c 2007 Society for Industrial and Applied Mathematics

Transcription

1 SIAM J. NUMER. ANAL. Vol. 45, No. 3, pp c 2007 Socity for Industrial and Applid Mathmatics NEW FINITE ELEMENT METHODS IN COMPUTATIONAL FLUID DYNAMICS BY H (DIV) ELEMENTS JUNPING WANG AND XIU YE Abstract. In this papr, th authors prsnt two formulations for th Stoks problm which mak us of th xisting H(div) lmnts of th Raviart Thomas typ originally dvlopd for th scond-ordr lliptic problms. In addition, two nw H(div) lmnts ar constructd and analyzd particularly for th nw formulations. Optimal-ordr rror stimats ar stablishd for th corrsponding finit lmnt solutions in various Sobolv norms. Th finit lmnt solutions fatur a full satisfaction of th continuity quation whn xisting Raviart Thomas-typ lmnts ar mployd in th numrical schm. y words. finit lmnt mthods, Stoks problm AMS subjct classifications. Primary, 65N15, 65N30, 76D07; Scondary, 35B45, 35J50 DOI / Introduction. This papr is concrnd with numrical solutions of incomprssibl fluid flow problms by finit lmnt mthods. Our objctiv is to introduc a finit lmnt schm with attntion paid to th discrtization of th mass continuity quation. For illustrativ purposs, w show how th mthod works for th Stoks problm, which sks a pair of unknown functions (u; p) satisfying (1.1) (1.2) (1.3) ν u + p = f in Ω, u = 0 in Ω, u = 0 on Ω, whr ν dnots th fluid viscosity;,, and dnot th Laplacian, gradint, and divrgnc oprators, rspctivly; Ω R d is th rgion occupid by th fluid; f = f(x) (L 2 (Ω)) d is th unit xtrnal volumtric forc acting on th fluid at x Ω. Th commonly usd finit lmnt mthods for th Stoks problm (1.1) (1.3) ar basd on a variational quation which is obtaind by tsting th momntum quation (1.1) by functions in (H0 1 (Ω)) d and th continuity quation (1.2) by functions in L 2 (Ω) (s sction 2 for thir dfinition). Th corrsponding finit lmnt mthod rquirs a pair of finit lmnt spacs which ar conforming in (H0 1 (Ω)) d L 2 (Ω) and satisfy th inf-sup condition of Babu ska [2] and Brzzi [3]. Ths constraints rsult in finit lmnt approximations, dnotd by (u h ; p h ), which hardly satisfy th continuity quation (1.4) u h (x) = 0 x Ω. Radrs ar rfrrd to [8, 19, 21] for mor dtails rgarding th approximation mthods and thir proprtis. Rcivd by th ditors January 5, 2006; accptd for publication (in rvisd form) Fbruary 1, 2007; publishd lctronically May 22, Division of Mathmatical Scincs, National Scinc Foundation, Arlington, VA (jwang@ nsf.gov). This author s rsarch was supportd in part by th NSF IR/D program. This rsarch was initiatd whil th author was visiting NASA Cntr for Computational Scincs (NCCS) at GSFC through th NASA/ASEE Faculty Fllowship Program. Dpartmnt of Mathmatics, Univrsity of Arkansas at Littl Rock, Littl Rock, AR (xxy@ualr.du). 1269

2 1270 JUNPING WANG AND XIU YE Th rcnt dvlopmnt in discontinuous Galrkin mthods [1, 4, 5, 6, 10, 11, 13] provids nw mans of solving th Stoks quations numrically. Howvr, th corrsponding finit lmnt solutions ar usually totally discontinuous and fail to satisfy th continuity quation (1.4) in th classical sns [12, 22, 24, 27]. Th continuity quation (1.4) rquirs th numrical solution u h to b a mmbr of th Sobolv spac H(div; Ω). Thrfor, th discontinuous Galrkin mthods [12, 22, 24, 27] appar to b noncomptitiv whn (1.4) nds to b satisfid. On th othr hand, th (H0 1 ) d L 2 conforming finit lmnt mthods rquir th total continuity of u h, which is too much to satisfy for (1.4). Thrfor, it sms that th H(div) lmnts of Raviart Thomas typ [25, 7, 8, 17] might b good candidats for producing nw numrical schms that satisfy (1.4). Th goal of this papr is to prsnt a mthod that dmonstrats th us of H(div) lmnts in solving th Stoks problm. Our main contribution is on th dvlopmnt of a nw formulation for th Stoks problm which maks us of th xisting H(div) lmnts in numrical schms. Optimal-ordr rror stimats ar drivd for th rsulting H(div) finit lmnt approximations. In addition, two nw familis of H(div) lmnts ar proposd and analyzd in this articl. This papr is organizd as follows. In sction 2, w introduc som prliminaris and notations for Sobolv spacs. A nw variational formula is prsntd in sction 3 for th Stoks problm. In sction 4, w prsnt a H(div) finit lmnt mthod by using th variational formula dvlopd in sction 3. In sction 5, w stablish som optimal-ordr rror stimats for th nw finit lmnt approximations in H 1 and L 2 norms. Finally, in sction 6, w rviw som rprsntativs of H(div) lmnts, followd with a dtaild dscription of two nw H(div) lmnts. 2. Prliminaris and notations. Lt D b any domain in R d, d = 2, 3. For simplicity, th mthod will b prsntd for two-dimnsional problms only. An xtnsion to highr-dimnsional problms can b mad formally for gnral polyhdral domains. 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 0. For xampl, for any intgr s 0, th sminorm s,d is givn by with th usual notation v s,d = Th Sobolv norm m,d is givn by α =s D α v 2 dd α = (α 1, α 2 ), α = α 1 + α 2, α = α 1 x 1 α 2 x 2. m v m,d = j=0 v 2 j,d Th spac H 0 (D) coincids with L 2 (D), for which th norm and th innr product ar dnotd by D and (, ) D, rspctivly. Whn D = Ω, w shall drop th subscript D in th norm and innr product notation. W also us L 2 0(Ω) to dnot th subspac of L 2 (Ω) consisting of functions with man valu zro ,

