A ROBUST NUMERICAL METHOD FOR THE STOKES EQUATIONS BASED ON DIVERGENCE-FREE H(DIV) FINITE ELEMENT METHODS

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1 A ROBUST NUMERICAL METHOD FOR THE STOKES EQUATIONS BASED ON DIVERGENCE-FREE H(DIV) FINITE ELEMENT METHODS JUNPING WANG, YANQIU WANG, AND XIU YE Abstract. A computational procdur basd on a divrgnc-fr H(div) mthod is prsntd for th Stoks and Navir-Stoks quations in this articl. This mthod is dsignd to find vlocity approximation in an xact divrgnc-fr subspac of th corrsponding H(div) finit lmnt spac. That is, th continuity quation is strongly nforcd a priori and th prssur is liminatd from th calculation. A strngth of this approach is that th saddl point problm for th Stoks quations is rducd to a symmtric positiv dfinit problm in a subspac for which basis functions ar radily availabl. Th rsulting discrt systm can thn b solvd by using xisting sophisticatd solvrs. Th aim of this articl is to dmonstrat th fficincy and robustnss of H(div) finit lmnt mthods for th Stoks and Navir-Stoks quations. Th rsults not only confirm th xisting thortical prdictions, but also rval additional advantags of th mthod in daling with discontinuous boundary conditions. Ky words. finit lmnt mthods, divrgnc-fr, th Stoks problm, th Navir-Stoks quations AMS subjct classifications. Primary, 65N5, 65N3, 76D7; Scondary, 35B45, 35J5. Introduction. On of th main difficultis in solving th Navir-Stoks quations is that th vlocity and th prssur variabls ar coupld in a mixd systm with saddl point proprtis. Rcnt study has rsultd in svral fficint mthods (.g., projction mthods and Uzawa typ itrativ mthods) in ordr to ovrcom this difficulty. In this articl, w ar concrnd with a divrgnc-fr approach which ssntially dcoupls th variabls by computing an approximat vlocity solution of th Stoks and Navir-Stoks quations in a divrgnc-fr subspac, wakly or xactly. Th main objctiv of this articl is to dmonstrat th fficincy and numrical robustnss for a nwly dvlopd H(div) finit lmnt mthods [?,?]. In standard finit lmnt mthods for th Navir-Stoks quations, both prssur and vlocity variabls ar approximatd simultanously [?] by using finit lmnt functions satisfying a stability condition known as th inf-sup condition. This mthod, known as primitiv variabl approach, rsults in a larg saddl point problm for which most xisting numrical solvrs ar lss ffctiv and robust than for dfinit systms. Whil such saddl-point systms can b rducd to dfinit problms for th vlocity unknown dfind in wakly divrgnc-fr subspacs, it is gnrally challnging, if not impossibl, to construct computationally fasibl bass for th rsulting (wakly) divrgnc-fr subspacs. This difficulty significantly limitd th advantags of divrgnc-fr approach in solving th Stoks and Navir-Stoks quations. Division of Mathmatical Scincs, National Scinc Foundation, Arlington, VA 2223 (jwang@ nsf.gov). Th rsarch of Wang was supportd by th NSF IR/D program, whil working at th Foundation. Howvr, any opinion, finding, and conclusions or rcommndations xprssd in this matrial ar thos of th author and do not ncssarily rflct th viws of th National Scinc Foundation. Dpartmnt of Mathmatics, Oklahoma Stat Univrsity, Stillwatr, OK 7475 (yqwang@math.okstat.du). Dpartmnt of Mathmatics, Univrsity of Arkansas at Littl Rock, Littl Rock, AR 7224 (xxy@ualr.du). This rsarch of Y was supportd in part by National Scinc Foundation Grant DMS-62435

