UNIFIED A POSTERIORI ERROR ESTIMATOR FOR FINITE ELEMENT METHODS FOR THE STOKES EQUATIONS

UNIFIED A POSTERIORI ERROR ESTIMATOR FOR FINITE ELEMENT METHODS FOR THE STOKES EQUATIONS JUNPING WANG, YANQIU WANG, AND XIU YE Abstract. This papr is concrnd with rsidual typ a postriori rror stimators for finit lmnt mthods for th Stoks quations. In particular, th authors stablishd a unifid approach for driving and analyzing a postriori rror stimators for vlocity-prssur basd finit lmnt formulations for th Stoks quations. A gnral a postriori rror stimator was prsntd with a unifid mathmatical analysis for th gnral finit lmnt formulation that covrs conforming, nonconforming, and discontinuous Galrkin mthods as xampls. Th ky bhind th mathmatical analysis is th us of a lifting oprator from discontinuous finit lmnt spacs to continuous ons for which all th trms involving jumps at intrior dgs disappar. Ky words. A postriori rror stimat, finit lmnt mthods, Stoks quations AMS subjct classifications. Primary, 65N15, 65N30, 76D07; Scondary, 35B45, 35J50 1. Introduction. A postriori rror stimator rfrs to a computabl formula that offrs a masur for judging th rliability and fficincy of a particular numrical schm mployd for approximating th solution of partial diffrntial quations or alik. With a mathmatically justifid a postriori rror stimator, on would b abl to gnrat a msh that is tailord at rducing computational rrors at placs of grat nd. This procss is commonly known as adaptiv msh rfinmnt which has bcom a usful and important tool in today s scintific and nginring computing. Th goal of this papr is to offr a systmatic framwork for dvloping and analyzing a postriori rror stimators for finit lmnt mthods for modl Stoks quations. This papr is concrnd with rsidual typ a postriori rror stimators. In othr words, th computabl formula for judging th fficincy and rliability of numrical schms shall b givn by functions of rsiduals. Along this avnu, svral fin rsults hav bn dvlopd for finit lmnt mthods for th Stoks quations. For conforming finit lmnt mthods, som a postriori rror stimators hav bn drivd for mini-lmnts by Vrfurth [1] and Bank-Wlfrt [4]. Ainsworth-Odn [3] and Nobil [18] hav considrd mor gnral conforming finit lmnts in thir study. For nonconforming finit lmnts, a postriori rror stimation for th Crouzix- Raviart lmnt [8] has bn dvlopd by svral rsarchrs such as Vrfurth [], Dari-Durán-Padra [9] and Dorflr-Ainsworth [10]. Carstnsn, Gudi, and Jnsn [5] proposd and analyzd an a postriori rror stimator for discontinuous Galrkin mthods by using a strss-vlocity-prssur formulation for th Stoks quations. Kay and Silvstr [16] stablishd a postriori rror stimation for th stabilizd finit lmnt formulation. Th rcovry basd a postriori rror stimat for th Stoks quations is invstigatd in [1]. Division of Mathmatical Scincs, National Scinc Foundation, Arlington, VA 30 (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 74075 (yqwang@math.okstat.du). Dpartmnt of Mathmatics, Univrsity of Arkansas at Littl Rock, Littl Rock, AR 704 (xxy@ualr.du).this rsarch was supportd in part by National Scinc Foundation Grant DMS- 0813571 1

In both [9] and [10], th analysis for thir a postriori rror stimators was basd on a Hlmholtz dcomposition for dcomposing th Crouzix-Raviart lmnt into two parts: an xactly divrgnc-fr part and th scond as its orthogonal complmnt. Whil th Hlmholtz dcomposition offrs an applaudabl approach for analyzing th fficincy and rliability of a postriori rror stimators for th Stoks quations, th mthod has difficulty in bing xtndd to finit lmnt approximations arising from discontinuous Galrkin mthods. Th main difficulty coms from th fact that th approximat vlocity fild from th discontinuous finit lmnt mthods is not divrgnc-fr in th classical sns. Thrfor, othr analytical tchniqus hav bn dvlopd for discontinuous finit lmnts; but most of thm rquirs spcial and unncssary proprtis about th finit lmnt msh. For xampl, Houston, Schötzau and Wihlr [14] hav dvlopd an a postriori rror analysis for th discontinuous Q k Q k 1 lmnt on partitions consisting of paralllograms only. In this papr, w stablish a unifid approach for driving and analyzing a postriori rror stimators of rsidual typ for vlocity-prssur basd formulations of th Stoks quations. In particular, w shall dvlop a gnral finit lmnt formulation that covrs conforming, non-conforming, and discontinuous Galrkin mthods as xampls. Thn, a gnral a postriori rror stimator shall b prsntd with a unifid mathmatical analysis. Th ky bhind th analysis is th us of a lifting oprator from discontinuous finit lmnt spacs to continuous ons for which all th trms involving jumps at intrior dgs disappar. A similar lifting oprator was mployd by Karakashian and Pascal [15] for analyzing a postriori rror stimats for a discontinuous Galrkin approximation to scond ordr lliptic quations. Th papr is organizd as follows. In Sction, a modl Stoks problm and som notations ar introducd. In Sction 3, w shall first prsnt a gnral finit lmnt formulation for th Stoks quations, and thn illustrat how most xisting conforming, nonconforming, and discontinuous Galrkin mthods b rprsntd by th gnral framwork. In Sction 4, w stablish an analytical tool for analyzing th gnral a postriori rror stimator of rsidual typ. Finally in Sction 5, w prsnt som numrical rsults to confirm th thory dvlopd in prvious sctions.. Prliminaris and notations. Lt Ω b an opn boundd domain in R d,d =,3. Dnot by Ω th boundary of Ω. Th modl problm sks a vlocity function u and a prssur function p satisfying u + p = f in Ω, (.1) u = 0 in Ω, (.) u = 0 on Ω, (.3) whr,, and dnot th Laplacian, gradint, and divrgnc oprators, rspctivly, and f is th xtrnal volumtric forc acting on th fluid. For simplicity, th algorithm and its analysis will b prsntd for th modl Stoks problm (.1)-(.3) only in two-dimnsional spacs (i..; d = ) with polygonal domains. An xtnsion to th Stoks problm in thr dimnsions can b mad formally for gnral polyhdral domains. For any givn polygon D Ω, w us th standard dfinition of Sobolv spacs H s (D) with s 0 (.g., s [1, 6] for dtails). Th associatd innr product, norm, and sminorms in H s (D) ar dnotd by (, ) s,d, s,d, and r,d,0 r s, rspctivly. Whn s = 0, H 0 (D) coincids with th spac of squar intgrabl functions L (D). In this cas, th subscript s is supprssd from th notation of norm, smi-norm, and

