2 h J t k 1 σ m 2 ds 1/2. Proposition 6.6. Thr xists a constant C>0 such that 6.15 η m C k 1/2 β m 0ω η me C k 1/2 β m 0Ω C k 1/2 E m 0Ω whr ω is th union of all lmnts that shar dg. Proof. A similar proof as thos of Lmmas in [10] shows that thr xists a constant C>0indpndnt of k such that h J t k 1 σ m 2 ds C k 1 β m 2 0ω. Hnc η 2 m C k1 β m 2 0ω = C k 1 k 1/2 β K m 2 0K C k 1/2 β m 2 0ω k 1 k 1/2 β K m 2 0K which provs th first inquality in Summing it ovr all dgs Elads to th scond inquality in Thorm 6.7. Th following local fficincy bound for th xplicit rror stimator ˆη mk holds: 6.16 ˆη 2 mk C k 1/2 β m 0ωK whr ω K is th union of lmnts sharing a common dg with K. Th following global fficincy bound holds for all rror stimators: 6.17 η m ˆη m C k 1/2 β m 0Ω C k 1/2 E m 0Ω. Proof follows straightforward from To show th validity of 6.16 by Proposition 6.6 it suffics to prov that for any lmnt K T 6.18 ˆη 2 mk C ɛ K η 2 m = C K\ Ω h 2 J t k 1 σ m 2 ds. To do so for any dg K without loss of gnrality lt K b K lt K b th adjacnt lmnt with th common dg. Sincτ = k 1 σ m is picwis linar w hav τ K = τ1k ψ 1 τ 2K ψ 2 K

16 RECOVERY ERROR ESTIMATORS FOR INTERFACE PROBLEMS 45 whr τ ik = si1 τ t K ds is th i 1th momnt of th tangntial componnt of τ on ψ i is th nodal basis function of ND 2. For any x K ˆρ m τ = ˆρ 1 τ 1K ψ 1 ˆρ 2 τ 2K ψ 2 K = K\ Ω = K\ Ω = K\ Ω 1 γ 1 τ 1K τ 1K ψ 1 τ 2K τ 2K ψ 2 γ 1 1 ψ 1 J t τ ds ψ 2 sj t τ ds γ 1 1 j 1 ψ 1 j 2 ψ 2 with j i = si1 J t τ ds. SincJ t τ is a linar function on Lmma 4.2 givs J t τ =j 1 ψ 1 t j 2 ψ 2 t J t τ 2 ds = j1 2 ψ 1 2 ds j2 2 ψ 2 2 ds. Now by th triangl inquality th fact that K ψ i 2 dx Ch ψ i 2 ds w hav ˆη mk = k 1/2 ˆρ m τ 0K C C K\ Ω K\ Ω 1 γ 1 j1 2 h K J t τ 2 ds This provs 6.18 hnc th thorm. ψ 1 2 dx j 2 2 1/2 C ɛ K\ Ω K ψ 2 2 dx η 2 m Efficincy on nonconforming lmnts. A wightd dg rror stimator for nonconforming lmnts is dfind by η nce := E η 2 nc 1/2 1/2 with 6.19 ηnc 2 = 2h J n k h u nc 2 ds h J t h u nc 2 ds. Thorm 6.8. Thr xists a constant C>0 such that 6.20 ηnc 2 C k 1/2 h nc 2 0ω K f f h 2 0K K T ω h2

