AN UPSTREAM PSEUDOSTRESS-VELOCITY MIXED FORMULATION FOR THE OSEEN EQUATIONS

Transcription

1 Bull. Koran Math. Soc ), No. 1, pp AN UPSTEAM PSEUDOSTESS-VELOCITY MIXED FOMULATION FO THE OSEEN EQUATIONS Eun-Ja Park and Boyoon So Abstract. An upstram schm basd on th psudostrss-vlocty mxd formulaton s studd to solv convcton-domnatd Osn quatons. Lagrang multplrs ar ntroducd to trat th trac-fr constrant and th lowst ordr avart-thomas fnt lmnt spac on rctangular msh s usd. Error analyss for svral quantts of ntrst s gvn. Partcularly, frst-ordr convrgnc n L norm for th vlocty s provd. Fnally, numrcal xprmnts for varous cass ar prsntd to show th ffcncy of ths mthod. 1. Introducton W consdr th followng Osn quatons αu ν u+b u+ p = f n Ω, 1) dv u = 0 n Ω, u = 0 on Ω, whr Ω s an axs paralll doman n wth Lpschtz contnuous boundary Ω. Lt f = f 1,f ) and ν > 0 b th gvn xtrnal body forc and knmatc vscosty, rspctvly. Dnot u and p to b th vlocty vctor and prssur, rspctvly. For smplcty, w assum that α > 0 and dvb = 0 and b W 1, Ω). Hr w us th standard Sobolv spacs. Th Osn problm occurs as lnarzd Navr-Stoks quatons or oftn arss from an tratv procdur such as Pcard s traton [15]. Gnrally, th Osn quatons ar convcton-domnatd and standard cntrd dffrnc schms or pcws lnar approxmatons produc spurous numrcal oscllatons. In th cas of convcton problm, upwnd or upstram wghtng cvd March 0, Mathmatcs Subjct Classfcaton. Prmary 58B34, 58J4, 81T75. Ky words and phrass. psudostrss-vlocty formulaton, upstram schm, mxd fnt lmnt, Osn quatons. Ths rsarch was supportd by Basc Scnc sarch Program through th Natonal sarch Foundaton of Kora NF) fundd by th Mnstry of Educaton, Scnc and Tchnology NF-011AAA c 014 Th Koran Mathmatcal Socty

2 68 EUN-JAE PAK AND BOYOON SEO tchnqu can b usd [18]. Thr ar varous ways to trat ths problm s, for xampl, [, 5, 6]). Th mxd fnt lmnt mthod has bn succssfully appld to svral aras of ntrst, n partcular, flud flows n porous mda. Ths s manly du to th fact that th mxd mthod satsfs local mass consrvaton proprty and provds accurat fluxs whos normal componnts ar contnuous across ntr-lmnt boundars. Mxd mthods for lnar and nonlnar scond ordr llptc problms ar studd n [4, 0,, 3]. Th standard mxd fnt lmnt mthod to convcton-domnatd dffuson problms gvs solutons wth spurous oscllatons. Hnc w nd to xplot an upstram wghtng schm for th convctontrm n th contxt of th mxd fnt lmnt mthod. Ths da was dvlopd by Jaffr [16] and by Dawson [1] for th scalar convctondffuson problm. Th a postror rror analyss of th upstram wghng mxd schm was analyzd n [17]. Thr, t s shown that th a postror rror stmator s not only rlabl and ffcnt, but also computatonally robust for svral tst problms. Th psudostrss-vlocty formulaton allows to us th avart-thomas mxd fnt approxmaton of th Stoks problm [8, 9, 10, 14]. In ths papr, w propos and analyz th upstram schms basd on th trac-fr psudostrss and vlocty formulaton of th Osn problm. Lagrang multplrs ar ntroducd to trat th trac-fr constrant. W us som of th das prsntd n [19] to obtan rror bounds for vlocty and psudostrss varabls on rctangular msh. Th rmandr of ths artcl s organzd as follows. Th psudostrssvlocty formulaton s drvd n th nxt scton. In Scton 3, w ntroduc th upstram mxd lmnt mthod for th Osn problm. Th convrgnc analyss s gvn n Scton 4. Fnally, numrcal xprmnts ar prsntd n th last scton.. Psudostrss-vlocty formulaton Lt us dscrb som notatons and thn drv wak formulaton. Dnot M to b th fld of matrx functons. For vctor functons v = v 1,v ) T and b = b 1,b ) T, dfn ts gradnt v M as a tnsor and b v as a vctor: v = v 1 x v x v 1 y v y, b v = v 1 b 1 x +b v 1 y v b 1 x +b v y For a tnsor functon τ = τ j ), lt τ = τ 1,τ ) dnot ts th-row for = 1, and dfn ts dvrgnc, normal, and trac by ) ) dvτ1 n τ1 dvτ =, n τ = τn =, trτ = τ dvτ n τ 11 +τ,.

