A POSTERIORI ERROR ESTIMATION FOR AN INTERIOR PENALTY TYPE METHOD EMPLOYING H(DIV) ELEMENTS FOR THE STOKES EQUATIONS

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1 A POSTERIORI ERROR ESTIMATION FOR AN INTERIOR PENALTY TYPE METHOD EMPLOYING H(DIV) ELEMENTS FOR THE STOKES EQUATIONS JUNPING WANG, YANQIU WANG, AND XIU YE Abstract. This papr stablishs a postriori rror analysis for th Stoks quations discrtizd by an intrior pnalty typ mthod using H(div) finit lmnts. Th a postriori rror stimator is thn mployd for dsigning two grid rfinmnt stratgis: on is locally basd and th othr is globally basd. Th locally basd rfinmnt tchniqu is blivd to b abl to captur local singularitis in th numrical solution. Th numrical formulations for th Stoks problm mak us of H(div) conforming lmnts of th Raviart Thomas typ. Thrfor, th finit lmnt solution faturs a full satisfaction of th continuity quation (mass consrvation). Th rsult of this papr provids a rigorous analysis for th mthod s rliability and fficincy. In particular, an H -norm a postriori rror stimator is obtaind, togthr with uppr and lowr bound stimats. Numrical rsults ar prsntd to vrify th nw thory of a postriori rror stimators. Ky words. finit lmnt mthods, Stoks problm, a postriori stimats AMS subjct classifications. Primary, 65N5, 65N3, 76D7; Scondary, 35B45, 35J5. Introduction. In this papr, th authors ar concrnd with a postriori rror analysis for numrical solutions of th Stoks quations discrtizd by an intrior pnalty typ mthod using H(div) finit lmnts. Th Stoks quations undr study sks a vlocity u and a prssur p satisfying (.) (.2) (.3) u + p = f in Ω, u = in Ω, u = on Ω, whr,, and dnot th Laplacian, gradint, and divrgnc oprators, rspctivly; Ω R d, d = 2,3 is th rgion occupid by th fluid; f (L 2 (Ω)) d is th unit xtrnal volumtric forc acting on th fluid. For simplicity, th mthod will b prsntd for two-dimnsional problms (d = 2) on polygonal domains. An xtnsion to thr dimnsions can b mad formally for gnral polyhdral domains. In th nginring socity, it is rquird for numrical schms to rtain th original physical proprtis, such as mass and nrgy consrvation. For th Stoks quations, such a rquirmnt translats to a discrtization schm satisfying th incomprssibl constraint quation (.2) xactly on th computational domain. Howvr, constructing such a finit lmnt spac in H is quit challnging, and th rsulting spacs ar oftn not computationally frindly. Rcntly, a nw approach, which uss H(div) conforming lmnts, has bn dvlopd for th Stoks [22] and Navir-Stoks quations [23]. Numrical solutions of this mthod satisfy th continuity quation (.2) Division of Mathmatical Scincs, National Scinc Foundation, Arlington, VA 2223 (jwang@ nsf.gov). Th rsarch of Wang was supportd by th NSF IR/D program, whil working at th Foundation. Howvr, any opinion, finding, and conclusions or rcommndations xprssd in this matrial ar thos of th author and do not ncssarily rflct th viws of th National Scinc Foundation. Dpartmnt of Mathmatics, Oklahoma Stat Univrsity, Stillwatr, OK 7475 (yqwang@math.okstat.du). Dpartmnt of Mathmatics, Univrsity of Arkansas at Littl Rock, Littl Rock, AR 7224 (xxy@ualr.du). This rsarch was supportd in part by National Scinc Foundation Grant DMS- 8357

2 2 JUNPING WANG, YANQIU WANG AND XIU YE xactly. Th discrt vlocity, which lis in an H(div) conforming finit lmnt spac, has continuous normal componnt across intrnal dgs. Th tangntial continuity is imposd wakly, using th ida similar to th on in intrior pnalty mthods. Th ida of mploying H(div) conforming lmnts to th Stoks quations has bn xplord by svral rsarchrs in th last two dcads. All th xisting work applis th H(div) conforming lmnts to ithr a strss-vlocity or a strss-vlocityprssur formulation of Stoks and Navir-Stoks quations, whr th strss is in H(div). In [5, 6], a psudostrss-vlocity formulation has bn proposd and solvd by H(div) lmnts. Its xtnsion to th psudostrss-vlocity-prssur formulation has bn considrd in [], togthr with a priori and a postriori rror analysis. In [], an augmntd formulation using th H(div) conforming lmnt mthod has bn dvlopd. Also, th dual-mixd mthod has bn studid in [2, 3]. In all ths work, th H(div) lmnt was usd to approximat th strss or strss-typ dual variabls. W would lik to point out that our mthod is diffrnt from th xisting ons in that w us th vlocity-prssur formulation and th H(div) lmnt is usd to approximat th vlocity dirctly. Th goal of this papr is to obtain an a postriori rror stimator for th H(div) finit lmnt mthod dvlopd in [22], and to provid its uppr and lowr bound stimats. In th analysis, w borrow many idas from som prvious a postriori rror analysis for Stoks quations, including [2], which uss conforming finit lmnts, and [8, 9, 7, 4, 5, 2], which us nonconforming finit lmnts. Our contributions in this papr ar: () succssfully stablishd a postriori rror stimator for th H(div) finit lmnt mthod, (2) proposd and tstd a nw grid rfinmnt mthod using local information in ordr to captur local singularitis, (3) conductd a sris of numrical xprimnts for th rfinmnt stratgis. W hop that th numrical rsults prsntd in this papr will shin som light on a furthr dvlopmnt of computational tchniqus in fluid dynamics. Th papr is organizd as follows. In Sction 2, w introduc som notations for scalar, vctor, and matrix Sobolv spacs. In Sction 3, th H(div) finit lmnt formulation and its a priori rror stimat ar statd. Sction 4 is ddicatd to an stablishmnt and analysis of an a postriori rror stimator. In Sction 5, w shall prsnt two grid rfinmnt stratgis: on is locally basd and th othr is globally basd. Finally in Sction 6, w prsnt som numrical rsults and offr som of our own obsrvations. 2. Notations. Lt D b a polygon. W us standard dfinitions for th Sobolv spacs H s (D) and thir associatd innr products (, ) s,d, norms s,d, and sminorms s,d, for s. Th spac H (D) coincids with L 2 (D), for which th norm and th innr product ar also dnotd by D and (, ) D, rspctivly. For convninc, whn D = Ω, w usually supprss D in th subscript. Dnot L 2 (Ω) to b th subspac of L 2 (Ω) consisting of functions with man valu zro. Notic that all abov dfinitions can b xtndd to th cas of vctor-valud or matrix-valud functions, through product spacs. W us th sam notation for thir norms and innr products. Also, all ths dfinitions can b transportd from a polygon D to an dg. Similar notation systm will b mployd, for xampl, s, and. Throughout th papr, w follow th convntion that a bold fac Latin charactr

