A NEW FINITE VOLUME METHOD FOR THE STOKES PROBLEMS

Transcription

1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volum -, Numbr -, Pags 22 c - Institut for Scintific Computing and Information A NEW FINITE VOLUME METHOD FOR THE STOKES PROBLEMS Abstract. JUNPING WANG, YANQIU WANG, AND XIU YE A nw finit volum mthod for solving th Stoks quations is dvlopd in this papr. Th finit volum mthod maks us of th BDM mixd lmnt in approximating th vlocity unknown, and consquntly, th finit volum solution faturs a full satisfaction of th divrgnc-fr constraint as rquird for th xact solution. Optimal-ordr rror stimats ar stablishd for th corrsponding finit volum solutions in various Sobolv norms. Som prliminary numrical xprimnts ar conductd and prsntd in th papr. In particular, a post-procssing procdur was numrically invstigatd for th prssur approximation. Th rsult shows a suprconvrgnc for a local avraging post-procssing mthod. Ky Words. finit volum mthods, Stoks problms, discontinuous Galrkin mthod. Introduction In scintific computing for scinc and nginring problms, finit volum mthods ar widly usd and apprciatd by usrs du to thir local consrvativ proprtis for quantitis which ar of practical intrst (.g., mass or nrgy). Among many rfrncs, w would lik to cit som which addrsss thortical issus such as stability and convrgnc [5, 6,,, 5, 6, 2, 2, 22, 8, 9,, 28, 29]. Th goal of this papr is to invstigat a finit volum mthod for th Stoks quations by using th wll-known BDM lmnts [3] originally dsignd for solving scond ordr lliptic problms. W intnd to dmonstrat how th BDM lmnt can b mployd in constructing finit volum mthods for th modl Stoks quations. Th ida to b prsntd in th papr can b xtndd to problms of Stoks and Navir-Stoks typ without any difficulty. Mass consrvation is a proprty that numrical schms should sustain in computational fluid dynamics. This proprty is oftn charactrizd as an incomprssibility constraint in th modling quations. To sustain th mass consrvation proprty for th Stoks quations, svral finit lmnt schms hav bn dvlopd to gnrat locally divrgnc-fr solutions [2, 23]. In particular, a rcnt approach by using H(div) conforming finit lmnts has bn proposd and studid for a numrical approximation of incomprssibl fluid flow problms [3, 25, 26]. Th main 99 Mathmatics Subjct Classification. Primary, 65N5, 65N3, 76D7; Scondary, 35B45, 35J5. Th rsarch of Junping 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. Th rsarch of Xiu Y was supportd in part by National Scinc Foundation Grant DMS

2 2 J. WANG, Y. WANG, AND X. YE advantag of using H(div) conforming lmnts is that th discrt vlocity fild is xactly divrgnc-fr. Anothr advantag of using H(div) conforming lmnts is that th rsulting linar or nonlinar algbraic systms can b asily dcoupld btwn th vlocity and th prssur unknowns, largly du to th availability of a computationally fasibl divrgnc-fr subspac for th vlocity fild. Th purpos of this papr is to furthr xplor th H(div) conforming lmnts in a finit volum contxt. Our modl Stoks quations ar dfind on a two-dimnsional domain Ω. Th standard Dirichlt boundary condition is imposd on th vlocity fild. Th Stoks problm sks a vlocity u and a prssur p such that () (2) (3) u + p = f in Ω, u = in Ω, u = g on Ω, whr th symbols,, and dnot th Laplacian, gradint, and divrgnc oprators, rspctivly. f is th xtrnal volumtric forc, and g is th vlocity fild on th boundary. For simplicity, w shall assum g = in th algorithmic dscription of th finit volum mthod. But th numrical xprimnts of Sction 6 will b conductd for non-homognous data. This papr is organizd as follows. In Sction 2, w introduc som notations that hlp us to giv a tchnical prsntation. In Sction 3, a wak formulation is prsntd for th Stoks problm. Sction 4 is ddicatd to a prsntation of a finit volum schm by using th BDM lmnt. In Sction 5, w provid a thortical justification for th finit volum schm by stablishing som rror stimats in various norms. In addition to th standard H and L 2 rror stimats, w shall includ an stimat for th prssur rror in a ngativ norm, which nsurs a crtain suprconvrgnc for th prssur whn appropriat postprocssing mthods ar applid. In Sction 6, a divrgnc-fr finit volum formulation is discussd. Finally in Sction 7, w prsnt som numrical rsults that dmonstrat th fficincy and accuracy of th nw schm. 2. Prliminaris and notations W us standard notations for th Sobolv spacs H s (K) and thir associatd innr products (, ) s,k, norms s,k, and smi-norms s,k, s on a domain K. Th spac H (K) coincids with L 2 (K), in which cas th norm and innr product ar dnotd by K and (, ) K, rspctivly. Th subscript K is supprssd whn K = Ω. Dnot by L 2 (Ω) th subspac of L 2 (Ω) consisting of functions with man valu zro. Lt H(div,Ω) b th spac of all vctor functions in (L 2 (Ω)) 2 whos divrgnc is also in L 2 (Ω), and H (div,ω) b th spac of all functions v H(div,Ω) such that v n = on Ω, whr n is th unit outward normal vctor. Throughout th papr, w adopt th convntion that a bold charactr in lowr cas stands for a vctor. For simplicity, th Stoks problm () (3) is assumd to hav a full rgularity of u (H 2 (Ω)) 2 and p H (Ω). In addition, w us ( ) to dnot lss than (gratr than) or qual to up to a constant indpndnc of th msh siz or othr variabls appard in th inquality. Lt T h b a quasi-uniform triangulation of Ω with charactristic msh siz h. Dnot E h to b th st of all dgs in T h and E h = E h\ Ω to b th st of all intrior dgs. Each triangl T T h is furthr dividd into thr subtriangls by conncting th barycntr C to th vrtics A k, k =,2,3, as shown in Figur.