3 H(DIV) FINITE ELEMENT METHOD 1271 Th spac H(div; Ω) is dfind as 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) 1 2. Lt Ω b a triangl or quadrilatral. For any smooth vctor-valud functions w and v, it follows from th divrgnc thorm that w (2.1) ( w) vd = ( w, v) v ds, n whr ds rprsnts th boundary lmnt, n is th outward normal dirction on, and ( w, v) = 2 i,j=1 w i x j v i x j d. Lt τ b th tangntial dirction to so that n and τ form a right-hand coordinat systm. It follows from th rprsntation that (2.2) v = (v n )n + (v τ )τ w v = (w n ) (v n ) + (w τ ) (v τ ). n n n 3. A variational formula. For simplicity, w lt ν = 1 for th fluid viscosity in th Stoks quation (1.1). Furthrmor, w assum that Ω is a plan polygonal domain without cracks. Lt T h b a finit lmnt partition of th domain Ω with msh siz h. Assum that th partition T h is shap rgular so that th routin invrs inquality in finit lmnts holds tru (s [9]). 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 V r () T h ; v n Ω = 0} T h W h = {q L 2 0(Ω) : q W m () T h }, whr n is th outward normal dirction on th boundary of Ω, V r () is a spac of vctor-valud polynomials on th lmnt with indx r 1, and W m () is a st of polynomials on th lmnt with indx m 0. Exampls of V r () and W m () will b givn in sction 6. To driv a wak formulation, w multiply th quation (1.1) by any v V h and us (2.1) to obtain ( ) u (3.1) ( u, v) v ds (p, v) = (f, v), n

4 1272 JUNPING WANG AND XIU YE whr w hav also usd th intgration by parts to dduc p vdω = (p, v). T h Ω Th fact that v V h implis that v n is continuous across ach intrior boundary. Thus, it follows from (2.2) that u v ds = (u τ ) (3.2) v τ ds. n n Introduc th following notation: T h ( h u, h v) = By substituting (3.2) into (3.1) w obtain ( h u, h v) (p, v) (3.3) T h T h ( u, v). (u τ ) n v τ ds = (f, v), which is th basis of our first quation in th nw variational form. Our scond quation can b drivd from tsting (1.2) against any q W h, yilding (3.4) ( u, q) = 0. W now rformulat th boundary intgrals in (3.3). Lt b an intrior dg shard by two lmnts 1 and 2, and lt n 1 and n 2 b unit normal vctors on pointing xtrior to 1 and 2, rspctivly. Dnot by τ 1 and τ 2 th two tangntial dirctions which mak th right-hand coordinat systms with n 1 and n 2, rspctivly. W dfin th avrag { } and jump [[ ]] on for vctor-valud functions w as follows: {ε(w) } = 1 2 (n 1 (w τ 1 ) 1 + n 2 (w τ 2 ) 2 ), [[w ] = w 1 τ 1 + w 2 τ 2. For boundary dg = 1 Ω, th abov two oprations must b modifid by {ε(w) } = n 1 (w τ 1 ) 1, [[w]] = w 1 τ 1. Lt E h dnot th union of th boundaris of all lmnts in T h. For sufficintly smooth u (.g., u H 3 2 +ɛ (Ω) for som ɛ > 0), it is not hard to s that (u τ ) v τ ds = {ε(u) }[[v]]ds. n E h T h Substituting th abov into (3.3) w obtain ( h u, h v) ( v, p) (3.5) E h {ε(u) }[[v]]ds = (f, v). Lt V (h) = V h + (H s (Ω) H0 1 (Ω)) 2, with s > 3 2. Dnot by a o (u, v) = ( h u, h v) {ε(u) }[v]]ds E h

