A LOCALLY CONSERVATIVE LDG METHOD FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

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1 A LOCALLY CONSERVATIVE LDG METHOD FOR THE INCOMPRESSIBLE NAVIER-STOES EQUATIONS BERNARDO COCBURN, GUIDO ANSCHAT, AND DOMINI SCHÖTZAU Abstract. In this papr, a nw local discontinuous Galrkin mthod for th incomprssibl stationary Navir-Stoks quations is proposd and analyzd. Four important faturs rndr this mthod uniqu: Its stability, its local consrvativity, its high-ordr accuracy, and th xact satisfaction of th incomprssibility constraint. Although th mthod uss compltly discontinuous approximations, a globally divrgnc-fr approximat vlocity in H(div; ) is obtaind by a simpl, lmnt-by-lmnt post-procssing. Optimal rror stimats ar provn and an itrativ procdur usd to comput th approximat solution is shown to convrg. This procdur is nothing but a discrt vrsion of th classical fixd point itration usd to obtain xistnc and uniqunss of solutions to th incomprssibl Navir-Stoks quations by solving a squnc of Osn problms. Numrical rsults ar shown which vrify th thortical rats of convrgnc. Thy also confirm th indpndnc of th numbr of fixd point itrations with rspct to th discrtization paramtrs. Finally, thy show that th mthod works wll for a wid rang of Rynolds numbrs. Math. Comp., Vol. 74, pp , Introduction In this papr, w propos and analyz a local discontinuous Galrkin (LDG) mthod for th stationary incomprssibl Navir-Stoks quations (1.1) ν u + (u u) + p = f in, u = 0 in, u = 0 on Γ =. Hr, ν is th kinmatic viscosity, u th vlocity, p th prssur, and f th xtrnal body forc. For th sak of simplicity, w tak to b a polygonal domain of R 2. This papr is th fourth in a sris ([9], [8] and [7]) dvotd to th study of th LDG mthod as applid to incomprssibl fluid flow problms. In [9], w considrd 2000 Mathmatics Subjct Classification. 65N30. y words and phrass. Finit lmnt mthods, discontinuous Galrkin mthods, incomprssibl Navir-Stoks quations. This work was carrid out in part whil th authors wr at th Mathmatischs Forschungsinstitut Obrwolfach for th mting on Discontinuous Galrkin Mthods in April 21-27, 2002 and whil th scond and third authors visitd th School of Mathmatics, Univrsity of Minnsota, in Sptmbr Th first author was supportd in part by th National Scinc Foundation (Grant DMS ) and by th Univrsity of Minnsota Suprcomputing Institut. 1 c 2002 Amrican Mathmatical Socity

2 2 B. Cockburn, G. anschat and D. Schötzau th Stoks quations (1.2) ν u + p = f in, u = 0 in, u = 0 on Γ, and focusd on th problm of how to dal with th incomprssibility condition. Latr, in [8], w considrd th Osn quations (1.3) ν u + (w ) u + p = f in, u = 0 in, u = 0 on Γ, whr th convctiv vlocity w was takn to b a smooth function, and focusd on th problm of how to incorporat th linar convctiv trm. Th rsulting mthod was shown to b optimally convrgnt and robust for a wid rang of Rynolds numbrs. A succinct rviw of this work can b found in [7]. In this papr, w continu our study of LDG mthods for incomprssibl flows and considr thir application to th Navir-Stoks quations (1.1). Our main concrn is to dvis LDG mthods that ar locally consrvativ and can b provn to b stabl. Th local consrvativity, a proprty highly valud by practitionrs of computational fluid dynamics, is a discrt vrsion of th idntity (1.4) ( ν u n + (u n ) u + p n ) ds = f dx, whr is an arbitrary sub-domain of with outward normal unit vctor n. This proprty can b asily nforcd by LDG mthods as soon as th quations ar writtn in divrgnc form. Howvr, to dvis an LDG mthod that can b provn to b stabl, that is, that satisfis a discrt vrsion of th following stability stimat for th continuous cas, (1.5) u 1 C P f 0, ν whr th constant C P is th Poincaré constant, is xtrmly difficult. Th rason for this is that, in ordr to obtain th stability stimat (1.5), th incomprssibility condition must b usd. It is wll known that, for many numrical mthods for th Stoks and th Osn quations, a wakly nforcd incomprssibility is nough to guarant stability. Howvr, this is not so for th Navir-Stoks quations bcaus of th prsnc of th non-linar convction. Morovr, th now standard solution to this problm, [21], [22], which is basd on a suitabl modification of th non-linarity of th Navir-Stoks quations, cannot b usd. This happns bcaus such a modification dos not hav divrgnc form and hnc prvnts LDG mthods from bing locally consrvativ. Th main contribution of this papr is to show how to ovrcom this difficulty. In fact, w show that this can b don in two ways. Th first on focuss on th convctiv non-linarity and is basd on a nw modification in divrgnc form of th non-linarity; it will b xplord lswhr in dtail. Th scond on, which constituts th main subjct of this papr, focuss on th incomprssibility constraint and is basd on discrtizing th Osn quations (1.3) whr th convctiv vlocity w is takn to b a projction of th approximat vlocity u h, (1.6) w = Pu h,

