A SPLITTING METHOD USING DISCONTINUOUS GALERKIN FOR THE TRANSIENT INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

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1 ESAIM: MAN Vol. 39, N o 6, 005, pp DOI: 0.05/man: ESAIM: Mathmatical Modlling and Numrical Analysis A SPLITTING METHOD USING DISCONTINUOUS GALERKIN FOR THE TRANSIENT INCOMPRESSIBLE NAVIER-STOKES EQUATIONS Vivtt Girault,Béatric Rivièr and Mary F. Whlr 3 Abstract. In this papr w solv th tim-dpndnt incomprssibl Navir-Stoks quations by splitting th non-linarity and incomprssibility, and using discontinuous or continuous finit lmnt mthods in spac. W prov optimal rror stimats for th vlocity and suboptimal stimats for th prssur. W prsnt som numrical xprimnts. Mathmatics Subct Classification. 65M, 65M5, 65M60. Rcivd: Sptmbr, 004. Introduction Th Navir-Stoks quations charactriz a varity of flows, which play an important rol in many nginring applications. For incomprssibl flows, th momntum and continuity quations ar: u t µ u + u u + p = f, u =0, whr u is th fluid vlocity, p th prssur, µ>0 th constant viscosity, and f a givn xtrnal forc. Ths quations ar compltd by adquat boundary and initial conditions. Ths quations ar difficult to solv numrically bcaus on on hand, thy ar nonlinar and on th othr hand, th vlocity is coupld with th prssur. In this papr, w study a particular oprator splitting tchniqu introducd by Blasco t al. [5] in 997, for dcoupling th convction and prssur trms. It is convnint to dscrib th gnral ida of this splitting tchniqu at th smi-discrt tim lvl; givn an approximation U of th vlocity ut attimt andanapproximationf of ft, th computation of th discrt vlocity and prssur at tim t procds in two stps: Linarizd convction stp: solv for an intrmdiat vlocity Ũ satisfying t Ũ U µ Ũ + U Ũ = f. 0. Kywords and phrass. Oprator splitting, tim-dpndnt Navir-Stoks, SIPG. Univrsité Pirr t Mari Curi, Paris VI, Laboratoir Jacqus-Louis Lions, 4, plac Jussiu, 755 Paris Cdx 05, Franc. girault@ann.ussiu.fr Dpartmnt of Mathmatics, Univrsity of Pittsburgh, 30 Thackray, Pittsburgh, PA 560, USA. rivir@math.pitt.du 3 Cntr for Subsurfac Modling, Institut for Computational Enginring and Scincs, Univrsity of Txas, 0 E. 4th St., Austin TX 787, USA. mfw@ics.utxas.du c EDP Scincs, SMAI 005 Articl publishd by EDP Scincs and availabl at or

2 6 V. GIRAULT ET AL. Incomprssibility stp: solv for U and P satisfying t U Ũ µ U Ũ + P = 0, 0. U = If th spac discrtization is wll chosn, this splitting tchniqu has th advantags that th first stp rducs to a systm of scalar quations, that can b solvd in paralll; th discrt vlocity obtaind from th scond stp is locally consrvativ; and 3 th boundary condition can b nforcd at ach stp. Thr ar svral stratgis for spac discrtizations that bnfit from part or all of ths advantags. For instanc, both stps can b solvd by a symmtric or non-symmtric discontinuous Galrkin mthod; th scond stp can b solvd by a discontinuous Galrkin mthod whil th first stp can b solvd by a continuous finit lmnt mthod in som appropriat rgion possibly th ntir rgion and by a discontinuous Galrkin mthod in othr rgions; 3 th domain can b subdividd into rgions in which both stps ar solvd ithr by a discontinuous or by a nonconforming finit lmnt mthod. Th ida of dcoupling th nonlinarity from th incomprssibility condition dats back to th work of Chorin [7] and Tmam [7]. This mthod was known as th proction mthod and sinc thn, it has bn studid and modifid by svral authors. Th radr can find a good historical account in th introduction of [4] by Blasco and Codina. Without bing xhaustiv, lt us quot th work of Yannko [3] on fractional stp mthods, th work of Frnandz-Cara and Bltram [], Rannachr [5], Turk [9], Gurmond and Quartapll [7], Quartroni t al. [4], Almgrn t al. [], Gurmond and Shn [8, 9], all on proction mthods. W rfr to th rcnt book of Glowinski [5] for a vry comprhnsiv tratmnt of fractional stp mthods and proction mthods. Th splitting tchniqu of [4] studid hr can b viwd as a vry particular cas of fractional stp mthods in which th tim advancs by a full tim stp. On th othr hand, it diffrs from th abov-mntiond proction mthods bcaus it procts in H instad of L. Thus it is mor complx than th standard proction mthod, but in contrast it prsrvs th boundary condition and producs no artificial boundary layr. To our knowldg, thr is vry littl in th litratur on th analysis of discontinuous Galrkin mthods for Navir-Stoks quations. Th Symmtric Intrior Pnalty Galrkin SIPG mthod originally calld intrior pnalty mthod and Non-symmtric Intrior Pnalty Galrkin NIPG mthod wr first introducd for lliptic problms by Whlr [30] and Rivièr t al. [6]. Th ida of using a non-symmtric form without pnalty was introducd by Baumann and Odn [3]. Th NIPG and SIPG mthods for th stady-stat Navir-Stoks quations wr first formulatd and analyzd in Girault t al. [4]. In Kaya and Rivièr [0] both NIPG and SIPG mthods coupld with a subgrid ddy viscosity mthod ar applid to th tim-dpndnt Navir- Stoks problm. Th mthod w propos hr mploys th SIPG or NIPG mthods, i.. th bilinar form that approximats th viscous trm is ithr symmtric or non-symmtric. Our numrical xprimnts with both mthods in Sction 7 giv accurat rsults. Th discontinuous Galrkin mthods prsnt svral advantags: thy ar asily usd on highly unstructurd mshs, thy ar locally consrvativ and thy lnd thmslvs wll to domain dcomposition. Furthrmor, th approach w propos satisfis a compatibility condition, which is important in air and watr quality modling s Rm..5. Morovr, as far as th Navir-Stoks quations ar concrnd, discontinuous Galrkin mthods lnd thmslvs asily to an upwinding of th convction trm, as was studid by Lsaint and Raviart [] in nutron transport. Th coupling of th continuous rgions with th discontinuous rgions is usful in many applications such as surfac flow s th application to shallow watr in [9], whr th cost of using fully discontinuous Galrkin mthods can b rducd. Howvr, in th forthcoming analysis, w shall s that w los optimality if th

3 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 7 finit lmnts chang whn w pass from stp to stp. In this rspct, combining a simpl continuous finit lmnt mthod with a discontinuous Galrkin mthod rquirs lss dgrs of frdom but is lss attractiv than combining an appropriat nonconforming mthod with a discontinuous Galrkin mthod. Th nonconforming approach, which is locally consrvativ, appars to b a good compromis btwn stratgis and. It is intrsting to not that non of th analysis blow rquirs a quasi-uniform triangulation. Although this mthod and most of its analysis apply to 3D, for simplicity, w shall only approximat th Navir-Stoks quations in D. Th full problm is: u t µ u + u u + p = f, in Ω 0,T, 0.4 u =0, in Ω 0,T, 0.5 u = 0, on Ω 0,T, 0.6 u = u 0, in Ω {0}, 0.7 whr Ω is a domain in IR with Lipschitz-continuous boundary Ω. As usual, w writ formally: u v = i= u i v x i and u = It is wll known that if f L 0,T; H Ω andu 0 Hdiv, Ω, thn this problm has a uniqu solution u L 0,T; H 0 Ω L 0,T; L Ω, p W, 0,T; L 0 Ω and u t L 0,T; V s Lions [3], Tmam [8], Girault and Raviart [3]. Hr, L Ω is th classical spac of squar-intgrabl functions with th innrproduct f,g = Ω fg, L 0Ω is th subspac of functions of L Ω with zro man valu: L 0Ω = and H Ω dnots th classical Sobolv spac: i= { } v L Ω : v =0, Ω H Ω = {v L Ω : v L Ω }. By dfinition, H 0 Ω is th closur of DΩ in H Ω, whr DΩ is th spac of infinitly diffrntiabl functions with compact support, H Ω is th dual of H 0 Ω, V is th spac of functions of H 0 Ω with zro divrgnc: V = {v H 0 Ω : v =0}, 0.8 and V is its dual spac. It is wll known that H 0 Ω is charactrizd as th subspac of functions of H Ω that vanish on Ω. Mor gnrally, w shall us th spacs u i x i W,r Ω = {v L r Ω : v L r Ω }, quippd with th smi-norm /r v W,r Ω = v r, Ω and norm for which it is a Banach spac: v W,r Ω = v r L r Ω + v r W,r Ω /r. Ths dfinitions ar xtndd in th usual way to r =. W shall also us H Ω = {v H Ω : v H Ω },