2 In addition to th primitiv variabl approach with standard Galrkin mthods, attntion was rcntly paid to numrical mthods by using discontinuous finit lmnts [?,?,?,?,?,?,?]. But, onc again, most of thm rsult in a numrical schm in which th vlocity is approximatd in a wakly divrgnc-fr subspac. Furthrmor, th construction of computationally fasibl basis functions was only possibl for som spcial lmnts with crtain particular ordrs [?,?,?,?,?,?,?,?]. To th author s knowldg, thr is no systmatic approach for constructing basis functions for (wakly) divrgnc-fr finit lmnt subspacs in th litratur. An altrnativ way in approximating th Stoks and Navir-Stoks quations is to us H(div) conforming finit lmnts [?,?,?]. Th main motivation and advantag of using H(div) conforming lmnts for fluid flow problms is that th discrt vlocity fild will b globally xactly divrgnc-fr, assuming that on is approximating th incomprssibl Stoks or Navir-Stoks quations. Sinc xactly divrgnc-fr functions can b writtn as th curl of a potntial/stram function, divrgnc-fr subspacs can thn b constructd from taking curl of corrsponding potntial spacs. In two dimnsional spac, th potntial functional spac, also calld stram functions, is wll undrstood with basis functions radily availabl for computational purposs. In thr dimnsional spac, vctor potntials nd to b considrd and a larg krnl of th curl oprator adds to th complxity of th problm. In this papr, w focus on two-dimnsional problms, with possibl xtnsion to thr-dimnsional cass. Anothr important fatur of using H(div) conforming lmnts is that th mthod offrs a mor dynamic tratmnt of boundary conditions than th standard Garlkin mthods. For xampl, th normal componnt of th vlocity is st as ssntial boundary conditions and is strongly nforcd, but th tangntial componnt of th vlocity is tratd as a natural boundary condition and is wakly imposd. Thus, this approach mpowrs th H(div) mthod for problms with discontinuous boundary conditions. This nic fatur is numrically illustratd in Sction?? for a lid drivn cavity flow problm whr th boundary condition is discontinuous. This papr is organizd as follows. In sction 2, w introduc som standard notation for Sobolv spacs. A variational formulation is givn for th Stoks problm. In sction 3, w prsnt a divrgnc-fr H(div) finit lmnt mthod by using th variational formulation dvlopd prviously. A dtaild dscription of th divrgncfr subspacs for H(div) conforming lmnts in two and thr dimnsions is also givn. An xtnsion to th Navir-Stoks quations is discussd in Sction 4. In Sction??, w prsnt som numrical rsults for thr tst problms, ach with a diffrnt configuration of boundary condition for th vlocity. Finally in Sction??, w conduct som numrical tst for a lid drivn cavity problm. 2. Prliminaris. W start from th Stoks problm: find th vlocity u and th prssur p such that (2.) (2.2) (2.3) u + p = f in Ω, u = in Ω, u = on Ω, whr,, and dnot th Laplacian, gradint, and divrgnc oprators, rspctivly; Ω R d 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. Discussion for inhomognous boundary problms will b givn in Sction 3. For simplicity, th mthod will b prsntd for two-dimnsional problms only. 2

3 Extnsion to thr-dimnsional problms is straight forward. Furthrmor, w assum that Ω is a plan polygonal domain without cracks. 2.. Notation. Lt D b a boundd domain in R 2. W us standard dfinitions for th Sobolv spacs H s (D) and thir associatd innr-products (, ) s,d, norms s,d, and smi-norms s,d for s [?]. Th spac H (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 (Ω) to dnot th subspac of L 2 (Ω) consisting of functions with man valu zro. Throughout th papr, w adopt th convntion that a bold-fac charactr dnots a vctor. Dfin H(div;Ω) as th spac of vctor-valud functions by and with th norm H(div;Ω) = { v : v (L 2 (Ω)) 2, v L 2 (Ω) }, v H(div;Ω) = ( v 2 + v 2) 2. Lt K Ω b a triangl or quadrilatral. For any smooth vctor-valud functions w and v, it follows from th divrgnc thorm that w (2.4) ( w) vdk = ( w, v) K v ds, n K K whr ds rprsnts th boundary lmnt, n K is th outward normal dirction on K, and ( w, v) K = 2 i,j= K K w i x j v i x j dk. Lt τ K b th tangntial dirction to K so that n K and τ K form a right-hand coordinat systm. It follows from th rprsntation that (2.5) v = (v n K )n K + (v τ K )τ K w n K v = (w n K) n K (v n K ) + (w τ K) n K (v τ K ) A variational formulation. To solv th Stoks systm (2.)-(2.3), A discontinuous Galrkin typ formulation and finit lmnt discrtization was introducd in [?]. W follow thir ida and for th radr s convninc, som dtails xtractd from thir papr ar givn in blow. Lt T h b a quasi-uniform triangulation of Ω with charactristic msh siz h. Dfin finit lmnt spacs V h and W h for th vlocity and prssur, rspctivly, by V h = {v H(div;Ω) : v K V k (K) K T h ; v n Ω = } W h = {q L 2 (Ω) : q K W k (K) K T h }, whr n is th outward normal dirction on th boundary of Ω, (V k (K), W k (K)) can b any xisting H(div) conforming finit lmnt pairs with ordr k. For 3