3 A NEW FINITE VOLUME METHOD FOR THE STOKES PROBLEMS 3 Dnot th subtriangls by S k, k =,2,3. Associatd with ach intrior dg, th two subtriangls which shar th dg form a quadrilatral. Similarly, ach boundary dg is associatd with on subtriangl. Dfin th dual partition Th to b th union of ths intrior quadrilatrals and th bordr triangls. A A 3 S S C 2 S 3 A 2 Figur. Subtriangls in T and th dual volum associatd with intrior dg. Lt P k (T) b th st of all polynomials on T, with dgr lss than or qual to k. W us th lowst ordr Brzzi-Douglas-Marini lmnt (BDM ) to approximat th vlocity. This lmnt consists of picwis linars on ach triangl and th dgrs of frdom ar th zroth and th first ordr momntum on ach dg [3, 4]. Dfin th trial spac V h and th tst spac W h for th vlocity, rspctivly, by V h = {v H (div,ω) : v T BDM (T), T T h }, W h = {ξ L 2 (Ω) 2 : ξ K P (K) 2, K Th }. Notic that W h is dfind on th dual partition Th, which is a common fatur of finit volum mthods. Lt th discrt spac for prssur b dfind by Q h = {q L 2 (Ω) : q T P (T), T T h }. For vctors v and n, lt v n dnot th matrix whos ijth componnt is v i n j as in [3]. For two matrix valud variabls σ and τ, w dfin σ : τ = 2 i,j= σ ijτ ij. Lt b an intrior dg shard by two lmnts K and K 2 in T h, and lt n and n 2 b unit normal vctors on pointing xtrior to K and K 2, rspctivly. W dfin th avrag { } and jump [ ] on for scalar q, vctor w and matrix τ, rspctivly, by and {q} = 2 (q K + q K2 ), [q] = q K n + q K2 n 2, {w} = 2 (w K + w K2 ), [w] = w K n + w K2 n 2, {τ} = 2 (τ K + τ K2 ), [τ] = τ K n + τ K2 n 2. W also dfin a matrix valud jump [[ ]] for a vctor w as [[w]] = w K n + w K2 n 2 on. If is an dg on th boundary of Ω, dfin {q} = q, [w] = w n, {τ} = τ, [[w]] = w n. Lt V (h) = V h + (H 2 (Ω) H(Ω)) 2 and Q(h) = Q h + (H (Ω) L 2 (Ω)). Dfin a mapping γ : V (h) W h by γv K = {v}ds for all K Th, h whr E h is th dg associatd with th dual volum K and h is th lngth of.

4 4 J. WANG, Y. WANG, AND X. YE Finally, w introduc thr bilinar forms which will b usd to dscrib a finit volum schm in th coming sction. A (v,w) = v K Th K n (γw)ds, B(v,q) = vqdx, T T T h (4) C(v,q) = q(γv) nds. K T h 3. A wak formulation K Th objctiv of this sction is to driv a discrt wak formulation that will lad to a finit volum schm for th Stoks problm. Th main ida is to apply th consrvativ principl on ach dual lmnt K Th and thn sk for an approximat solution from th rgular trial finit lmnt spac V h Q h. To this nd, w multiply () and (2) by ξ W h and q Q h, rspctivly, and using intgration by parts to obtain u n ξds + (5) pξ nds = (f, ξ) (6) K T h K K T h K T uqdx =, whr n is th unit outward normal vctor on K. With th hlp of notation (4), it is asily sn that th solution (u,p) of ()-(3) satisfis (7) (8) A (u,v) + C(v,p) = (f, γv) for all v V (h), B(u,q) = for all q Q h. Whn rstricting all th functions in (7) and (8) into appropriat finit lmnt spacs, th bilinar forms A (, ) and C(, ) can b rprsntd in a way that rsmbls thos from th standard finit lmnt mthods. Such a rprsntation can shd light on a finit volum schm that is stabl and accurat. Th rst of this sction shall discuss various rprsntations for th bilinar forms A (, ) and C(, ). Dnot by (, ) Th th discrt L 2 innr-product as th summation of L 2 innrproducts ovr all lmnt T T h. Th following two lmmas giv quivalnt rprsntations for A (, ) and C(, ). Thir proof will b givn in Appndix A. Lmma. For v,w V (h), w hav A (v,w) =( v, w) Th { v} : [[w]]ds E h (9) + E h Furthrmor, if v V h and w V h, thn () A (v,w) = ( v, w) Th E h [ v] {γw w}ds + ( v,w γw) Th. { v} : [[w]]ds. Lmma 2. For (v,q) V (h) Q(h), w hav () C(v,q) = ( v,q) Th + (v γv) nq ds + ( q,γv v) Th. T

5 A NEW FINITE VOLUME METHOD FOR THE STOKES PROBLEMS 5 Furthrmor, if q Q h, thn (2) C(v,q) = ( v,q) Th = B(v,q). 4. Numrical schms In this sction, w propos and study thr discrt numrical schms basd on a stabilization of th wak formulation (7) (8). To this nd, w introduc a bilinar form as follows: A (v,w) = A (v,w) + α h [[v]] : [[w]]ds, E h whr α > is a paramtr to b dtrmind latr. Not that th bilinar form A (, ) has th rprsntation () whn th argumnts fall into th finit lmnt spac V h. Thus, a skw-symmtric and a symmtric variation of A (, ) can b dfind as follows: (skw-symmtric) A 2 (v,w) = A (v,w) + { w} : [[v]]ds, E h (symmtric) A 3 (v,w) = A (v,w) { w} : [[v]]ds. E h Notic that if u is smooth nough (.g., u (H 3/2+ε (Ω)) 2 for ε > ), thn A i (u,v) = A (u,v) for all v V h and i =,2,3. Dfin a bilinar form D(r,q) as D(r,q) = β E h h [r][q]ds, whr β. Th finit volum schm sks (u h,p h ) V h Q h such that (3) (4) A(u h,v) + C(v,p h ) = (f, γv) for all v V h, B(u h,q) + D(p h,q) = for all q Q h, whr A(, ) is on of A i (, ), i =,2,3. This finit volum mthod is consistnt, i.. th solution (u, p) of th Stoks quations () (3) also satisfis th systm: (5) (6) A(u,v) + C(v,p) = (f, γv) for all v V h, B(u,q) + D(p,q) = for all q Q h. It can b shown that th discrt systm (3) (4) is wll-posd. In fact, it follows from Lmma 2 that C(v,p h ) = B(v,p h ). In addition, it is wll known that for BDM lmnts, th divrgnc oprator from V h to Q h is onto. Hnc by th mixd finit lmnt thory [4, 24], on only nds to vrify that A(, ) and B(, ) ar continuous and A(, ) is corciv with rspct to a suitably dfind norm. Spaking