5 H(DIV) FINITE ELEMENT METHOD 1273 and b(v, q) = ( v, q) two bilinar forms on V (h) V (h) and V (h) L 2 0(Ω). With th conditions spcifid in this papr, it can b provd that th xact solution (u; p) of th Stoks problm in 2D blongs to V (h) for som s > 3 2. Radrs ar rfrrd to [20, 15, 14, 23] for dtails. As a rsult, it follows from (3.5) and (3.4) that th xact solution of th 2D Stoks problm satisfis th following variational quations: (3.6) (3.7) a o (u, v) b(v, p) = (f, v) v V h, b(u, q) = 0 q W h. Howvr, it is not clar if th sam statmnt can b mad for th Stoks problm in thr-dimnsional spac without assuming a smooth boundary Ω or a convx polyhdral domain Ω [16, 18]. 4. Finit lmnt schms. Our goal of this sction is to propos two finit lmnt schms basd on two modifications of th wak formulation (3.6) (3.7) for th Stoks problm (1.1) (1.3). To this nd, lt us introduc a symmtric bilinar form on V (h) V (h) as follows: a s (w, v) = a o (w, v) + ( αh 1 [[w]][[v]] {ε(v) }[[w]] ) ds, E h whr α > 0 is a paramtr to b dtrmind latr, and h is th lngth of th dg. For th xact solution (u; p) of th Stoks problm, w clarly hav a s (u, v) = a o (u, v) v V h. Thrfor, it follows from (3.6) and (3.7) that (4.1) (4.2) a s (u, v) b(v, p) = (f, v) v V h, b(u, q) = 0 q W h. Th corrsponding finit lmnt schm for (1.1) (1.3) sks (u h ; p h ) V h W h such that (4.3) (4.4) a s (u h, v) b(v, p h ) = (f, v) v V h, b(u h, q) = 0 q W h. To invstigat th proprtis of th abov numrical schm, w introduc two norms 1 and for th st V (h) as follows: (4.5) (4.6) v 2 1 = v 2 1,h + E h h 1 [[v]] 2, v 2 = v E h h {ε(v) } 2, whr v 2 1,h = T h v 2 1, and v 2 = v vds.

6 1274 JUNPING WANG AND XIU YE Lt b an lmnt with as an dg. It is wll known that thr xists a constant C such that for any function g H 1 () (4.7) In particular, for any v V h, w hav g 2 C ( h 1 g 2 + h g 2 ). (4.8) h {ε(v) } 2 C ( v 2 + h 2 2 v 2 ). Th standard invrs inquality can b mployd to th last trm of th abov inquality, yilding (4.9) h {ε(v) } 2 C v 2 for som constant C indpndnt of th msh siz h. Consquntly, thr is a constant C such that (4.10) v C 0 v 1 v V h. Th following rsult is concrnd with th llipticity of th bilinar form a s (, ) in V h V h. Lmma 4.1. Thr xists a constant α 0 indpndnt of h such that for any v V h w hav (4.11) a s (v, v) α 0 v 2, providd that α is sufficintly larg. Proof. It follows from th Cauchy Schwarz inquality that thr is a constant C such that E h ( {ε(w) }[[v]]ds C h {ε(w) } 2 E h ( C w 1,h h 1 [[v]] 2 E h ) 1 2 ( ) 1 2 h 1 E h [[v ] 2 ) w 2 1,h + C E h h 1 [[v]] 2, whr w hav usd th stimat (4.9) in th scond lin. Using th abov inquality and (4.10), w obtain a s (v, v) = ( h v, h v) + α h 1 [v]] 2 ds 2 {ε(v) }[[v ]ds E h E h v 2 1,h + α E h h 1 [[v]] v 2 1,h C h 1 [[v ] 2 E h = 1 2 v 2 1,h + (α C) E h h 1 [[v ] 2 α 1 v 2 1 α 0 v 2, with α 1 = min( 1 2, α C) and α 0 = α 1 /C 0. For xampl, on may hav α 0 = 1/(2C 0 ) if th paramtr α is chosn so that α C

7 H(DIV) FINITE ELEMENT METHOD 1275 In th rst of th papr, w assum that th paramtr α is chosn so that (4.11) holds tru for th symmtric bilinar from a s (, ). Th proof of Lmma 4.1 indicats that th valu of α dpnds upon th constant in th invrs inquality for finit lmnt functions. Thrfor, th valu of α for which a s (, ) is corciv is mshdpndnt. Existing rsults for saddl-point problms indicat that it is thortically and computationally important to hav th corcivity (4.11). Thrfor, th msh dpndnc of th paramtr α maks th finit lmnt schm (4.3) (4.4) conditionally intrsting in practical computation. To ovrcom th difficulty on th paramtr slction, w introduc a scond finit lmnt schm which is paramtr-insnsitiv. To this nd, w dfin a nonsymmtric bilinar form on V (h) V (h) as follows: a ns (w, v) = a o (w, v) + E h ( αh 1 [[w ][[v ] + {ε(v) }[[w]] ) ds. Similar to th bilinar form a s (, ), for th xact solution (u; p) of th Stoks problm w hav a ns (u, v) = a o (u, v) v V h. Consquntly, th solution of th Stoks problm satisfis th following variational quations: (4.12) (4.13) a ns (u, v) b(v, p) = (f, v) v V h, b(u, q) = 0 q W h. Our scond finit lmnt schm for (1.1) (1.3) sks (u h ; p h ) V h W h such that (4.14) (4.15) a ns (u h, v) b(v, p h ) = (f, v) v V h, b(u h, q) = 0 q W h. To s th corcivity of th bilinar form a ns (, ), w us its dfinition and (4.10) to obtain a ns (v, v) = ( h v, h v) + α h 1 [[v]] 2 ds E h min(1, α) v 2 1 min(1, α)c 1 0 v 2, whr v V h. Thus, th corcivity (4.11) holds tru for th bilinar form a ns (, ) with any valu of α > 0. Th following is a rsult on th bounddnss of th bilinar forms a s (, ) and a ns (, ). Lmma 4.2. Thr xists a constant C indpndnt of h such that (4.16) a i (w, v) C w v w, v V (h), whr i = s, ns.