innr products. Furthrmor, th subscript D is also supprssd whn D = Ω. Dnot by L 0(D) th subspac of L (D) consisting of functions with man valu zro. Th abov dfinition/notation can asily b xtndd to vctor-valud and matrixvalud functions. Th norm, smi-norms, and innr-product for such functions shall follow th sam naming convntion. In addition, all ths dfinitions can b transfrrd from a polygonal domain D to an dg, a domain with lowr dimnsion. Similar notation systm will b mployd. For xampl, s, and would dnot th norm in H s () and L () tc. Throughout th papr, w follow th convntion that a bold Latin lttr dnots a vctor. Lt u = [u i ] 1 i, v = [v i ] 1 i b two vctors, and σ = [σ ij ] 1 i,j, τ = [τ ij ] 1 i,j b two matrics, dfin It is not hard to s that ( v1 x v x v 1 y v y ) v =, v = v 1 x + v y, ( ) u1 v u v = 1 u 1 v, σ : τ = σ u v 1 u v ij τ ij, ( ) σ11 v v σ = 1 + σ 1 v, v σ u = σ 1 v 1 + σ v σ : (u v) = v σ u. i,j=1 σ ij u i v j. i,j=1 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 by h T th diamtr of triangl T T h, and h th maximum of all h T. W assum that T h is shap rgular in th sns that for ach T T h, th ratio btwn h T and th diamtr of th inscribd circl is boundd from abov. Th shap rgularity of T h nsurs a validity of th invrs inquality for finit lmnt functions. In addition, shap rgularity allows on to apply th routin scaling argumnts in finit lmnt analysis. Lt us introduc two finit dimnsional spacs V s and P t as follows: V s = {v V : v T [P s (T)], for all T T h }, P t = {q L 0(Ω) : q T P t (T), for all T T h }, whr V = [L (Ω)] or [H 1 0(Ω)], s and t ar non-ngativ intgrs, and P k (T) is th st of polynomials of dgr no mor than k on T. A finit lmnt mthod usually sks discrt vlocity and prssur approximations in som subspacs V h V s and Q h Q t. Crtain continuity condition may b imposd on V h and Q h, dpnding on th typ of finit lmnts. For a postriori rror stimats only, V h and Q h ar not rquird to satisfy th discrt inf-sup condition. Howvr, w do nd to mak th following assumption {v [H 1 0(Ω)] : v T [P 1 (T)], for all T T h } V h, (.4) in th analysis of th a postriori rror stimats. For a finit lmnt partition T h, with triangls only, (.4) is satisfid as long as k, ordr of th polynomials, is no lss 3

than 1. Thrfor, th assumption (.4) is rasonabl and xampls that satisfy it will b givn latr. Dnot by E h th st of all dgs in T h, and dnot by E 0 h := E h\ Ω th collction of all intrior dgs. Nxt, w dfin th avrag and jump on dgs for scalar-valud function q, vctor-valud function w, and matrix-valud function τ, rspctivly. For any intrior dg E 0 h, lt T 1, T b two triangls sharing and n 1 and n b th unit outward normal vctors on, associatd with T 1 and T, rspctivly. Dfin th avrag { } and jump [ ] on by {q} = 1 (q T 1 + q T ), [[q]] = q T1 n 1 + q T n, {w} = 1 (w T 1 + w T ), [[w]] = w T1 n 1 + w T n, {τ} = 1 (τ T 1 + τ T ), [[τ]] = n 1 τ T1 + n τ T. W also dfin a matrix-valud jump [ ] for w on by [w] = w T1 n 1 + w T n. If is a boundary dg, th abov dfinitions nd to b adjustd accordingly so that both th avrag and th jump ar qual to th on-sidd valus on. That is, {q} = q, {w} = w, {τ} = τ, [[q]] = q n, [[w]] = w n, [[τ]] = n τ, [w] = w n, whr n is th unit outward normal of Ω. Lt q, v and τ b scalar-, vctor-, and matrix-valud functions that ar rgular nough to mak all involving trms wll-dfind, thn th following idntitis ar standard []: q v nds = [[q]] {v}ds + {q}[[v]]ds, (.5) T T T h Eh 0 E h n τ v ds = [[τ]] {v}ds + {τ} : [v]ds. (.6) T T T h Eh 0 E h 3. Finit lmnt formulation. W first driv a gnral wak formulation for th Stoks quations that covrs a st of xisting numrical mthods including discontinuous Galrkin, conforming finit lmnts, and som nonconforming finit lmnt schms. To this nd, lt v and w b vctor-valud functions, and w introduc th following notation (v, w) Th := v w dx, K T h K (v, w) Eh := E h v w ds. Tsting th momntum quation (.1) by v V h yilds ( u, v) + ( p, v) = (f, v). (3.1) 4