6 6 J. WANG, Y. WANG, AND X. YE of norms, lt us introduc th following norms and smi-norms on V (h): v 2,h = v 2,T, v 2 = v vds, T T h v 2 = v 2,h + h [[v]] 2, v 2 = v 2 + h 2 T v 2 2,T, E h [q] 2 = β E h h [q] 2, whr h T is th maximum dg lngth of T. Th standard invrs inquality implis that (7) v v v V h. Lt b an dg of any triangl T. It is wll known [] that for any function g H (T), (8) g 2 ( h T g 2 T + h T g 2,T). Th corcivity of th bilinar form A(, ) is givn in two coming lmmas. Th proof of ths lmmas rquirs th following inqualitis on th approximability of γ, whos proof will b givn in Appndix B: (9) (2) T v γv Th h v (v γv) n 2 ds h v 2 Lmma 3. For v,w V (h) and r, q Q(h), w hav (2) (22) (23) A i (v,w) v w for i =,2,3, ( C(v,q) v q 2 + ) /2 h 2 T q 2,T, D(r,q) [r] [q] [r] ( Furthrmor, if (v,q) V h Q h, thn (24) C(v,q) v q. q 2 + for all v V (h), for all v V (h). h 2 T q 2,T ) /2, Proof. It follows from th Cauchy-Schwarz inquality and inquality (8) that ( ) /2 ( { v} : [[w]]ds E h { v} 2 h [[w]] 2 h E h E h ( v 2,h + ) /2 ( /2 (25) h 2 T v 2 2,T h [[w]] ) 2 v w. E h ) /2 Similarly, by using th Cauchy-Schwarz inquality and inqualitis (8), (9), it can b shown that [ v] {γw w}ds v w. E h

7 A NEW FINITE VOLUME METHOD FOR THE STOKES PROBLEMS 7 Thrfor, by using th rprsntation (9) of Lmma, inqualitis (8), (9), and th Cauchy-Schwarz inquality, w obtain A (v,w) v,h w,h + v w + v Th h w ( ) /2 ( +α h [[v]] 2 h [[w]] 2 E h E h v w. ) /2 Th sam argumnt can b applid to handl th bounddnss of A 2 (, ) and A 3 (, ). As to th bounddnss of th bilinar form C(, ), by using th rprsntation () of Lmma 2, inqualitis (8), (9) and (2), w hav ( C(v,q) v,h q + h q 2 + ) /2 h q 2,T h /2 v + q h v Th ( v q 2 + ) /2 h 2 T q 2,T. This complts th proof of th bounddnss stimat (22). (23) can b obtaind similarly. Th rquird inquality (24) follows immdiatly from th standard invrs inquality for finit lmnt functions. Lmma 4. For all v V h, w hav th following corcivity rsults: (26) A i (v,v) v 2 for i =,2,3, providd that α > for i = 2 and sufficintly larg valus of α for i =, 3. Proof. Using th rprsntation () of Lmma, th inquality (25), and th standard invrs inquality for finit lmnt functions, w obtain A (v,v) = ( v, v) Th { v} : [[v]]ds + α h [[v]] 2 ds E h E h v 2,h + α E h h [[v]] 2 C v,h ( E h h [[v]] 2 ) /2 v 2 v 2, whr C is a positiv constant rlatd to th invrs inquality. Hr, th last lin is obtaind by using th arithmtic-gomtric man inquality and by choosing α larg nough. Th corcivity of A 3 (, ) can b stablishd in a similar mannr. For A 2 (, ), w hav A 2 (v,v) = ( v, v) Th + α h [[v]] 2 ds E h min(,α) v 2 min(,α) v 2. Th proof of Lmma 4 indicats that, for i =,3, th valu of α dpnds upon th constant in th invrs inquality. Thrfor, th minimum valu of α for which A (, ) is corciv is msh dpndnt. Although this dpndnc is thortically wak, it may still impos som inconvninc in practical computation bcaus

8 8 J. WANG, Y. WANG, AND X. YE th valu of th paramtr α has to b accuratly stimatd in ordr to hav a mathmatically wll justifid numrical schm for th Stoks problm. To avoid th difficulty of slcting paramtrs, on is rcommndd to us th bilinar form A 2 (, ) which is paramtr insnsitiv. 5. Error stimats In this sction, w assum that α is proprly slctd so that A(, ) is corciv. To stablish rror stimats, w first nd to vrify th discrt inf-sup condition [4]. Lmma 5. Th bilinar form B(, ) satisfis th discrt inf-sup condition B(v,q) (27) sup q for all q Q h. v V h v Proof. Lt Π : (H (Ω)) 2 V h b th local intrpolation with rspct to th dgrs of frdom of th BDM lmnt. It has th following proprtis [4]: B(v Π v,q) =, for all q Q h v Π v s,t h t s v t,t, for all T T h, s =,, and t 2. Hnc it is not hard to s that v Π v v for all v (H (Ω)) 2. Thn, it follows from v = v v and th triangl inquality that (28) Π v v. To vrify (27), w first us th oprator Π to obtain B(v,q) B(Π v,q) B(v,q) (29) sup sup = sup v V h v v (H Π (Ω))2 v v (H Π (Ω))2 v. Obsrv that by using (28), and (7), w hav for all v (H (Ω)) 2 (3) Π v Π v v. Thus, substituting (3) into inquality (29) givs B(v,q) B(v,q) sup sup q, v V h v v (H v (Ω))2 whr w hav usd th inf-sup condition for th continuous cas [4, 9]. 5.. Error stimat in H L 2. W first stablish an optimal-ordr rror stimat for th vlocity in -norm and for th prssur in th L 2 -norm. Obsrv that th solution of th Stoks problm () (3) satisfis quations (5) (6) and th discrt solution satisfis (3) (4). Thus, th rror stimat in H L 2 can b drivd by following a routin procdur in th thory for mixd finit lmnt mthods [4]. Thorm. Lt (u h,p h ) V h Q h b th solution of (3) (4) and (u,p) (H 2 (Ω) H (Ω)) 2 (L 2 (Ω) H (Ω)) b th solution of () (3). Thn, on has (3) u u h + p p h u Π u + p Π 2 p + h p.