8 1276 JUNPING WANG AND XIU YE Proof. For simplicity, w shall prsnt th analysis for a s (, ) only. By th dfinition of a s (w, v) and th Schwarz inquality, thr xists a constant C such that ( ) 1 ( ) a s (w, v) C w 1,h v 1,h + h {ε(w) } 2 h 1 [[v]] 2 E h E h ( ) 1 ( ) h {ε(v) } 2 h 1 [w]] 2 E h E h ( ) 1 ( ) α h 1 [[w[] 2 h 1 [v[] 2 E h E h C w v, which provs th dsird bounddnss. 5. Error stimats. Th first goal of this sction is to driv an optimal-ordr rror stimat for th prssur in L 2 (Ω) and th vlocity in th norm givn by (4.6). Th scond goal is to driv an optimal-ordr rror stimat for th vlocity approximation in th L 2 -norm for th symmtric schm (4.3) (4.4). Assumption 1. Thr xists an oprator Π h : (H 1 0 (Ω)) 2 V h such that (5.1) b(v Π h v, q) = 0 q W h. In addition, th oprator Π h is assumd to satisfy th following: (5.2) v Π h v s, Ch t s v t, T h, s = 0, 1, whr th constant C dpnds only on th shap of and 1 t r + 1. From (5.2) and th inquality (4.7) it is not hard to s that v Π h v 1 C v 1 v (H 1 0 (Ω)) 2. Thus, it follows from v 1 = v 1 v 1 and th triangl inquality that (5.3) Π h v 1 C v 1. For our finit lmnt formulations, th inf-sup condition givn in Brzzi s framwork would rad as follows: Thr xists a positiv constant β, indpndnt of h, such that b(v, q) (5.4) sup β q q W h. v V h v To vrify (5.4), w first us th oprator Π h to obtain (5.5) b(v, q) b(π h v, q) b(v, q) sup sup = sup v V h v v (H0 1 Π (Ω))2 h v v (H0 1 Π (Ω))2 h v. Obsrv that by using (5.3), and (4.10), w hav for all v (H 1 0 (Ω)) 2 (5.6) Π h v C Π h v 1 C v 1. Thus, substituting (5.6) into th inquality (5.5) givs b(v, q) sup v V h v C 1 b(v, q) sup β q, v (H0 1 v (Ω))2 1 whr w hav usd th inf-sup condition for th continuous cas [19, 8].

9 H(DIV) FINITE ELEMENT METHOD Error stimats in H 1 L 2. Th rror analysis rquirs th L 2 projction from L 2 0(Ω) to th finit lmnt spac W h, which is dnotd by Q h. In addition, th following rror quations turn out to b usful: (5.7) (5.8) a s (u u h, v) b(v, p p h ) = 0 v V h, b(u u h, q) = 0 q W h. Ths rror quations can b obtaind by subtracting (4.3) (4.4) from (4.1) (4.2). Similar rror quations hold tru for th nonsymmtric schm (4.14) (4.15) with a s (, ) bing rplacd by a ns (, ). Now w ar in a position to prsnt an rror stimat for th nw finit lmnt approximations. Thorm 5.1. Lt (u; p) b th solution of (1.1) (1.3) and (u h ; p h ) V h W h b obtaind from ithr (4.3) (4.4) or (4.14) (4.15). Assum that Assumption 1 holds tru. Thn, thr xists a constant C indpndnt of h such that (5.9) u u h + p p h C ( u Π h u + p Q h p ). Proof. Lt ξ h = u h Π h u, η h = p h Q h p b th rror btwn th finit lmnt solution (u h ; p h ) and th projction (Π h u; Q h p) of th xact solution. Dnot by ξ = u Π h u, η = p Q h p th rror btwn th xact solution (u; p) and its projction. It follows from th rror quations (5.7) and (5.8) that (5.10) (5.11) a(ξ h, v) b(v, η h ) = a(ξ, v) b(v, η), b(ξ h, q) = b(ξ, q) = 0 for any v V h and q W h. Hr and in what follows of this sction, a(, ) dnots ithr a s (, ) or a ns (, ). By ltting v = ξ h in (5.10) and q = η h in (5.11), th sum of (5.10) and (5.11) givs a(ξ h, ξ h ) = a(ξ, ξ h ) b(ξ h, η). Thus, it follows from th corcivity (4.11) and th bounddnss (4.16) that which implis th following: Th abov stimat can b rwrittn as α 0 ξ h 2 C( ξ ξ h + η ξ h ), ξ h C ( ξ + η ). (5.12) u h Π h u C ( u Π h u + p Q h p ). Now using th triangl inquality and th rror stimat (5.12) w gt (5.13) u u h C ( u Π h u + p Q h p ), which complts th stimat for th vlocity approximation.