4 n 2 τ K τ 2 n K 2 Fig. 2.. Normal and tangntial vctors for nighbouring triangls. xampl, th Raviart-Thomas lmnt of ordr k (RT k ) [?] or th Brzzi-Douglas- Marini lmnt of ordr k (BDM k ) [?]. Thr nw H(div) conforming lmnts hav bn obtaind by Wang and Y [?]. Thy wr cratd spcially to solv th Stoks and Navir-Stoks quations. In this papr, w focus on th RT k and BDM k lmnts. Both of thm satisfy th discrt inf-sup condition (LBB condition) [?]. Dtails of ths lmnts ar skippd sinc thy can b found in numrous sourcs. Multiplying quation (2.) by any tst function v V h, thn using intgration by parts and quation (2.4), w gt (2.6) ( ( u, v) K K K ) u v ds (p, v) = (f,v). n K Sinc v V h, its normal componnt v n K is continuous across ach intrior dg. Thrfor, it follows from (2.5) that u v ds = (u τ K ) v τ K ds. n K n K By dfining ( h u, h v) = ( u, v) K and substituting th abov quation into (2.6), w obtain (2.7) ( h u, h v) (p, v) (u τ K ) v τ K ds = (f,v). n K W now rformulat th boundary intgrals in (2.7). Lt b an intrior dg shard by two lmnts K and K 2. Dnot unit normal vctors n, n 2 and tangntial dirctions τ, τ 2, rspctivly, on for K and K 2 (as shown in Figur 2.). Dfin th avrag { } and jump [[ ]] on for vctor-valud functions w as follows: {ε(w) } = 2 (n (w τ ) K + n 2 (w τ 2 ) K2 ), K K [[w]] = w K τ + w K2 τ 2. For boundary dg = K Ω, th abov two oprations must b modifid by {ε(w) } = n (w τ ) K, [[w]] = w K τ. Lt E h b th st of all dgs, including boundary dgs, in T h. For sufficintly smooth u (.g., u H 3 2 +ɛ (Ω) for som ɛ > ), it is not hard to s that (u τ K ) v τ K ds = {ε(u) }[[v]]ds. K T K n K h E h 4

5 Substituting th abov into (2.7), w gt (2.8) ( h u, h v) ( v,p) E h {ε(u) }[[v]]ds = (f,v). This givs th first quation in th variational form. For th scond, tsting (2.2) against any q W h yilds (2.9) ( u,q) =. Finally, lt V (h) = V h + (H s (Ω) H(Ω)) 2, with s > 3 2. Dnot by a o (u,v) = ( h u, h v) (2.) {ε(u) }[[v]]ds, E h (2.) b(v,q) = ( v,q), two bilinar forms on V (h) V (h) and V (h) L 2 (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 [?,?,?,?] for dtails. As a rsult, it follows from (2.8) and (2.9) that th xact solution of th 2D Stoks problm satisfis th following variational quations: (2.2) (2.3) a o (u,v) b(v,p) = (f,v) v V h, b(u,q) = q W h. 3. Divrgnc-fr finit lmnt mthod. In this sction, divrgnc-fr finit lmnt schms basd on th wak formulation (2.2) (2.3) will b prsntd. 3.. Finit lmnt discrtization and rror stimats. First, w introduc a symmtric and a skw symmtric bilinar forms on V (h) V (h) as follows: a s (w,v) = a o (w,v) + ( αh [[w]][[v]] {ε(v) }[[w]] ) ds, E h a ns (w,v) = a o (w,v) + E h ( αh [[w]][[v]] + {ε(v) }[[w]] ) ds. whr α > is a paramtr, and h is th lngth of th dg. Thn th discrt problm can b writtn as [?]: Algorithm. Find (u h ;p h ) V h W h such that (3.) (3.2) a(u h,v) b(v,p h ) = (f,v) v V h, b(u h,q) = q W h, whr a(v,w) = a s (v,w) or a(v,w) = a ns (v,w) Rmark. As is usual in th practic of discontinuous Galrkin mthods, in a(, ), th trm αh [[w]][[v]] ds is dsignd to nsur stability of th formulation. Th paramtr α should b larg nough to guarant a good convrgnc rat, and as small as possibl in ordr to kp th condition numbr of th discrt systm low. 5