9 A NEW FINITE VOLUME METHOD FOR THE STOKES PROBLEMS 9 Proof. Lt Π b dfind as in th proof of Lmma 5, and Π 2 b th L 2 projction from L 2 (Ω) to th finit lmnt spac Q h. Lt (32) ǫ h = u h Π u, η h = p h Π 2 p b th rror btwn th finit volum solution (u h,p h ) and th projction (Π u,π 2 p) of th xact solution. Dnot by (33) ǫ = u Π u, η = p Π 2 p th rror btwn th xact solution (u, p) and its projction. Subtracting (3) and (4) from (5) and (6), rspctivly, w hav (34) (35) A(ǫ h,v) + C(v,η h ) = A(ǫ,v) + C(v,η), B(ǫ h,q) + D(η h,q) = B(ǫ,q) + D(η,q), for all v V h and q Q h. By stting v = ǫ h in (34) and q = η h in (35), and using Lmma 2 and th fact B(ǫ,η h ) =, th sum of (34) and (35) givs (36) A(ǫ h,ǫ h ) + D(η h,η h ) = A(ǫ,ǫ h ) + C(ǫ h,η) + D(η,η h ). Thus, it follows from th corcivity (26) and th bounddnss (2), (22), (23) that ( ǫ h 2 + [η h ] 2 ǫ ǫ h + η 2 + ) /2 h 2 T η 2,T ( ǫ h + [η h ] ), which implis that ǫ h + [η h ] ǫ + η + ( h 2 T η 2,T ) /2. Th abov stimat, combind with th triangl inquality and th wll-known H stability of th L 2 projction Π 2, givs (37) u u h + [p p h ] u Π u + p Π 2 p + h p, which complts th stimat for th vlocity approximation. As to th rror for th prssur approximation, it follows from th discrt infsup condition (27), th rprsntation (2) of Lmma 2, inqualitis (2), (22), and (37) that B(v,p h Π 2 p) C(v,Π 2 p p h ) p h Π 2 p sup = sup v V h v v V h v = A(u h u,v) + C(v,p Π 2 p) sup v V h v u u h + p Π 2 p + ( h 2 T p Π 2 p 2,T u Π u + p Π 2 p + h p. ) /2 Thn th dsird stimat (3) follows by using th standard triangl inquality.

10 J. WANG, Y. WANG, AND X. YE 5.2. An L 2 rror stimat for th vlocity approximation. W first introduc a dual problm: find (w,λ) H (Ω) 2 L 2 (Ω) satisfying (38) (39) (4) w + λ = u u h in Ω, w = in Ω, w = on Ω. As assumd arlir, this Stoks problm also has H 2 (Ω) H (Ω)-rgularity and th following a priori stimat holds tru: (4) w 2 + λ u u h. Lt w I V h b th continuous picwis linar intrpolation of w, thn th jump trm [[w w I ]] is zro and it is not hard to s that (42) w w I + λ Π 2 λ h u u h. For convninc, dnot (v,w) Eh = E h v w ds. W hav th following optimal ordr L 2 stimat for th vlocity. Thorm 2. Lt (u,p) (H 2 (Ω) H (Ω)) 2 (L 2 (Ω) H (Ω)) and (u h,p h ) V h Q h b th solutions of ()-(3) and (3)-(4), rspctivly, with A(, ) = A 3 (, ). Assuming th body forc f is in (H (Ω)) 2, thn u u h h( u u h + p p h + h f ). Proof. Tsting (38) by u u h, thn using quation (59) and th fact that (u u h ) = and (u u h ) n is continuous across intrnal dgs of T h, w hav (43) u u h 2 = (u u h, w) + (u u h, λ) = ( (u u h ), w) Th ( w n, u u h ) T = ( (u u h ), w) Th ({ w},[[u u h ]]) Eh. Manwhil, subtracting (3) (4) from (5) (6), and stting th tst function to b (w I,Π 2 λ), yilds (44) (45) A(u u h,w I ) + C(w I,p p h ) =, B(u u h, Π 2 λ) =. Using Lmma, th dfinition of γ, and th fact that w I is continuous, w hav A(u u h,w I ) =( (u u h ), w I ) Th + ( (u u h ), w I γw I ) Th ({ w I }, [[u u h ]]) Eh. Similarly, using Lmma 2, th facts that ((w I γw I ) n, p p h ) T = and T (w I γw I )dx =, w hav C(w I, p p h ) = ( w I, p p h ) Th + ((w I γw I ) n, p p h ) T ( (p p h ), w I γw I ) Th = ( w I, p p h ) Th ( p, w I γw I ) Th.

11 A NEW FINITE VOLUME METHOD FOR THE STOKES PROBLEMS Substituting th abov two quations into (44), subtracting (44) from (43), and using w =, w hav u u h 2 = ( (u u h ), w) Th ({ w}, [[u u h ]]) Eh ( (u u h ), w I ) Th ( u, w I γw I ) Th + ({ w I }, [[u u h ]]) Eh + ( w I, p p h ) Th + ( p, w I γw I ) Th = ( (u u h ), (w w I )) Th + ( u + p, w I γw I ) Th ( (w w I ), p p h ) Th ({ (w w I )}, [[u u h ]]) Eh I + I 2 + I 3 + I 4. W stimat I i, i =,...,4, on by on. Using th Schwarz inquality and inquality (42), w hav I = ( (u u h ), (w w I )) Th u u h w w I h u u h u u h. An lmntary calculation shows that T (w I γw I )dx = for all T T h. By inqualitis (9), (42), w hav I 2 = ( u + p, w I γw I ) Th = (f f, w I γw I ) Th h 2 f w I = h 2 f w I h 2 f u u h, whr f is th picwis constant avrag of f ovr ach triangl. It is clar that I 3 = ( (w w I ), p p h ) Th h u u h p p h Finally, using th Schwartz inquality and inquality (8), w hav ( ) ( 2 I 4 h { (w w I )} 2 h [[u u h ]] 2 E h E h h u u h u u h. Combining all th abov givs This complts th proof. u u h h( u u h + p p h + h f ) An rror stimat for th prssur in ngativ norms. Error stimats in ngativ norms oftn rval important approximation proprtis that th standard, and commonly usd L 2 or H norms ar built to ignor. For xampl, suprconvrgnc is on such proprty that can b idntifid through an rror analysis with ngativ norms. Th goal of this subsction is to stablish som rror stimat for th prssur approximation in th H norm. Th rsult to b prsntd hr suggsts a suprconvrgnc for th prssur approximation whn corrct postprocssing tchniqus ar applid. Considr th following dual problm: find (ω,ξ) H (Ω) 2 L 2 (Ω) such that (46) (47) (48) ω + ξ = in Ω, ω = φ in Ω, ω = on Ω, whr φ H (Ω) is a givn function with th corrct compatibility condition. W assum th H 2 (Ω) H (Ω)-rgularity for th solution of th problm (46)-(48): (49) ω 2 + ξ φ. ) 2