17 46 ZHIQIANG CAI AND SHUN ZHANG 6.21 η nce C k 1/2 h nc 0Ω C hk 1/2 f f h 0Ω. Proof. A similar proof as that for th conforming linar lmnts in [24] shows that th first trm of ηnc 2 dfind in 6.19 satisfis It thn suffics to show that 6.20 holds for th scond trm of ηnc. 2 To do so w rcall th following inquality stimat 3.3 in [12]: 6.22 h J t h u nc 2 ds C h nc 2 0ω which holds with a constant C>0indpndnt of th jump of k. Hnc h k K J t h u nc 2 ds C k K h nc 2 0ω k K k K C k 1/2 h 2 k 0K k 1/2 h K nc 2 k K k 0K C k 1/2 h K nc 2 0ω. This provs Th global bound 6.21 is a dirct rsult of Thorm 6.9. Lt ω K b th union of lmnts sharing a common dg with K thn th xplicit rror stimator ˆη nck has th following local fficincy bound: 6.23 ˆη nck 2 C k 1/2 h nc 2 0ω K T f f h 2 0T. k T T ω K h2 For th stimators η nc η nc w hav th following global fficincy bound: 6.24 η nc η nc C k 1/2 h nc 0Ω C hk 1/2 f f h 0Ω. Proof is a dirct consqunc of To show th validity of 6.23 by Thorm 6.8 it suffics to prov that for any lmnt K T 6.25 ˆη nck 2 C ηnc 2. ɛ K To this nd for any dg K without loss of gnrality lt n b th outward unit vctor normal to K dnot by K th adjacnt lmnt with th common dg. Ltτ 2 = k h u nc. Lmma 4.1 implis τ 2 K = K τ 2K φ. Hnc by ˆσ nc τ 2 = ˆσ τ 2K φ = γ 2 1 τ 2K τ 2K φ K = K\ Ω K\ Ω γ 2 1J n τ 2 φ for any x K. Similarly for τ 3 = h u nc any x K whav ˆρ nc τ 3 = ˆρ τ 3K ψ x = γ 3 1 τ 3K τ 3K ψ K = K\ Ω γ 3 1J t τ 3 ψ. K\ Ω

18 RECOVERY ERROR ESTIMATORS FOR INTERFACE PROBLEMS 47 Now it follows from th triangl inquality th facts that φ 2 dx Ch 2 ψ 2 dx Ch that K ˆη nck 2 = c 2 k 1/2 ˆσ nc τ 2 2 0K 1c 2 k 1/2 ˆρ nc τ 3 2 0K C k 1 K ˆσ nc τ 2 2 0K ˆσ nc τ 3 2 0K C 2k 1 K 1 γ 2 2 h 2 J nτ γ 3 2 h 2 J tτ 3 2 = C C K\ Ω K\ Ω K\ Ω h 2 k 1/2 K k1/2 K K K\ Ω ds 2 J nτ 2 2 ds J tτ 3 2 ds h 2 J nτ 2 2 ds J tτ 3 2 = C K\ Ω which provs 6.25 hnc This complts th proof of th thorm. Thorm Thr xists a positiv constant C such that 6.26 η nc C k 1/2 h nc 0Ω k 1/2 hf f h 0Ω. Proof is a dirct consqunc of Thorm 6.9 th following inquality Thorm 3.2 in [1]: ˆη ncfk C k 1/2 α nc 0K k 1/2 hf f h 0K. 7. Numrical xprimnts. In this sction w rport som numrical rsults for an intrfac problm with intrscting intrfacs usd by many authors.g. [20] which is considrd as a bnchmark tst problm. For this tst problm w show numrically that th rcovry-basd a postriori rror stimators introducd in [11] for both mixd nonconforming lmnts ovrrfin rgions along th intrfacs hnc fail to rduc th global rror. For th sam tst problm numrical rsults show that th stimators introducd in this papr ar accurat gnrat mshs with optimal dcay of th rror with rspct to th numbr of unknowns. To this nd lt Ω = ur θ =r β μθ in th polar coordinats at th origin with μθ bing a smooth function of θ [20 9]. Th function ur θ satisfis th intrfac quation in 2.1 with Γ N = f =0 { R in kx = 1 in Ω\ [0 1] 2 [1 0] 2. Th β dpnds on th siz of th jump. For xampl β =0.1 is corrsponding to R 161. η 2 nc