Nxt, using intgration by parts, Equation (.6) and th fact that [[ u]] = 0 on all Eh 0, w hav ( u, v) = ( u, v) Th n u v ds K T h K = ( u, v) Th ({ u}, [v]) Eh. Analogously, sinc [[p]] = 0 on all Eh 0, thn by using Equation (.5) w arriv at ( p, v) = ( v, p) Th + v np ds Thus, (3.1) can b rwrittn as K T h K = ( v, p) Th + ({p}, [[v]]) Eh. ( u, v) Th ({ u}, [v]) Eh ( v, p) Th + ({p}, [[v]]) Eh = (f, v). (3.) Th mass consrvation quation (.) can b tstd by using any q Q h, yilding ( u, q) Th = 0. (3.3) In ordr to driv a gnral wak formulation, w introduc th following bilinar forms: a(u,v) := ( u, v) Th ( γ ({ u}, [v]) Eh + δ({ v}, [u]) Eh α(h 1 [u], [v]) Eh ), (3.4) b(v,p) := ( v, p) Th + γ([[v]], {p}) Eh, (3.5) whr γ = 0 or 1, δ = 1, 1 or 0, and α 0 ar paramtrs with various valus. Th gnral wak formulation for th Stoks quations is thn givn as follows: Algorithm G: Find (u h ;p h ) V h Q h such that for all (v;q) V h Q h. a(u h,v) + b(v,p h ) = (f,v), (3.6) b(u h, q) = 0 (3.7) Whn γ = 1, it is not hard to s that systm (3.6)-(3.7) is consistnt with th systm (3.)-(3.3), as th xact solution of th Stoks problm satisfis [[u]] = 0 and [u] = 0 on all E h. Whn γ = 0, ths two systms ar in gnral not consistnt. It should b pointd out that by choosing diffrnt valus of γ and th spacs V h and Q h, Algorithm G rprsnts diffrnt typs of finit lmnt formulations. For illustrativ purpos, w shall list thr most important xampls; all of which satisfy th assumption (.4) on V h. Exampl 1: Discontinuous Galrkin. St γ = 1 and V h {v [L (Ω)] : v T [P s (T)], for all T T h }, Q h {q L 0(Ω) : q T P t (T), for all T T h }. 5

Whn δ = 1, th corrsponding formulation is symmtric. On xampl of such a discontinuous Galrkin formulation was discussd in [5], whr V h, Q h ar chosn to b H(div) conforming lmnts and th wll-known inf-sup condition was inhritd to b valid. Exampl : Conforming finit lmnt mthod. St γ = 1 and V h {v [H 1 0(Ω)] : v T [P s (T)], for all T T h }, Q h {q L 0(Ω) : q T P t (T), for all T T h }. By th continuity rquirmnt on V h, w asily s that th two bilinar forms ar simplifid to a(v,w) = ( v, w), b(v,q) = ( v, q). Th conforming finit lmnt schm is on of th most wll-studid formulations for th Stoks problms. Latr in th sction for numrical xprimnts, w shall considr on such lmnt, namly, th Taylor-Hood lmnt [13]. Exampl 3: Nonconforming finit lmnt mthod. St γ = 0 and chos a finit lmnt spac V h such that V h [H 1 0(Ω)]. This givs som nonconforming finit lmnt mthods, dpnding on th slction of V h and Q h. In this papr, w only considr nonconforming finit lmnts which satisfis th following condition: Assumption (H) ({ u pi}, [v]) Eh = 0 for all u, v V h, p Q h, whr I is th idntity matrix. It is not hard to s that th Crouzix- Raviart typ nonconforming lmnts [8, 11, 7] satisfy this assumption, bcaus by its dfinition, { u pi} is a polynomial on dgr lowr than [v] and [v] vanishs at all Gaussian points on ach dg E h. Th goal of this manuscript is to provid a tool that can b mployd to analyz a postriori rror stimators for th unifid formulation (3.6)-(3.7). To this nd, w first driv an orthogonality proprty of th rror btwn th xact solution and its finit lmnt approximation: = u u h, ǫ = p p h. By subtracting (3.6) from (3.) and using ({p h }, [[v]]) Eh = ({p h I}, [v]) Eh, w hav ( ) (, v) Th ({ }, [v]) Eh + γ δ({ v}, [u h ]) Eh α(h 1 [u h ], [v]) Eh ( v,ǫ) Th + ({ǫ}, [[v]]) Eh (1 γ)({ u h p h I}, [v]) Eh = 0. Not that for all of th abov-mntiond discontinuous Galrkin, conforming and nonconforming cass, w must hav (1 γ)({ u h p h I}, [v]) Eh = 0. (3.8) In fact, (3.8) follows from γ = 1 for th discontinuous Galrkin and th conforming cass, and from Assumption (H) for th nonconforming cas. Combining this and th fact that [u] = 0 on all dgs, th abov quation can b rwrittn as 6