10 1278 JUNPING WANG AND XIU YE It rmains to stimat th prssur approximation p h. To this nd, w us th discrt inf-sup condition (5.4) to obtain p h Q h p 1 β sup b(v, p h Q h p) v V h v which, togthr with (5.13), implis that = 1 β sup b(v, p h p) + b(v, p Q h p) v V h v = 1 β sup a(u u h, v) + b(v, p Q h p) v V h v 1 C sup v V h v v ( u u h + p Q h p ) C( u u h + p Q h p ), p h Q h p C ( u Π h u + p Q h p ). Th rror stimat for th prssur approximation is thn compltd by combining th abov inquality with th standard triangl inquality An L 2 -rror stimat for th vlocity approximation. Considr only th finit lmnt approximat solutions arising from th symmtric finit lmnt schm. To driv an L 2 -rror stimat for th vlocity approximation, w sk (w; λ) (H0 1 (Ω)) 2 L 2 0(Ω) satisfying w + λ = u u h in Ω, w = 0 in Ω, w = 0 on Ω. Not that for any (v; q) V (h) L 2 0(Ω) th solution (w; λ) satisfis (5.14) (5.15) a s (w, v) b(v, λ) = (u u h, v), b(w, q) = 0. Assum that th Stoks problm has th H 2 (Ω) H 1 (Ω)-rgularity proprty in th sns that th solution (w; λ) (H 2 (Ω)) 2 H 1 (Ω) and th following a priori stimat holds tru: (5.16) w 2 + λ 1 C u u h. In addition, w assum that th finit lmnt spac V h and th projction oprator Π h hav th following proprty: (5.17) w Π h w Ch w 2. With ths assumptions, it is not hard to s that thr xists a constant C indpndnt of h such that (5.18) w Π h w + λ Q h λ Ch u u h. It must b pointd out that th H 2 H 1 -rgularity proprty assumption statd as abov rquirs that th polygonal domain Ω b convx. For nonconvx but smooth

11 H(DIV) FINITE ELEMENT METHOD 1279 domains, th rgularity (5.16) can b provd to b valid. Howvr, isoparamtric lmnts would nd to b mployd in th finit lmnt schm in ordr to maintain optimal-ordr rror stimats in ithr H 1 - or L 2 -norms. Thorm 5.2. Lt (u h ; p h ) V h W h and (u; p) b th solutions of (4.3) (4.4) and (1.1) (1.3), rspctivly. Assum that Assumption 1 and th stimat (5.17) hold tru and that th Stoks problm (1.1) (1.3) has th H 2 (Ω) H 1 (Ω)-rgularity proprty. Thn thr xists a constant C indpndnt of h such that (5.19) u u h Ch( u Π h u + p Q h p ). Proof. By ltting v = u u h in (5.14) w arriv at (5.20) a s (u u h, w) b(u u h, λ) = u u h 2. Notic that (5.21) b(u u h, λ) = b(u u h, λ Q h λ) and (5.22) a s (u u h, w) = a s (u u h, w Π h w) + a s (u u h, Π h w) = a s (u u h, w Π h w) + b(π h w, p p h ) = a s (u u h, w Π h w) + b(π h w w, p p h ). Substituting (5.21) and (5.22) into (5.20) w obtain u u h 2 = a s (u u h, w Π h w) + b(π h w w, p p h ) b(u u h, λ Q h λ). Thus, u u h 2 C ( u u h + p p h ) ( w Π h w + λ Q h λ ). Substituting (5.18) into th abov stimat w obtain which implis that u u h 2 Ch ( u u h + p p h ) u u h, u u h Ch ( u u h + p p h ). Th abov inquality and th rror stimat (5.9) imply This complts th proof of th thorm. u u h Ch( u Π h u + p Q h p ). 6. Exampls of H(div) lmnts. Lt us rcall that th rror stimats stablishd in sction 5 ar basd on th following thr proprtis: B1. V h H(div; Ω), B2. Assumption 1 and th stimat (5.17) as dscribd in sction 5, and B3. th H 2 H 1 -rgularity proprty assumption for th Stoks problm.