12 2 J. WANG, Y. WANG, AND X. YE Lt ω I V h b th continuous picwis linar intrpolation of ω, thn th jump trm [[ω ω I ]] is zro and (5) ω ω I + ξ Π 2 ξ h φ. Thorm 3. Lt (u,p) (H 2 (Ω) H (Ω)) 2 (L 2 (Ω) H (Ω)) and (u h,p h ) V h Q h b th solutions of ()-(3) and (3)-(4), rspctivly, with A(, ) = A 3 (, ) and β =. Assuming th body forc f is in (H (Ω)) 2, thn p p h h( u u h + p p h + h f ). Proof. Subtracting (3) (4) from (5) (6), and stting th tst function to b (ω I,Π 2 ξ), yilds (5) (52) First, tsting (47) by p p h givs A(u u h,ω I ) + C(ω I,p p h ) =, B(u u h, Π 2 ξ) =. (53) (p p h,φ) = B(ω,p p h ) = B(ω ω I,p p h ) + B(ω I,p p h ). Using Lmma 2 and th fact that ((ω I γω I ) n, p p h ) T and T (ω I γω I )dx ar both zro, w hav (54) B(ω I,p p h ) = C(ω I,p p h ) ( p, ω γω) Th. Using (5) and (54), (53) bcoms (55) (p p h,φ) = B(ω ω I,p p h ) + A(u u h,ω I ) ( p, ω γω) Th. Tsting (46) by u u h, thn using quation (59) and th fact that (u u h ) = and (u u h ) n is continuous across intrnal dgs of T h, w hav (56) = (u u h, ω) + (u u h, ξ) = ( (u u h ), ω) Th ( ω n, u u h ) T = ( (u u h ), ω) Th ({ ω},[[u u h ]]) Eh. Using Lmma, th dfinition of γ, and th fact that ω I is continuous, w hav A(u u h,ω I ) =( (u u h ), ω I ) Th + ( (u u h ), ω I γω I ) Th ({ ω I }, [[u u h ]]) Eh. Substituting th abov quation into (55), thn subtracting (56) from (55), w hav (p p h,φ) = B(ω ω I,p p h ) + ( (u u h ), ω I ) Th + ( u, ω I γω I ) Th ({ ω I }, [[u u h ]]) Eh ( (u u h ), ω) Th + ({ ω}, [[u u h ]]) Eh ( p, ω γω) Th = B(ω ω I,p p h ) ( (u u h ), (ω ω I )) Th + ( u p, ω I γω I ) Th + ({ (ω ω I )}, [[u u h ]]) Eh = I + I 2 + I 3 + I 4. Using th Schwarz inquality and inquality (5), w hav and I = B(ω ω I,p p h ) p p h ω ω I h φ p p h. I 2 = ( (u u h ), (ω ω I )) Th u u h ω ω I h φ u u h.

13 A NEW FINITE VOLUME METHOD FOR THE STOKES PROBLEMS 3 Using th fact T (ω I γω I )dx = for all T T h and th inqualitis (9), (5), w hav I 3 = ( u p, ω I γω I ) Th = (f f, γω I ω I ) Th h 2 f ω I = h 2 f ω I h 2 f φ, whr f is th picwis constant avrag of f ovr ach triangl. Using th Schwartz inquality and inquality (8), w hav ( ) ( 2 I 4 h [[u u h ]] 2 E h h { (ω ω I )} 2 E h h φ u u h. Combining all th abov givs This complts th proof. p p h h( u u h + p p h + h f ). ) 2 6. A divrgnc-fr finit volum formulation Obsrv that th vlocity fild in discrt quations (3) (4) is xactly divrgncfr. Thus, it is natural to solv for th vlocity from a divrgnc fr subspac D h : D h = {v V h ; v = }. In particular, by rstricting th tst function to th subspac D h, th discrt formulation (3)-(4) can b rducd into th following divrgnc-fr finit volum schm: find u h D h such that (57) A(u h,v) = (f,γv) for all v D h. Th abov formulation has svral advantags in practical computation. First, it liminats th prssur from a coupld systm and thrfor avoids th solution of a saddl point problm of vry larg scal. Scondly, th problm (57) is symmtric and positiv dfinit if th form A(v,w) = A 3 (v,w) was usd. Consquntly, th rsulting matrix problm can b solvd by som xisting conjugat gradint mthods. In addition, thr ar mthods availabl for dvloping fficint prconditionrs for symmtric and positiv dfinit problms. Th formulation (57) is particularly attractiv in cass whr th vlocity is th primary variabl of intrst, such as th application of groundwatr flow simulation. In th computational implmntation for (57), on nds to know th structur of th subspac D h. In fact, a computabl basis for th divrgnc-fr subspac D h can b drivd by using th potntial from th Hlmholtz dcomposition. For problms with two spacial variabls, a divrgnc-fr vctor v admits a potntial function φ and ( ) y φ v = curlφ :=. x φ For th Brzzi-Douglas-Marini (BDM) lmnts, th following rsult is wll-known [4, 7, 8]: Thorm 4. Thr xists a on-to-on map curl : S h D h, whr th stramfunction spac S h is dfind as following: (58) S h = {φ H (Ω); φ T P 2 (T), T T h }.