6 Th wll-posdnss of Algorithm coms from th wll-known discrt inf-sup condition [?] and th following lmma [?]: Lmma 3.. Th symmtric bilinar form a s (, ) is corciv for sufficintly larg α, and th skw symmtric bilinar form a ns (, ) is corciv for any α >. Error stimats for Algorithm ar also givn in [?]. W first introduc two norms and on V (h) as follows: (3.3) (3.4) v 2 = v 2,h + E h h [[v]] 2, v 2 = v 2 + E h h {ε(v) } 2, whr v 2,h = v 2,K and v 2 = v vds. Thorm 3.2. Lt (u;p) b th solution of (2.) (2.3) and (u h ;p h ) V h W h b obtaind from (3.) (3.2). Assum α larg nough if a(v,w) = a s (v,w). Thn, thr xists a constant C indpndnt of h such that (3.5) u u h + p p h Ch k ( u k+ + p k ). Furthrmor, if th Stoks problm has th (H 2 ) 2 H -rgularity proprty, thn (3.6) u u h Ch k+ ( u k+ + p k ) providd that (u;p) (H k+ (Ω)) 2 H k (Ω) with k Divrgnc-fr schm. Th Algorithm, as dscribd in th prvious sction, introducs a coupld saddl point systm. In ordr to solv this systm fficintly, nxt, w will focus on dcoupling it by using th divrgnc-fr finit lmnt mthod. Dfin th divrgnc-fr subspac D h of V h by D h {v V h ; b(v,q) =, q W h }. By proprtis of th BDM k and RT k finit lmnt spacs, w hav V h = W h [?]. Thrfor, it is asy to s that D h = {v V h ; v = }. In othr words, D h is globally xactly divrgnc-fr. W point out that this is usually not tru for H conforming vlocity discrtizations, and functions in D h will thn only b wakly divrgnc-fr. By taking th tst function in D h, th discrt formulation (3.)-(3.2) can b rducd into th following divrgnc-fr finit lmnt schm: Algorithm 2. Find u h D h such that for all v D h (3.7) a(u h,v) = (f,v). Problm (3.7) is symmtric positiv dfinit if w choos a(v,w) = a s (v,w), which brings grat advantag in numrical simulation sinc it can b solvd by th fficint conjugat gradint mthod. Also, dvloping prconditionrs for symmtric positiv dfinit quations is considrably asir than for othr systms. 6