12 1280 JUNPING WANG AND XIU YE Th last proprty (B3) is rquird only for th L 2 -rror stimat for th vlocity approximation. This mans that any finit lmnt pair V h W h satisfying proprtis B1 B2 is applicabl for th formulations prsntd arlir in this manuscript. Dnot by P k () th spac of polynomials of dgr k and P k1,k 2 () = p(x 1, x 2 ) : p(x 1, x 2 ) = a ij x i 1x j 2. 0 i k 1,0 j k 2 P k1,k 2,k 3 () is dfind similarly in thr-dimnsional spacs. Dfin Q k () as follows: { Pk,k () for d = 2, Q k () = P k,k,k () for d = 3. Obsrv that th finit lmnt pair V h W h is constructd from local lmnts V r () and W m () as dscribd in sction 3. Thrfor, it suffics to spcify th local pair V r () W m () for ach xampl to b prsntd Existing lmnts. All of th xisting H(div) lmnts dsignd for th scond-ordr lliptic problms (.g., s [25, 8, 7, 17, 19]) satisfy proprtis B1 B2, xcpt th stimat (5.17) for th lowst-ordr Raviart Thomas lmnt on triangls and quadrilatrals. Thrfor, thr ar plnty of finit lmnt spacs applicabl to th nw formulation of th Stoks problm. For illustrativ purposs, w mntion thr xampls. Radrs ar rfrrd to th book by Brzzi and Fortin [8] for mor xampls of th H(div) lmnt Raviart Thomas lmnts on triangls or ttrahdra: RT k (). Lt k 1 b any intgr. For any triangular or ttrahdral lmnt, th local lmnt V r () W m () is dfind by V k () = (P k ()) d xp k (), W k () = P k (), whr d = 2 if is a triangl and d = 3 if is a ttrahdron. Th projction oprator Π h satisfying all of th rquird proprtis is givn locally on ach lmnt. For xampl, th rstriction of Π h on th lmnt, dnotd by Π, is dfind as follows: (v Π v) nqds = 0 q P k ( ), (v Π v) qd = 0 q (P k 1 ()) d, k BDM lmnts on triangls or ttrahdra: BDM k () [8]. Lt k 1 b any intgr. For any triangular or ttrahdral lmnt, th local lmnt V r () W m () is dfind by stting r = m + 1 = k and V k () = (P k ()) d, W k 1 () = P k 1 (). On a triangular lmnt, th local projction oprator Π : (H 1 ()) 2 V k () is dfind by (v Π v) nqds = 0 q P k ( ), (v Π v) qd = 0 q P k 1 (), (v Π v) curl(b q)d = 0 q P k 2 (), k 2,

13 H(DIV) FINITE ELEMENT METHOD 1281 whr b is th bubbl function dfind on. On a ttrahdral lmnt, th corrsponding local projction Π is givn by (v Π v) nqds = 0 q P k ( ), (v Π v) qd = 0 q P k 1 (), (v Π v) qd = 0 q Φ k (), whr Φ() = {φ (P k ()) 3 : φ = 0, φ n = 0 on } BDM lmnts on quadrilatrals: BDM [k] (). It is sufficint to dscrib th lmnt on th unit squar. Lt k 1 b any intgr. Th local lmnt V r () W m () is dfind by V k () = (P k ()) 2 curl(x k+1 1 x 2 ) curl(x 1 x k+1 2 ), W k 1 () = P k 1 (). On th unit squar lmnt, th local projction oprator Π : (H 1 ()) 2 V k () is dfind by (v Π v) nqds = 0 q P k ( ), (v Π v) wd = 0 w (P k 2 ()) 2, k Error stimats for th xisting lmnts. Rcall that th vlocity V h and th prssur spac W h ar dfind, rspctivly, by (6.1) V h = {v H(div; Ω) : v V r () T h ; v n Ω = 0} and (6.2) W h = {q L 2 0(Ω) : q W m () T h }. For th xisting H(div) lmnts listd abov, w hav V r () = RT k (), BDM k (), or BDM [k] () and W m () = P k (), P k 1 (), or P k 1 (), rspctivly. Th projction oprator Π h is givn by (6.3) (Π h v) = Π (v ). Th dfinition of Π h implis that (6.4) b(v Π h v, q) = 0 q W h. Furthrmor, it has bn provd in [8] that (5.2) and (5.17) hold tru for Π h dfind in (6.3). Thrfor, proprtis B1 B2 ar wll justifid. Lt Q h b th L 2 projction from L 2 0(Ω) to W h. It is not hard to s that W h has th following local approximation proprtis: For BDM k () and BDM [k] () (6.5) p Q h p s, Ch k s p k, T h, s = 0, 1,

14 1282 JUNPING WANG AND XIU YE and for RT k () (6.6) p Q h p s, Ch k+1 s p k+1, T h, s = 0, 1. Th constant C in (6.5) (6.6) dpnds only on k and th shap of. Th following rsult follows from (5.17), (6.5) (6.6), and Thorms 5.1 and 5.2. Proposition 6.1. Lt (u; p) b th solution of (1.1) (1.3) and (u h ; p h ) V h W h b obtaind from ithr (4.3) (4.4) or (4.14) (4.15). Assum that (u; p) (H t+1 (Ω)) 2 H t (Ω) for som 1 t k. Thn thr xists a constant C indpndnt of h such that for BDM k (), BDM [k] (), and RT k () (6.7) u u h + p p h Ch t ( u t+1 + p t ). Furthrmor, if th H 2 H 1 -rgularity proprty holds tru for th Stoks problm, thn thr is a constant C such that th finit lmnt approximation (u h ; p h ) from th symmtric formulation has th following rror stimat: u u h Ch t+1 ( u t+1 + p t ). W commnt that th abov rror stimats hold tru for all of th H(div) lmnts listd in [8] Nw lmnts. Stability and accuracy ar two main factors in th construction of nw finit lmnts. For th variational schms prsntd in this papr, th stability part is ralizd by a combination of th inf-sup condition and th corcivity for th corrsponding bilinar forms. Th accuracy part is charactrizd by Assumption 1 and a balancd prssur spac W h. For xampl, Proposition 6.1 indicats that th Raviart Thomas lmnt can b usd to approximat th solution of th Stoks quations, which is convrgnt as th msh siz dcrass (not that w do not know any convrgnc whn RT 0 () is usd). Howvr, th Raviart Thomas lmnts do not appar to b wll balancd, bcaus both th vlocity and th prssur unknowns ar approximatd by polynomials of ordr k. In contrast, th BDM lmnts offr a bttr/optimal combination for th solution of th Stoks problm. But th BDM [k] () lmnt on rctangls is constructd in an awkward way by involving th curl of som polynomials. W fl that bttr constructd lmnts should b xplord on rctangls and cubs for solving th Stoks problm. For this purpos, w would lik to propos som altrnativs on rctangls and cubs which ar suitabl for approximating th solution of th Stoks problm. Ths lmnts can b usd on quadrilatrals through local transformations as dscribd in [26] A nw lmnt on rctangls: NE1 k (). W illustrat th construction of th nw NE1 k () lmnt on th unit squar = [0, 1] [0, 1]. Lt k 1 b any intgr. W dfin local lmnts V r () W m () by V k () = (Q k ()) 2, W k 1 () = Q k 1 (). For th first componnt of v = (v 1, v 2 ), w dfin an oprator Π,1 : H 1 () Q k () as follows: (6.8) (v 1 Π,1 v 1 )φds = 0 φ P k (), = wst, ast, (6.9) (v 1 Π,1 v 1 )ψd = 0 ψ P k 2,k (),