19 48 ZHIQIANG CAI AND SHUN ZHANG Rmark 7.1. This problm dos not satisfy Hypothsis 2.7 in [6] th distribution of its cofficints is not quasi-monoton. Not that th solution ur θ isonlyinh 1βɛ Ω for any ɛ>0 hnc it is vry singular for small β at th origin. This suggsts that rfinmnt is cntrd around th origin. In this xampl w choos η nc η nc with constant c 2 =0.5 as th implicit xplicit rror stimators for th nonconforming mthod rspctivly. Starting with a coars triangulation T 0 a squnc of mshs is gnratd by using a stard adaptiv mshing algorithm that adopts th Dörflr s bulk marking stratgy: construct a minimal subst ˆT of T such that 7.1 η 2 θ2 K E η 2 K K ˆT K T with θ E =0.2 which is not critical 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. Dfin th ffctivity indx: ff-indx m := η k 1/2 u k 1/2 σ m 0Ω ff-indx nc := us th following stopping critria: η k 1/2 h u u nc 0Ω rl-rr m := k1/2 u k 1/2 σ m 0Ω tol rl-rr k 1/2 nc := k1/2 h u u nc 0Ω tol. σ 0Ω k 1/2 u 0Ω Dnot by l th numbr of lvls of rfinmnt by N th numbr of vrtics of triangulation. Th rror stimators introducd in [11] for both mixd nonconforming finit lmnt approximations to th Poisson quations rcovr th gradint in th continuous linar finit lmnt spac. A natural xtnsion of ths stimators to th intrfac problms is to rcovr ithr th gradint or th flux again in th continuous linar finit lmnt spac. Mor spcifically for th mixd mthod lt σ m b th solution of 3.1 lt ρ mf U 2 ρ mg U 2 satisfy th following problms: k 1 ρ mf τ =k 1 σ m τ τ U 2 kρ mg τ =σ m τ τ U 2 rspctivly. Thn th corrsponding rror stimators ar dfind by η mcbf = k 1/2 σ m ρ mf 0Ω η mcbg = k 1/2 σ m k 1/2 ρ mg 0Ω. For th nonconforming mthod lt u nc b th solution of 3.5 lt ρ ncf U 2 ρ ncg U 2 satisfy th following problms: k 1 ρ ncf τ = h u nc τ kρ ncg τ =k h u nc τ for all τ U 2 rspctivly. Thn th corrsponding rror stimators ar dfind by η nccbf = k 1/2 h u nc k 1/2 ρ ncf 0Ω η nccbg = k 1/2 h u nc ρ ncg 0Ω.

20 RECOVERY ERROR ESTIMATORS FOR INTERFACE PROBLEMS 49 Tabl 1 Comparison of stimators for rlativ rror lss than 0.1 for mixd mthods. l N rr rl-rr η ff-indx η m ˆη m η mcbf η mcbg Tabl 2 Comparison of stimators for rlativ rror lss than 0.1 for nonconforming mthods. l N rr rl-rr η ff-indx η nc η nc ˆη nc η nccbf η nccbg Fig. 1. Msh gnratd by η mcbf. Fig. 2. Msh gnratd by η mcbg. η mrt k 1/2 uk 1/2 σ m η / k 1/2 u mrt 0 k 1/2 uk 1/2 σ / k 1/2 u m rfrnc lin with slop 1/ numbr of nods Fig. 3. Msh gnratd by η m. Fig. 4. Error stimator η m.

21 50 ZHIQIANG CAI AND SHUN ZHANG η mrtxplicit k 1/2 uk 1/2 σ m rfrnc lin with slop 1/2 η / k 1/2 u mrtxplicit 0 k 1/2 uk 1/2 σ / k 1/2 u m numbr of nods Fig. 5. Msh gnratd by ˆη m. Fig. 6. Error stimator ˆη m. Fig. 7. Msh gnratd by η nccbf. Fig. 8. Msh gnratd by η nccbg. η nc k 1/2 u h u nc η nc k 1/2 u u / k 1/2 u h nc rfrnc lin with slop 1/ numbr of nods Fig. 9. Msh gnratd by η nc. Fig. 10. Error stimator η nc. W start with th coarsst triangulation T 0 obtaind from halving 16 congrunt squars by conncting th bottom lft uppr right cornrs. W rport numrical rsults with th stopping critrion tol = 0.1. From Tabls 1 2 Figurs it is clar that th CB stimators introduc unncssary rfinmnts along th intrfacs. Mshs gnratd by η m ˆη m η nc ˆη nc η nc shown in Figurs ar similar. Th comparisons of rrors stimators ar dpictd