( ) (, v) Th ({ }, [v]) Eh γ δ({ v}, []) Eh α(h 1 [], [v]) Eh ( v,ǫ) Th + ({ǫ}, [[v]]) Eh = 0. (3.9) Morovr, if v V h [H 1 0(Ω)], thn (3.9) bcoms (, v) Th + γδ({ v}, [u h ]) Eh ( v,ǫ) Th = 0. (3.10) Nxt, w introduc a norm for th spac [H 1 0(Ω)] + V h as follows v = v 1,T + h 1 [v]. (3.11) E h For a wll-craftd numrical schm, w usually xpct to hav an a priori rror stimat lik th following + ǫ Ch k+1 ( u k+1 + p k ), (3.1) whr C is a positiv constant and k is dtrmind by th ordr of th corrsponding finit lmnts and th rgularity of th xact solution (u; p). For som lmnts, th a priori rror stimat (3.1) may hav variations on th right-hand sid, but this dos not affct our analysis to b prsntd. A priori rror stimation lik (3.1) is wll known for conforming finit lmnt mthods, for which v = v 1. Such an stimat is also known for som discontinuous Galrkin formulations;.g., s [4, 5] for an approach with H(div)-lmnts. For nonconforming finit lmnts, such as th Crouzix-Raviart typ lmnts, similar rror stimats hold tru du to th fact that th jump trm, [u h ], vanishs at all Gaussian points on th dg. In fact, for any intrior dg shard by two lmnts T 1 and T, lt ē b th avrag valu of on dg. Sinc [] = [u h ] vanishs at all Gaussian points on th dg, thn ē has th sam valu whn th trac of was takn from ithr T 1 or T. Hnc, h 1 [] = h 1 [ ē] h 1 ē T i T 1,T C 1,T i. T 1,T Consquntly, w hav [] C h 1 1,T. E h Thrfor, is in th sam ordr as 1,T for th Crouzix-Raviart typ lmnts. Th analysis hr shows that th norm as dfind in (3.11) is a rasonabl on to us in th a priori and latr th a postriori rror stimats for th unifid formulation (3.6)-(3.7). 7

4. A postriori rror stimation. Th goal of this sction is to driv an a postriori rror stimation for th Algorithm G by using a unifid framwork. For simplicity of notation, w shall 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. To this nd, dfin { [[ uh p J 1 ( u h p h I) = h I]] on Eh 0, 0 on boundary dgs, and { [uh ] on E J (u h ) = h 0, u h n on boundary dgs. Our rsidual-basd global rror stimator is givn by η = whr η T, (4.1) ηt =h T f h + u h p h T + u h T + 1 ( h J 1 ( u h p h I) + h 1 J (u h ) ) ds, E h with h bing th lngth of dg, and f h th L projction of th load function f onto V h. It is also convnint to introduc an oscillation quantity for th load function f on T h as follows osc(f) = ( h T f f h T ) 1/. Our ultimat goal is to stablish th following rsult. Thorm 4.1. Lt (u;p) b th solution of (.1)-(.), and (u h ;p h ) b its finit lmnt approximation arising from (3.6)-(3.7). Thn, on has ( ) 1/ + p p h η + osc(f) (4.) (u u h ) T and η ( (u u h ) T ) 1/ + p p h + osc(f). (4.3) For convninc, th rlation (4.) is rfrrd to as a rliability stimat and (4.3) as an fficincy stimat. 4.1. A lifting oprator and som tchnical stimats. Dfin H 1 (T h ) = H 1 (T) and S k = P k (T). For any triangl T T h, dnot by T (T) th st of all triangls in T h having a nonmpty intrsct with T, including T itslf. Dnot by E(T) th st of all dgs in E h having a nonmpty intrsction with T, 8

including all thr dgs of T. Similarly, for any point x in Ω, dnot by E(x) th st of all dgs that pass through x. Not that E(x) is nonmpty only whn x lis on E h. Lt b an dg of triangl T. For any v S k, lt us dfin a lifting oprator L k : v L k (v) S k H 1 0(Ω) (4.4) that lifts a discontinuous picwis polynomial to a continuous picwis polynomial with vanishing boundary trac as follows. Lt G k (T) b th st of all Lagrangian nodal points for P k (T). At all intrnal Lagrangian nodal points x j G k (T), w st L k (v)(x j ) = v(x j ). At boundary Lagrangian points x j G k (T) T, w lt L k (v)(x j ) b ithr a trac of v from any sid or a prscribd wightd avrag of all possibl tracs. At global Lagrangian points x j T Ω, w st L k (v)(x j ) = 0. Not that for any function g H 1 (T) th following stimat holds: g h 1 T g T + h T g T. (4.5) Lmma 4.. For any v S k, k 1, th lifting oprator L k as dfind in (4.4) satisfis th following stimat: v L k (v) T + h T (v L k (v)) T E(T) h [[v]] T T h. (4.6) Proof. Th proof follows from a routin scaling argumnt and th fact that all norms on finit dimnsional spacs ar quivalnt. To this nd, w obsrv that v L k (v) T + h T (v L k (v)) T h T x j G k (T) (v L k (v))(x j ), whr G k (T) is th st of all Lagrangian nodal points for P k (T). For simplicity, lt L k (v) b dfind by taking a random on-sidd trac on th boundary Lagrangian points. It follows from th dfinition of L k that v L k (v) vanishs at all intrnal Lagrangian points in T. At Lagrangian points on T, thr ar two possibilitis: (1) x j is in th intrior of an dg ; () x j is a vrtx of T. In th first cas, w s that (v L k (v))(x j ) is ithr 0 or [[v]] (x j ), whr [[v]] dnots th jump on. In th scond cas, w can us th triangl inquality to travrs through all dgs E(x j ) and to obtain (v L k (v))(x j ) [[v]] (x j ). E(x j) Th abov analysis, togthr with th shap rgularity of T h, implis that x j G k (T) (v L k (v))(x j ) E(T) x j G k () [[v]] (x j ), whr G k () was usd to dnot th corrsponding Lagrangian points on dg. Thn, using th routin scaling argumnt on dgs, inquality (4.6) follows immdiatly. Th following is anothr rsult that turns out to b vry usful in th forthcoming analysis. 9