15 H(DIV) FINITE ELEMENT METHOD 1283 whr = wst mans that is th wst dg (i.., = {(0, x 2 ) : x 2 [0, 1]}) of th unit squar; th ast dg is dfind accordingly. Th systm (6.8) involvs xactly 2(k + 1) linar quations and (6.9) involvs (k 1)(k + 1) linar quations. Th total numbr of quations is givn by 2(k + 1) + (k 1)(k + 1) = (k + 1) 2, which is th sam as th total numbr of dgrs of frdom for a polynomial in Q k (). Th following proposition shows that th linar systms (6.8) and (6.9) uniquly dtrmin th projction Π,1 v 1. Proposition 6.2. Lt v Q k () b such that (6.10) vφds = 0 φ P k (), = wst, ast, (6.11) vψd = 0 ψ P k 2,k (). Thn w must hav v 0. Proof. Th condition (6.10) implis that v = 0 at th ast and wst dgs of th unit squar. Thus, thr is a polynomial g = g(x 1, x 2 ) P k 2,k () such that v = x 1 (1 x 1 )g. Substitut v = x 1 (1 x 1 ) into (6.11), and thn lt ψ = g. It follows that g 0. This shows that v 0. Th projction of th scond componnt of v, dnotd by Π,2 v 2, can b dfind in a similar fashion. Th local projction oprator is thn givn by Π v = (Π,1 v 1, Π,2 v 2 ). It is not hard to show that such a dfind projction satisfis all of th conditions rquird in th prvious sctions. As a rsult, th lmnt NE1 k () can b usd to approximat th Stoks problm A nw lmnt on cubs: NE2 k (). Again, w shall dscrib dtails only on th unit cub = [0, 1] 3. Lt k 1 b an intgr. A straightforward xtnsion of th NE1 k lmnt to thr-dimnsional spac is givn by V k () = (Q k ()) 3, W k 1 () = Q k 1 (). Our goal hr is to show that th abov xtnsion actually works. To this nd, it suffics to construct a projction oprator Π which satisfis th rquird proprtis. Lt v = (v 1, v 2, v 3 ) ( H 1 (Ω) ) 3 b a vctor-valud function. For ach componnt v i, w dfin its projction to Q k () as follows: (6.12) (v i Π,i v i )φds = 0 φ Q k ( i ), i (6.13) (v i Π,i v i )ψd = 0 ψ P k1,k 2,k 3 (), whr i = {(x 1, x 2, x 3 ) : x j [0, 1], j i; x i = 0 or 1} ar th two facs of th cub which ar orthogonal to th x i -axis, and k i = k 2, k j = k for j i. Thr ar 2(k + 1) 2 linar quations from th condition (6.12) and (k 1)(k + 1) 2 linar quations from th condition (6.13). Th total numbr of linar quations is thn givn by 2(k + 1) 2 + (k 1)(k + 1) 2 = (k + 1) 3,