7 Nxt, w giv a computabl basis for th divrgnc-fr subspac D h by using th potntial from Hlmholtz dcomposition. In two-dimnsion, a divrgnc-fr vctor v admits a potntial function φ and ( ) y φ v = curlφ :=. x φ Th 2D potntial φ is usually calld th stram-function in litratur. Th Hlmholtz dcomposition also holds for most 2D H(div) conforming finit lmnt spacs [?,?,?]. For th Raviart-Thomas (RT) and Brzzi-Douglas-Marini (BDM) lmnts, th following rsult is wll-known [?,?,?]: Thorm 3.3. Thr xists a on-to-on map curl : S h D h whr th stramfunction spac S h is dfind as following:. for triangular RT lmnt of ordr k or BDM lmnt of ordr k, S h = {φ H (Ω); φ K P k+ (K), K T h }; 2. for rctangular RT lmnt of ordr k, S h = {φ H (Ω); φ K Q k+ (K), K T h }; whr P k (K) = span{x i y j i + j k} and Q k (K) = span{x i y j i, j k}. According to th thorm, on can simply tak curl of th nodal basis of P k+ or Q k+ conforming spacs, and driv a computabl basis for D h. In thr-dimnsion, th divrgnc-fr vctor fild can similarly b idntifid as th curl of a vctor potntial. Hr curl is th usual vctor curl which maps 3D vctors into 3D vctors. Lt V h b th Raviart-Thomas discrtization of H(div) with ordr k and S h b th Ndlc dg discrtization of H(curl) with ordr k. Th following rsult is wll known [?]: Thorm 3.4. Lt D h b th globally divrgnc-fr subspac of V h, thn D h = curls h. Th thr-dimnsional cas is significantly mor complicatd sinc th 3D curl oprator has a fairly larg krnl containing all gradint vctors. Hnc it is usually not practical to driv a basis of D h from a basis of S h in thr dimnsion. Although thr ar som rsults in this dirction [?,?], thy ar ithr complicatd or carrying many limitations. For thr-dimnsional problm, anothr possibl solution is to solv th problm dirctly on curl{φ h }, whr {φ h } is a basis for S h. Notic that curl{φ h } is linarly dpndnt and hnc is not a basis for D h. Discrtization using a linarly dpndnt spanning st lads to a singular linar algbraic systm. Howvr, w know that many Krylov subspac itrativ solvrs can handl singular systms wll as long as th right-hand sid and th initial guss is orthogonal to th null spac of th matrix. With carful dsign, th problm may still b solvabl using Krylov subspac solvrs Inhomognous boundary conditions. In practical computation, many Stoks problms ar imposd with inhomognous boundary conditions for th vlocity. Hr w gnraliz th divrgnc-fr finit lmnt mthod to th following problm: u + p = f in Ω, u = in Ω, u = g on Ω. 7

8 Th boundary data can b sparatd into two parts: u n = g n and u τ = g τ. Th normal componnt u n part will b posd as an ssntial boundary condition. In othr words, w sk for discrt solutions from th following finit lmnt spac Ṽ h = {v H(div;Ω) : v K V k (K) K T h ; v n Ω = I h (g n)}, whr I h (g n) is a suitabl nodal valu intrpolation on Ω, basd on th dgrs of frdom of th discrt spac for v n on Ω. Th tangntial componnt u τ will b tratd as a natural boundary condition, that is, it will b imposd wakly through boundary intgrals. Lt E B h b th st of all boundary dgs in T h, thn Algorithm should b modifid as: find (u h ;p h ) Ṽh W h such that a(u,v) b(v,p) = (f,v) + (g τ)(αh v τ E B h (v τ) n )ds v V h, b(u,q) = q W h, whr th minus is takn whn using th symmtric form a s (, ) and th plus for a ns (, ). As usual in trating inhomognous boundary problms, th tst spac should still carris th homognous boundary condition, which mans v is in V h instad of Ṽh. Similar changs nd to b mad whn using th divrgnc-fr schm. Th right-hand sid in algorithm 2 should b modifid consquntly. Furthrmor, sinc th computational basis of D h is drivd from S h, w nd to b carful whn imposing th ssntial boundary condition u n = g n. Indd, w impos th ssntial boundary condition on S h. Lt u D h and φ S h satisfis u = curlφ. Thn φ τ = u n = g n on Ω. Thrfor, on only nds to impos th following ssntial boundary condition on S h and comput th basis accordingly: x φ(x) = φ(x ) + g nds x x Ω. Hr x is an arbitrary point on Ω and th intgral is takn countr-clockwisly along Ω. This boundary condition for φ is uniqu only up to a constant φ(x ). Howvr, th constant will go away aftr taking curl. In othr words, on is fr to choos any x and φ(x ) in th implmntation. 4. Extnsion to th Navir-Stoks quations. Th goal of this sction is to apply th divrgnc-fr finit lmnt mthod on th following 2D Navir-Stoks quations: (4.) (4.2) (4.3) ν u + u u + p = f in Ω, u = in Ω, u = on Ω, 8