3 A POSTERIORI ERROR ESTIMATION 3 dnots a vctor. For vctor function v R 2, dfin ) v = ( v x v 2 x v y v 2 y, curl v = ( v y v2 y v x v 2 x ), v = v x + v 2 y. Dfin th spac H(div;Ω) to b th st of vctor-valud functions on Ω which, togthr with thir divrgnc, ar squar intgrabl; i.., H(div;Ω) = { v : v (L 2 (Ω)) 2, v L 2 (Ω) }. Th norm in H(div;Ω) is dfind by v H(div;Ω) = ( v 2 + v 2) 2. Lt T h b a gomtrically conformal triangulation of th domain Ω, i.., th intrsction of any two triangls in T h is ithr mpty, a common vrtx, or a common dg. Dnot h K to b th diamtr of triangl K T h, and h to b th maximum of all h K. W also assum T h is shap rgular, that is, for ach K T h, th ratio btwn h K and th diamtr of th inscribd circl is boundd abov. This nsurs that th scaling argumnts and th invrs inqualitis work on ach triangl. Dfin th finit lmnt spacs V h and W h for th vlocity and prssur variabls, rspctivly, by V h = {v H(div;Ω) : v K V k (K) K T h ; v n Ω = } W h = {q L 2 (Ω) : q K W k (K) K T h }, whr n is th outward normal dirction, (V k (K), W k (K)) can b any xisting H(div) conforming finit lmnt pairs [4] of ordr k. For xampl, th Raviart Thomas lmnts (RT k ) [6] or th Brzzi Douglas Marini lmnts (BDM k ) [3]. For BDM lmnt, W h consists of all picwis constants on T h. Also, notic that for all v V h, it has continuous normal componnt v n across intrnal dgs, whil its tangntial componnt is not ncssarily continuous. For vctors v, n R 2, dnot v n = {v i n j } i,j 2 to b th vctor tnsor product. For matrics σ, τ R 2 2, dfin σ : τ = 2 i,j= σ ijτ ij. Notic that v (σn) = (v n) : σ. Latr w will us th abov quation without xplicit mntioning. Lt b an intrior dg shard by two lmnts K and K 2 in T h. Dnot unit normal vctors n, n 2 and tangntial dirctions t, t 2, rspctivly, on for K and K 2 (as shown in Figur 2.). Dfin th avrag { } and jump [ ] on for scalar function q, vctor function v and matrix function σ, rspctivly, by {q} = 2 (q K + q K2 ), [qn] = q K n + q K2 n 2, {v} = 2 (v K + v K2 ), {σ} = 2 (σ K + σ K2 ), [σn] = σ K n + σ K2 n 2, [σt] = σ K t + σ K2 t 2.

4 4 JUNPING WANG, YANQIU WANG AND XIU YE n 2 t K t 2 n K 2 Fig. 2.. Normal and tangntial vctors for nighboring triangls. W also dfin a scalar avrag {ε( ) } and a matrix valud jump [[ ]] for a vctor function v by {ε(v) } = 2 (n (v t ) K + n 2 (v t 2 ) K2 ), [[v]] = v K n + v K2 n 2. If is a boundary dg, th abov dfinitions should b modifid such that both th avrag and th jump ar qual to th on-sidd valus on. For xampl {q} = q, [qn] = q n. Othr trms should b modifid in th sam fashion. Dnot by E h th st of all dgs in T h, and Eh := { E h, Ω} th st of all intrior dgs. Lt V (h) = V h +(H s (Ω) H(Ω)) 2, with s > 3 2, whr th summation mans th mathmatical sum of functions from ach subspac. For v V (h), dfin h v to b th function whos rstriction to ach lmnt K T h is givn by th standard gradint v. 3. Finit lmnt schm and a priori rror stimat. W us th numrical schm proposd and analyzd in [22], whr dtails on convrgnc analysis can b found. For simplicity of prsntation, this papr uss a slightly diffrnt notation in dscribing th numrical schms. To this nd, w introduc two bilinar forms on V (h) V (h) as follows a s (w,v) = ( h w, h v) + ( αh [[w]] : [[v]] { w} : [[v]] { v} : [[w]] ) ds, E h a ns (w,v) = ( h w, h v) + E h ( αh [[w]] : [[v]] { w} : [[v]] + { v} : [[w]] ) ds, whr α > is a paramtr to b dtrmind latr, and h is th lngth of th dg. It is not hard to vrify that th abov two bilinar forms ar xactly th sam as thos statd in [22]. For radr s convninc, a brif xplanation is givn in Appndix A for such a vrification. As usual, thr is a bilinar form on V (h) L 2 (Ω) givn by b(v,q) = ( v,q). Th H(div) finit lmnt schm for (.) (.3) sks (u h ;p h ) V h W h such that (3.) (3.2) a(u h,v) b(v,p h ) = (f,v) v V h, b(u h,q) = q W h,