Lmma 4.3. For any v H 1 (T h ), thr xists a v I S k H0(Ω), 1 k 1, satisfying v v I T + h T (v v I ) T h T v T + h [[v]] T T h. T T (T) E(T) (4.7) Proof. First of all, thr xists a picwis constant v 0 S k (.g., th cll avrag of v) such that v v 0 T + h T (v v 0 ) T h T v T T T h. (4.8) Furthrmor, by th approximation proprty, inquality (4.5) and th shap rgularity of T h, w hav for ach Eh 0 h [[v 0 ]] = h v 0 T1 n 1 + v 0 T n ds h (v 0 v) T1 n 1 ds + h (v 0 v) T n ds + h [[v]] (4.9) h T 1 v T 1 + h T v T + h [[v]], whr T 1 and T ar th two triangls sharing as a common dg. On boundary dgs, similar rsult can obviously b obtaind without any difficulty. Sinc v 0 S k with k = 1, according to Lmma 4., on may lift v 0 to a continuous picwis linar function L 1 (v 0 ) S 1 H 1 0(Ω) which satisfis (4.6). By taking v I = L 1 (v 0 ), w obtain from th usual triangl inquality, (4.8), and (4.9) that v v I T + h T (v v I ) T v v 0 T + h T (v v 0 ) T which complts th proof. + v 0 L 1 (v 0 ) T + h T (v 0 L 1 (v 0 )) T h T v T + h [[v 0 ]] T T (T) E(T) h T v T + E(T) h [[v]], 4.. Rliability stimat. W first stablish an stimat for th prssur rror in trms of th a priori rror stimator and th vlocity rror. Th rsult can b statd as follows. Lmma 4.4. Lt (u;p) b th solution of (.1)-(.) and (u h ;p h ) b its finit lmnt approximation arising from (3.6)-(3.7). Thn, w hav p p h η + osc(f) + ( (u u h ) T ) 1/. (4.10) Proof. Lt v [H 1 0(Ω)] and v I V h [H 1 0(Ω)] b an intrpolation of v such that both componnts satisfy (4.7). Obsrv that such an intrpolation v I is possibl if V h satisfis th assumption (.4). Not that [[v]] = 0 on vry dg sinc v [H 1 0(Ω)]. 10

Lt = u u h and ǫ = p p h b th rror for vlocity and prssur approximations, rspctivly. Using intgration by parts, (.6), (3.10), (4.5), (4.7), and th fact that both v and v I ar continuous across ach intrior, w arriv at ( v, ǫ) = ( (v v I ), ǫ) + ( v I, ǫ) = ( (v v I ), ǫ) + (, v I ) Th + γδ({ v I }, []) Eh = ( (v v I ), ǫi ) Th + (, v) Th + γδ({ v I }, []) Eh = (f + u h p h, v v I ) Th + ({v v I }, [[ u h p h I]]) E 0 h γδ({ v I }, [u h ]) Eh + (, v) Th ( v 1 ( h T f + u h p h T) 1/ + ( h [[ u h p h I]] ) 1/ +( E h h 1 [u h ] ) 1/ + ( which, togthr with th following inf-sup condition p p h yilds th rquird stimat (4.10). E 0 h T) 1/ ), ( v, p p h ) sup (4.11) v [H0 1 v (Ω)] 1 Th nxt rsult is concrnd with an stimat for th vlocity approximation, which can b statd as follow. Thorm 4.5. Lt (u;p) and (u h ;p h ) b th solutions of (.1)-(.) and (3.6)- (3.7). Thn on has th following global rliability stimat: ( ) 1/ η + osc(f). (4.1) (u u h ) T Substituting (4.1) into (4.10) yilds th following stimat for th prssur approximation: Thus, it follows from th dfinition of that p p h η + osc(f). (4.13) + ǫ η + osc(f). Proof. Lt I V h [H 1 0(Ω)] b an intrpolation of satisfying (4.7). Again, such a choic of I is possibl bcaus V h was assumd to satisfy (.4). Obsrv that [[]] = [[u h ]]. Thus, it follows from (4.7) and th msh rgularity that I T +h T ( I ) T h T T T (T) 11 T + E(T) J (u h ). (4.14) h 1