14 4 J. WANG, Y. WANG, AND X. YE According to this thorm, on can driv a computabl basis for D h by simply taking curl of th nodal basis of P 2 conforming lmnts. 7. Numrical xampls In this sction, w prsnt som numrical rsults for th nw discrtization schm. All th numrical xprimnts ar conductd on th Stoks quation dfind on th unit squar domain Ω = (,) (,) with uniform triangulations. Th triangulations ar constructd as follows: () partition th domain into an n n rctangular msh, and (2) divid ach squar lmnt into two triangls by th diagonal lin with a ngativ slop. W us T h to dnot th uniform triangular msh with msh siz h = /n. Hr w only rport rsults for β =. Th rsults for β > ar similar. Th tst problm assums th Stoks problm has an xact solution of u = (u,u 2 ) and p whr and u = 2x 2 (x ) 2 y(y )(2y ), u 2 = 2y 2 (y ) 2 x(x )(2x ) p = x 2 + y 2 2/3. It can b sn that th xact solution satisfis th following conditions: u Ω =, pdx =. Th tstd numrical schm uss th form A(, ) = A 3 (, ) with a paramtr valu of α =. Th rsulting linar systm, which is symmtric but indfinit, was solvd by using th MINRES (minimum rsidual) mthod. Th itration was stoppd whn th rlativ rsidual rachs 2. Th rror on diffrnt mshs ar rportd in Tabl, in which ( /2 u u h Eh = h u u h ) 2. E h Th numrical rsult indicats a convrgnc of ordr O(h 2 ) for th vlocity in th standard L 2 norm, O(h) for th vlocity in an quivalnt H norm, and O(h) for th prssur in th standard L 2 norm. Th numrical rsults ar in accordanc with th thortical prdiction as stablishd in prvious sctions. It was obsrvd that th numrical approximation for th prssur unknown somtims posssss a crtain oscillation around th xact solution (s Figur 2 for th prssur plot and th lft on in Figur 3). To obtain a prssur approximation with bttr accuracy, w invstigatd a simpl postprocssing mthod for th prssur approximation by using a local avraging mthod. Mor prcisly, th postprocssing mthod allows us to construct a nw prssur approximation at ach intrior nodal point by taking th avrag of th prssur approximations at six triangls that shar th nod as a vrtx point. This post-procssd prssur approximation is dnotd by p h and an rror in a discrt maximum norm (at all intrior nodal points) was computd and rportd in th tabl. This rror is dnotd as p p h max in all th tabls. It is clar that th post-procssd prssur has smallr rrors with a much fastr convrgnc. Th numrical plot (s th right plot in Figur 3) shows a much improvd prssur approximation. Th numrical algorithm prsntd in this papr can b xtndd to problms with nonhomognous Dirichlt boundary conditions without any difficulty. For Ω

15 A NEW FINITE VOLUME METHOD FOR THE STOKES PROBLEMS 5 Tabl. Numrical rsults for tst problm : u n =,u t =,α =, with form A 3 and xact solution u = 2x 2 y(x ) 2 (2y )(y ), u 2 = 2xy 2 (2x )(x )(y ) 2, p = x 2 + y 2 2/3 msh siz h u u h u u h u u h Eh p p h p p h max / / / / / / / / / / / / / Asym. Rat xampl, th standard Dirichlt boundary condition of u = g on Ω. can b implmntd by imposing both u n = g n and u t = g t as boundary conditions, whr n is th normal dirction and t is th tangntial dirction of Ω. It should b pointd out that such an implmntation should trat u n = g n as an ssntial boundary condition and u t = g t as a natural boundary condition. Th part of th natural boundary condition corrsponds to a modification of th original right-hand sid, which is (f,γv h ), as follows: A (, ) : (f,γv h ) + α h (g t)[[v h ]]ds, A 2 (, ) : (f,γv h ) + α h (g t)[[v h ]]ds + { v h}(g t)ds, A 3 (, ) : (f,γv h ) + α h (g t)[[v h ]]ds { v h}(g t)ds. Th numrical schm was tstd on svral othr xampls. Th scond tst problm assums th following xact solution for th Stoks problm: ( ) sin(2πx)cos(2πy) u =, p = x 2 + y 2 2/3. cos(2πx) sin(2πy) It is not hard to s that for tst problm 2, th boundary condition is homognous for u n, but non-homognous for u t. Th third tst problm assums th following xact solution ( ) cos(2πx)sin(2πy) u =, p =, sin(2πx) cos(2πy) for which on has u n and u t = on Ω. Again, th tst problms 2 and 3 wr approximatd by using th form A 3 (, ) with α = and th linar solvr assumd a stopping critrion with rlativ rsidual 2. Th rsults ar rportd in Tabls 2 and 3.

16 6 J. WANG, Y. WANG, AND X. YE Tabl 2. Numrical rsults for tst problm 2: u n =,u t,α =, with form A 3 and xact solution of u = sin(2πx)cos(2πy), u 2 = cos(2πx)sin(2πy), p = x 2 + y 2 2/3 msh siz h u u h u u h u u h Eh p p h p p h max / / / / / / / / / / / / / Asym. Ordr In all thr tst cass, w xamind th rror for th post-procssd prssur p h whos valu at ach intrior nod is calculatd by taking avrag of nighboring triangls. Th numrical rsults dmonstrat a convrgnc of ordr btwn O(h.5 ) and O(h 2 ) for th post-procssd prssur approximation. Th prssur approximation p h was thortically and numrically known to b of ordr O(h) accurat. Thrfor, th post-procssd prssur approximation p h is vry likly of suprconvrgnt. Th rror analysis for th prssur shows an accuracy of O(h 2 ) in th H norm, which is blivd to indicat a suprconvrgnc for th prssur approximation with ordr of O(h 2 ) at som spacial locations yt to b dtrmind. Intrstd radrs ar ncouragd to invstigat this suprconvrgnc from a thortical point of viw. W mphasiz that this possibl suprconvrgnc should b dpndnt of th msh uniformity. W also xplord th ffct of diffrnt valus of α on th rrors for a tst problm with xact solution givn by ( ) x(x )(2y ) u =, q = x 2 + y 2 2/3. y(2x )(y ) Obsrv that this tst problm has non-homognous boundary conditions for both th ssntial and th natural data. Again, th numrical tst rsults wr computd by using th form A 3 (, ) with msh siz h = /6. Th MINRES itration stops at a rlativ rsidual of 2. Th rrors at various norms for both th vlocity and th prssur approximation ar rportd in Tabl 4. Sinc A 3 (, ) is only corciv for α larg nough, th xtrmly larg itration numbr (which is not rportd in this tabl) for th cas of α = might indicat that A 3 (, ) is not corciv. It can b sn that th rror in L 2 and H norm for th vlocity approximation is not ffctd much by varying α. Howvr, th rror quantity u u h Eh, which masurs th jump of th vlocity along dgs, is supposd to gt smallr as α incrass. This thortical prdiction is clarly dmonstratd by th numrical xprimnts as shown in Tabl 4. It is also intrsting to not that th prssur