3 A locally consrvativ LDG mthod for th incomprssibl Navir-Stoks quations 3 into th spac of globally divrgnc-fr functions. This projction is a slight modification of wll-known projctions Π h with th proprty P h = Π h, whr P h is an L 2 -projction; s [5]. Its implmntation is vry fficint as it can b computd in an lmnt-by-lmnt fashion. Thus, givn th convctiv vlocity w = Pu h, th rsulting schm is nothing but th LDG mthod [8] alrady studid for th Osn quations (1.3). Sinc that mthod is stabl, high-ordr accurat and locally consrvativ, so is th LDG mthod undr considration. Morovr, th approximation to th vlocity givn by w has continuous normal componnts across lmnts and is globally divrgncfr in H(div; ) = {v L 2 () 2 : v L 2 ()}. To th knowldg of th authors, no othr numrical schm for th incomprssibl Navir-Stoks quations has all ths proprtis. Of cours, th convctiv vlocity w dpnds on th discrt vlocity fild u h through (1.6) and, hnc, w nd to us an itrativ mthod to comput it. To do so, w not that if S(u h ) is th LDG approximat vlocity of th Osn problm with convctiv vlocity w = Pu h, thn th approximat vlocity u h of th LDG mthod undr considration is a fixd point of S. If S is provn to b a contraction, to comput th approximat solution u h, w can us th fixd point itration u l+1 h := S(u l h). This is nothing but a discrt vrsion of th argumnt usd to prov th xistnc and uniqunss of th xact solution of (1.1). It nsurs th xistnc and uniqunss of th xact solution (u, p) H 1 0 () 2 L 2 ()/R of (1.1) undr a smallnss condition of th typ C C P f 0 (1.7) ν 2 < 1, whr C > 0 only dpnds on ; s [17, Thorm ] and th rfrncs thrin. W mimic this argumnt to show that th approximat solution of th LDG mthod xists and is uniqu undr a similar condition. Lt us point out that xact incomprssibility can b achivd trivially if th LDG mthod has a vlocity spac that is div-conforming, i.., that is includd in H(div; ). In this particular cas, wak incomprssibility implis xact incomprssibility, providd that th discrt spacs ar matchd corrctly. As will b discussd blow, this approach can b viwd as a particular LDG mthod for which th oprator P is chosn to b th idntity. Consquntly, all th rsults of this papr hold tru vrbatim for mthods that ar basd on div-conforming vlocity spacs. Furthrmor, w not that, although w hav usd th LDG mthod to discrtiz th trms associatd with th viscosity ffcts, any othr DG discrtization whos primal form is both corciv and continuous could hav bn usd to that ffct; s th discussions in [2] and [19]. Th organization of th papr is as follows. In Sction 2, w discuss th idas that motivat th dvising of th LDG mthod w propos in this papr. In Sction 3, w prsnt th LDG discrtization in dtail and vrify its local consrvativity. In Sction 4, w stat and discuss th main rsults, namly, th stability of th mthod, th convrgnc of th fixd point itration and th a-priori rror stimats, and in Sction 5, w prsnt thir proofs. In Sction 6, w prsnt numrical xprimnts vrifying th thortical rsults. W nd in Sction 7 with som concluding rmarks.