5 A POSTERIORI ERROR ESTIMATION 5 whr a(, ) can b takn as ithr a s (, ) or a ns (, ). It has bn provd in [22] that th abov systm (3.) (3.2) is wll posd for non-symmtric bilinar form a ns (, ) with α >, and for symmtric bilinar form a s (, ) with α larg nough. It is not hard to s that th solution (u; p) of (.) (.3) also satisfis (3.3) (3.4) a(u,v) b(v,p) = (f,v) v V h, b(u,q) = q W h. Subtracting (3.) (3.2) from (3.3) (3.4) givs th following rror quations (3.5) (3.6) a(u u h,v) b(v,p p h ) = v V h, b(u u h,q) = q W h. To invstigat th approximation proprty of th abov numrical schm, w introduc a norm on V (h) as follows: v 2 = h v 2 + E h h [[v]] 2 + E h h {ε(v) } 2. Lt Π h b th intrpolation into V h associatd with th usual dgrs of frdom (s [4] for dtails), and Q h b th L 2 projction from L 2 (Ω) onto W h. It is wll-known that ( )Π h = Q h ( ). Furthrmor, th following a priori rror stimat has bn provd in [22]: Thorm 3.. Lt (u;p) b th solution of (.) (.3) and (u h ;p h ) V h W h b obtaind from (3.) (3.2). Thn, thr xists a constant C indpndnt of h such that (3.7) u u h + p p h C ( u Π h u + p Q h p ). For xampl, Thorm 3. implis an rror stimat of u u h + p p h = O(h) whn th BDM lmnt is usd in th numrical discrtization with an xact solution (u; p) (H 2 H ) 2 (H L 2 ). 4. A postriori rror stimator. Th goal of this sction is to driv an a postriori rror stimator for th finit lmnt formulation (3.) (3.2). A dtaild prsntation will b givn only for th symmtric formulation a s (, ); th non-symmtric cas of a ns (, ) can b handld analogously without any difficulty. For simplicity of notation, w us to dnot lss than or qual to up to a constant indpndnt of th msh siz, variabls, or othr paramtrs apparing in th inquality. On ach dg, w introduc th following jumps : { [ uh n] [p J ( u h n p h n) = h n], if Eh,, othrwis, and { [[uh ]], if E J 2 (u h ) = h, 2u h n, othrwis. Dfin a local rror stimator on ach lmnt K T h by (4.) ηk 2 = h 2 K f h + u h p h 2 K + ( h J ( u h n p h n) 2 + h J 2 (u h ) 2 2 ), K

6 6 JUNPING WANG, YANQIU WANG AND XIU YE and dfin a global rror stimator η 2 = K T h ηk 2. Hr and in what follows of this papr, f h is th L 2 projction of th load function f into th vlocity spac dfind locally on ach lmnt. It will b sn that f h can b a projction of f into any polynomial spac dfind on ach individual lmnt K. For any K T K and on of its dgs, it can b provd by using th trac thorm and th scaling argumnt that for vry q H (K), w hav th following stimat (s Thorm 3. in [] or Equation (2.4) in [2]) (4.2) q 2 h K q 2 K + h K q 2 K. Anothr usful stimat can b statd as follows. Lmma 4.. Lt v V h, thn (4.3) h [ vt] 2 h [[v]] 2. Proof. First, lt b an intrior dg shard by triangls K and K 2 in T h. On dg, dfin q = v K v K2. Thn, by th invrs inquality, w hav h [ vt] 2 = h q 2 h q 2 = h [[v]] 2. Th proof for boundary dgs is similar. 4.. Rliability of th stimator. Lt = u u h and ǫ = p p h. It has bn provd in [8] that vry matrix-valud function in (L 2 (Ω)) 4, and hnc h, admits th following dcomposition: (4.4) h = r qi + curl s, whr r H (Ω) 2 is divrgnc-fr, s H (Ω) 2, q L 2 (Ω), and I is th 2 2 idntity matrix. Furthrmor, th following bound holds [8] (4.5) r + s h Ω. Sinc is xactly divrgnc-fr, it follows that (4.6) ( h, qi) = (,q) =. Thrfor, w hav from (4.4) and (4.6) that (4.7) ( h, h ) = ( h, r) + ( h, curl s). For any vctor-valud function v (H (Ω)) 2, dnot by v I th Clémnt typ intrpolation onto continuous picwis linars on T h, prsrving th homognous boundary condition; dtails for such an intrpolation can b found from [7]. Whn RT k or BDM k lmnts, with k, ar usd in th numrical discrtization schm, w s that th abov mntiond intrpolation satisfis v I (H (Ω)) 2 V h. Clarly, th jump trm [[v I ]] = vanishs on vry E h. Furthrmor, w hav th following approximation proprty: (4.8) v I,K v,k, v v I K h K v,k, v v I h /2 v /2,.

7 A POSTERIORI ERROR ESTIMATION 7 Thorm 4.2. Lt (u;p) and (u h ;p h ) b th solution of (.)-(.3) and (3.)- (3.2), rspctivly. Thn w hav th following uppr bound for h : (4.9) h η + ( K T h h 2 K f f h 2 K) /2. Proof. Rcall th dcomposition (4.4). By stting v = r I in th rror quation (3.5) and using [[r I ]] =, w hav (4.) ( h, r I ) { r I } : [[u h ]] ( r I, ǫ) =. E h Using (4.), intgration by parts, and th fact that r is divrgnc-fr, w hav ( h, r) = ( h, r r I ) + { r I } : [[u h ]]ds + ( r I, ǫ) E h = ( u + u h,r r I ) K + ( n) (r r I )ds K T h K T K h + { r I } : [[u h ]]ds ( (r r I ), ǫ) E h = ( u + u h,r r I ) K [ u h n] (r r I )ds + { r I } : [[u h ]]ds K T h Eh E h + ( p p h, r r I ) K (r r I ) n(p p h ) ds K T h K T K h = (f + u h p h, r r I ) K ([ u h n] [p h n]) (r r I )ds K T h Eh + { r I } : [[u h ]]ds E h Applying quations (4.2), (4.5), (4.8), th trac thorm, and th standard invrs inquality to th abov givs ( h, r) ( η + ( K T h h 2 K f f h 2 K) /2 Nxt, it follows from th intgration by parts that ( h, curl s) = ( h u h, curl s) ) h. = ( h u h, curl (s s I )) ( h u h, curl s I ) = ( ( u h t) (s s I ) u h (curl s I n)) ds K T h K = ( [ u h t] (s s I ) [[u h ]] : {curl s I }) ds. E h