3 AN UPSTEAM PSEUDOSTESS-VELOCITY FOMULATION 69 rspctvly. Lt A : M M b a lnar map dfnd by Aτ = τ 1 trτ)i, whr I s dntty matrx. Introducng th psudostrss varabl ) σ = ν u pi, systm 1) can b wrttn as { κaσ u = 0, 3) dvσ b u αu = f, whr κ = 1/ν. Indd, notc that Aσ s trac fr, hnc th ncomprssblty constrant dvu = 0 s satsfd through dvu = tr u) = 0. Also, th prssur p = 1 trσ s unqu up to a constant. It s clar that th Osn quatons hav a unqu soluton provdd that Ωp = 0, whch mpls 4) trσ = 0. So, w us th followng functon spacs: Ω H := Hdv;Ω) = Hdv;Ω), Σ := {τ H trτ = 0}, Ω V := L Ω) = L Ω), whr Hdv;Ω) = {v L Ω) dvv L Ω)}. Th smpl varatonal problmofthpsudostrss-vloctyformulatonstofndaparσ,u) Σ V such that { κaσ,τ)+dvτ,u) = 0, τ Σ, 5) dvσ,v) Gu,v) = f,v), v V, whr Gu,v) = b u,v)+αu,v). Hr, th nnr product σ,τ) for tnsor functons s Ω σ:τ and u,v) = u v for vctor functons. But, ths wak Ω formulaton gv us dffcults n rror analyss and takng th bass for fnt lmnt spac of Σ. So, w ntroduc a Lagrang multplr to satsfy th tracfr condton 4). Th followng wak form s quvalnt to 5): Fnd a par σ,u,l) H V such that κaσ,τ)+dvτ,u)+dτ,l) = 0, τ H, 6) dvσ,v) Gu,v) = f,v), v V, dσ,µ) = 0, µ, whr dτ,l) = l Ω trτ. W wll us th followng lmma whos proof can b found n [4, 7].

4 70 EUN-JAE PAK AND BOYOON SEO Lmma.1. For any τ Σ, w hav Not that trτ C Aτ + dvτ 1 ). τ = Aτ + 1 trτ, whch, togthr wth Lmma.1, mpls 7) τ C Aτ + dvτ 1 ) C Aτ + dvτ ). 3. Th upstram mxd lmnt mthod Lt h = {,j : 0 n 1, 0 j m 1} b a quas-unform partton of th doman Ω = a,b) c,d) nto a unon of rctangl,j := [x,x +1 ] [y j,y j+1 ] basd on axs parttons: a = x 0 < x 1 < < x n = b, c = y 0 < y 1 < < y m = d. Lt h x = x +1 x, h y j = y j+1 y j and ts ara as,j and dnot four dgs as follows x = {x,y) : y j < y < y j+1 }, x +1 = {x +1,y) : y j < y < y j+1 }, y j = {x,y j) : x < x < x +1 }, y j+1 = {x,y j+1) : x < x < x +1 }. Lt h b th largst msh sz of th rctangluaton,.., h = max,j {h x,hy j }. Th partton h s quas-unform, whch mans that thr xst two constants C 1, C such that C 1 h,j C h. W dfn mxd fnt lmnt spac H h V h H V basd on th lowst avart-thomas spac. For = 1,j,j f 1,j or,j xsts, dfn scalar functon φ x on Qx as follows x x 1 h x f x,y) 1,j, φ x 1 x,y) = x +1 x h x f x,y),j, 0 othrws. Smlarly, for Q y j =,j 1,j f,j 1 or,j xsts, dfn functon φ y j on as follows y y j 1 h y f x,y),j 1, φ y j 1 j x,y) = y j+1 y h y f x,y),j, j 0 othrws. Q y j