4 4 B. Cockburn, G. anschat and D. Schötzau 2. Dvising th LDG mthod In this sction, w discuss th idas that ld us to th dvising of an LDG mthod that is both stabl and locally consrvativ. To kp th discussion as simpl and clar as possibl, w do not work with th numrical mthod. Instad, w work dirctly with th quations (1.1) and infr, from thir structur, th proprtis of th corrsponding LDG mthod A locally consrvativ LDG mthod. Sinc th incomprssibl Navir- Stoks quations (1.1) ar writtn in divrgnc form, a locally consrvativ LDG mthod can b asily constructd. Howvr, it is vry difficult to prov its stability. Lt us illustrat this difficulty by using th quations for th xact solution. If w multiply th first quation of (1.1) by u, intgrat by parts and us th boundary conditions, w gt ν u : u dx + 1 u 2 u dx p u dx = f u dx. 2 W s that w must us th incomprssibility condition to obtain th quation ν u : u dx = f u dx, from which th stability stimat (1.5) immdiatly follows. In gnral, sinc xact incomprssibility is vry difficult to achiv aftr discrtization, it is usually only nforcd wakly. This wak incomprssibility is nough, in a wid varity of cass, to guarant that th discrt vrsion of th trm p u dx is xactly zro, as for most mixd mthods for th Stoks and Navir-Stoks quations, or non-ngativ, as for th LDG mthods considrd for th Stoks [9] and Osn [8] problms. Unfortunatly, this is not tru for th discrt vrsion of th trm 1 u 2 u dx, 2 bcaus th squar of th modulus of th approximat vlocity dos not ncssarily blong to th spac of th approximat prssur Th classical modification of th non-linarity. A solution to this impass can b obtaind by using a now classical tchniqu proposd back in th 60 s; s [21] and [22]. From our prspctiv, it consists in modifying th non-linarity of th quations as follows: ν u + (u u) 1 ( u) u + p = f 2 in, u = 0 in, u = 0 on Γ. To s that this solvs th problm, multiply th first quation by u, intgrat by parts and us th boundary conditions to gt ν u : u dx p u dx = f u dx.

5 A locally consrvativ LDG mthod for th incomprssibl Navir-Stoks quations 5 As a consqunc, stability can follow from wak incomprssibility. This is th cas for most mixd mthods for th Navir-Stoks quations. It is also th cas for th first discontinuous Galrkin mthod for th incomprssibl Navir-Stoks quations [13], a mthod which uss locally divrgnc-fr polynomial approximations of th vlocity, and for th mor rcnt discontinuous Galrkin mthod dvlopd in [10]. Th only problm with this approach is that local consrvativity cannot b achivd bcaus th first quation is not writtn in divrgnc form A nw modification of th non-linarity. To ovrcom this difficulty, it is nough to tak anothr glanc at th first quation in this sction to raliz that instad of working with th kinmatic prssur p, w should work with th nw variabl P = p 1 2 u 2. If w incorporat this unorthodox prssur into th Navir-Stoks quations, w gt ν u + (u u) u 2 + P = f in, u = 0 in, u = 0 on Γ. W now s that, sinc th abov modification is in divrgnc form, locally consrvativ LDG mthods can asily b constructd. Morovr, sinc w hav ν u : u dx P u dx = f u dx, w also s that th stability of th LDG mthod can follow from wak incomprssibility. Th LDG mthod obtaind with this approach can indd b provn to hav thos proprtis; it is going to b studid thoroughly in a forthcoming papr Enforcing xact incomprssibility. As w hav sn, suitabl modifications of th non-linarity can b introducd which allow obtaining stability by using only wakly incomprssibl approximations to th vlocity. Th approach w considr in this papr dos not rly on a modification of that typ. Instad, it is basd on nforcing xact incomprssibility in th spac H(div; ) := {v L 2 () 2 : v L 2 ()}. Th ida that allows this to happn is basd on two obsrvations: Th first is that w can rwrit th Navir-Stoks quations as th Osn problm ν u + (w ) u + p = f in, u = 0 in, u = 0 on Γ, whr, of cours, w = u. If w multiply th first quation by u, intgrat by parts and us th boundary conditions, w gt ν u : u dx 1 u 2 w dx p u dx = f u dx. 2 This suggsts to considr an LDG mthod with two diffrnt (but strongly rlatd) approximations to th vlocity: On approximation for u and anothr for w. Th stability for th LDG mthod would thn b achivd if th approximation to u