8 8 JUNPING WANG, YANQIU WANG AND XIU YE Thus, by using Lmma 4., w obtain ( ) /2 ( ) /2 ( h, curl s) h [ u h t] 2 + h [[u h ]] 2 h E h E h η h. Combining (4.7) with th abov stimats complts a proof of th lmma. As to th prssur rror, w hav th following rsult. Thorm 4.3. Lt (u;p) and (u h ;p h ) b th solution of (.)-(.3) and (3.)- (3.2), rspctivly. Thn w hav th following uppr bound for ǫ : (4.) ǫ η + ( K T h h 2 K f f h 2 K) /2. Proof. First, w rcall th following continuous vrsion of th inf-sup condition: (4.2) p p h sup v [H (Ω)]2 b(v,p p h ) v. For v H (Ω) 2, using intgration by parts, quations (3.), (4.2), (4.5), (4.8), (4.9), th trac thorm, and th invrs inquality, w hav ( v,p p h ) = (f,v) + ( u, v) ( v,p h ) = (f,v) + ( h, v) + ( h u h, v) ( v,p h ) + (f,v I ) a(u h,v I ) + b(v I,p h ) = (f,v) + ( h, v) + ( h u h, v) ( v,p h ) + (f,v I ) ( h u h, v I ) + { v I } : [[u h ]]ds + ( v I,p h ) E h K T h K = (f,v v I ) + ( h, v) ( u h,v v I ) K + ( p h, v v I ) K K T h K T h + ( u h n p h n) (v v I )ds + { v I } : [[u h ]]ds E h = (f + u h p h,v v I ) + ( h, v) K T h + ([ u h ] n [p h ]) (v v I )ds + { v I } : [[u h ]]ds. Eh E h Thus, w obtain from th standard Cauchy-Schwarz inquality that ( ( v,p p h ) η + ( ) v, K T h h 2 K f f h 2 K) /2 which, togthr with th inquality (4.2), complts th proof of Thorm 4.3. By th dfinition of f h and th approximation proprty of L 2 projction, it is not hard to s that ( K T h h 2 K f f h 2 K )/2 has highr ordr in h K than h + ǫ, as long as f is smooth nough. Hnc, thorms 4.2 and 4.3 imply that th a postriori rror stimator η is rliabl in that th rror trm h + ǫ must b small whn η is small. Morovr, th formr is controlld by th lattr in thir magnitud.

9 A POSTERIORI ERROR ESTIMATION Efficincy of th stimator. For ach triangular lmnt K T h, dnot by φ K th following bubbl function { 27λ λ 2 λ 3 in K, φ K = in Ω\K, whr λ i, i = ; 2; 3, ar barycntric coordinats on K. Ndlss to say, th finit lmnt partition T h is assumd to contain triangular lmnts only. It is clar that φ K H (Ω) and satisfis th following proprtis (Sction I.2.2 in [9]): For any polynomial q with dgr at most m, thr xist positiv constants c m and C M, dpnding only on m, such that (4.3) c m q 2 K q 2 φ K dx q 2 K, (4.4) K (qφ K ) K C m h K q K. For ach Eh, w can analogously dfin an dg bubbl function φ. Lt K and K 2 b two triangls sharing th dg. To this nd, dnot by Ω = K K 2 th union of th lmnts K and K 2. Assum that in K i, i =,2, th barycntric coordinats associatd with th two nds of ar λ Ki and λ Ki 2, rspctivly. Th dg bubbl function can b dfind as follows 4λ K λk 2 in K, φ = 4λ K2 λk2 2 in K 2, in Ω\Ω. It is obviously that φ H(Ω) and satisfis th following proprtis (Sction I.2.2 in [9]): For any polynomial q with dgr at most m, thr xist positiv constants d m, D M and E m, dpnding only on m, such that (4.5) d m q 2 q 2 φ ds q 2, (4.6) (4.7) (qφ ) Ω D m h /2 q, qφ Ω E m h /2 q. Rgarding th fficincy of th numrical schms undr considration, w hav th following rsult. Thorm 4.4. For vry K T h or E h, (4.8) (4.9) h K f h + u h p h K K + ǫ K + h K f f h K, h /2 [ u h n] [p h n] ( K + ǫ K + h K f f h K ). K Ω Proof. Lt w K = (f h + u h p h )φ K. It is clar that (f, w K ) = ( u, w K ) ( w K, p),

10 JUNPING WANG, YANQIU WANG AND XIU YE which implis (f f h, w K ) + (f h, w K ) ( u h, w K ) + ( w K, p h ) = (, w K ) ( w K, ǫ). Using intgration by parts, th abov quation bcoms (f h + u h p h, w K ) = (, w K ) ( w K, ǫ) (f f h, w K ). By th dfinition of w K, th Cauchy-Schwartz inquality, and inqualitis (4.3)- (4.4), w hav f h + u h p h 2 K (, w K ) ( w K, ǫ) (f f h, w K ) ( h K K + h K ǫ K + f f h K ) fh + u h p h K. This complts th proof of inquality (4.8). As to an stimat for th jump trm [ u h n] [p h n], w considr a similar function w dfind by w = ([ u h n] [p h n])φ. Hr, in th multiplication, th function [ u h n] [p h n] should b xtndd as a constant function in th normal dirction of th dg ; i.., w taks th polynomial [ u h n] [p h n] on and all lins paralll with. It is not hard to s that K Ω ((f, w ) K ( u h, w ) K + ( w,p h )) = K Ω ((, w ) K ( w,ǫ) K ). Using intgration by parts, w hav ((f+ u h p h, w ) K ([ u h n] [p h n]) w ds K Ω = ((, w ) K ( w,ǫ) K ). K Ω Thrfor, by th dfinition of w, th Cauchy-Schwartz inquality, and inqualitis (4.5)-(4.7), w hav [ u h n] [p h n] 2 (h /2 f f h Ω + h /2 f h + u h p h Ω + h /2 h Ω + h /2 ǫ Ω ) [ u h n] [p h n]. Inquality (4.9) follows from th abov, inquality (4.8), and th fact that T h is shap rgular. This complts th proof of th thorm. Summing ovr all th lmnts yilds th following lowr bound for th rror trm + ǫ. Thorm 4.5. Lt (u;p) and (u h ;p h ) b th solution of (.)-(.3) and (3.)- (3.2), rspctivly. Thn w hav th following lowr bound stimat: ( ) /2 η + ǫ + h 2 f f h 2 K. K T h