4 8 V. GIRAULT ET AL. with th smi-norm v H Ω = v H Ω and th norm v H Ω = v H Ω + v H Ω /. W rfr to Adams [], Lions and Magns [] for ths spacs and for xtnding thm to fractional xponnts. As usual, for handling tim-dpndnt problms, it is convnint to considr functions dfind on a tim intrval a, b with valus in a functional spac, say Y s []. Mor prcisly, lt Y dnot th norm of Y ;thn for any numbr r, r, w dfin { } b L r a, b; Y = f masurabl in a, b : ft r Y dt< quippd with th norm f L r a,b;y = b a ft r Y dt/r, with th usual modification if r =. ItisaBanach spac if Y is a Banach spac. Th outlin of th papr is as follows. First, w prsnt th discontinuous Galrkin mthod and th first splitting tchniqu. In Sctions and 4, a priori stimats and suboptimal rror stimats ar drivd. Sction 3 contains L r stimats. Improvd optimal rror stimats ar provd in Sction 5. Th scond and third splitting mthods ar brifly prsntd in Sction 6. Th papr nds with numrical xprimnts in Sction 7. a. Discontinuous Galrkin for both stps Problm has th following wak formulation, valid a.. on 0,T: v H0 Ω, u t t, v+µ ut, v+ut ut, v pt, v =ft, v,. q L 0Ω, ut,q=0,. u0 = u 0, in Ω..3 To discrtiz this problm, w introduc a rgular family of triangulations of Ω, E h, consisting of triangls of maximum diamtr h. Lth E dnot th diamtr of a triangl E and ρ E th diamtr of its inscribd circl. By rgular, w man s Ciarlt [6] that thr xists a paramtr ζ>0, indpndnt of h, such that E E h, h E ρ E = ζ E ζ..4 W shall us this assumption throughout this work. W dnot by Γ h th st of all dgs of E h, i.. th st of all dgs in th domain Ω. Lt dnot a sgmnt of Γ h shard by two triangls E k and E l of E h ; w associat with a spcific unit normal vctor n dirctd from E k to E l and w dfin formally th ump and avrag of a function φ on by: [φ] =φ E k φ E l, {φ} = φ E k + φ E l. If is adacnt to Ω, thn n is th unit normal n xtrior to Ω and th ump and th avrag of φ on coincid with th trac of φ on. Thn, w dfin th spacs of discontinuous functions X = {v L Ω : E E h, v E W,4/3 E },.5 M = {q L 0Ω : E E h, q E W,4/3 E},.6

5 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 9 and th brokn norm, for any vctor or tnsor v: v 0,Ω = v L E W associat with th spacs X and M th following norms: whr /. v X = v 0,Ω + J 0 v, v /,.7 q M = q L Ω,.8 J 0 u, v = σ Γ h [u] [v]..9 Hr dnots th masur of and σ is a ump cofficint boundd blow by a sufficintly larg constant σ 0 and boundd abov by a constant σ m, both constants bing indpndnt of h, but dpndnt on th mthod usd. For th symmtric mthod, SIPG, ach constant σ is adustd in ordr to guarant llipticity of th form a + J 0, s.. For th non-symmtric mthod, NIPG, it is wll-known that it suffics to tak ach constant qual to on for instanc. Howvr, w shall s in Sction, that bcaus of th splitting, ach constant σ has to b adustd in ordr to prov stability of th algorithm. Nvrthlss, our numrical rsults in Sction 7 tnd to show that in th xampls w hav chosn, th rror is not vry snsitiv to th choic of σ whn using NIPG. On this triangulation, w dfin two finit-dimnsional subspacs X h X and M h M: X h = {v h L Ω : E E h, v h IP E },.0 M h = {q h L 0Ω : E E h, q h IP 0 E}.. For simplicity, w driv th analysis for picwis linar vlocity and picwis constant prssur. This is consistnt with th fact that w shall us a first-ordr discrtization in tim. W could considr a highrordr approximations in spac, but this would hav to b matchd by a highr-ordr approximation in tim or an appropriatly small tim stp as dmonstratd in th numrical xampls in Sction 7. To simplify th discussion, w shall analyz in dtail th standard discontinuous symmtric mthod SIPG and brifly sktch th analysis for th non-symmtric mthod. In both mthods, th incomprssibility condition is nforcd by mans of th bilinar form b : X M IR bv,p= p v + {p}[v] n,. E that is simply obtaind by applying Grn s formula in ach lmnt to th lft-hand sid of.. In particular if p H Ω, thn v X, bv,p= p v..3 Thus, w approximat th spac V dfind in 0.8 by Γ h Ω V h = {v h X h : q h M h,bv h,q h =0}..4 Finally th nonlinar convction trm u u is approximatd by th following variant of Lsaint-Raviart upwinding s [] that was introducd in [4]. In thory, it is difficult to prov that it brings an improvmnt,

6 0 V. GIRAULT ET AL. bcaus th Navir-Stoks quation is not purly a transport quation, but in practic, upwinding is usful whn th convction is dominant. Th approximation w propos is: u, v, w, z X, whr c z u; v, w = v w + Eu uv w E [u] n {v w} + {u} n E v int v xt w int,.5 E Γ h E = {x E : {z} n E < 0}, th suprscript z dnots th dpndnc of E on z and th suprscript int rsp. xt rfrs to th trac of th function on a sid of E coming from th intrior of E rsp. coming from th xtrior of E on that sid. Whn th sid of E blongs to Ω, th convntion is th sam as for dfining th ump and avrag, i.., th ump and avrag coincid with th trac of th function. Not that c z u; v, w can also b writtn as c z u; v, w = u v w + E {u} n E v int v xt w int E Thus if v is continuous in Ω or blongs to H Ω,whav cu; v, w = u v w bu, v w. E bu, v w. Th suprscript z is droppd sinc th intgral on E disappars. It is provn in [4] that for all u, v, w X, w hav c u u; v, w = c u u; w, v,.6 whr c u u; w, v := w v + Eu + This implis that for all u, v X, c u u; v, v = whr dnots th Euclidan norm... Approximation with SIPG E E \ Ω uw v Γ h \ Ω [u] n {v w} {u} n E w int w xt v xt u n v w..7 E {u} n E [v] + E Ω Ω u n E v,.8 In SIPG, th diffusion oprator is approximatd by th bilinar form a : X X IR au, v = u : v { u}n [v] { v}n [u]..9 E Γ h Γ h