9 whr ν is th fluid viscosity, and u u should b viwd as a row vctor u tims a matrix u from lft with [ ] x u u = x u 2. x2 u x2 u 2 Multiply quation (4.) by any v V h and us intgration by parts, w obtain ( ) u ν( u, v) K ν v ds + (u u,v) K (p, v) = (f,v). K n K W prsnt a tratmnt of th nonlinar trm (u u,v) K by adding som stabilization trms. Dfin a trilinar form ( ) a sk (u,v,w) := (u v,w) K (u w,v) K. 2 It is skw symmtric in th last two variabls. Through a straight forward us of intgration by parts, on arrivs at th following idntity (s,.g. [?]): (u v,w) K = a sk (u,v,w) (( u)v,w) K 2 + (u n)(v w)ds 2 for all u,v,w V h. In particular, if u,v (H(Ω)) 2 and u =, thn (4.4) (u v,w) K = a sk (u,v,w) + (u n)(v w)ds. 2 K T K h K E h Sinc v = on th boundary of Ω, w hav (4.5) n)(v w)ds = 2 K(u u n(v L w L v R w R )ds, 2 whr E h is th collction of all intrior dgs, n is a fixd orintation of E h, w L is th trac of w on as sn from th lft, and w R is th trac of w on as sn from th right. Mor prcisly, w L and w R ar dfind as follows: w L (x) = lim t + w(x tn), w R (x) = lim t + w(x + tn). Not that v (H (Ω)) 2 implis v L = v R on ach intrior dg. It follows that v L w L v R w R = v R w L v L w R. Substituting th abov into (4.5) yilds n)(v w)ds = 2 K(u u n(v R w L v L w R )ds, 2 E h 9

10 which, togthr with (4.4), implis (4.6) (u v,w) K = a sk (u,v,w) + u n(v R w L v L w R )ds 2 K T h E h for any w V h and u,v (H (Ω)) 2 with u =. Th right-hand sid of (??) can b furthr stabilizd as follows: (4.7) (u v,w) K = a sk (u,v,w) + u n(v R w L v L w R )ds + γ u n [[v]][[w]]ds, 2 E h E h whr γ > is a stabilization paramtr. Th right-hand sid of (??) provids a suitabl wak form for th nonlinar inrtial trm of th Navir-Stoks quations. W dnot it by a (u,v,w). Thn th finit lmnt schm for (4.) (4.3) can b prsntd as: Algorithm 3. Find (u h ;p h ) V h W h such that (4.8) (4.9) a(u h,v) + a (u h,u h,v) b(v,p h ) = (f,v) v V h, b(u h,q) = q W h, whr a(v,w) and b(v,q) ar dfind in (3.) and (2.) rspctivly. Similarly, th divrgnc-fr finit lmnt approximation for th Navir-Stoks quations can b writtn as: Algorithm 4. Find u h D h such that for all v D h (4.) a(u h,v) + a (u h,u h,v) = (f,v). 5. Numrical xprimnts for th Stoks quations. Numrical rsults for th two dimnsional Stoks quations ar prsntd in this sction. Th divrgncfr finit lmnt schm introducd in Algorithm 2 is usd. Th main objctiv hr is to numrically xamin th accuracy and fficincy of th H(div) schm. For simplicity, a rctangular computational domain with uniform rctangular grids ar usd in this numrical study. Th Raviart-Thomas finit lmnt of ordr k = (RT ) is mployd in th finit lmnt discrtization. Only th symmtric bilinar formulation a s (, ) is tstd. Th discrt systm is solvd fficintly by th conjugat gradint mthod with rlativ rsidual ε =. 8. Lt u b th xact vlocity and u h b its divrgnc-fr finit lmnt approximation obtaind from Algorithm 2. Th rror is calculatd by computing various norms or smi-norms of P h u u h, whr P h is th nodal valu intrpolation into Q 2 conforming finit lmnt spacs. This provids an accurat and ffctiv mthod for computing th rror undr various norms. To b mor prcis, w introduc th

11 following notations ( ) /2 E = ( (P h u u h ), (P h u u h )) K, ( /2 ( ) /2 E 2 = h [[(P h u u h )]] ) 2, E 3 = h {ε(p h u u h )} 2. E h E h Clarly, w hav P h u u h 2 = E 2, P h u u h 2 = E2 + E 2 2, P h u u h 2 = E 2 + E E 2 3. Th domain is givn by Ω = (,) (,), which is partitiond into uniform rctangular grids along th x- and y- axs. W xamin th rror for th symmtric formulation a s ( ; ), with various valus for th stabilization paramtr α. 5.. Som numrical rsults for tst problm. Th tst problm is a Stoks systm with xact solutions givn as follows: ( ) x(x )(2y ) u = 2xy(x )(y ), p =. y(y )(2x ) It is clar that homognous boundary condition is satisfid by this vlocity. Tabl 5. Numrical prformanc for tst problm, using th symmtric formulation a s with paramtr valu α =. msh itration E E 2 E 3 L 2 L asym. ordr O(h k ), k = Tabl 5.2 Error information for tst problm, using th symmtric formulation a s with α =. msh itration E E 2 E 3 L 2 L asym. ordr O(h k ),k = Th numrical rsults in tabls?? and?? show an O(h) convrgnc for th rror norm and O(h 2 ) convrgnc for th L 2 rror norm, which agrs wll with th thortical rsults in Thorm 3.2. Th rat of convrgnc is illustratd in figur??. It also shows that E 2 and E 3, which ar rlatd to th jump on intrnal dgs, ar