5 AN UPSTEAM PSEUDOSTESS-VELOCITY FOMULATION 71 Thn, w dfn fnt lmnt spacs as H h : = T 0 = {τ Hdv;Ω) : τ T 0 ), h } { ) φ x = span 0 0 φ y ) ), j , φ x, 0 0 φ y j )} and { 1 V h : = span 0 ), 0 1 )}. Not that ach row of tnsor functon n H h satsfs th contnuty of th normal componnt of vctor fld at ntrfacs of lmnts. For xposton of upstram schms som notatons ar n ordr. W wll rprsnt normal vctor as two ways. Frst, for gvn dg of an lmnt h wassgnauntnormalvctorn, whchs thsamasth x drcton or y drcton. Thn, gvn a par, n ) wth an ntror dg, on can unquly dfn th nghborng lmnts + and wth common dg so that n ponts toward +. Scond, for gvn lmnt a vctor n wll b consdrd outward to an undrlyng lmnt as n Fgur 1. Fgur 1. a) dg-basd normal vctor and lmnts b) lmnt-basd outward unt normal vctor W dfn b n) + = max{b n,0}, b n) = mn{b n,0}. For gvn lmnt, u h ) nt stands for th trac of u h on from th ntror of and u h ) xt s that from th xtror of. W st th xtror trac on Ω to b 0. Now, w dfn an upstram mxd fnt lmnt approxmaton of Gu, v) n 6) as follows: G h u,v) = b n) + u h ) nt +b n) u h ) xt) v h ds+αu,v). h

6 7 EUN-JAE PAK AND BOYOON SEO Our upstram mxd fnt lmnt approxmaton of th wak formulaton 6) s to fnd a par σ h,u h,l h ) H h V h such that κaσ h,τ h )+dvτ h,u h )+dτ h,l h ) = 0, τ h H h, 8) dvσ h,v h ) G h u h,v h ) = f,v h ), v h V h, dσ h,µ h ) = 0, µ h. Lmma 3.1. Lt ε b th collcton of ntror dgs. Thn th blnar form G h u,v) can b rwrttn as follows: G h u,v) = 1 b n [u] [v])ds+αu, v) ε + 1 b n )u + +u ) v v + )ds, v V h, ε whr [u] dnots th jump of u on dg. Proof. Th da s to rwrt th corrspondng sums ovr th dgs. For gvn ntrordg and apr-assgndunt normalvctorn, thr artwo lmnts + and. W assum that + contrbut wth th vctor n n and contrbut wth th vctor n n as rfrnc to Fgur 1. S [11] for a complt proof. Lmma 3.. For any v V h, th blnar form G h v,v) satsfs: whr v = ε G h v,v) = αv,v) + 1 v, b n [v] [v])ds. Proof. It follows from Lmma 3.1 that 1 b n )v + +v ) v v + )ds ε = 1 v b n )[ ] v + ds ε = 1 b n)v vds = 1 h h dvbv v)dxdy = 0. Th last dntty follows from th ncomprssblty condton dvb = 0. Now w ar rady to prov unqu solvablty of our dscrt systm. Thorm 3.3. For suffcntly small h, thr xsts a unqu soluton σ h, u h, l h ) n H h V h for systm 8).

7 AN UPSTEAM PSEUDOSTESS-VELOCITY FOMULATION 73 Proof. It s suffcnt to prov that th problm has just trval soluton whn f = 0. Slctng τ h = σ h, v h = u h and µ h = l h n 8), w gt { κaσ h,σ h )+dvσ h,u h ) = 0, dvσ h,u h ) G h u h,u h ) = 0. So, w hav that κaσ h,σ h )+G h u h,u h ) = 0. Snc κaσ h,σ h ) = κ Aσ h and G h u h,u h ) = α u h + 1 uh, w hav Aσ h = 0 and u h = 0. Nxt, f w choos v h = dvσ h n scond quaton of 8), w gt from 7) Fnally, from th frst quaton n 8) σ h = 0. dτ h,l h ) = κaσ h,τ h ) dvτ h,u h ) = 0. Choosng τ h H h wth Ω trτh 0, w hav l h = 0 as rqurd. mark 3.4. If w choos a constant tnsor τ h = 1 Ω blongs to H h and satsfs that l l h = trτ h. Notng that Aτ h = 0 and from rror quaton w must hav l = l h. l l h ) = dτ h,l l h ) Ω l l h = κaσ σ h ),τ h ) dvτ h,u u h ) ) 0 0 l l h = κσ σ h ),Aτ h ) dvτ h,u u h ) = 0, 4. Error analyss, thn τ h To stmat rrors, w dfn a projcton usng man valu of ntgraton. For a gvn,j and scalar functon px,y), put px ) = 1 h y j yj+1 and dfn an ntrpolaton π x p for x,y),j y j px,y)dy π x px,y) = px )φ x x,y)+ px +1 )φ x +1x,y), whch s pcws constant n y and pcws lnar n x. Smlarly w dfn an ntrpolaton π y px,y), whch s pcws constant n x and pcws lnar n y. Dfn Π h : Hdv;) T 0 )