7 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES Considring this form a, it will b usful to introduc anothr msh-dpndnt norm: v X, [ v ] = v X + Γ h { v} n L /. Thn, w hav u, v X, au, v [ u ][ v ]..0 Not that whn v X h, th quivalnc of norms in finit-dimnsional spacs on th rfrnc lmnt implis that thr xists a constant C indpndnt of h such that v X h, Γ h { v} n L / C v 0,Ω.. As far as th llipticity of th form a + J 0 is concrnd, it is stablishd in [30] that, if th cofficints σ ar sufficintly larg but indpndnt of h, thr xists a constant K>0, also indpndnt of h, such that v h X h, av h, v h +J 0 v h, v h K v h X.. In th squl, w shall always assum that. holds so that a + J 0 is lliptic. As far as th inf-sup condition is concrnd, it is provn in [4] that th pair of spacs dfind by.0,. satisfis a uniform discrt inf-sup condition. Mor prcisly, with th spac w hav th following rsult: X h = { v h X h : Γ h, Lmma.. Thr xists a constant β > 0, indpndnt of h, such that inf p h M h } [v h ]=0,.3 bv h,p h sup v h X β..4 v h X p h M h Discrtization with rspct to tim is don on a uniform subdivision of th intrval [0,T]. Lt N ban intgr, t = T N and t = t, 0 N. Sinc th approximation both in spac and tim ar of ordr on, w assum that h and t ar of th sam ordr, i.. thr xist constants γ 0 and γ indpndnt of h and t such that γ 0 t h γ t..5 Th discrt schm consists of two stps. First, knowing U V h, find Ũ X h solution of v h X h, t Ũ U, v h +µ aũ, v h +J 0 Ũ, v h + c U U ; Ũ, v h =ˇf, v h..6 Scond, find U X h and P M h,solutionof v h X h, t U Ũ, v h +µ au Ũ, v h +J 0 U Ũ, v h + bv h,p =0,.7 q h M h, bu,q h =0..8

8 V. GIRAULT ET AL. At tim t =0,U 0 is a suitabl approximation of u 0 that w spcify latr. Th trm ˇf dnots an appropriat approximation of f at tim t. To simplify th analysis, w choos ˇf = t t f, but this is only a mattr of convninc. As far as xistnc is concrnd, givn U,.6 has a uniqu solution owing to th llipticity proprty. and th positivity.8 of c. Similarly, givn Ũ,.7,.8 has a uniqu solution owing to th llipticity proprty. and th inf-sup condition.4. By summing thtwostps,thconsistncyofthschmfollowsfromthfollowinglmma. Wskipthproof,whichis straightforward. Lmma.. Formally, th solution u,p of satisfis a.. on 0,T: v h X h, u t, v h +µ au, v h +J 0 u, v h + cu; u, v h +bv h,p=f, v h..9 Now w rcall som approximation proprtis of th spacs X h and M h.forx h,ltr h LH Ω ; X h b th oprator dfind by v H Ω, Γ h, R h v v =0..30 It is asy to s that.30 dfins a uniqu function R h v X h s [8] and implis that v H Ω, R h v v =0,.3 E v H0 Ω, Γ h, [R h v]=0..3 Thus, w hav v H0 Ω, q h M h, bv R h v,q h =0..33 Furthrmor, sinc R h prsrvs th polynomials of IP in ach lmnt, it satisfis th rror bounds: E E h, s [, ], r, v W s,r E,m=0,, v R h v W m,r E Ch s m E v W s,r E..34 For M h,ltr h LL 0 Ω; M hbdfindinache E h by: E E h, r h q q =0. E Thn q H s E, s [0, ], q r h q L E Ch s E q H s E..35 From.3 and.34 with s = m =andr =, w asily driv th nxt lmma. Lmma.3. Th oprator R h satisfis th following stability proprty: thr xists a constant C indpndnt of h such that, u H0 Ω, R h u X C u H Ω..36 W hav th following consistncy rror for a: Lmma.4. Thr xists a constant C, indpndnt of h, such that for all u in H Ω H 0 Ω and all v h in X h : au R h u, v h Ch u H Ω v h X..37

9 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 3 Proof. In viw of.0 and. it suffics to prov that This follows asily from.34. u H Ω H 0 Ω, [ u Rh u ] Ch u H Ω..38 Rmark.5. Th proposd splitting tchniqu satisfis th compatibility condition of zro accuracy dscribd in [0] whr picwis discontinuous linars ar usd in a discontinuous Galrkin transport schm. In othr words, constants ar rproducd whn an approximat vlocity dfind by.7,.8 is usd in transport. Th compatibility condition is: >0, E E h, This condition follows immdiatly from.8... Approximation with NIPG E {U } n E =0. In NIPG, th form a is rplacd by au, v = u : v { u}n [v]+ { v}n [u],.39 E Γ h and for th momnt, w tak ach constant σ arbitrary. All th othr trms ar unchangd and th formulation of th discrt problm is givn, with this nw form a, by.6.8. Clarly, all th proprtis listd abov ar prsrvd and. is improvd sinc Γ h v h X h, av h, v h +J 0 v h, v h = v h X..40. A PRIORI stimats In this sction, w prov that th schm.6.8 is unconditionally stabl. Th proof uss th discrt Poincaré inquality 3.4 in th particular cas whr r =... Approximation with SIPG Proposition.. If th llipticity. holds, th squncs U and Ũ dfind by.6.8 satisfy th following a priori stimat: U N L Ω + + t U Ũ L Ω + Ũ U L Ω + E Ω t N U n E Ũ + µk t U X + = {U } n E [Ũ ] E N Ũ N X + U Ũ X = U 0 L Ω + C 0 µk whr K is th constant of. and C 0 is th constant of 3.4 with r =. = N t ˇf L Ω,. =

10 4 V. GIRAULT ET AL. Proof. First taking v = Ũ in.6 and using.8, w obtain: Ũ L t Ω U L Ω + Ũ U L Ω + {U } n E [Ũ ] + E + µ aũ, Ũ +J 0 Ũ, Ũ U n E Ũ =ˇf, Ũ.. E Ω Nxt taking v = U in.7 and using th symmtry of a and.8, w obtain U L t Ω Ũ L Ω + U Ũ L Ω + µ au, U aũ, Ũ +au Ũ, U Ũ + µ J 0 U, U J 0 Ũ, Ũ +J 0 U Ũ, U Ũ =0..3 Summing. and.3, and using th llipticity., w obtain U L t Ω U L Ω + U Ũ L Ω + Ũ U L Ω + {U } n E [Ũ ] + U n E Ũ E E E h E E E h Ω + µk U X + Ũ X + U Ũ X ˇf, Ũ..4 W now driv an stimat that is proportional to T and ssntially invrsly proportional to th viscosity. First from 3.4, th right hand-sid of.4 is boundd as follows, for any ɛ>0: ˇf, Ũ ˇf L Ω Ũ L Ω C 0 ˇf L Ω Ũ X ɛ Ũ X + C 0 ɛ ˇf L Ω. Choos ɛ = µk, thn.4 bcoms U L t Ω U L Ω + U Ũ L Ω + Ũ U L Ω + {U } n E [Ũ ] + U n E Ũ E E E h E E E h Ω + µk U X + Ũ X + U Ũ X C 0 µk ˇf L Ω. Th rsult follows by multiplying by t and summing from =0to = N. Proposition. has in particular th following corollary. Corollary.. If th llipticity. holds, th quantitis: sup U L Ω, sup Ũ L Ω, N µk t U X = /, µk N t Ũ X = /

11 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 5 ar all boundd by N U 0 L Ω + C 0 t ˇf L µk Ω = /... Approximation with NIPG Th schm.6.8 is also unconditionally stabl for NIPG providd ach constant σ is sufficintly larg, but indpndnt of h. Mor prcisly, w assum that ach σ is chosn so that: v X h, { v}n σ Γ v 0,Ω..5 h It is asy to chck that.5 holds providd that Γ h, σ 0 σ σ m, for σ 0 andσ m both indpndnt of and h but possibly diffrnt from th constants of SIPG. Thn Lmma. is rplacd by Proposition.3. Assum that.5 holds. Thn, th squncs U and Ũ dfind by.6.8, withth form a dfind by.39, satisfy th following a priori stimat: U N L Ω + U Ũ L Ω + Ũ U L Ω + t + E Ω whr C 0 is th constant of 3.4 with r =. U n E Ũ + µ = t E {U } n E [Ũ ] N t U X + Ũ X U 0 L Ω + C 0 µ N t ˇf L Ω,.6 Proof. As in Lmma., w start with., but in.7 w cannot us th symmtry of a, so instad of.3, w hav: U L t Ω Ũ L Ω + U Ũ L Ω + µ U X Ũ X + U Ũ X + µ { Ũ }n [U ] { U }n [Ũ ] =0,.7 Γ h Γ h =