Using (3.10), intgration by parts, (.6), and th fact that I, u, u, p ar continuous across all intrior dgs, w obtain (, ) Th = (, ( I )) Th + (, I ) Th = (, ( I )) Th + ( I, ǫ) γδ({ I }, []) Eh = ( ( I ), ǫi) Th + (, ǫ) Th γδ({ I }, []) Eh = (f + u h p h, I ) Th + (, ǫ) Th γδ({ I }, []) Eh ({ I }, [[ u h p h I]]) E 0 h ({ ǫi}, [u h ]) Eh = (f + u h p h, I ) Th + (, ǫ) Th γδ({ I }, []) Eh ({ I }, [[ u h p h I]]) E 0 h ({ ǫi}, [u h χ]) Eh = I 1 + I + I 3 + I 4 + I 5 ; (4.15) whr χ = L k (u h ) V h [H0(Ω)] 1 is a continuous intrpolation of u h such that (4.6) is satisfid, and I j rprsnts th corrsponding trm from th prvious quation. Th first trm I 1 can b stimatd by using th inquality (4.14) as follows ( ) 1 ( I 1 h T f + u h p h T h T I T ( ) 1 ( ) 1 h T f + u h p h T T + h 1 J (u h ) E h ( ) 1/ (η + osc(f)) η + T. (4.16) Th trm I can b handld by using (4.10) and th fact that = u h as follows ( ) 1/ I u h T p p h ( ) 1/ η η + osc(f) + T. (4.17) Using [] = [u h ] and th stimat (4.14), th trm I 3 can b boundd as follows ( ) 1 ( ) 1 I 3 γδ h I h 1 [u h ] E h E h ( ) 1 ( ) 1 I T h 1 [u h ] E h ( ) 1 ( ) 1 T + h 1 [u h ] h 1 [u h ] E h E h ( ) 1 T + η η. (4.18) 1 ) 1

Th sam argumnt can b applid to provid an stimat for th trm I 4 : ( ) 1 I 4 T + η η. (4.19) As to I 5, w first us (3.9) to obtain I 5 =({ ǫi}, [u h χ]) Eh =({ }, [u h χ]) Eh ({ǫ}, [[u h χ]]) Eh =(, (u h χ)) Th γδ({ (u h χ)}, []) Eh + γα(h 1 [u h χ], []) Eh ( (u h χ), ǫ) Th =(, (u h χ)) Th γδ({ (u h χ)}, []) Eh γα(h 1 [u h ], [u h ]) Eh ( (u h χ), ǫ) Th =J 1 + J + J 3 + J 4. Rcall that χ = L k (u h ) is a lift of u h so that th stimat (4.6) holds tru. It follows from (4.6) that J 1 η Nxt, w us [] = [u h ] to obtain ( T ) 1 J γδ E h (u h L k (u h )) [u h ], which, with th hlp of (4.5) and (4.6), can b boundd by J η. It is obvious that J 3 η, and it follows from (4.10) and (4.6) that ) J 4 η (η + osc(f) + (, ) 1. Th Collcting all th stimats for J s yilds ) I 5 η (η + osc(f) + (, ) 1. (4.0) Th Now w substitut all th stimats for I s into (4.15) to obtain (, ) Th (η + osc(f)) η + osc(f) + ( 3 (η + osc(f)) + 1 (, ) T h. T ) 1/ This complts th proof of (4.1). 13

4.3. Efficincy. W first dfin two bubbl functions, which ar widly usd in a postriori rror stimations [3]. For ach triangl T T h, dnot by φ T th following bubbl function { 7λ 1 λ λ 3 in T, φ T = 0 in Ω\T, whr λ i, i = 1,,3 ar barycntric coordinats on T. It is clar that φ T H 1 0(Ω) and satisfis th following proprtis [3]: For any polynomial q with dgr at most m, thr xist positiv constants c m and C m, dpnding only on m, such that c m q T q φ T dx q T, (4.1) T (qφ T ) T C m h 1 T q T. (4.) For ach Eh 0, w can analogously dfin an dg bubbl function φ. Lt T 1 and T b two triangls sharing th dg. To this nd, dnot by ω = T 1 T th union of th lmnts T 1 and T. Assum that in T i, i = 1,, th barycntric coordinats associatd with th two nds of ar λ Ti 1 bubbl function can b dfind as follows 4λ T1 1 λt1 in T 1, φ = 4λ T 1 λt in T, 0 in Ω\ω. and λti, rspctivly. Th dg Thn φ H0(Ω) 1 and satisfis th following proprtis [3]: 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 d m q q φ ds q, (4.3) Thn w hav th following fficincy bound. (qφ ) ω D m h 1/ q, (4.4) qφ ω E m h 1/ q. (4.5) Thorm 4.6. Lt (u;p) and (u h ;p h ) b th solutions of (.1)-(.) and (3.6)- (3.7), rspctivly. Thn for all T T h and Eh 0, w hav h T f h + u h p h T T + ǫ T + h T f f h T, (4.6) ( ) 1/ h 1/ J 1 ( u h p h I) h f f h ω + T + ǫ, (4.7) ω T ω u h T T. (4.8) By summing th abov stimats ovr all lmnt T T h and dgs E h, w obtain th following fficincy stimat: η + ǫ + osc(f). 14