8 74 EUN-JAE PAK AND BOYOON SEO by Π h p = π x p 1,π y p ) for p = p 1,p ) Hdv;). Thn, th projcton satsfs th followngs 9) p Π h p) nds = 0 for ach dg. σ11 σ1 Thus, w xtnd ths projcton to tnsor σ = σ 1 σ ) H, w dfn Π h σ as follows Π h σ ) πx σ = 11 π y σ 10) 1. π x σ 1 π y σ Thn, Π h σ := Π h σ. h Nxt, w dfn a pcws constant ntrpolaton P h v for v L ) as follows P h v = 1 vx, y) dxdy. So, for u = u1 u ) L Ω), dfn an ntrpolaton P h u V h as follows P h u = ) Ph u 11) 1. P h u h From 9), t s asy to chck th valdty of th commutatvty proprty dvπ h τ = P h dvτ, τ H. Thus, w obtan convrgnc rsults for projctons: Lmma 4.1. For σ H, w hav th followng approxmaton proprts 1) 13) σ Π h σ Ch σ 1, dvσ Π h σ) Ch dvσ 1. Lmma 4.. For σ H, th ntrpolaton Π h σ satsfs th followngs 14) and 15) κaσ Π h σ),τ) κch Aσ 1 τ, dvσ Π h σ),v) = 0, v V h. τ H h Proof. It follows from Lmma 4.1 and th commutng proprty AΠ h σ = Π h Aσ that Aσ Π h σ) Aσ Π h Aσ Ch Aσ 1. Snc v V h s constant on, t follows from th dfnton of Π h σ that w hav v dvπ h σ)dxdy = v n Π h σ) ds

9 AN UPSTEAM PSEUDOSTESS-VELOCITY FOMULATION 75 = Now, for th rror analyss, dfn v n σ) ds = v dvσ)dxdy. ξ σ = Π h σ σ h, ξ u = P h u u h, η σ = σ Π h σ, η u = u P h u. Lmma 4.3. Whn h s suffcntly small thr s a postv constant C ndpndnt of h such that 1 ξu,,j ξ u, 1,j 16) κc Aξ σ +κch σ 1, x 17) y j ξu,,j ξ u,,j 1 1 κc Aξ σ +κch σ 1, ) ξu1 ξσ11 ξ whr ξ u = ξ u and ξ σ = σ1 ξ σ1 ξ σ ). Proof. Lt = 1,j,j. Slctng τ = ) φ x n 6) and 8) w hav that κ σ 11 1 ) trσ φ x φ x 18) = Q x x u 1 l φ x = h y j P hu 1, 1,j P h u 1,,j ) l and 19) Not that κ κ σ h11 1 ) trσh φ x = φ x x uh 1 l h φ x = h y j uh 1, 1,j u h 1,,j) l h σ 11 σ11 h ) 1 trσ trσh ) )φ x κ κ Snc l = l h, by subtractng 18) from 19), w hav that ξ u1,,j ξ u1, 1,j κ 1 h y Aξ σ11 +κc 1 j h y j φ x. Aσ 11 Aσ h 11 φ x Aξ σ11 + Aη σ11 ). Aη σ11.