12 Fig. 5.. Rat of convrgnc for tst problm, using th symmtric formulation a s with α = Error for tst problm 2 E E rror E 2 L 2 L O(h) O(h 2 ) 2 Tabl 5.3 Numrical prformanc for tst problm 2, using th symmtric formulation a s with α =. h msh itration E E 2 E 3 L 2 L asym. ordr O(h k ),k = usually of highr ordr accuracy than th discrt smi H -norm E. This phnomna was not prdictd by any xisting convrgnc thory. Furthrmor, th L norm of th rror sms to b of ordr O(h 2 ) accuracy, though no thortical proof can b sn in any xisting litratur Som numrical rsults for tst problm 2. Th tst problm 2 is a Stoks systm with xact solutions givn as follows: u = ( ) sin (2π x) cos (2π y), p = x 2 + y 2. cos (2 π x) sin(2π y) It is not hard to s that th following natural boundary conditions ar satisfid by this vlocity: u n Ω =, u τ Ω. Not that th scond boundary condition (along th tangntial dirction) only indicats that a natural boundary condition should b imposd for this problm whn using th H(div) finit lmnt mthods. On would nd to comput u τ Ω from th rprsntation of th vlocity u in th numrical tst. Th rror information and rat of convrgnc for th numrical schm ar illustratd in Tabls?? and??. 2

13 Tabl 5.4 Error information for tst problm 2, using a s, α =. msh itration E E 2 E asym. ordr O(h k ),k = Tabl 5.5 Numrical prformanc for tst problm 3, using th symmtric formation a s with α =. msh itr. E E 2 E 3 L 2 L asym. ordr O(h k ),k = Som numrical rsults for tst problm 3. In tst problm 3, th xact solution of th Stoks quations is givn by ( ) cos (2 π x) sin (2π y) u =, p = x 2 + y 2. sin (2π x) cos (2π y) Th following ssntial boundary conditions ar satisfid by this vlocity: u n Ω, u τ Ω =. Again, on would nd to comput u n Ω from th rprsntation of this vlocity in doing numrical computation. Th convrgnc and rror profil for diffrnt msh configuration ar illustratd in Tabls?? and??. Tabl 5.6 Error information for tst problm 3, using th symmtric formulation a s with α =. msh itr. E E 2 E asym. ordr O(h k ),k = Condition numbr and rror dpndncy on α. W also tstd how th condition numbr of th discrt systm and th rror dpnd on various valus of th stabilization paramtr α. Th xisting thory prdictd that th numrical 3

14 Tabl 5.7 Condition numbr and rror for diffrnt valus of α for tst problm on th 6 6 grid, with symmtric formulation a s α itration condition E E 2 E 3 L mthod is stabl and accurat for sufficintly larg valus of α. Sinc th discrt systm is symmtric and positiv dfinit, th condition numbr of th discrt systm can b convnintly calculatd using stimats for xtrm ignvalus from th conjugat gradint (Lanczos-typ) procss. To this nd, w solv th tst problm with diffrnt valus of α, and compar th rsults in Tabl??. Th condition numbr sms to dpnd linarly on α. As α bcoms largr, th tangntial jump across intrnal dgs will bcom smallr. Hnc it is xpctd that E 2 gts bttr as α incrass. From Tabl??, it sms othr rror norms ar not affctd much by α whn α is larg nough. Fig. 6.. Stramlin portrait of th lid drivn cavity problm, obtaind from th symmtric formulation a s with various valus of th stabilization paramtr α stram contour, alpha= stram contour, alpha= stram contour, alpha=28 stram contour, alpha=