17 A NEW FINITE VOLUME METHOD FOR THE STOKES PROBLEMS 7 Tabl 3. Numrical rsults for tst problm 3: u n,u t =,α =, with form A 3 and xact solution of u = cos(2πx)sin(2πy), u 2 = sin(2πx)cos(2πy), p = msh siz h u u h u u h u u h Eh p p h p p h max / / / / / / / / / / / / / Asym. Ordr Tabl 4. Numrical rsults for tst problm 4: u n,u t, msh siz 6x6, with form A 3 and xact solution of u = x(x )(2y ), u 2 = y(2x )(y ), p = x 2 + y 2 2/3. α u u h u u h u u h Eh p p h p p h max approximation is somwhat snsitiv to th valus of α. For xampl, p p h sms to first dcras and thn incras as th valu of α incrass. Th numrical rsults shown in Tabl 4 suggst that th bst rsult allowd for a givn msh may hav alrady bn rachd at α = 4. Tabl 5 shows th prformanc of th numrical schm with α = 3.5. Th rsults in Tabl 5 should b compard with thos in Tabl 3.

18 8 J. WANG, Y. WANG, AND X. YE Tabl 5. Numrical rsults for tst problm 3: u n,u t =,α = 3.5, with form A 3 and xact solution of u = cos(2πx)sin(2πy), u 2 = sin(2πx)cos(2πy), p = msh siz h u u h u u h u u h Eh p p h p p h max / / / / / / / / / / / / / Asym. Ordr Figur 2. Numrical solution for u (top-lft), u 2 (top-right), and p (bottom-lft) for tst problm 3: α = 3.5, msh=6 6 with form A 3 and xact solution of u = cos(2πx)sin(2πy), u 2 = sin(2πx)cos(2πy), p = Appndix A. Proof of Lmma and 2 A straightforward computation givs v (τn)ds = [τ] {v}ds + (59) {τ} : [[v]]ds. T T T h Eh E h

19 A NEW FINITE VOLUME METHOD FOR THE STOKES PROBLEMS 9 Figur 3. Numrical solution for th prssur (lft) and its postprocssd vrsion p h (right) for tst problm 3: α = 3.5, msh=64 64 with form A3 and xact solution of u = cos(2πx) sin(2πy), u2 = sin(2πx) cos(2πy), p =. Prssur Post procssd Prssur W first prov lmma. Proof. From Figur, it is not hard to s that for v, w V (h), (6) A (v, w) = 3 Z X X T Th j= Aj+ CAj X Z v v γwds γwds. n n Ω Hr w dnot vrtx A4 = A for convninc. Th intgral on Aj+ CAj should b undrstood as intgration along th joint of lin sgmnts Aj+ C and CAj, and th outward normal n is st with rspct to ach subtriangl Si, i =, 2, 3. Using th divrgnc thorm on ach subtriangl Sj shown in Figur, and notic that γw is a constant on ach Sj, w hav 3 Z X X T Th j= 3 Z X X X X v γwds ( v, γw)sj n T Th j= Aj Aj+ T Th Sj T X Z X Z v v (γw w) = w ds + ds ( v, γw)th n n T Th T T Th T X Z v (γw w) =( v, w)th + ds + ( v, w γw)th. n T = (6) Aj+ CAj v γwds n T Th Using (59) and th dfinition of γ, th scond trm bcoms X Z v ds (γw w) n T Th T XZ XZ X Z v γwds { v} : [[w]]ds + = [ v] {γw w}ds. n Ω Eh Eh

20 2 J. WANG, Y. WANG, AND X. YE Combining (6), (6) and th abov givs Equation (9). For v V h, both Eh [ v] {γw w}ds and ( v,w γw) Th vanishs, by th proprtis of V h and γ. This complts th proof of Lmma. Nxt w prov Lmma 2. Proof. Notic that all v V (h) satisfy th boundary condition v n = on Ω. By th dfinition of γ, w hav γv n = on Ω. Hnc, using th divrgnc thorm on ach subtriangl S j for v V (h), = = C(v,q) = 3 j= = T T 3 j= A j+ca j γv nqds A ja j+ γv nqds + (v γv) nqds S j T T ( q, γv) Tj v nqds + ( q,γv) Th (v γv) nqds + ( q,γv) Th ( q,v) Th ( v,q) Th = ( v, q) Th + T (v γv) nqds + ( q,γv v) Th. If q Q h, th third trm in th abov xprssion obviously vanishs. Th scond trm also vanishs sinc (v γv) n is continuous across th dgs of T h and hnc by dfinition w hav (v γv) nds = for all E h. This complts th proof of Lmma 2. Appndix B. Proof of inqualitis (9) and (2) W first prov inquality (9). Sinc ach T T h is composd of thr subtriangls, w hav v γv 2 T h = 3 v γv 2 S j. j= Th proof will b don on ach subtriangl. Lt S b a subtriangl and, with lngth h, b th dg of S that blongs to E h. By th Brambl-Hilbrt lmma, it is asy to s that for all v (H 2 (S)) 2, w hav (62) v v ds S h v,s. h Thrfor, on ach subtriangl whos chosn dg is on Ω, th statmnt in inquality (9) is automatically tru. W only nd to considr subtriangls associatd with intrnal dgs. Lt T,T 2 T h b two triangls sharing dg, with outward normal n,n 2, rspctivly. For v V (h), dnot its valu on T,T 2 by v,v 2. Now w xamin