10 76 EUN-JAE PAK AND BOYOON SEO So, usng a+b) a +b ) and th Höldr nqualty, ξ u1,,j ξ u1, 1,j) x C x κ Qx h y Aξ j ) σ11 +κ Qx ) Q x h y Aη j ) σ11 κ C Aξ σ +κ Ch σ 1. If w slct τ = 0 0) φ x, w hav that 0 ξ u,,j ξ u, 1,j) κ C Aξ σ +κ Ch σ 1, x whch complts th proof of 16). Th proof of 17) follows smlarly. Now w stmat Gu,v h ) G h u h,v h ). W assum that u s contnuous n th whol doman for smplcty. Not that for v h V h, w hav Gu,v h ) = b u) v h ds+αu,v h ) = b n)u v h ds+αu,v h ). Thus, w fnd that Gu,v h ) = b n)u v,j h ds+αu,v h ),j,j = b 1 n 1 ) + +b 1 n 1 ) )u v 1,j h v,j)ds h x x + b n ) + +b n ) )u v,j 1 h v,j)ds+αu,v h h ) y y j j and G h u h,v h ) = b n) + u h ) nt +b n) u h ) xt) v,j h ds+αuh,v h ),j,j = b 1 n 1 ) + u h 1,j +b 1n 1 ) u h,j ) vh 1,j vh,j )dy x x + b n ) + u h,j 1 +b n ) u h,j ) vh,j 1 vh,j )dx+αuh,v h ). y y j j Wrtng u = P h u+u P h u) and usng P h u 1,j +u P h u 1,j ) = P h u,j +u P h u,j ),

11 AN UPSTEAM PSEUDOSTESS-VELOCITY FOMULATION 77 w arrv at 0) Gu,v h ) G h u h,v h ) G h ξ u,v h ) = αu P h u),v h ) b1 n 1 ) + u P h u 1,j )+b 1 n 1 ) u P h u,j ) ) v 1,j h v,j)dy h + x + y j x y j b n ) + u P h u,j 1 )+b n ) u P h u,j ) ) v h,j 1 v h,j)dx. Lmma 4.4. Undr th assumpton of Lmma 4.3 w hav that Gu,ξ u ) G h u h,ξ u ) G h ξ u, ξ u ) κch u 1 Aξ σ +κch σ 1 u 1 +αch u 1 ξ u. Proof. Not that th unt outward normal vctor s n = ±1,0) on x and n = ±0,1) on y j. By dfnton of projcton, for k = 1, th L -projcton P h u k s constant on,j. So, t follows that x u k P h u k )dy = 1 h x By takng v h = ξ u n 0), w hav that 1) x x C x u k x,y) u k x,y)) dxdy,j,j u k x dxdy. [b 1 n 1 ) + u 1,j P h u 1,j )+b 1 n 1 ) u,j P h u,j )] v h 1,j v h,j)dy,j Ch u 1 [ x u ) x dxdy ξu,j ξ u 1,j ξ u,j ξ u 1,j ] 1/ κch u 1 Aξ σ +κch σ 1 u 1. Smlarly by takng v h = ξ u n 0), w hav ) y j y j C y j [b n ) + u,j 1 P h u,j 1 )+b n ) u,j P h u,j )] v h,j 1 v h,j)dx,j u ) y dxdy ξu,j ξ u,j 1 κch u 1 Aξ σ +κch σ 1 u 1.

12 78 EUN-JAE PAK AND BOYOON SEO Snc u P h u Ch u 1, addng th quaton 1) and ), w complt th proof from 0). W ar now rady to prov th frst ordr convrgnc of th vlocty and trac-fr psudostrss varabls. Thorm 4.5. For h suffcntly small, thr xsts a constant C ndpndnt of h such that 3) 4) Aσ σ h ) Ch σ 1 + u 1 ), u u h Ch σ 1 + u 1 ). Proof. Not that σ σ h = η σ +ξ σ. Subtractng 8) from 6) and usng 15), w hav that 5) κaξ σ,τ h )+dvτ h,ξ u )+dτ h,l h l) = κaη σ,τ h ), τ h H h, dvξ σ,v h )+G h u h,v h ) Gu,v h ) = 0, v h V h, dξ σ,µ h ) = 0, µ h. Takng τ h = ξ σ, v h = ξ u and µ = l h l = 0, w hav that { κaξ σ,ξ σ )+dvξ σ,ξ u ) = κaη σ,ξ σ ), 6) dvξ σ,ξ u )+G h u h,ξ u ) Gu,ξ u ) = 0. So, t follows that 7) κaξ σ,ξ σ )+Gu,ξ u ) G h u h,ξ u ) = κaη σ,ξ σ ) Not that from Lmma 3., α ξ u G h ξ u,ξ u ). Applyng 7) and Lmma 4.4, w hav that κch Aξ σ σ 1. κ Aξ σ +α ξ u κaξ σ,ξ σ )+G h ξ u,ξ u ) [ κaξ σ,ξ σ )+Gu, ξ u ) G h u h,ξ u ) ] Thus, [ Gu,ξ u ) G h u h,ξ u ) G h ξ u,ξ u ) ] κch Aξ σ σ 1 +κch u 1 Aξ σ +κch σ 1 u 1 +αch u 1 ξ u 1 κ Aξ σ + 1 α ξ u +Ch κ σ 1 +κ+α) u 1 +κ σ 1 u 1 ). κ Aξ σ +α ξ u Ch κ σ 1 +κ+α) u 1 +κ σ 1 u 1 ).