12 6 V. GIRAULT ET AL. and w must find an uppr bound for this last factor of µ. This trm can b writtn: { Ũ }n [U ] { U }n [Ũ ]= { Ũ U }n [U ] Γ h Γ h Γ h + { U }n [U Ũ ] Γ h Γ h σ { Ũ U }n L Γ h σ { U }n L + Γ h σ [U ] L Γ h σ [U Ũ ] L 4 U X + Ũ U X, whr w hav usd.5 in th last inquality. Thn substituting this bound into.7 and adding., w driv U L t Ω U L Ω + Ũ U L Ω + U Ũ L Ω + µ U X + Ũ X + {U } n E [Ũ ] E E E h + U n E Ũ ɛ Ũ X + C 0 ɛ ˇf L Ω. E Ω Finally,.6 is obtaind by choosing ɛ = µ and summing ovr. Proposition.3 has a corollary similar to Corollary.. Rmark.4. Th stimat.6 is slightly bttr than. bcaus it dos not involv th factor K that is likly to b smallr than on half. On th othr hand, it dos not giv an stimat for N = t U Ũ X, but w shall not us this trm furthr on. 3. L r stimats In th squl, w shall rquir som stimats in L r and intrpolation stimats for th functions of X h. Ths ar a rfinmnt of th L r stimats provn in Lmma 6. of [4]. W first dfin a postprocssing tchniqu: with any function u h in X h, w want to associat a function ū h in X h s.3. To this nd, givn an intrior dg Γ h common to two lmnts E and E in E h,such that n is outward to E, w construct a picwis IP function λ as follows. Lt b dnot th midpoint of and lt λ IP E for all E E h b dfind by λ b E =, λ b E =0, λ b =0 Γ h,. Thus, λ vanishs ovr all triangls othr than E and [λ ]b =, [λ ]=, [λ ]=0, Γ h,.

13 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 7 If Γ h lis on Ω thn w simply st Now, for any u h X h, dfin ū h X h by: λ b =, λ b =0, Γ h,. ū h = u h Γ h Thn, Γ h, [ū h ]= [u h ] [u h ] [λ ]=0, and hnc ū h X h. Th nxt two lmmas show that ū h and u h ar closly rlatd. Lmma 3.. Thr xists a constant C, indpndnt of h, such that [u h ]λ. 3. u h X h, u h ū h X CJ 0 u h, u h /. 3. Proof. It suffics to considr a componnt of u h, dnotd by u h. Lt us first study th gradint part of th norm. By.4, w hav for any E E h : u h ū h L E E [u h ] λ L E C E / ρ E E / [u h] L ˆ ˆλ L Ê C E / [u h] L, whr hr and in th squl, th hat suprscript dnots th rfrnc lmnt and quantitis rlatd to th rfrnc lmnt. Thus, / u h ū h L E C 3 J 0 u h,u h /. 3.3 Nxt, w considr th ump trm. For any intrior Γ h, th st of dgs Γ h for which [λ ] 0isa subst of th union of and th dgs of E E, and if Ω E, it is a subst of E ;wdnotthis st by S. Thrfor, [u h ū h ] L C 4 / S / [u h] L [ˆλ ] L ê. Hnc, / [u h ū h ] L C 5 [u h] / L, S and thus J 0 u h ū h,u h ū h / C 6 J 0 u h,u h /. This complts th proof. Lmma 3.. For any r [, ], thr xists a constant C r dpnding on r but not on h, such that u h X h, u h ū h Lr Ω C r h /r J 0 u h, u h /. 3.4

14 8 V. GIRAULT ET AL. Proof. Lt E E h and lt r<. As in th proof of 3.3, w writ for any componnt u h of u h : u h ū h L r E C E /r [u h] / L. Thn summing ovr all E and applying Jnsn s inquality that is valid sinc r, w obtain E u h ū h L r Ω C h /r J 0 u h,u h /. Whn r =, lte b any lmnt whr th maximum valu of u h ū h is attaind. Thn, max u hx ū h x C 3 x E E / [u h] L, and w rcovr again 3.4. Lmma 3.3. Thr xists a constant C, indpndnt of h, such that ū h X h, J 0 ū h, ū h / C ū h L Ω. 3.5 Proof. Lt u h b a componnt of ū h X h. For any Γ h,asu h has th sam man valu, dnotd by m, coming from E and from E, w can writ for simplicity, w assum that ê =: [u h ] L = / û h Ê û h Ê L ê = / û h m Ê û h m Ê L ê / û h m Ê L ê + û h m Ê L ê. Similarly, if lis on Ω, sinc th man valu m =0,whav [u h ] L / û h m L ê. As th man-valu is prsrvd by th transformation that maps onto ê, wobtain [u h ] L C / ˆ û h L Ê + ˆ û h L Ê. Hnc, [u h] / L C u h L E + u h L E, thus implying 3.5. Now w dfin a lifting function ūh ofū h, as in [4]: ūh H0 Ω is th only solution of v H0 Ω, ūh : v = ū h : v. 3.6 From 3.6 and 3.3, w immdiatly driv that Ω E ūh L Ω ū h L Ω C u h X. 3.7 Lmma 3.4. For any r [,, thr xists a constant C r, dpnding on r but not on h, such that u h ūh L r Ω C r h /r J 0 u h, u h / if r 4, 3.8 u h ūh L r Ω C r h / J 0 u h, u h / if r<4. 3.9

15 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 9 Proof. Again, w considr on componnt u h of u h. Sinc w hav 3.4, it suffics to prov that 3.8, 3.9 hold for ū h ūh. Th proof is similar, but sharpr than that of Lmma 6. of [4]. W procd by duality and writ with /r +/r =: ū h ūh Lr Ω = For a fixd g in L r Ω, lt φ H 0 Ω solv: sup g L r Ω Whn r 4, thn r 4/3 and it follows from [6] that φ W,r Ω with Ω ū h ūhg 3.0 g L r Ω φ = g, φ Ω =0. 3. φ W,r Ω C r g L r Ω. 3. Whn r<4, thn r > 4/3 andg blongs always to L 4/3 Ω. Thrfor w also hav φ W,4/3 Ω with From 3., w driv ū h ūhg = φū h ūh Ω Ω = φ ū h ūh E φ W,4/3 Ω C 4 g L 4/3 Ω C r g L r Ω. 3.3 E φ n E ū h ūh = φ n [ū h ], owing to 3.6 and th rgularity of φ and ūh. Thus, th zro man-valu of [ū h ]onach implis that for any constants c,whav: ū h ūhg = φ n c [ū h ]. Ω Γ h Lt E b a triangl adacnt to and tak c = c n,whr c = φ. E Lt 4 r<. Thn<r 4/3 andthtracof φ on ach dg blongs to L s with /s =/r and <s. Thn, with s + s =, φ c n [ū h ] φ c L s [ū h] L s. On on hand, passing to th rfrnc lmnt, applying th trac thorm with this valu of s and using th dfinition of c, w hav φ c L s C 3 /s φ W,r Ê C 4 /s h E E /r φ W,r E C 5 φ W,r E. On th othr hand, a local quivalnc of norms givs: [ū h ] L s C 6 /s [ū h ] L ê C 6 /s / [ū h ] L. E Γ h

16 30 V. GIRAULT ET AL. Combining ths two inqualitis, w obtain φ c n [ū h ] C 7h /r φ W,r E [ū h] / L. Thn, summing ovr, applying Jnsn s inquality and 3., w obtain for r 4: ū h ūhg C 8h /r J 0 ū h, ū h / g L r Ω. Ω Whn r<4, w apply th abov rsult with th xponnt r =4/3 and w us 3.3: ū h ūhg C 9h / J 0 ū h, ū h / g L r Ω. Ω This concluds th proof. Rmark 3.5. Of cours, by combining 3.7, 3.8 and 3.9 w rcovr th L r stimats of [4]: for any r [,, thr xists a constant C r, dpnding on r but not on h, such that u h X h, u h Lr Ω C r u h X. 3.4 Rmark 3.6. Whn r<4, w can improv 3.9 by rstricting th angls of Ω. In particular, if r = and Ω is convx, w rcovr a full powr of h: This follows from th fact that 3.3 is rplacd by: u h ūh L Ω ChJ 0 u h, u h /. 3.5 φ H Ω C g L Ω. Rmark 3.7. W asily driv from th abov rsults that thr xists a constant C r, that dpnds on r but not on h, such that for all u in H 0 Ω and all u h in X h,whav u h u L r Ω C r u h u X. Indd, sinc ū = u, w associat ūh toū h u as in 3.6. Hnc, Sobolv s imbdding givs u h u L r Ω u h u ūh L r Ω + C ūh H Ω. Thn, w asily chck that 3.4, 3.8 and 3.9 hold for u h u and it suffics to bound ūh H Ω. But Thn Lmma 3. givs th rsult. ūh H Ω ū h u L Ω. Finally, whn r = 4, w driv an analogu of th wll-known intrpolation inquality, that is valid in H 0 Ω: v H 0 Ω, v L 4 Ω /4 v / L Ω v / L Ω. 3.6