11 A POSTERIORI ERROR ESTIMATION Combining thorms 4.2, 4.3 and 4.5, and noticing that ( K T h h 2 K f f h 2 K )/2 is a highr ordr trm, w can conclud that, thortically, η is a good indicator for + ǫ. 5. Two stratgis in local grid rfinmnt. Th local a priori rror stimator η K as dfind in (4.) can b usd to provid algorithms for local grid rfinmnt. Two diffrnt rfinmnt stratgis ar considrd in this study. Th first on is basd on a comparison of ach rror η K with th maximum valu of all th rror stimators. Th stratgy can b dscribd as follow. Local Rfinmnt by Maximum Stratgy :. Givn a currnt triangular msh, rror stimators η K on ach triangl, and a thrshold θ (,) (.g., θ =.5). On computs th maximum rror η max = max η K. 2. For ach triangl K, if η K θ η max, rfin this triangl uniformly by conncting th cntr of thr dgs. 3. Th prvious stp will gnrat hanging nods. Us bisction to gt a conformal msh. Mor prcisly, on nds to chck vry unrfind triangl K and prform th following modifications: If K has on hanging nod, thn bisct it onc. If K has two hanging nods, thn bisct it twic. Tak a spcial car to prvnt th occurrnc of dgnratd triangls; i.., to guarant that th nw msh prsrvs th shap rgularity. This can b don by adding xtra hanging nods if th currnt bisction rsults in dgnratd triangls. Th scond rfinmnt stratgy is basd on a comparison of η K with thos for its nighbors. To xplain th main ida, lt ρ > b a prscribd distanc paramtr and st T ρ,k = { K : < K K ρ}, whr K K stands for th distanc of th cntrs of K and K. With a givn thrshold θ >, w mark a triangl K for uniform rfinmnt if η K θ η N(K), whr η N(K) is th avrag of th local rror indicator on all th nighboring triangls K T ρ,k. Th following is such a rfinmnt stratgy that was numrically invstigatd in this study. Local Rfinmnt by Local Stratgy :. Givn a currnt triangular msh, rror stimators η K on ach triangl, and a thrshold θ >. (.g., θ =.3). On computs an rror indicator η N(K) as th avrag of th local rror indicator on nighboring triangls that shar a vrtx or an dg with K, not including K itslf. 2. For ach triangl K, if η K θ η N(K), rfin this triangl uniformly by conncting th cntr of thr dgs. 3. Th prvious stp will gnrat hanging nods. Us bisction to gt a conformal msh. Mor prcisly, on nds to chck ach unrfind triangl K and prform th following modifications:

12 2 JUNPING WANG, YANQIU WANG AND XIU YE If K has on hanging nod, thn bisct it onc. If K has two hanging nods, thn bisct it twic. Tak a spcial car to prvnt th occurrnc of dgnratd triangls; i.., to guarant that th nw msh prsrvs th shap rgularity. This can b don by adding xtra hanging nods if th currnt bisction rsults in dgnratd triangls. Th rfinmnt mthod by th local stratgy is basd on th obsrvation that local stimators on th xact location of th singularity tnd to b much largr than on surrounding rgions. Sinc th singularity usually occurs on a lowr dimnsional manifold, for xampl, a point or an dg in two dimnsion, it is saf to us such a local stratgy to locat singularitis. In our numrical xprimnts to b prsntd in th coming sction, w hav found that θ =.3 is a practical choic. 6. Som numrical rsults. Th goal of this sction is to rport som numrical rsults for th rfinmnt stratgis as discussd in th prvious sction. Without loss of gnrality, w discuss only numrical rsults arising from th non-symmtric formulation with a ns (, ). As mntiond bfor, th non-symmtric formulation is wll-posd for any α >, whil th symmtric formulation is so only for α sufficintly larg. Although numrical tsts in [24] hav indicatd that th symmtric form works for α gratr than a modrat numbr (usually btwn and 5), thy hav also suggstd that th non-symmtric formulation is mor stabl than th symmtric on with rspct to th slction of α valus (s xampls givn in [24]). For th numrical rsults to b prsntd in this sction, th paramtr α is st to b 5, and th Stoks quations ar discrtizd by th BDM lmnt for th vlocity. Th GMRES itrativ mthod is usd for solving th rsulting linar algbraic systm, and a rlativ rsidual of 8 is st to b th stopping critria. It should b pointd out that, for th tst problms and th valu of α chosn in this papr, numrical rsults show that th a postriori rror indicators for both th symmtric and nonsymmtric formulations bhav almost xactly th sam. Hnc th rsults for th symmtric formulation ar omittd. Howvr, w did not xplor th cass whn α is too small or too larg. Th numrical formulation in this papr can b asily xtndd to problms with non-homognous Dirichlt boundary conditions; dtails hav bn givn in [24]. Accordingly, changs nd to b mad for th a postriori rror stimator η K. For th Dirichlt boundary condition u = g on Ω, w must modify J 2 (u h ) as follows: { [[uh ]] if E J 2 (u h ) = h, (u h g) n othrwis. Anothr small modification is that, in th dfinition of η K, th trm h 2 K f h + u h p h 2 K must b computd as 2 K f h + u h p h 2 K, whr K is th ara of th triangl K. This modification is purly for implmntation purpos sinc th two formulas ar quivalnt mathmatically. For th BDM lmnt, w also hav u h p h =. In th xprimnt, f h K is calculatd by computing f K using a high ordr numrical intgration, which is xact for up to 7th ordr polynomials. Th ffct is quivalnt to considring f h to b a picwis cubic intrpolation of f. 6.. Tst problms. W considr thr tst problms, all ar dfind on Ω = (,) (,). Two of thm hav xact solutions givn as: ( ) 2x Tst problm : u = 2 y(x ) 2 (2y )(y ) xy 2 (2x )(x )(y ) 2, p =,