6 6 B. Cockburn, G. anschat and D. Schötzau is wakly incomprssibl and if th approximation to w is xactly incomprssibl. Furthr, local consrvativity can b radily achivd for such an LDG mthod. Indd, th fact that th quations ar not writtn in consrvativ form can b compnsatd by th fact that th approximation to w is globally divrgnc-fr. Th scond obsrvation is that, if th approximation to u givn by an LDG mthod, u h, is wakly incomprssibl, it is possibl to comput, in an lmnt-bylmnt fashion, anothr approximation, w = P(u h ), which is xactly incomprssibl. As w shall s, th post-procssing oprator P is a slight modification of th wll-known Brzzi-Douglas-Marini (BDM) intrpolation oprator; s [4]. W not that th post-procssing procdur can b omittd in th particular cas whr th vlocity spac is div-conforming. Indd, in this cas th fact that th approximation to th vlocity is wakly incomprssibl dos imply that it is xactly incomprssibl, providd th prssur spac is chosn suitably. Hnc, w can tak w = u h. W ar now rady to dscrib th LDG mthod in dtail. 3. Th LDG mthod In this sction, w introduc a locally consrvativ LDG discrtization for th Navir-Stoks quations (1.1) Mshs and trac oprators. W bgin by introducing som notation. W dnot by T h a rgular and shap-rgular triangulation of msh-siz h of th domain into triangls {}. W furthr dnot by Eh I th st of all intrior dgs of T h and by Eh B th st of all boundary dgs; w st E h = Eh I EB h. Nxt, w introduc notation associatd with tracs: Lt + and b two adjacnt lmnts of T h ; lt x b an arbitrary point of th intrior dg = + Eh I. Lt ϕ b a picwis smooth scalar-, vctor-, or matrix-valud function and lt us dnot by ϕ ± th tracs of ϕ on takn from within th intrior of ±. Thn, w dfin th man valu { } at x as {ϕ } := 1 2 (ϕ+ + ϕ ). Furthr, for a gnric multiplication oprator, w dfin th jump [ ] at x as [ϕ n] := ϕ + n + + ϕ n. Hr, n dnots outward unit normal vctor on th boundary of lmnt. On boundary dgs, w st accordingly {ϕ } := ϕ, and [ϕ n] := ϕ n, with n dnoting th outward unit normal vctor on Γ Th LDG mthod for th Osn quations. W now rcall th LDG mthod for th Osn quations (1.3). W assum that th convctiv vlocity fild w is in th spac J(T h ) = {v L 2 () 2 : v 0 and v H 1 () 2, T h }.

7 A locally consrvativ LDG mthod for th incomprssibl Navir-Stoks quations 7 W bgin by introducing th auxiliary variabl σ = ν u and rwriting th Osn quations as (3.1) σ = ν u in, σ + (w )u + p = f in, u = 0 in, u = 0 on Γ. Nxt, w introduc th spac Σ h V h Q h whr Σ h = { v L 2 () 2 2 : τ P k () 2 2, T h }, V h = { v L 2 () 2 : v P k () 2, T h }, Q h = { q L 2 () : q P k 1 (), T h, q dx = 0 }, for an approximation ordr k 1. Hr, P k () dnots th spac of polynomials of total dgr at most k on. For simplicity, w considr hr only so-calld mixdordr lmnts whr th approximation dgr in th prssur is of on ordr lowr than th on in th vlocity. Finally, w dfin th approximat solution (σ h, u, p h ) Σ h V h Q h by rqusting that for ach T h, (3.2) σ h : τ dx = ν u h τ dx + ν û σ h τ n ds, [ σh : v p h v ] ] (3.3) dx [ σh : (v n ) p h v n ds u h (v w) dx + w n û w h v ds = f v dx, (3.4) u h q dx + û p h n q ds = 0, for all tst functions (τ, v, q) Σ h V h Q h. Each of th abov quations is nforcd locally, that is, lmnt by lmnt, du to th apparanc of th so-calld numrical fluxs û σ h, σ h, p h, û w h and ûp h. Thanks to this structur of th LDG mthod, w immdiatly gt that ( σ h n + (w n ) û w h + p h n ) ds u h w dx = f dx, and sinc w is globally divrgnc-fr, w obtain a discrt vrsion of th proprty of local consrvativity (1.4), namly, ( σ h n + (w n ) û w h + p h n ) ds = f dx. In othr words, th LDG mthod is locally consrvativ. To nsur that th mthod is also stabl (and high-ordr accurat), th numrical fluxs, which ar nothing but discrt approximations to th tracs on th boundary of th lmnts, must b dfind carfully. As w shall prov, th numrical fluxs that dfin th LDG mthod for th Osn quations [8] do nsur stability. For th sak of clarity, w considr th fluxs in thir simplst form.