15 6. Numrical xprimnts for a lid drivn cavity problm. In this sction, w rport som numrical rsults on a lid drivn cavity problm, for which th xact solution is not known. Th 2D lid drivn cavity problm dscribs th flow in a rctangular containr which is drivn by th uniform motion of on lid [?]. Th lid drivn cavity problm is on of th most popular bnchmark problms for tsting numrical schms in fluid flow. On of th main difficultis of this problm is that it has a discontinuous vlocity boundary condition and th standard primitiv Galrkin mthods hav difficulty in daling with such discontinuitis without a furthr approximation of th boundary data. In two dimnsional cas, this boundary condition rsults in cornr singularitis for th solution. Fig Stramlin portrait for th lid drivn cavity problm, obtaind from th non-symmtric, but absolutly stabl formulation a ns with various valus of th stabilization paramtr α stram contour, alpha=. (NS) stram contour, alpha=. (NS) stram contour, alpha= (NS) stram contour, alpha=8 (NS) Th 2D lid drivn cavity problm was formulatd in Ω = (,) 2, with boundary condition u = (,) t on th top lid and u = (,) t lswhr. In most dirct numrical simulations of this problm, on has to choos xplicitly th ssntial boundary data on two top cornrs. Popular choics ar th laky typ, whr u = (,) t ; th non-laky typ, whr u = (,) t ; and th rgularizd typ, whr th boundary data on th top lid is rplacd by a smooth function which vanishs at two cornrs. W mphasiz that, in our divrgnc-fr H(div) mthod, th boundary data discontinuity no longr caus any difficulty in th numrical schm. This is so bcaus, as pointd out arlir, only th normal componnt of th boundary data u n (which quals zro and is thus continuous) is imposd as th ssntial boundary condition. Th tangntial componnt u τ, which carris th discontinuity, is imposd wakly through boundary intgrals. Thrfor, th H(div) finit lmnt mthod is a natural fit to th lid drivn cavity problm (of caus th mthod was not motivatd by th lid drivn cavity problm). 5

16 Fig Th first vlocity componnt profil for th lid drivn cavity problm, obtaind by using th symmtric formulation with various valus of α. vlocity u, alpha=8 vlocity u, alpha= vlocity u, alpha= Th computational rsults for th two-dimnsional lid drivn cavity problm ar obtaind by using a uniform rctangular msh with both symmtric and nonsymmtric finit lmnt formulations. W ar spcially intrstd in sing how th numrical solutions ar affctd by th chang of α valus. Th stramlin portraits ar shown in Figurs?? and??, and th vlocity profils ar illustratd in Figurs??,??,??,??, and??. Whil w would lik to lav radrs to draw conclusions, w do lik to point out th following obvious phnomna: Th symmtric schm is non-stabl whn valus of α ar not sufficintly larg. For xampl, th CG mthod did not convrg for th linar systm with α =. This mans that th rquird positiv dfinitnss of th linar systm may fail to b valid for this cas. Th non-symmtric schm is stabl rgardlss th valu of α. Of cours, th systm s corcivity gts wakr and wakr whn th paramtr α is gtting clos to zro from positiv. For small valus of α, th continuity of th vlocity approximation is lss nforcd. This can b sn from th numrical solution in Figurs?? and??. But th discontinuity is supprssd whn th valu of α gts larg as shown in Figur??. In th stramlin portrait, th primary ddy and two cornr ddis ar clarly visibl. Th non-symmtric finit lmnt formulation is quit stabl with rspct to th paramtr valu of α. But xtra car might b ndd for solving th rsulting non-symmtric matrix problm. GMRES was mployd in our numrical xprimnts and th mthod has an accptabl convrgnc for th siz of problm illustratd in this articl. 6

17 Fig Th scond vlocity componnt profil for th lid drivn cavity problm, obtaind by using th symmtric formulation with various valus of α. vlocity u2, alpha=8 vlocity u2, alpha= vlocity u2, alpha= Fig Th vlocity profil for th lid drivn cavity problm, obtaind by using th nonsymmtric formulation with α =.. vlocity u, alpha=. (NS) vlocity u2, alpha=. (NS) Fig Th vlocity profil for th lid drivn cavity problm, obtaind by using th nonsymmtric formulation with α =.. vlocity u, alpha=. (NS) vlocity u2, alpha=. (NS)

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