13 and Tst problm 2: u = A POSTERIORI ERROR ESTIMATION 3 ( 3 ( 2 r cos θ cos 3θ 2 ( r 3sin θ 2 sin 3θ 2 ) ) ), p = 6r /2 cos θ 2. Th third on is a lid drivn cavity problm. Clarly, th solution of tst problm is smooth and satisfis th homognous Dirichlt boundary condition. Th solution to tst problm 2 satisfis u + p =. Th Dirichlt boundary condition can b st by using th valu of u on th boundary. Tst problm 2 has a cornr singularity of ordr.5 at th origin (,). Th 2D lid drivn cavity problm dscribs th flow in a rctangular containr which is drivn by th uniform motion of th top lid [8]. Bcaus of th discontinuous vlocity boundary condition at two top cornrs, it is know that th xact solution (u; p) dos not vn blong to (H ) 2 L 2, if thr is any in a crtain sns. Indd, th wak formulation dos not hold for this problm. Howvr, th discrt problm is still wll-posd and provids a crtain approximation to th actual solution. In two dimnsional cas, th discontinuous boundary condition also rsults in cornr singularitis at th two top cornrs On uniform mshs. In this xprimnt, tst problms and 2 ar solvd on uniform mshs. Th msh is gnratd by dividing Ω into n n sub-rctangls, and thn dividing ach sub-rctangl into two triangls by conncting th diagonal lins with th ngativ slop. For tst problm, sinc th solution is smooth, w xpct thortically that η = O(h), = O(h), ǫ = O(h), and = O(h 2 ). For tst problm 2, only η and ar computd for ach msh. Du to th cornr singularity, w xpct thm to hav asymptotic ordr btwn and 2, rspctivly. Numrical rsults for ths two tst problms ar rportd in Tabls 6. and 6.2. Thy agr with th thortical prdiction. Tabl 6. Error norms for tst problm on n n uniform triangular mshs. n η h ǫ Asym. Ordr O(h k ), k = Adaptiv rfinmnts for tst problm 2. W prform adaptiv rfinmnts for tst problm 2, which has a cornr singularity. Th two rfinmnt stratgis as xplaind in Sction 5 ar mployd in this invstigation. Th rrors

14 4 JUNPING WANG, YANQIU WANG AND XIU YE Tabl 6.2 Error norms for tst problm 2 on n n uniform triangular mshs. n dofs GMRES itr. η Asym. Ordr O(h k ), k = from th Maximum Stratgy ar rportd in Tabl 6.3, and corrsponding mshs ar prsntd in Figur 6.. Th rrors from th Local Stratgy ar rportd in Tabl 6.4, and corrsponding mshs ar givn in Figur 6.2. W also draw th rsults from tabls in Figur 6.3, to provid a dirct imag of th rlation btwn η and th numbr of dgrs of frdom. Aftr liminating svral starting lvls, th computd asymptotic ordrs of η in trms of th numbr of dgrs of frdom ar O(N.597 ) and O(N.458 ), rspctivly, for th maximum stratgy and th local stratgy. By comparing Tabl 6.2, 6.3 and 6.4, it can b sn that th adaptiv rfinmnt is much mor fficint in rducing th rror for this problm. Th msh rfinmnts shown in Figur 6. and 6.2 indicat that th local rror indicator η K has locatd th singularity accuratly. It sms that th Local Stratgy givs similar rfinmnts to th Maximum Stratgy for this tst problm. Th valus of θ, whn lying in a rasonabl rang, controls th rfinmnt scal in both stratgis. Tabl 6.3 Tst problm 2, adaptiv rfinmnt using th Maximum stratgy. # Triangls dofs GMRES itr. η W also draw th graph of th vlocity and prssur on th finst msh from th Local Stratgy in Figur 6.4. Thy agr wll with th graph of th xact solution. Th graph of numrical solution using th Maximum Stratgy is similar, and thus omittd. It has bn known that an avraging post-procssing of th prssur usually improvs th approximation for th prssur. Hnc w would also lik to chck th

15 A POSTERIORI ERROR ESTIMATION 5 Fig. 6.. Tst problm 2, adaptiv rfinmnt using th Maximum stratgy. Initial msh and mshs aftr svral rfinmnts. Tabl 6.4 Tst problm 2, adaptiv rfinmnt using th Local stratgy, θ =.3. # Triangls dofs GMRES itr. η robustnss of our adaptiv msh rfinmnts undr such a prssur-avraging procss, for which th main ida can b dscribd as follows. Aftr a solution has bn computd on th currnt msh, for ach triangl K in th msh, idntify a nighborhood of K including triangls sharing on vrtx or on dg with K, including K itslf. Sum up th products of th computd prssur, which is picwis constant, with th ara of triangls ovr all triangls in th nighborhood of K. Thn divid th summation by th total ara of th involvd triangls. Assign this valu to th post-procssd prssur on K. This prssur-avraging procss is applid bfor on computs th rror indicators. Th rst of th adaptiv msh rfinmnt rmains unchangd. W rport th rsults for tst problm 2 with prssur-avraging, using both maximum and local stratgis, in Tabls 6.5, 6.6 and Figurs 6.5, 6.6, 6.7. W point out that aftr th prssur-avraging procss, th a postriori rror η is diffrnt from th on without post-procssing, whil th L 2 norm of th rror for th vlocity rmains unchangd. Our numrical rsults show that th rror indicator works wll with th prssur-avraging mthod Adaptiv rfinmnts for th drivn cavity problm. Sinc th xact solution for drivn cavity problm is not vn in (H ) 2 L 2, w anticipat that η dos not dcras whn th msh is rfind. Our numrical xprimnts show that th local rror indicator η K is abl to locat both cornr singularitis for this problm. Th mshs ar plottd in Figur 6.8, 6.9, 6., 6., and 6.2. Notic that whn using th Local Stratgy, changing th valu of θ clarly affcts th rfinmnt dramatically. Indd, small oscillations of local rror indicators hav bn obsrvd nar th cornr singularitis such that th Local Stratgy with θ =.3 oftn locats an oscillation instad of th singularity. W pointd out that such oscillations might b inhritd from th numrical discrtization schm du to