13 AN UPSTEAM PSEUDOSTESS-VELOCITY FOMULATION 79 Thrfor, usng Aσ Π h σ) Ch σ 1 and u P h u Ch u 1 and th trangl nqualty, w hav Aσ σ h ) Ch σ 1 + u 1 ), u u h Ch σ 1 + u 1 ). In th nxt thorm, w prov stablty of σ h n th Hdv;Ω)-norm. Not that n th scalar convcton-dffuson problm, a wakr stablty rsult was obtand [16]; s also Thorm 3. n [17]. Thorm 4.6. For h suffcntly small, thr xsts a constant C ndpndnt of h such that 8) σ σ h Hdv;Ω) C σ 1 + u 1 ). Proof. Consdr rror quaton 5). Takng τ h = ξ σ, v h = dvξ σ and µ = l h l = 0, w gt that { κaξ σ,ξ σ )+dvξ σ,ξ u ) = κaη σ,ξ σ ), 9) dvξ σ,dvξ σ )+G h u h,dvξ σ ) Gu,dvξ σ ) = 0. Addng two quatons n 9) lads to 30) κaξ σ,ξ σ )+dvξ σ,dvξ σ ) = κaη σ,ξ σ ) dvξ σ,ξ u )+Gu,dvξ σ ) G h u h,dvξ σ ). To stmat 30), usng th rlaton Gu,v h ) = b n)u v h ds+αu,v h ) = b n) + u+b n) u) v h ds+αu,v h ), w hav Gu,v h ) G h u h,v h ) = αu u h ),v h ) + b n) + u u h ) nt ) v h ds + b n) u u h ) xt ) v h ds = αu u h ),v h )+I+II. Not that v h s a constant vctor wth support. I = b n) + u u h ) nt ) v h ds C v h u u h ds

14 80 EUN-JAE PAK AND BOYOON SEO C Ch 1/ v h h 1/ ) 1/ u u h ds h v h ) u u h 0, ) Ch 1/ h v h ) 1/ u u h 0, ) 1/. By th trac thormc.f., [3]) and Thorm 4.5, I Ch 1/ v h 0 u u h 0, u u h 1, ) 1/ Ch 1/ u u h 0 u 1 ) 1/ v h 0 C σ 1 + u 1 ) v h 0. Smlarly, w gt II C σ 1 + u 1 ) v h 0. Thus, takng v h = dvξ σ, Gu,dvξ σ ) G h u h,dvξ σ ) C σ 1 + u 1 ) dvξ σ 0. Thrfor, w gt th followng stmat from 7) and 30) ξ σ Hdv;Ω) = ξ σ + dvξ σ By th trangl nqualty C Aξ σ + dvξ σ ) ) C Aη σ ξ σ + ξ u dvξ σ + σ 1 + u 1 ) dvξ σ ) C Aη σ + ξ u + σ 1 + u 1 ξ σ Hdv;Ω). σ σ h Hdv;Ω) η σ Hdv;Ω) + ξ σ Hdv;Ω) C σ 1 + u 1 ). 5. Numrcal rsults In ths scton, w prform varous numrcal xprmnts to tst th upstram schm basd on th psudostrss-vlocty formulaton. All xprmnts wr run n Matlab s [1]) Exampl 1 W solv Osn quatons 1) n th unt squar Ω = 0,1) wth α = and b =,3) T. Th functon f s dtrmnd by th followng xact soluton, u = πsnπx) snπy), px,y) = cosπx)cosπy). πsnπx)snπy)