22 RECOVERY ERROR ESTIMATORS FOR INTERFACE PROBLEMS 51 η ncxplicit k 1/2 u h u nc η ncxplicit k 1/2 u u / k 1/2 u h nc rfrnc lin with slop 1/ numbr of nods Fig. 11. Msh gnratd by ˆη nc. Fig. 12. Error stimator ˆη nc. η ncfxplicit k 1/2 u h u nc η ncfxplicit k 1/2 u u / k 1/2 u h nc rfrnc lin with slop 1/ numbr of nods Fig. 13. Msh gnratd by η nc. Fig. 14. Error stimator η nc. in Figurs By inspcting th ffctivity indx all th rror stimators introducd in this papr ar accurat. Morovr th slop of th logdof- logrlativ rror for all stimators is 1/2 which indicats th optimal dcay of th rror with rspct to th numbr of unknowns. REFERENCES [1] M. Ainsworth Robust a postriori rror stimation for nonconforming finit lmnt approximation SIAM J. Numr. Anal pp [2] M. Ainsworth A postriori rror stimation for lowst ordr Raviart Thomas mixd finit lmnts SIAM J. Sci. Comput pp [3] M. Ainsworth J. T. Odn A Postriori Error Estimation in Finit Elmnt Analysis Pur Appl. Math. Wily-Intrscinc Nw York [4] A. Alonso Error stimators for a mixd mthod Numr. Math pp [5] C. Brnardi F. Hcht Error indicators for th mortar finit lmnt discrtization of th Laplac quation Math. Comp pp [6] C. Brnardi R. Vrfürth Adaptiv finit lmnt mthods for lliptic quations with non-smooth cofficints Numr. Math pp [7] F. Brzzi M. Fortin Mixd Hybrid Finit Elmnt Mthods Springr-Vrlag Nw York [8] Z. Cai S. Zhang Rcovry-basd rror stimator for intrfac problms: Mixd nonconforming lmnts xtndd vrsion manuscript purdu.du/ zcai/pdf-papr/cazh08b.pdf.

23 52 ZHIQIANG CAI AND SHUN ZHANG [9] Z. Cai S. Zhang Rcovry-basd rror stimator for intrfac problms: Conforming linar lmnts SIAM J. Numr. Anal pp [10] C. Carstnsn A postriori rror stimat for th mixd finit lmnt mthod Math. Comp pp [11] C. Carstnsn S. Bartls Each avraging tchniqu yilds rliabl a postriori rror control in FEM on unstructur grids. Part I: Low ordr conforming nonconforming mixd FEM Math. Comp pp [12] C. Carstnsn S. Bartls S. Jansch A postriori rror stimats for nonconforming finit lmnt mthods Numr. Math pp [13] P. G. Ciarlt Th Finit Elmnt Mthod for Elliptic Problms North-Holl Amstrdam [14] E. Dari R. Duran C. Padra V. Vampa A postriori rror stimators for nonconforming finit lmnt mthods RAIRO Modl Math. Anal. Numr pp [15] V. Girault P.-A. Raviart Finit Elmnt Mthods for Navir-Stoks Equations Springr-Vrlag Brlin [16] K.-Y. Kim A postriori rror analysis for locally consrvativ mixd mthods Math. Comp pp [17] C. Lovadina R. Stnbrg Enrgy norm a postriori stimats for mixd finit lmnt mthods Math. Comp pp [18] R. Luc B. I. Wohlmuth A local a postriori rror stimator basd on quilibratd fluxs SIAM J. Numr. Anal pp [19] L. D. Marini An inxpnsiv mthod for th valuation of th solution of th lowst ordr Raviart Thomas mixd mthod SIAM J. Numr. Anal pp [20] P. Morin R. H. Nochtto K. G. Sibrt Convrgnc of adaptiv finit lmnt mthods SIAM Rv pp [21] J. C. Ndlc Mixd finit lmnts in R 3 Numr. Math pp [22] J. C. Ndlc A nw family of mixd finit lmnts in R 3 Numr. Math pp [23] J. S. Ovall Fixing a bug in rcovry-typ a postriori rror stimators Max-Planck- Institut fur Mathmatick in dn Naturwissnschaftn Lipzig Tchnical rport [24] M. Ptzoldt A postriori rror stimators for lliptic quations with discontinuous cofficints Adv. Comput. Math pp [25] R. Vrfürth A Rviw of a Postriori Error Estimation Adaptiv Msh-Rfinmnt Tchniqus Wily-Tubnr Stuttgart Grmany [26] M. Vohralik A postriori rror stimats for lowst-ordr mixd finit lmnt discrtizations of convction-diffusion-raction quations SIAM J. Numr. Anal pp [27] B. I. Wohlmuth R. H. W. Hopp A comparison of a postriori rror stimators for mixd finit lmnt discrtizations by Raviart-Thomas lmnts Math. Comp pp