16 1284 JUNPING WANG AND XIU YE which is th sam as th total numbr of dgrs of frdom for a polynomial in th spac V k (). A similar argumnt as in th prvious subsction for NE1 k () can b applid to show that th projction Π,i v i is uniquly dtrmind by (6.12) and (6.13). Furthrmor, th local projction givn by Π v = (Π,1 v 1, Π,2 v 2, Π,3 v 3 ) can b vrifid to satisfy all of th proprtis rquird by th convrgnc thory dvlopd in prvious sctions for th nw finit lmnt mthods Anothr nw lmnt on rctangls: NE3 k (). Again for simplicity, w shall dscrib th nw lmnt on th unit squar = [0, 1] 2. This lmnt will b a simplifid vrsion of NE1 k () but with th sam ordr of accuracy. Lt k 1 b any intgr, and dfin V k () = (P k () {x 1 x k 2}) (P k () {x 2 x k 1}), W k 1 () = P k 1 (). For th first componnt of v = (v 1, v 2 ), w dfin its projction Π,1 v 1 P k () {x 1 x k 2} by using th following quations: (6.14) (v 1 Π,1 v 1 )φds = 0 φ P k (), = wst, ast, (6.15) (v 1 Π,1 v 1 )ψd = 0 ψ P k 2 (). Thr ar 2(k + 1) quations from th condition (6.14) and 1 2 (k 1)k quations from th condition (6.15). Th total numbr of linar quations is givn by 2(k + 1) (k 1)k = 1 (k + 1)(k + 2) + 1, 2 which is th sam as th total numbr of dgrs of frdom for functions in th spac P k () {x 1 x k 2}. Using th sam tchniqu as in th analysis for NE1 k (), it can b provd that Π,1 v 1 is uniquly dtrmind by (6.14) and (6.15). Th projction of th scond componnt of v can b dtrmind in a similar way. Th rsulting local projction Π v = (Π,1 v 1, Π,2 v 2 ) satisfis all of th proprtis rquird in th convrgnc thory Error stimats for th nw lmnts. First, w dfin th vlocity spac V h by (6.16) V h = {v H(div; Ω) : v V r () T h ; v n Ω = 0} and th prssur spac W h by (6.17) W h = {q L 2 0(Ω) : q W m () T h }, whr V r () = NE1 k (), NE2 k (), or NE3 k () and W m () = Q k 1 (), Q k 1 (), or P k 1 (), rspctivly. For any v (H 1 0 (Ω)) d, with d = 2, 3, dfin Π h v V h by (6.18) (Π h v) = Π v T h,

17 H(DIV) FINITE ELEMENT METHOD 1285 whr Π is th corrsponding local projction oprator on ach lmnt. From th construction of Π, it is asy to s that it is indd tru that Π h v V h, and, morovr, on has (6.19) b(v Π h v, q) = 0 q W h and that proprtis B1 B2 ar satisfid for th thr nw lmnts NE1 k (), NE2 k (), and NE3 k (). Similar to Proposition 6.1, w hav th following convrgnc stimats. Proposition 6.3. Lt (u; p) b th solution of (1.1) (1.3) and (u h ; p h ) V h W h b obtaind from ithr (4.3) (4.4) or (4.14) (4.15) by using th nw lmnts dscribd in this subsction. Assum that (u; p) (H t+1 (Ω)) d H t (Ω) for som < t k. Thn thr xists a constant C indpndnt of h such that 1 2 u u h + p p h Ch t ( u t+1 + p t ), and for th symmtric formulation w also hav u u h Ch t+1 ( u t+1 + p t ), providd that th H 2 H 1 -rgularity proprty holds tru for th Stoks problm. W point out that, unlik th xisting H(div) lmnts, th nw H(div) lmnts dscribd in this sction do not yild numrical vlocitis that satisfy th continuity quation (1.4) in th classical sns. Howvr, th numrical approximations arising from th nw lmnts indd consrv mass locally on ach lmnt. REFERENCES [1] D. N. Arnold, F. Brzzi, B. Cockburn, and L. D. Marini, Unifid analysis of discontinuous Galrkin mthods for lliptic problms, SIAM J. Numr. Anal., 39 (2002), pp [2] I. Babu ska, Th finit lmnt mthod with Lagrangian multiplir, Numr. Math., 20 (1973), pp [3] F. Brzzi, On th xistnc, uniqunss, and approximation of saddl point problms arising from Lagrangian multiplirs, RAIRO, Anal. Numér., 2 (1974), pp [4] I. Babu ska and M. Zlámal, Nonconforming lmnts in th finit lmnt mthod with pnalty, SIAM J. Numr. Anal., 10 (1973), pp [5] F. Bassi and S. Rbay, A high-ordr accurat discontinuous finit lmnt mthod for th numrical solution of th comprssibl Navir-Stoks quations, J. Comput. Phys., 131 (1997), pp [6] C.E. Baumann and J.T. Odn, A discontinuous hp finit lmnt mthod for convctiondiffusion problms, Comput. Mthods Appl. Mch. Engrg., 175 (1999), pp [7] F. Brzzi, J. Douglas, and L. Marini, Two familis of mixd finit lmnts for scond ordr lliptic problms, Numr. Math., 47 (1985), pp [8] F. Brzzi and M. Fortin, Mixd and Hybrid Finit Elmnts, Springr-Vrlag, Nw York, [9] P.G. Ciarlt, Th Finit Elmnt Mthod for Elliptic Problms, North-Holland, Nw York, [10] B. Cockburn, S. Hou, and C.-W. Shu, TVB Rung-utta local projction discontinuous Galrkin finit lmnt mthod for consrvation laws IV: Th multidimnsional cas, Math. Comp., 54 (1990), pp [11] B. Cockburn, G. anschat, and D. Schotzau, A locally consrvativ LDG mthod for th incomprssibl Navir-Stoks quations, Math. Comp., 74 (2005), pp [12] B. Cockburn, G. anschat, D. Schötzau, and C. Schwab, Local discontinuous Galrkin mthods for th Stoks systm, SIAM J. Numr. Anal., 40 (2002), pp [13] B. Cockburn and C.-W. Shu, Th local discontinuous Galrkin mthod for tim-dpndnt convction-diffusion systms, SIAM J. Numr. Anal., 35 (1998), pp [14] M. Costabl and M. Daug, Crack singularitis for gnral lliptic systms, Math. Nachr., 235 (2002), pp

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