8 8 B. Cockburn, G. anschat and D. Schötzau Th convctiv numrical flux. For th convctiv flux û w h in (3.3), w tak th standard upwind flux introducd in [15, 18]: For an lmnt T h, w st { û w h (x) = lim ɛ 0 u h (x ɛw(x)), x \ Γ, (3.5) 0, x Γ, whr Γ is th inflow part of Γ givn by Γ = { x Γ : w(x) n(x) < 0 }. Th diffusiv numrical fluxs. If a fac lis insid th domain, w tak (3.6) σ h = {σ h } κ[u h n], û σ h = {u h }, and, if lis on th boundary, w tak (3.7) σ h = σ h κu h n, û σ h = 0. As will b shown latr, th rol of th paramtr κ is to nsur th stability of th mthod; s also [6]. Th numrical fluxs rlatd to th incomprssibility constraint. Th numrical fluxs associatd with th incomprssibility constraint, û p h and p h, ar dfind by using an analogous rcip. If th fac lis on th intrior of, w tak (3.8) û p h = {u h }, p h = {p h }. On th boundary, w st (3.9) û p h = 0, p h = p h. This complts th dfinition of th LDG mthod for th Osn problm in (3.1). Rmark 3.1. Notic that on can tak th div-conforming vlocity spac (3.10) Ṽ h = {v V h : v L 2 ()}, whil kping th othr spacs and dfinitions abov unchangd Th post-procssing oprator. To complt th dfinition of th LDG mthod for th Navir-Stoks quations (1.1), it only rmains to introduc what w rfr to as th post-procssing oprator P and to st w = Pu h in th approximation (3.2) (3.4). For a picwis smooth vlocity fild u, w dfin th oprator P by Pu = P ( u, û p), T h, whr û p is th numrical flux (3.8) (3.9) rlatd to th incomprssibility constraint. For ach lmnt, th local oprator P is givn via th following momnts: P u n ϕ ds = û p n ϕ ds ϕ P k (), for any dg, P u ϕ dx = u ϕ dx ϕ P k 1 (), P u Ψ dx = u Ψ dx Ψ Ψ k (), whr Ψ k () = {Ψ L 2 () 2 : DF t Ψ F Ψ k ( )}.

9 A locally consrvativ LDG mthod for th incomprssibl Navir-Stoks quations 9 Hr, F : dnots th lmntal mapping and DF its Jacobian. On th rfrnc triangl = {( x 1, x 2 ) : x 1 > 0, x 1 + x 2 < 1}, th spac Ψ k ( ) is dfind by Ψ k ( ) = {Ψ P k ( ) 2 : Ψ = 0 in, Ψ n b = 0 on }. Th post-procssing oprator P is wll-dfind and can b computd in an lmnt-by-lmnt fashion. Morovr, if u h V h satisfis th quations (3.4), that is, if it is wakly incomprssibl, thn w = Pu h is xactly incomprssibl. Ths rsults ar gathrd in th nxt rsult and ar givn in trms of th Piola transformation, which maps any vctor fild v on th rfrnc triangl into P v = dt(df ) 1 DF v F 1, T h, and of th BDM projction on, P BDM ; s [4]. b Proposition 3.2. W hav th following rsults. (1) Pu is wll-dfind and Pu is in th spac Ṽh = {v V h : v L 2 ()}. (2) If u H0 1 () 2 and T h, thn P u = P P BDM b P 1 u. (3) If u V h satisfis (3.4), thn Pu = 0 in and Pu J(T h ). Proof. Th proof of th first assrtion is straightforward and that of th scond asily follows from th dfinitions of th projctions and from th fact that, if u H0 1 () 2, thn û p = u. To prov th third assrtion, w first obsrv that Pu Q h. This is du to th fact that Pu P k 1 () for all T h and Pu dx = Pu n ds = û p n ds = 0, in viw of th dfinitions of P and û p in (3.9). Now, lt u V h satisfy (3.4). For q Q h, w obtain Pu q dx = ( Pu q dx + T h = = 0. T h Γ ( Γ ) Pu n q ds ) u q dx + û p n q ds Hr, w hav usd intgration by parts, th proprtis of P and (3.4). Thus, w hav Pu 0 in. It follows that Pu J(T h ). Rmark 3.3. For th LDG mthod using th div-conforming spac Ṽh in (3.10), it can b radily sn that a fild u Ṽh satisfying (3.4) alrady is xactly incomprssibl and blongs to J(T h ). Hnc, for this particular LDG mthod, w can tak P as th idntity Th mixd stting of th LDG mthod. Nxt, w rcast th LDG mthod undr considration in a classical mixd stting, not only to facilitat its analysis, but to b abl to stat our main rsults in a mor prcis way. Thus, w liminat th auxiliary variabl σ h and show that th approximation (u h, p h ) V h Q h