Proof. Lt w T = (f h + u h p h )φ T. Sinc f = u + p and w T vanishs on T, w clarly hav (f f h, w T ) T + (f h + u h p h, w T ) T = (, w T ) T ( w T, ǫ ) T. Thn, by inqualitis (4.1)-(4.), f h + u h p h T (f h + u h p h, w T ) T = (, w T ) T ( w T, ǫ ) T (f f h, w T ) T ) ( T + ǫ T w T T + f f h T w T T ( h 1 T T + h 1 T ǫ T + f f h T ) f h + u h p h T. This complts th proof of (4.6). Similarly, for any Eh 0, lt w = ([[ u h p h I]])φ. Using intgration by parts and th fact that w = 0 on ω, w hav ( u h, w ) ω = ( u h,w ) T + [[ u h ]] w ds, (4.9) T ω and ( w, p h ) ω = ( p h,w ) T + [[p h I]] w ds. (4.30) T ω Tsting (.1) by using w ovr ω and thn using intgration by parts giv (f, w ) ω = ( u, w ) ω ( w, p) ω. (4.31) Using th proprtis of φ and quations (4.9)-(4.31), w hav [[ u h p h I]] [[u h p h I]] w ds = ) ((f f h, w ) T + (f h + u h p h, w ) T (, w ) T + (ǫ, w ) T T ω ( [[ u h p h I]] h 1/ f f h ω + h 1/ f h + u h p h ω + h 1/ ( ) T) 1/ + h 1/ ǫ ω T ω Combining th abov with (4.6) givs (4.7). Finally, th stimat (4.8) holds tru as u = 0 and clarly u h T = u h u T = T T. This complts th proof of th thorm. 15

5. Numrical rsults. Som numrical rsults for such a postriori rror stimators hav bn rportd in [10] for th Crouzix-Raviart lmnt, and in [4] for an H(div) basd discontinuous Galrkin formulation. In this sction, w would lik to prsnt som computational rsults for th conforming finit lmnt mthod in ordr to vrify th thory stablishd in prvious Sctions. To this nd, w considr th Taylor-Hood lmnt [13] whn applid to th Stoks problm. W us th Bi- Conjugat Gradint (BICG) itrativ solvr for solving th rsulting linar algbraic quations, with a rlativ rsidual = 10 8 bing st as a stopping critria. 5.1. Tst problms. Thr problms ar considrd in th numrical tst; all ar dfind on th domain of unit squar Ω = (0,1) (0,1). Two of thm hav xact solutions givn by: Tst Problm 1: A first Stoks problm with xact solution ( ) x u = y(x 1) (y 1)(y 1) xy (x 1)(x 1)(y 1), p = sin π(y x), Tst Problm : A scond Stoks problm with xact solution in th polar coordinat systm ( 3 ( r cos θ u = cos ) ) 3θ 3 ( r 3sin θ sin ), 3θ p = 6r 1/ cos θ. Obsrv that th solution has a cornr singularity of ordr 0.5 at th origin (0,0). Tst Problm 3: D lid drivn cavity. This is a Stoks problm which dscribs th flow of fluid in a rctangular containr drivn by a uniform motion of th top lid [0]. Not that th boundary condition faturs a discontinuity at two top cornrs. Th xact solution (u; p), if it xists, should not b sufficintly smooth so that (u; p) [H 1 ] L. Howvr, th discrt problm is still wll-posd and should provid an approximation to th actual solution. It must b pointd out that th finit lmnt formulation was only prsntd for th homognous Dirichlt boundary condition in th prvious Sctions. But th formulation can b asily xtndd to problms with non-homognous Dirichlt boundary conditions such as u = g on Ω. In this cas, th trm J (u h ) in th a postriori rror stimator nds to b modifid as follows { [uh ] on intrior dg E J (u h ) = h 0, (u h g) n on boundary dgs. Also, in th computational implmntation, w had rplacd h T f h + u h p h T by T f h + u h p h T, whr T stands for th ara of triangl T. 5.. Rsults with uniform mshs. In this xprimnt, w solv tst problms 1 and on uniform mshs and comput th asymptotic ordr of th rror stimator η. Th coarsst msh is gnratd by dividing Ω into 4 4 sub-rctangls, 16