17 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 3 Thorm 3.8. Thr xist constants C i, i 3, indpndnt of h, such that u h X h, u h L4 Ω C u h / L Ω u h / X + C h /4 J 0 u h, u h /4 u h / X + C 3h / J 0 u h, u h /. 3.7 Whn Ω is convx, th abov inquality can b improvd u h X h, u h L4 Ω C u h / L Ω u h / X + C h / J 0 u h, u h /4 u h / X + C 3h / J 0 u h, u h /. Proof. Considr on componnt u h of u h. From 3.8 and 3.6 w infr u h L 4 Ω u h ūh L 4 Ω + ūh L 4 Ω C h / J 0 u h,u h / + /4 ūh / L Ω ūh / L Ω. Thn 3.7 and 3.9 giv u h L 4 Ω C h / J 0 u h,u h / + C u h / X u h / L Ω + C 3h /4 J 0 u h,u h /4. This implis 3.7. Whn Ω is convx, th rsult follows by applying 3.5 instad of First rror stimat for vlocity In this sction, w obtain a first rror stimat for th vlocity that is suboptimal in tim, namly of th ordr Oh + t /. An improvd optimal stimat is obtaind in Sction 5. W shall nd th following stimats for th trilinar form c. Th proof is similar to that of Lmma 6.4 of [4], but w writ it hr for th radr s convninc. Proposition 4.. i Assum that u W,r Ω for som r>. Thr xists a constant C that is indpndnt of h, such that for all v h V h and w h X h, cv h ; u, w h C u W,r Ω w h X v h L Ω. 4. ii If u H Ω, thn for any z X, v h V h and w X h w hav c z v h ; R h u u, w h C R h u u W,r Ω + h / u H Ω v h L Ω w h X. 4. iii Finally, for any z X, v h, u h and w h in X h, w hav c z v h ; u h, w h C v h X u h X w h X. 4.3 Proof. i Sinc u has no umps, w can writ: cv h ; u, w h = v h u w h bv h, u w h. E E E h Th first trm is boundd by virtu of 3.4: v h u w h v h L Ω u Lr Ω w h L r Ω C r v h L Ω u W,r Ω w h X, E E E h

18 3 V. GIRAULT ET AL. whr /r +/r =/, r>, r >. To bound th scond trm, w us an argumnt of Girault and Lions []. Dnot by c th picwis constant that is, in ach lmnt E, th scalar product of two constant vctors c c. In viw of.4, w can writ bv h, u w h =bv h, u w h c c =bv h, u c w h +bv h, c w h c. Lt us choos in ach E: From th dfinition of r,whav c = u, c = w h. E E E E u c w h L E u c L r E w h L r E C h w h L r E u W,r E. 4.4 Similarly, c w h c L E c w h c L E C h u L E w h L E. Hnc, using locally an invrs inquality in ach E, whav: v h u w h c c C 3 v h L E u W,r E w h L r E + u L E w h L E. E To stimat th dg trms in b, w considr on lmnt E adacnt to and w apply th trac thorm: u c w h E L C 4 / E / u c w h L E + u c w h L E. W apply 4.4 to th first trm and for th scond trm w writ u c w h L E u L r E w h L r E + C 5 u L E w h L E. Hnc, using a local quivalnc of norms and dnoting E = E E,wobtain {u c w h }[v h ] n C 6 v h L E u W,r E w h L r E + u L E w h L E. Th scond dg trm is asir sinc it only involvs quivalnt norms: {c w h c }[v h ] n C 7 u L E w h L E v h L E. Thn summing ovr all lmnts and dgs, applying Holdr s inquality, 3.4 and Sobolv imbdding, w obtain: v h V h, w h X h, bv h, u w h C 8 v h L Ω u W,r Ω w h X. 4.5 ii To stablish 4., obsrv that th abov argumnt applis to R h u u instad of u for all xcpt th upwind trm. Using th approximation proprtis of R h and 3.4, th upwind trm is boundd by C 9 v h L 4 w h L 4 [R h u u] L Γ h C 0 /4 E / v h L E /4 E /4 w h L 4 E / E / R h u u L E + R h u u L E Γ h C h / E v h L E w h L4 E u H E C h / u H Ω v h L Ω w h X.

19 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 33 iii W skip th proof of 4.3 bcaus it is straightforward. Th proof of a sharpr vrsion is givn in [4] whn v h blongs to V h. W dnot th rrors btwn th numrical solutions and th approximation by = U R h u and ẽ = Ũ R h u,whru stands for ut,. Th following lmma shows xactly whr th loss of optimality occurs. It is valid for both SIPG and NIPG. Lmma 4.. Assum that p L 0,T; H Ω. Thn for ach η>0, thr xists a constant C that dpnds on η but not on h, such that: bẽ,p 4 ẽ L Ω + η t X +Ch + t p L t,t ;L Ω. 4.6 t Proof. Sinc V h,wcanwrit bẽ,p=bẽ,p+b,p r h p. Now,.3 implis t bẽ,p ẽ L Ω t / p L t,t ;L Ω 4 ẽ L Ω + t p L t,t ;L Ω. On th othr hand, th approximation proprty.35 of r h implis b,p r h p C X h p L Ω. Thrfor, for any η>0, b,p r h p t η t X + Ch p L t,t ;L Ω. This concluds th proof of 4.6. Th nxt thorm stablishs an apriorirror stimat for SIPG. Thorm 4.3. Assum that u L 0,T; H Ω, u t L 0,T; H Ω, p L 0,T; H Ω, u 0 V and U 0 = R h u 0. If th llipticy. holds, thr xists a constant C, indpndnt of h and t, such that max L Ω + t + ẽ L Ω + E Ω ẽ L Ω + U n E ẽ + Kµ t E {U } n E [ẽ ] t X + ẽ X + ẽ X Ch + t + t. 4.7

20 34 V. GIRAULT ET AL. Proof. Error quations Intgrating.6 btwn t and t and using.9, w driv: v h X h, Ũ U, v h +µ taũ, v h +J 0 Ũ, v h + =u u, v h +µ au, v h +J 0 u, v h + t t c U U ; Ũ, vh t cu; u, v h + t bv h,p. Now insrting th approximations R h u and R h u and choosing v = ẽ, w obtain a first rror quation ẽ, ẽ +µ t aẽ, ẽ +J 0 ẽ, ẽ + =u R h u u R h u, ẽ +µ + c U U ; u R h u, ẽ + t t c U U ; ẽ, ẽ + t c ; u, ẽ t au Rh u, ẽ +J 0 u R h u, ẽ t cu R h u ; u, ẽ + t bẽ,p. Applying.8, this implis ẽ L Ω L Ω + ẽ L Ω +µ taẽ, ẽ +J 0 ẽ, ẽ + {U } n E [ẽ ] + U n E ẽ + t E t E Ω = t u R h u t, ẽ +µ t au R h u, ẽ +J 0 u R h u, ẽ t c ; u, ẽ + c U U ; u R h u, ẽ + cu R h u ; u, ẽ + bẽ,p. 4.8 t t t Similarly, insrting R h u in.7, w gt a scond rror quation: v h X h, t ẽ, v h +µa ẽ, v h +J 0 ẽ, v h + bv h,p =0. Choosing v h =, intgrating btwn t and t and using.8 and th symmtry of a, wdriv L Ω ẽ L Ω + ẽ L Ω +µ t a, +J 0, aẽ, ẽ J 0 ẽ, ẽ +a ẽ, ẽ +J 0 ẽ, ẽ =