16 6 JUNPING WANG, YANQIU WANG AND XIU YE Fig Tst problm 2, adaptiv rfinmnt using th Local stratgy, θ =.3. Initial msh and mshs aftr svral rfinmnts. Fig Tst problm 2, adaptiv rfinmnt, rlation btwn η and th numbr of dgrs of frdom, using both maximum and local stratgy. η vs. dofs O(N /2 ) Maximum stratgy Local stratgy η 3 N: dgrs of frdom th svr singularity of th boundary conditions. By stting θ =.5, th mthod can focus mor on th ral singularity, as shown in Figur 6.. Nxt, w draw th vlocity and prssur profils for th drivn cavity problm computd by using diffrnt rfinmnt stratgis. Th solutions in Tabls 6.3, 6.4, 6.5 and 6.6 ar drawn on th finst rportd msh of ach adaptiv stratgy. W would lik radrs to draw conclusions from rading th rsults. Finally, w compar th prsnt solutions with th numrical solution as computd by using th popular Taylor-Hood finit lmnt mthod. Figur 6.7 givs th vlocity profils, which contain th plot of u on th vrtical cntrlin of th domain and u 2 on th horizontal cntrlin. Th solution of th Taylor-Hood lmnt is computd on a triangular msh, and is dnotd by stars in th figur. Th vlocity profil of th solutions from adaptiv msh rfinmnts using diffrnt stratgis ar drawn in curvs. From th plots, on can s that th solutions match vry wll with ach othr. This is of cours a comparison of th solution at smooth aras. But it is vidnt that th proposd local rfinmnt mthods do provid comptitiv numrical solutions for th Stoks quation. Appndix A. Equivalnc of th wak formulation to th on statd in [22]. For simplicity, w only considr th cas for th symmtric bilinar form a s (, ). In [22], a s (, ) is dfind as following: a s (w,v) = ( h w, h v) + ( αh [w t][v t] {ε(w) }[v t] {ε(v) }[w t] ) ds, E h

17 A POSTERIORI ERROR ESTIMATION 7 Fig Tst problm 2, adaptiv rfinmnt using th Local Stratgy, θ =.3. Vlocity and prssur. Vlocity u Vlocity u Prssur Tabl 6.5 Tst problm 2, adaptiv rfinmnt using th Maximum stratgy, with prssur-avraging. # Triangls dofs GMRES itr. η whr on intrnal dg, [v t] = v K t + v K2 t 2, and on boundary dgs it only taks th valu on on sid. Th dfinitions for V h, W h and b(, ) rmain th sam. Thrfor, w only nd to show that for all w, v V (h) and E h, (A.) (A.2) [[w]] : [[v]] = [w t][v t], { w} : [[v]] = {ε(w) }[v t]. For v V (h), dnot { v K v K2 on intrnal dg, δv = v on boundary dg, { v = 2 (v K + v K2 ) on intrnal dg, v on boundary dg. Sinc vctors in V (h) hav continuous normal componnts across, clarly δv n = on intrnal dgs. By th homognous boundary condition, w also know that both δv n and v n vanish on boundary dgs. Thrfor, on vry E h, w hav and [[w]] : [[v]] = δw δv = (δw t )(δv t ) = [w t][v t], {ε(w) }[v t] = ( wn ) δv = ( w) : (δv n ) = { w} : [[v]].

18 8 JUNPING WANG, YANQIU WANG AND XIU YE Fig Tst problm 2, adaptiv rfinmnt using th Maximum stratgy, with prssuravraging. Initial msh and mshs aftr svral rfinmnts. Tabl 6.6 Tst problm 2, adaptiv rfinmnt using th Local stratgy, θ =.3, with prssur-avraging. # Triangls dofs GMRES itr. η Combining th abov, th wak form givn in in this papr is quivalnt to th on from [22]. Hnc Thorm 3. follows dirctly from [22]. REFERENCES [] S. Agmon, Lcturs on lliptic boundary problms, Van Nostrand, Princton, Nw Jrsy, 965. [2] D.N. Arnold, An intrior pnalty finit lmnt mthod with discontinuous lmnts, SIAM J. Numr. Anal., 9 (982), pp [3] F. Brzzi, J. Douglas, and L. Marini, Two familis of mixd finit lmnts for scond ordr lliptic problms, Numr. Math., 47 (985), pp [4] F. Brzzi and M. Fortin, Mixd and Hybrid Finit Elmnts, Springr-Vrlag, Nw York, 99. [5] Z. Cai, C. Tong, P.S. Vassilvski and C. Wang, Mixd finit lmnt mthods for incomprssibl flow: Stationary Stoks quations, Numr. Mth. Part. Diff. Eq., 26 (29), pp [6] Z. Cai, C. Wang and S. Zhang, Mixd Finit Elmnt Mthods for Incomprssibl Flow: Stationary NavirStoks Equations, SIAM J. Numr. Anal., 48 (2), pp [7] C. Carstnsn and S. Funkn A postriori rror control in low-ordr finit lmnt discrtisations of incomprssibl stationary flow problms, Math. Comp., 7 (2), pp [8] E. Dari, R. Durán and C. Padra Error stimators for nonconforming finit lmnt approximations of th Stoks problm, Math. Comp., 64 (995), pp [9] W. Dörflr and M. Ainsworth Rliabl a postriori rror control for nonconforming finit lmnt approximation of Stoks flow, Math. Comp., 74 (25), pp [] L. Figuroa, G.N. Gatica, A. Marquz, Augmntd mixd finit lmnt mthods for th stationary Stoks quations, SIAM J. Sci. Comput., 3 (28), pp [] G.N. Gatica, A. Marquz, and M.A. Sanchz, Analysis of a vlocity-prssur-psudostrss formulation for th stationary Stoks quations, Computr Mthods in Applid Mchanics and Enginring, 99 (2), pp [2] J.S. Howll, Dual-mixd finit lmnt approximation of Stoks and nonlinar Stoks problms using trac fr vlocity gradints, J. Comp. Appl. Math., 23 (29), pp [3] J.S. Howll, Approximation of gnralizd Stoks problms using dual-mixd finit lmnts without nrichmnt, Int. J. Numr. Mth. Fluids, (2), DOI.2/fld [4] C. Padra, A postriori rror stimators for nonconforming approximation of som quasi- Nwtonian flows, SIAM J. Numr. Anal., 34 (997), pp