15 AN UPSTEAM PSEUDOSTESS-VELOCITY FOMULATION 81 By th dfnton of th psudostrss n ), w hav σ = ν u pi νπ = snπx)snπy) px,y) νπ sn πx)cosπy) νπ cosπx)sn πy) νπ snπx)snπy) px,y) Partton th doman Ω = 0, 1) by unform rctangular lmnts,j = h,jh) for,j = 0,1,...,n wth h = 1/n. Whn n = 3, Fgur shows th xact vctor fld of vlocty and contours of prssur, rspctvly. If w wrt ) Fgur. Exact vctor fld of vlocty and contours of prssur th psudostrss σ and vlocty u as M N σ = Σ j Ψ j, u = U j φ j, j=1 whr M = 4nn+1), N = n. Th dscrt wak form 8) has th followng matrx form: B CT E T C G 0 Σ U = 0 F E 0 0 l 0 Not that rankb) = M 1 and G s postv dfnt by Lmma 3.. If w choos ν smallr, th Osn quatons bcom mor convcton-domnatd. Th tabl 1 dsplays th L -norm rrors of Aσ and vlocty u and thr convrgnc ordrs C.O.) compard wth L -norm and Hdv;Ω)-norm rrors for σ. From th tabl w confrm our thory prsntd n ths papr. 5.. Ld-drvn cavty flow Th nxt problm s that of ld-drvn flow n a squar cavty. Ths s a classc tst problm usd n flud dynamcs, known as drvn-cavty flow. Our am hr s to chck th prformanc of th schm wth b = 0. W j=1

16 8 EUN-JAE PAK AND BOYOON SEO Tabl 1. Errors and Convrgnc Ordrs N Aσ σ h ) C.O. u u h C.O. σ σ h C.O. σ σ h Hdv;Ω) C.O. ν = ν = ν = ν = mpos no-slp boundary condtons, that s, u 1 x,1) = 1 for 1 < x < 1 and u 1 1,1) = u 1 1,1) = 0. W solv th followng Stoks quatons wth unform msh. { u+ p = 0 n 1, 1) 1, 1), dv u = 0 n 1, 1) 1, 1). W plot xponntally spacd stramlns to llustrat th Moffatt dds n th bottom cornrs. Ths stramlns ar computd from th psudostrss soluton by solvng th followng Posson quaton numrcally subjct to a zro Drchlt boundary condton. 31) φ = u x u 1 := ω, y whr φ s a scalarstram functon and ω s th vortcty s [13, 1]). Bcaus our mthod s basd on th psudostrss-vlocty formulaton w calculat th psudostrss drctly. Snc ) κaσ = u = u1 x u x u 1 y u y,

17 AN UPSTEAM PSEUDOSTESS-VELOCITY FOMULATION Fgur 3. Contours of th stram functon wth xponntally dstrbutd w can us mor accurat approxmaton of ω from th computd psudostrss so that w shall solv quaton 31) mor ffcntly Osn flow ovr a stp Th fnal tst problm s an Osn flow ovr a backward facng stp. { 1 u+b u+ p = 0 n Ω, 3) dv u = 0 n Ω. Th doman Ω s L-shap as Fgur 4. Th ynolds numbr s 100 and w mpos a constant lft-to-rght wnd b = 1, 0) for y 0 and b = 0,0) for y < 0. Inflow vlocty s u 1 1,y) = y1 y) for 0 < y < 1. Outflow boundary condton s { 33) p+ 1 u x = 0. u 1 x = 0, Ths condton s quvalnt to σn = 0. Th othr boundary vlocts ar all zro. All abov condtons ar dpctd n Fgur 4. b=1,0) b=0,0) Fgur 4. Doman and boundary condtons