21 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 35 Summing 4.8 and 4.9, w obtain a third rror quation: L Ω L Ω + ẽ L Ω + ẽ L Ω + µ ta, +J 0, + aẽ, ẽ +J 0 ẽ, ẽ +a ẽ, ẽ +J 0 ẽ, ẽ + t {U } n E [ẽ ] + t U n E ẽ + c ; u, ẽ E E Ω t = t u R h u t, ẽ +µ t au R h u, ẽ +J 0 u R h u, ẽ + c U U ; u R h u, ẽ + cu R h u ; u, ẽ + bẽ,p. 4.0 t t t By virtu of 4., for any ɛ>0, w hav c ; u, ẽ t ɛ t ẽ X + C u L 0,T,W,r Ω t L Ω ; in this proof, C dnots various constants that dpnd on ɛ, but not on h and t. Thrfor, from., th lft-hand sid of 4.0 is boundd blow by L Ω L Ω + ẽ L Ω + ẽ L Ω + t + t E Ω E {U } n E [ẽ ] U n E ẽ + Kµ t X + ẽ X + ẽ X ɛ t ẽ X C t L Ω u L 0,T,W,r Ω. 4. Uppr bound of linar trms W now bound th linar trms in th right-hand sid of 4.0; first.34 givs u R h u t, ẽ t Ch ẽ L Ω u t L Ω t Ch t / ẽ L Ω u t L t,t,h Ω ɛ t ẽ L Ω + Ch u t L t,t,h Ω. 4. W rwrit th scond trm as follows: ar h u u, ẽ =ar h u u, ẽ +ar h u u, ẽ. Applying.0,. and.36 to th first trm, it is boundd by ar h u u, ẽ C R h u u X ẽ X C u u H Ω ẽ X Ct t / u t L t,t ;H Ω ẽ X. Nxt, by Lmma.4, ar h u u, ẽ Ch u H Ω ẽ X.

22 36 V. GIRAULT ET AL. Thus, ar h u u, ẽ t ɛ t ẽ X + C t u t L t,t,h Ω + Ch u L t,t,h Ω. 4.3 Bcaus of th rgularity of u, th ump trm satisfis: J 0 R h u u, ẽ =J 0 R h u u, ẽ. Hnc, by.38 J 0 R h u u, ẽ t Ch t u H Ω ẽ X ɛ t ẽ X + Ch t u L 0,T ;H Ω Uppr bound of nonlinar trms Now, w stimat th nonlinar trms. Th first nonlinar trm is split as follows: c U U ; R h u u, ẽ =c U ; R h u u, ẽ +c U R h u ; R h u u, ẽ +c U R h u ; R h u u, ẽ. 4.5 By noting that th upwind trm involving R h u u is th sam as th on involving R h u u,wcan apply Proposition 4. to th first trm in 4.5 and obtain c U ; R h u u, ẽ C L Ω ẽ X R h u u W,r Ω + h / u H Ω C u L 0,T ;H Ω L Ω ẽ X. For th scond trm, by applying 4.3 and.36: c U R h u ; R h u u, ẽ Ct t / u H Ω u t L t,t ;H Ω ẽ X. 4.6 Th third trm is boundd straightforwardly as follows: c U R h u ; R h u u, ẽ Ch u H Ω u H Ω ẽ X. Thus, combining all trms in 4.5: c U U ; R h u u, ẽ t ɛ t ẽ X + C t u L 0,T ;H Ω L Ω + C t u L 0,T ;H Ω u t L t,t ;H Ω + Ch u L 0,T ;H Ω u L t,t ;H Ω. 4.7 Th othr nonlinar trm is rwrittn as: cr h u u; u, ẽ = cr h u u ; u, ẽ + t t A slight variant of th argumnt in Proposition 4. givs cr h u u ; u, ẽ Ch u H Ω u W,4 Ω ẽ X. t cu u; u, ẽ.

23 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 37 Th scond trm rducs to: cu u; u, ẽ = t E u τ u u u ẽ = L Ω u W,4 Ω ẽ L4 Ω. t E u τ u ẽ Thus, cr h u u; u, ẽ t C t/ ẽ X t u L 0,T ;W,4 Ω u t L t,t ;L Ω + Ch u H Ω u L t,t ;W,4 Ω ɛ t ẽ X + C t u L 0,T ;W,4 Ω u t L t,t ;L Ω + Ch 4 u L 0,T ;H Ω u L t,t ;W,4 Ω. 4.8 Combining th bounds 4., 4.3, 4.4, 4.7 and 4.8 and using th fact that u L 0,T; H Ω, th right-hand sid of 4.0 is boundd by ɛ t ẽ X + Ch + t u t L t,t ;H Ω + Ch u L t,t ;H Ω + Ch t u L 0,T ;H Ω + C t L Ω u L 0,T ;H Ω + t bẽ,p. 4.9 Th last trm is stimatd by Lmma 4.. Thn, for an appropriat choic of ɛ and η in Lmma 4., w obtain: L Ω L Ω + ẽ L Ω + ẽ L Ω + Kµ t X + ẽ X + ẽ X+ t + t E Ω E {U } n E [ẽ ] U n E ẽ C t L Ω u L 0,T ;H Ω + Ch + t u t L t,t ;H Ω + Ch u L t,t ;H Ω + Ch t u L 0,T ;H Ω +Ch + t p L t,t,l Ω. 4.0 Sinc U 0 = R h u 0,whav 0 L Ω Ch u 0 H Ω and hnc applying Gronwall s lmma, w hav: max L Ω + + whnc 4.7. t ẽ L Ω + E Ω ẽ L Ω + U n E ẽ + Kµ t E {U } n E [ẽ ] t X + ẽ X + ẽ X C h + t + t CN t, Th following thorm stablishs an rror stimat for NIPG. W skip th proof which is a straightforward combination of th proofs of Thorm 4.3 and Proposition.3.

24 38 V. GIRAULT ET AL. Thorm 4.4. W rtain th assumptions of Thorm 4.3, but w rplac. by.5. Thn thr xists a constant C, indpndnt of h and t, such that max L Ω + + ẽ L Ω + t ẽ L Ω E {U } n E [ẽ ] + t + µ E Ω t X + ẽ X U n E ẽ Ch + t + t. 4. Ths two thorms imply immdiatly th nxt rsult. Corollary 4.5. Undr th assumptions of Thorm 4.3 forsipgorthorm4.4 for NIPG, thr xists a constant C indpndnt of h and t such that max ẽ L Ω Ch + t + t. 4. max X C, max ẽ X C Furthr rror stimat for vlocity and stimat for prssur Th following thorm sharpns th rsults of Thorm 4.3 for SIPG. Thorm 5.. Undr th assumptions of Thorm 4.3 and if u 0 H 3/ Ω, thr xists a constant C and a constant δ>0 indpndnt of h and t such that for all t δ, w hav max L Ω + L Ω + µk t X Ch + t. Proof. Nowthatwhavafirststimatfor and ẽ, w can sharpn th stimat for by liminating ẽ from th rror quation. This is achivd by summing th two quations.6 and.7 and intgrating btwn t and t : v h X h, U U, v h +µ au, v h +J 0 U, v h t + t bv h,p + t c U U ; Ũ, vh = t ˇf, vh. 5.