and thn dividing ach sub-rctangl into four triangls by conncting its two diagonal lins. W thn us th routin uniform rfinmnt procdur, which divids ach triangl into four sub-triangls by conncting th cntr of its thr dgs, to gnrat svral lvls of fin mshs. For tst problm 1, w know thortically that η = O(h ). For tst problm, th ordr of η is xpctd to b O(h 0.5 ). Numrical rsults for ths two tst problms ar rportd in tabls 5.1 and 5.. For tst problm, th rror of prssur is not calculatd sinc th prssur gos to infinity at th point (0,0). Tabl 5.1 Error profils for tst problm 1 on uniform triangular mshs. h dofs η (u u h ) u u h p p h 331 1.70-0 4.15-03 1.10-04 3.08-03 3 135 4.94-03 1.07-03 1.38-05 7.88-04 4 4771 1.30-03.71-04 1.70-06 1.96-04 5 18755 3.33-04 6.79-05.13-07 4.89-05 Asym. Ordr O(h k ), k = -1.943 1.8956 1.9796 3.0064 1.9931 Tabl 5. Error profils for tst problm on uniform triangular mshs. h ndofs η (u u h ) u u h 331.39+00 4.76-01 1.54-0 3 135 1.63+00 3.30-01 5.83-03 4 4771 1.1+00.33-01.17-03 5 18755 7.97-01 1.64-01 8.09-04 Asym. Ordr O(h k ), k = -1.943 0.595 0.5094 1.4183 5.3. Rsults with adaptiv rfinmnt. W prform adaptiv rfinmnts for tst problms and 3, which hav cornr singularitis. Two diffrnt rfinmnt stratgis ar mployd in this study. Th first on is basd on a comparison of ach rror η T with th maximum valu of all th rror stimators. Th stratgy can b dscribd as follows: Local Rfinmnt by Maximum Stratgy : 1. Givn a currnt triangular msh, rror indicators η T on ach triangl, and a thrshold θ (0,1) (.g., θ = 0.5). On computs th maximum rror η max = max η T.. For ach triangl T, if η T θ η max, mark this triangl for rfinmnt. 3. Th actual rfinmnt is don by th nwst nod bisction mthod [17, 19]. It has bn provd that this mthod will not caus msh dgnration. Th only rquirmnt is that th nwst nods for th coarsst msh must b assignd carfully such that vry triangl is compatibly divisibl. This can b asily chckd. Th scond rfinmnt stratgy is basd on a comparison of η T with thos for its nighbors. To xplain th main ida, lt ρ > 0 b a prscribd distanc paramtr 17

Tabl 5.3 Error profils for tst problm using adaptiv msh rfinmnts. Stratgy Rfinmnt tims dofs η (u u h ) u u h 0 331.3955 0.4763 0.0154 Maximum 8 43 0.9453 0.1968 0.009 16 764 0.401 0.0844 0.001 0 331.3955 0.4763 0.0154 Local 8 545 0.7596 0.1519 0.001 16 1117 0.505 0.0618 0.0006 and st T ρ,t = { T : 0 < T T ρ}, whr T T stands for th distanc of th cntrs of T and T. With a givn thrshold θ > 1, w mark a triangl T for rfinmnt if η T θ η N(T), whr η N(T) is th avrag of th local rror indicator on all th nighboring triangls T T ρ,t. Th following is such a rfinmnt stratgy that was numrically invstigatd in this study. Local Rfinmnt by Local Stratgy : 1. Givn a currnt triangular msh, rror stimators η T on ach triangl, and a thrshold θ > 1.0 (.g., θ = 1.5). On computs an rror indicator η N(T) as th avrag of th local rror indicator on nighboring triangls that shar a vrtx or an dg with T, not including T itslf.. For ach triangl T, if η T θ η N(T), mark this triangl for rfinmnt. 3. Th actual rfinmnt is again don by th nwst nod bisction mthod [17, 19]. Th rsidual stimator η and rrors btwn th tru solution and its finit lmnt approximations for tst problm ar rportd in Tabl 5.3. By comparing tabls 5. and 5.3, on clarly ss th powr of adaptiv rfinmnts in numrical mthods for PDEs. For xampl, th rror in L and H 1 norms ar givn by 6.0 10 4 and 6.18 10 with only 1117 dgr of frdoms whn th local adaptiv rfinmnt stratgy was usd, whil th uniform partition rquirs mor than 18755 dgr of frdoms in ordr to rach a comparabl accuracy. Th rfind mshs aftr 0, 8, and 16 rfinmnts ar illustratd in Figurs 5.1 and 5., which show that our rsidual stimator rally capturs th corrsponding cornr singularity corrctly, undr both rfinmnt stratgis. Furthrmor, in Figurs 5.3 and 5.4, w xamin th rlation of η, (u u h ), and u u h with th dgr of frdoms N during th procss of th adaptiv rfinmnt. Th plots start from th coarsst msh and nds aftr 16 rfinmnts. For th drivn cavity problm, sinc th xact solution may not b sufficintly as smooth as in [H 1 ] L, th rsidual rror η is not xpctd to dcras whn th msh is rfind. Our numrical xprimnts show that th rror indicator η T is abl to locat both cornr singularitis for this problm. Th mshs aftr 0, 8, and 16 rfinmnts ar plottd in Figurs 5.5 and 5.6. Radrs ar invitd to mak thir own conclusions for th numrical rsults illustratd in this Sction. 18

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Fig. 5.1. Tst problm, maximum stratgy adaptiv rfinmnt. Mshs aftr 0, 8, and 16 rfinmnts. Fig. 5.. Tst problm, local stratgy adaptiv rfinmnt. Mshs aftr 0, 8, and 16 rfinmnts. Fig. 5.3. Tst problm, maximum stratgy adaptiv rfinmnt. Plot of η, (u u h ) and u u h, vrsus th dgrs of frdom N. 10 0 10 1 10 O(1/N) η H1 smi norm L norm 10.6 10.7 10.8 N: dgrs of frdom 0

Fig. 5.4. Tst problm, local stratgy adaptiv rfinmnt. Plot of η, (u u h ) and u u h, vrsus th dgrs of frdom N. 10 0 10 1 10 O(1/N) η H1 smi norm L norm 10 3 N: dgrs of frdom 10 3 Fig. 5.5. Tst problm 3, maximum stratgy adaptiv rfinmnt. Mshs aftr 0, 8, and 16 rfinmnts. Fig. 5.6. Tst problm 3, local stratgy adaptiv rfinmnt. Mshs aftr 0, 8, and 16 rfinmnts. 1