18 84 EUN-JAE PAK AND BOYOON SEO W comput th approxmaton soluton of quatons3) wth psudostrssvlocty formulaton nvolvng our upstram mthod. From th vlocty computd alrady, th stramlns ar plottd by Matlab automatcally Fgur 5. Th stramlns whn s 100 frncs [1] C. Bahrawat and C. Carstnsn, Thr Matlab mplmntatons of th lowst-ordr avart-thomas MFEM wth a postror rror control, Comput. Mthods Appl. Math ), no. 4, [] M. Braack and E. Burman, Local projcton stablzaton for th Osn problm and ts ntrprtaton as a varatonal multscal mthod, SIAM J. Numr. Anal ), no. 6, [3] S. Brnnr and L.. Scott, Th Mathmatcal Thory of Fnt Elmnt Mthods, Sprngr-Vrlag, Nw York, [4] F. Brzz and M. Fortn, Mxd and Hybrd Fnt Elmnt Mthods, Sprngr-Vrlag, Nw York, [5] A. N. Brooks and T. J.. Hughs, Stramln upwnd/ptrov-galrkn formulatons for convcton domnatd flows wth partcular mphass on th ncomprssbl Navr- Stoks quatons, Comput. Mthods Appl. Mch. Engrg ), no. 1-3, [6] E. Burman, M. A. Frnandz, and P. Hansbo, Contnuous ntror pnalty fnt lmnt mthod for Osn s quatons, SIAM J. Numr. Anal ), no. 3, [7] Z. Ca and G. Stark, Frst-ordr systm last squars for th strss-dsplacmnt formulaton: lnar lastcty, SIAM J. Numr. Anal ), no., [8] Z. Ca, C. Tong, P. S. Vasslvsk, and C. Wang, Mxd fnt lmnt mthods for ncomprssbl flow: Statonary Stoks quatons, Numr. Mthods Partal Dffrntal Equatons 6 010), no. 4, [9] Z. Ca and Y. Wang, Psudostrss-vlocty formulaton for ncomprssbl Navr-Stoks quatons, Intrnat. J. Numr. Mthods Fluds ), no. 3, [10] C. Carstnsn, D. Km, and E.-J. Park, A pror and a postror psudostrss-vlocty mxd fnt lmnt rror analyss for th Stoks problm, SIAM J. Numr. Anal ), no. 6, [11] S. H. Chou, Mxd upwndng covolum mthods on rctangular grds for convctondffuson problms, SIAM J. Sc. Comput ), no. 1, [1] C. Dawson, Analyss of an upwnd-mxd fnt lmnt mthod for nonlnar contamnant transport quatons, SIAM J. Numr. Anal ), no. 5,

19 AN UPSTEAM PSEUDOSTESS-VELOCITY FOMULATION 85 [13] H. Elman, D. Slvstr, and A. Wathn, Fnt Elmnts and Fast Itratv Solvrs: Wth applcatons n ncomprssbl flud dynamcs, Oxford Unvrsty Prss, Nw York, 005. [14] G. Gatca, A. Marquz, and M. A. Sanchz, Analyss of a vlocty-prssur-psudostrss formulaton for th statonary Stoks quatons, Comput. Mthods Appl. Mch. Engrg ), no. 17-0, [15] V. Grault and P. A. avart, Fnt Elmnt Mthods for Navr-Stoks Equatons, Sprngr-Vrlag, Nw York, [16] J. Jaffr, Elémnts fns mxts t décntrag pour ls équatons d dffuson-convcton, Calcolo ), [17] D. Km and E.-J. Park, A postror rror stmators for th upstram wghtng mxd mthods for convcton dffuson problms, Comput. Mthods Appl. Mch. Engrg ), no. 6-8, [18] P. Lsant and P. A. avart, On a fnt lmnt mthod for solvng th nutron trasport quaton, Mathmatcal Aspct of Fnt Elmnts n Partal Dffrntal Equatons, Ed. Carl d Boor, Acadmc Prss, 1974, [19] Z. L, Convrgnc analyss of an upwnd mxd lmnt mthod for advcton dffuson problms, Appl. Math. Comput ), no., [0] F. A. Mlnr and E.-J. Park, A mxd fnt lmnt mthod for a strongly nonlnar scond-ordr llptc problm, Math. Comp ), no. 11, [1] S. Norburn and D. Slvstr, Stablsd vs. stabl mxd mthods for ncomprssbl flow, Comput. Mthods Appl. Mch. Engrg ), no. 1-, [] E.-J. Park, Mxd fnt lmnt mthods for nonlnar scond ordr llptc problms, SIAM J. Numr. Anal ), no. 3, [3] P. A. avart and J. Thomas, A mxd fnt lmnt mthod for nd ordr llptc problms, Mathmatcal aspcts of fnt lmnt mthods Proc. Conf., Consglo Naz. dll crch C.N..), om, 1975), pp Lctur Nots n Math., Vol. 606, Sprngr, Brln, Eun-Ja Park Dpartmnt of Mathmatcs and Dpartmnt of Computatonal Scnc and Engnrng Yons Unvrsty Soul , Kora E-mal addrss: jpark@yons.ac.kr Boyoon So Dpartmnt of Mathmatcs Yons Unvrsty Soul , Kora E-mal addrss: mathd@yons.ac.kr