19 A POSTERIORI ERROR ESTIMATION 9 Fig Tst problm 2, adaptiv rfinmnt using th Local stratgy, θ =.3, with prssuravraging. Initial msh and mshs aftr svral rfinmnts. Fig Tst problm 2, adaptiv rfinmnt, rlation btwn η and th numbr of dgrs of frdom, with prssur-avraging, using both maximum and local stratgy. η vs. dofs, with prssur avraging O(N /2 ) Maximum stratgy Local stratgy η 3 N: dgrs of frdom [5] M. Paraschivoiu and A.T. Patra, A postriori bounds for linar functional outputs of Crouzix-Raviart finit lmnt discrtizations of th incomprssibl Stoks problm, Int. J. Numr. Mthods Fluids, 32 (2), pp [6] P. Raviart and J. Thomas, A mixd finit lmnt mthod for scond ordr lliptic problms, Mathmatical Aspcts of th Finit Elmnt Mthod, I. Galligani, E. Magns, ds., Lcturs Nots in Math., Vol. 66, Springr-Vrlag, Nw York, 977. [7] L.R. Scott and S. Zhang, Finit lmnt intrpolation of nonsmooth function satisfying boundary conditions, Math. Comp., 54 (99), pp [8] P.N. Shankar and M.D. Dshpand, Fluid Mchanics in th drivn cavity, Annu. Rv. Fluid Mch., 32 (2), pp [9] R. Vrfürth, A rviw of a postriori rror stimation and adaptiv msh-rfinmnt tchniqus, Tubnr Skriptn zur Numrik. B.G. Willy-Tubnr, Stuttgart, 996. [2] R. Vrfürth, A postriori rror stimators for th Stoks quations, Numr. Math., 55 (989), pp [2] R. Vrfürth, A postriori rror stimators for th Stoks quations II. non-conforming disctizations, Numr. Math., 6 (99), pp [22] J. Wang and X. Y, Nw finit lmnt mthods in computational fluid dynamics by H(div) lmnts, SIAM J. Numr. Anal., 45 (27), pp [23] J. Wang, X. Wang and X. Y, Finit lmnt mthods for th Navir-Stoks quations by H(div) lmnts, J. Comput. Math., 26 (28), pp [24] J. Wang, Y. Wang and X. Y, A robust numrical mthod for Stoks quations basd on divrgnc-fr H(div) finit lmnt mthods, SIAM J. Sci. Comp., 3 (29), pp

20 2 JUNPING WANG, YANQIU WANG AND XIU YE Fig Drivn cavity problm, adaptiv rfinmnt using th Maximum stratgy. Initial msh and mshs aftr svral rfinmnts. Fig Drivn cavity problm, adaptiv rfinmnt using th Local stratgy, θ =.3. Initial msh and mshs aftr svral rfinmnts. Fig. 6.. Drivn cavity problm, adaptiv rfinmnt using th Local stratgy, θ =.5. Initial msh and mshs aftr ach rfinmnt. Fig. 6.. Drivn cavity problm, adaptiv rfinmnt using th Maximum stratgy, with prssur-avraging. Initial msh and mshs aftr svral rfinmnts.

21 A POSTERIORI ERROR ESTIMATION 2 Fig Drivn cavity problm, adaptiv rfinmnt using th Local stratgy, θ =.5, with prssur-avraging. Initial msh and mshs aftr svral rfinmnts. Fig Drivn cavity problm, adaptiv rfinmnt using th Maximum stratgy. Vlocity and prssur. Vlocity u Vlocity u Prssur Fig Drivn cavity problm, adaptiv rfinmnt using th Local Stratgy, θ =.5. Vlocity and prssur. Vlocity u Vlocity u Prssur Fig Drivn cavity problm, adaptiv rfinmnt using th Maximum stratgy, with prssur-avraging. Vlocity and prssur. Vlocity u Vlocity u Prssur

22 22 JUNPING WANG, YANQIU WANG AND XIU YE Fig Drivn cavity problm, adaptiv rfinmnt using th Local Stratgy, θ =.5, with prssur-avraging. Vlocity and prssur. Vlocity u Vlocity u Prssur Fig Vlocity profils of th drivn cavity problm. Th curvs ar vlocity profils of prsnt solutions aftr 5 rfinmnts, using diffrnt stratgis. Th stars dnot th vlocity profil of th solution using th Taylor-Hood lmnt. Maximum stratgy. Maximum stratgy with prssur avrag. Local stratgy, θ= Local stratgy, θ=.5. Local stratgy, θ=.5, prssur avrag