25 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 39 Insrting R h u and R h u, using.8 and.9 and choosing th tst function v h in V h in ordr to liminat th discrt prssur, giv: v h V h,, v h +µ a, v h +J 0, v h = u t, v h +R h u R h u, v h t t t + µ au Rh u, v h +J 0 u R h u, v h + cu; u, v h t t Taking v h = V h and applying., w obtain: t c U U ; Ũ, vh + t bv h,p r h p. L Ω L Ω + L Ω +µk t X t t t u R hu, + µ au Rh u, +J 0 u R h u, t + cu; u, c U U ; Ũ, t t + b,p r h p t. 5. Th first thr linar trms ar boundd as in Thorm 4.3. For th prssur trm, considring th dfinition of th approximation oprator r h,whav b,p r h p CJ 0, / h p H Ω. Th difficulty is to bound th nonlinar trms; w split thm as follows: cu; u, c U U ; Ũ, =cu R h u ; u, c ; u, + c U U ; u R h u, +c U U ; R h u u, c U U ; ẽ,. First, as in Thorm 4.3 cf. 4.8, w hav cu R h u ; u, C X u L 0,T ;W,4 Ω t t / u t L t,t ;L Ω + h u H Ω u W,4 Ω. Using Proposition 4., w hav, for som r> c ; u, C u W,r Ω X L Ω. In viw of.6, w writ c U U ; u R h u, = c U U ;, u R h u= c U ;, u R h u c U R h u ;, u R h u. For th first trm, using th approximation proprtis of R h and th fact that, according to Corollary 4.5, X is boundd by a constant indpndnt of, h and t, wobtain c U ;, u R h u Ch 3/ u H Ω X.

26 40 V. GIRAULT ET AL. Similarly, th approximation proprtis of R h imply that, for som r>, Nxt c U R h u ;, u R h u Ch u H Ω u W,r Ω X. c U U ; R h u u, =c U ; R h u u, +c U R h u ; R h u u,. For th first trm, w us 4.3,.36 and Corollary 4.5: c U ; R h u u, C X u u H Ω X Th scond trm is boundd lik 4.6. Thrfor C u u H Ω X Ct t / u t L t,t,h Ω X. c U U ; R h u u, Ct t / X u t L t,t ;H Ω u L 0,T ;H Ω. Finally, applying.6, w writ c U U ; ẽ, = c U U ;, ẽ = c U ;, ẽ c U R h u ;, ẽ. For th first trm, applying Thorm 3.8, Thorm 4.3, Corollary 4.5 and.5 c U ;, ẽ C X X ẽ L 4 Ω C X X C ẽ / L Ω ẽ / X + C h /4 J 0 ẽ, ẽ /4 ẽ / X + C 3h / J 0 ẽ, ẽ / C t /4 X X. For th scond trm, th approximation proprtis of R h and Corollary 4.5 imply that c U R h u ;, ẽ C X u W,4 Ω ẽ L Ω C t / u L 0,T ;W,r Ω X, considring that u L 0,T; W,4 Ω. Thus, intgrating all ths trms ovr t and t and summing ovr, th right-hand sid of 5. is boundd by ɛ t X + Ch + t +C t L Ω + C t /4 N t X. First, lt us choos δ such that C δ /4 = µk, i.. δ = 4 µk C, 5.3

27 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 4 and not that C dos not dpnd on ɛ. Nxt, tak ɛ = µk 4. Thn, 5. bcoms max L Ω + L Ω + µk t X 0 L Ω + Ch + t +C t L Ω + µk t 0 X Ch + t +C t L Ω, by applying th rgularity of u 0, th approximation proprtis of R h and.5. Thn, th rsult follows from Gronwall s lmma. Similarly, th nxt thorm sharpns th rsult of Thorm 4.4 for NIPG; its proof is almost idntical to that of Thorm 5., but 5.3 is rplacd by µ 4 δ =. 5.4 C Thorm 5.. Undr th assumptions of Thorm 4.4 and if u 0 H 3/ Ω, thr xists a constant C and a constant δ>0, indpndnt of h and t, such that for all t δ, w hav max L Ω + L Ω + µ t X Ch + t. Rmark 5.3. W can improv th stimat for ẽ L Ω by using a bootstrap argumnt in Thorms 4.3 and 4.4. Indd, in th cas of SIPG, lt th assumptions of Thorm 5. hold, and lt us rvisit th last trm of 4.9. Owing to th fact that blongs to V h, this trm can b writtn without th factor t bẽ,p = t bẽ,p+ b,p r h p t t p, ẽ + r h p}[ Γ h {p ] n t p L 0,T ;H Ω ẽ L Ω + Ch t / J 0, / p L t,t ;H Ω. To simplify, dnot C = p L 0,T ;H Ω. Thn, ithr or If 5.6 holds, thn t ẽ L Ω tc, 5.5 ẽ L Ω > tc. 5.6 bẽ,p ẽ L Ω + Ch t/ J 0, / p L t,t ;H Ω,

28 4 V. GIRAULT ET AL. and th first trm of this bound is absorbd by th lft-hand sid of 4.0. Sinc all th rmaining trms ar of th ordr Oh + t, thn th nd of th argumnt of Thorm 4.3 implis that ẽ L Ω is Oh + t. Othrwis, if 5.6 dos not hold, 5.5 holds and sinc w know from Thorm 5. that max L Ω = Oh + t, this implis that Hnc, in all cass, 5.7 holds. Th proof for NIPG is th sam. max ẽ L Ω = Oh + t. 5.7 Now, w stimat th prssur. Th bound w driv blow is not optimal, considring th dgr of th polynomials usd bcaus th argumnt of Thorm 5. dos not giv a sharpr stimat for ẽ X.Thonly rsult w hav coms from Sction 4; w only hav N t ẽ X = O t, 5.8 = whras, an optimal rror for th prssur rquirs N t ẽ X = O t. 5.9 = Indd, w shall s that th rror stimat for th prssur rquirs an L in spac and tim stimat for th discrt drivativ of U. Mor prcisly, w nd to show that t t L Ω = O t. 5.0 But, w cannot prov this bcaus it maks us of 5.9 in th tratmnt of th nonlinar trm. As it is, w only hav th following suboptimal stimat, which is an asy consqunc of Thorms 5. or 5.: t t With this, w prov th following bound for th prssur. L Ω = O t. 5. Thorm 5.4. Undr th assumptions of Thorm 5. for SIPG or Thorm 5. for NIPG, thr xists a constant C indpndnt of h and t such that for all t δ as dfind in 5.3 for SIPG or 5.4 for NIPG, w hav N t p P L Ω Ch + t. 5. = Proof. From.9 and 5., w hav an rror quation for p: bv h,p P =U U u u, v h +µ au u, v h +J 0 U u, v h t t t + c U U ; Ũ, v h cu; u, v h, v h X h. t

29 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 43 Insrting r h p, R h u and R h u and stting ξ = P r h p, this bcoms bv h,ξ = t + µ a, v h +J 0, v h + t t bv h,p r h p +, v h +R h u R h u u u, v h t c U U ; Ũ, v h cu; u, v h + µ arh u u, v h +J 0 R h u u, v h, v h X h. t From th inf-sup condition.4, it suffics to stimat th right-hand sid in trms of v h X for an arbitrary v h X h. This stimat is obtaind in much th sam way as in th proof of Thorm 5. for all trms xcpt th on involving. All th othr trms hav an optimal uppr bound. For,wsimplywrit, v h L Ω v h L Ω C L Ω v h L Ω, by 3.5. Whn summing ovr, it is clar in viw of 5. that th contribution to th rror of this trm is only Oh + t. 6. Coupling continuous or nonconforming and discontinuous mthods In this sction, w prsnt two possibl combinations of continuous IP, nonconforming IP and IP -SIPG or IP -NIPG in this splitting algorithm. From th computational point of viw, thy ar lss costly than th schm prsntd in Sction. To simplify th prsntation, w assum that Ω is a Lipschitz polygon partitiond into two Lipschitz polygonal subdomains Ω and Ω with intrfac γ, thatachω i is subdividd by a rgular family of triangulation E i h that match on γ. In othr words, w do not considr hanging nods. In th first mthod that corrsponds to th scond stratgy announcd in th introduction, for stp, w us continuous IP lmnts in Ω and SIPG rsp. NIPG in Ω and for stp, w us SIPG rsp. NIPG in th whol domain Ω. Thus, stting X h = {v h C 0 Ω : E E h, v h IP E, v h = 0 on Ω Ω }, dfining X h by and stting X h = {v h L Ω : E E h, v h IP E }, X h = {v h L Ω : v h Ω X h, v h Ω X h }, w rplac.6 by: knowing U V h, find Ũ X h solution of v h X h, t Ũ U, v h +µ aũ, v h +J 0 Ũ, v h + c U U ; Ũ, v h =ˇf, v h, 6. and kp.7 and.8 unchangd. Sinc th forms a, J 0 and c ar consistnt, w can dnot thm by th sam symbol in 6. although from a computational point of viw, thy simplify on Ω ; in particular, thr arnoumptrmsandnoupwindinω. It is asy to s that th stimats in Sction 4 rmain valid for this discrtization. Howvr th improvd stimats of Sction 5 do not appar to carry on hr, bcaus th spacs X h and X h ar diffrnt. Thus, this schm sms lss accurat but it rquirs fwr dgrs of frdom. Whn th spacs X h and X h ar diffrnt and mor prcisly X h is a propr subspac of X h, thn Equations.6 and.7 can no longr b summd bcaus thy ar not statd with th sam tst functions. Indd, on on hand.6 cannot b statd with discontinuous tst functions in Ω. And on th othr hand,.7 cannot b