21 A NEW FINITE VOLUME METHOD FOR THE STOKES PROBLEMS 2 S T 2 T Figur 4. Two triangls sharing dg with subtriangl S in T. th L 2 norm of v γv on a subtriangl S in T that is associatd with dg, as shown in Figur 4. By th dfinition of γ, γv S = h 2 (v + v 2 )ds = (v h 2 [[v]]n )ds. Thus, by inquality (62), th triangl inquality and th Schwartz inquality, v γv 2 S v v ds 2 S + h h 2 [[v]]n ds 2 S h 2 v 2,S + S ( )( ) 4h 2 [[v]]n 2 ds ds h 2 v 2,S + h [[v]] 2. Taking summation ovr all subtriangls givs inquality (9). Nxt, w prov inquality (2). Considr dg on T sid only. Again, sinc v has continuous normal componnt across and by using inquality (8), w hav (v γv) n 2 = v n v n ds 2 h h 2 v n 2, h v 2,S + h 3 v 2 2,S. Sum up ovr all dgs for all triangls in T h, w hav inquality (2). Rfrncs [] D. Arnold, F. Brzzi, B. Cockburn and D. Marini, Unifid analysis of discontinuous Galrkin mthods for lliptic problms, SIAM J. Numr. Anal., 39 (22), [2] Douglas N. Arnold, Richard S. Falk and Ragnar Winthr, Multigrid in H(div) and H(curl), Numr. Math., 85 (2) [3] F. Brzzi, J. Douglas and L.D. Marini, Two familis of mixd finit lmnts for scond ordr lliptic problm, Numr. Math., 47 (985) [4] F. Brzzi and M. Fortin, Mixd and Hybrid Finit Elmnts, Springr-Vrlag, Nw York, 99. [5] Z. Cai, J. Mandl and S. McCormick, Th finit volum lmnt mthod for diffusion quations on gnral triangulations, SIAM J. Numr. Anal., 28 (99), [6] Z. Cai and S. McCormick, On th accuracy of th finit volum lmnt mthod for diffusion quations on composit grids, SIAM J. Numr. Anal., 27 (99), [7] P. Chatzipantlidis, Finit volum mthods for lliptic PDE s: a nw approach, Mathmatical Modlling and Numrical Analysis, 36 (22), [8] S. H. Chou, Analysis and convrgnc of a covolum mthod for th gnralizd Stoks problm, Math. Comp. 27 (997), [9] S. H. Chou and D. Y. Kwak, A covolum mthod basd on rotatd bilinars for th gnralizd Stoks problm, SIAM J. Numr. Anal., 2 (998), [] S. H. Chou and P. S. Vassilvski, A gnral mixd co-volum framwork for constructing consrvativ schms for lliptic problms, Math. Comp., 68 (999), 99-.

22 22 J. WANG, Y. WANG, AND X. YE [] S. Chou and X. Y, Unifid analysis of finit volum mthods for scond ordr lliptic problms, SIAM Numrical Analysis, 45 (27), [2] B. Cockburn and J. Gopalakrishnan, Incomprssibl Finit Elmnts via Hybridization. Part I: Th Stoks Systm in Two Spac Dimnsions, SINUM 43, (25), [3] B. Cockburn, G. Kanschat, D. Schötzau, and C. Schwab, Local discontinuous Galrkin mthods for th Stoks systm, SIAM J. Numr. Anal., 4 (22), [4] M. Crouzix and P. A. Raviart, Conforming and nonconforming finit lmnt mthods for solving th stationary Stoks quation I, RAIRO Anal. Numr. 7 (973), [5] R. Ewing, T. Lin and Y. Lin, On th accuracy of th finit volum lmnt mthod basd on picwis linar polynomials, SIAM J. Numr. Anal., 39, (22), [6] R. Eymard, T. Gallout and R. Hrbin, Finit Volum Mthods, Handbook of Numrical Analysis, Vol. VII, Editd by P.G. Ciarlt and J.L. Lions (North Holland), 2. [7] R.E. Ewing and J. Wang, Analysis of th Schwarz algorithm for mixd finit lmnt mthods, R.A.I.R.O. Modlisation Mathmatiqu Analys Numriqu, 26 (992), [8] R.E. Ewing and J. Wang, Analysis of multilvl dcomposition itrativ mthods for mixd finit lmnt mthods, R.A.I.R.O. Mathmatical Modling and Numrical Analysis, 28 (994), [9] V. Girault and P.A. Raviart, Finit Elmnt Mthods for th Navir-Stoks Equations: Thory and Algorithms, Springr-Vrlag, Brlin, 986. [2] J. Huang and S. Xi, On th finit volum lmnt mthod for gnral slf-adjoint lliptic problms, SIAM J. Numr. Anal., 35 (998), [2] R. H. Li, Z. Y. Chn, and W. Wu, Gnralizd diffrnc mthods for diffrntial quations, Marcl Dkkr, Nw York, 2. [22] R. Lazarov, I. Michv and P. Vassilvski, Finit volum mthods for convction-diffusion problms, SIAM J. Numr. Anal., 33 (996), [23] F. Li and C.-W. Shu, Locally divrgnc-fr discontinuous Galrkin mthods for MHD quations, Journal of Scintific Computing, (25), [24] R.A. Nicolaids, Existnc, Uniqunss and approximation for gnralizd saddl point problms, SIAM J. Numr. Anal., 9 (982), [25] J. Wang and X. Y, Nw finit lmnt mthods in computational fluid dynamics by H(div) lmnts, SIAM Numrical Analysis, 45 (27), [26] J. Wang, X. Wang and X. Y, Finit lmnt mthods for th Navir-Stoks quations by H(div) Elmnts, Journal of Computational Mathmatics, 26, (28), [27] X. Y, On th rlationship btwn finit volum and finit lmnt mthods applid to th Stoks quations, Numr. Mthod for PDE, 7 (2), [28] X. Y, A nw discontinuous finit volum mthod for lliptic problms, SIAM J. Numrical Analysis, 42 (24), [29] X. Y, A discontinuous finit volum mthod for th Stoks problm, SIAM J. Numrical Analysis, 44 (26), Division of Mathmatical Scincs, National Scinc Foundation, Arlington, VA 2223, USA jwang@nsf.gov Dpartmnt of Mathmatics, Oklahoma Stat Univrsity, Stillwatr, OK 7475, USA yqwang@math.okstat.du Dpartmnt of Mathmatics, Univrsity of Arkansas at Littl Rock, Littl Rock, AR 7224, USA xxy@ualr.du