30 44 V. GIRAULT ET AL. statd with continuous IP functions, sinc thy do not satisfy th inf-sup condition whn combind with IP 0 prssurs. Hnc, 5., that is th starting point of Thorm 5., dos not hold. In th scond mthod that corrsponds to th third stratgy announcd in th introduction, w us th sam dcomposition of Ω and th sam spacs in both stps. In Ω, w rplac th continuous IP approximation of u by a IP nonconforming mthod. Mor prcisly, lt Γ h dnot th st of dgs of E h that do not li on th intrfac γ, ltγ h =Γ h \ Γ h and dfin X h = { v h L Ω : E Eh, v h E IP E, Γ h, which is vry similar to.3. Th spac X h is dfind as abov and w st X h = {v h L Ω : v h Ω X h, v h Ω X h }. } [v h ]=0, It is asy to s that th bilinar form a and th trilinar form c apply to all thr SIPG, NIPG and th IP - nonconforming mthod. On th othr hand, th ump J 0 is not rquird although it dos not ncssarily vanish in th nonconforming mthod. Thus w rplac.9 by J 0 u, v = Γ h σ [u] [v]. With this nw spac X h and nw form J 0, th formulation of this schm is givn again by.6,.7 and.8. As for th first mthod, th stimats of Sction 4 ar valid hr. In addition, sinc th sam spac is usd in both stps, th stimats of Sction 5 ar also valid and thrfor, as far as th vlocity is concrnd, this scond mthod has an optimal accurary. It rquirs lss dgrs of frdom than th SIPG or NIPG mthod and it rtains th proprty of local mass consrvation. 7. Numrical xprimnts Lt Ω =]0, [ ]0, [ and considr th transint Navir-Stoks quations with solution u = x 4 x 3 + x 4y 3 6y +yt, 4x 3 6x +xy 4 y 3 + y t, p =0. 7. W study hr th numrical convrgnc of th schm.6.8 introducd in Sction, but instad of rstricting th discussion to IP IP 0, w also comput th solution with IP IP that also satisfis th inf-sup condition, s [4]. Th tim stp t is chosn accordingly so that it is of th ordr h for th cas IP IP 0 and of th ordr h for th cas IP IP. Th domain is subdividd into an initial msh consisting of two lmnts. W thn succssivly rfin th msh and comput th rrors h on th msh of siz h and th numrical convrgnc rats by th ratio ln h / h/ / ln. W prsnt th numrical rrors of th vlocity in th nrgy norm and in th L norm and th numrical rror of th prssur in th L norm computd at th final tim of simulation. W choos a constant pnalty paramtr σ = 0 for SIPG and w considr thr cass for NIPG: σ =0,σ =andσ = 0. W did xplor th cas of SIPG with σ =, but th rsults wr inconclusiv. In th following tabls, th numbr aftr th nam SIPG or NIPG corrsponds to th valu of σ. Tabl shows th rrors and convrgnc rats for th cas whr th vlocitis ar approximatd by picwis linars and th prssur by picwis constants. As prdictd by th thory, w obsrv that th rror of u in th H 0 norm is Oh. Th first intrsting point in this tabl is that th rror of p in th L norm is Oh and that of u is Oh, much bttr than what th thory prdicts. Th scond intrsting point is that th rsults for NIPG ar also optimal, vn bttr in som cass than SIPG, and in this xprimnt ar not snsitiv to th choics of σ. Lt us rcall that usually, th advantag of NIPG is that th pnalty paramtr σ dos not hav to b adustd and can b kpt small, i.. σ =. Th third intrsting point is that NIPG with σ =0i..

31 OPERATOR SPLITTING AND DG FOR NAVIER-STOKES 45 Tabl. Numrical rrors and convrgnc rats, for IP IP 0 with t =0. Mthod h ut U N H 0 Ω rat ut U N L Ω rat pt P N L Ω rat SIPG 0 / / / / / / NIPG 0 / / / / / / NIPG / / / / / / NIPG 0 / / / / / / without umps givs good rsults, xcpt for th rror of u in L. This is surprising bcaus thr is no rror analysis for NIPG 0. Sinc this mthod is not adaptd to th IP discrtization of tim indpndnt lliptic problms, this good prformanc hr may b du to th ffct of th tim drivativ. W rpatd th xprimnts for th cas whr th vlocitis ar approximatd by picwis quadratics and th prssur by picwis linars. Th rsults ar shown in Tabl. All mthods convrg optimally in nrgy norm for vlocity and in L norm for prssur. SIPG 0 is also optimal in L for th vlocity, but th NIPG mthods ar suboptimal and only of th ordr Oh. This is consistnt with prvious rsults with NIPG for lliptic problms, namly optimal rsults in th L norm ar only obsrvd whn th dgr of th polynomial usd is odd. Concluding rmarks: In this work, w prsntd svral discrtizations basd on an oprator-splitting tchniqu. Bsids th advantags of th dcoupling of th incomprssibility condition and th nonlinarity, our proposd mthods bnfit from th advantags of th discontinuous Galrkin mthods: local mass consrvation, high ordr of

32 46 V. GIRAULT ET AL. Tabl. Numrical rrors and convrgnc rats, for IP IP with t =0 3. Mthod h ut U N H 0 Ω rat ut U N L Ω rat pt P N L Ω rat SIPG 0 / / / / / NIPG 0 / / / / / NIPG / / / / / NIPG 0 / / / / / approximation, robustnss and stability. It is to b notd that th SIPG vrsion might b prfrrd to th NIPG from a point of viw of bttr conditioning of th linar systm. Finally our multi-numrics approach coupling of continuous and discontinuous finit lmnts allows th us of fficint solvrs for th first stp, whil still obtaining a locally divrgnc-fr vlocity. Rfrncs [] R.A. Adams, Sobolv Spacs. Acadmic Prss, Nw York 975. [] A.S. Almgrn, J.B. Bll, P. Collla, L.H. Howll and M.L. Wlcom, A consrvativ adaptiv proction mthod for th variabl dnsity incomprssibl Navir-Stoks quations. Tchnical Rport LNBL-39075, UC [3] C.E. Baumann and J.T. Odn, A discontinuous hp finit lmnt mthod for convction-diffusion problms. Comput. Mthods Appl. Mch. Engrg [4] J. Blasco and R. Codina, Error stimats for an oprator-splitting mthod for incomprssibl flows. Appl. Numr. Math [5] J. Blasco, R. Codina and A. Hurta, A fractional-stp mthod for th incomprssibl Navir-Stoks quations rlatd to a prdictor-multicorrctor algorithm. Int. J. Numr. Mth. Fl [6] P.G. Ciarlt, Th finit lmnt mthods for lliptic problms. North-Holland, Amstrdam 978. [7] A.J. Chorin, Numrical solution of th Navir-Stoks quations. Math. Comp [8] M. Crouzix and P.A. Raviart, Conforming and non conforming finit lmnt mthods for solving th stationary Stoks quations. RAIRO Anal. Numér. R [9] C. Dawson and J.Proft, Discontinuous and coupld continuous/discontinuous Galrkin mthods for th shallow watr quations. Comput. Mthods Appl. Mch. Engrg