10 10 B. Cockburn, G. anschat and D. Schötzau givn by th LDG mthod satisfis (3.11) A h (u h, v) + O h (w; u h, v) + B h (v, p h ) = (3.12) B h (u h, q) = 0, for all (v, q) V h Q h whr (3.13) w = Pu h. f v dx, Hr, th forms A h, O h and B h ar associatd to th discrtization of th Laplacian, th convctiv trm and th incomprssibility constraint, rspctivly. W procd in svral stps. Stp 1: Solving for σ h in trms of u h. To b abl to liminat th auxiliary variabl σ h, w introduc th lifting oprator L : V h Σ h by L(v) : τ dx = [v n] : {τ } ds τ Σ h. E h It is now asy to s that th quation dfining σ h in trms of u h, quation (3.2), can b rwrittn as (3.14) σ h = ν [ h u h L(u h ) ], with h dnoting th lmnt-wis gradint. Not that, σ h can b computd from u h in an lmnt-by-lmnt fashion. Using this idntity, it is possibl to liminat σ h from th quations as w show nxt. Stp 2: Eliminating σ h. To liminat σ h from quation (3.3), w mak us of a scond lifting oprator M : V h Q h givn by M(v) q dx = {q }[v n] ds q Q h. E h If w insrt th xprssion of σ h into quation (3.3) and us th dfinitions of th numrical fluxs and th lifting oprators, w radily gt, s [2], [8] and [19], A h (u h, v) + O h (w; u h, v) + B h (v, p h ) = f v dx, whr A h (u, v) := ν [ h u L(u) ] : [ h v L(v) ] dx + κ [u n] : [v n] ds, E h O h (w; u, v) := u (v w) dx + T h B h (v, q) := q h v dx + qm(v) dx. \Γ w n û w v ds, This complts th limination of th auxiliary variabl σ h from th quations dfining th LDG mthod. Not that xactly th sam form B h is also usd in th mixd DG approachs of [11, 23, 19, 10].

11 A locally consrvativ LDG mthod for th incomprssibl Navir-Stoks quations 11 Stp 3: Rwriting th incomprssibility constraint. Finally, it is a simpl xrcis to s that quation (3.4) can b rwrittn as B h (u h, q) = 0 q Q h. This shows that th LDG mthod in (3.2) (3.4) can b cast in th form givn in (3.11) (3.12). 4. Th main rsults In this sction, w stat and discuss th main rsults of this papr Prliminaris. W considr LDG mthods with a vry spcific stabilization function κ. To dfin it, w introduc on th dgs th local msh-siz function h by h := h for all E h, with h dnoting th lngth of th dg. W thn st (4.1) κ := νκ 0 h 1, with κ 0 > 0 indpndnt of th msh-siz and th viscosity ν. Th rsults will b statd in trms of norms w introduc nxt. W considr th spac (4.2) V(h) := H 1 0 ()2 + V h, ndowd with th norm v 2 1,h := v 2 0, + T h E h For this norm, w hav th following Poincaré inquality (4.3) v 0 C p v 1,h v V(h), κ 0 h 1 [v n] 2 ds. for a constant C p > 0 indpndnt of th msh-siz; s,.g., [1, 3]. Finally, th spac Q h for th prssurs is quippd with th L 2 -norm Stability proprtis of th bilinar forms. Hr, w collct crucial stability proprtis of th forms that ar usd to dfin th LDG mthod. Our main rsults will b statd in trms of th corrsponding stability constants. Continuity. First, w study th continuity of th forms involvd in th LDG mthod. W obsrv that th lifting oprators L and M can b xtndd to oprators L : V(h) Σ h and M : V(h) Q h, rspctivly, using th sam dfinitions. It is thn wll-known from,.g., [16, Sction 3] and [19, Lmma 7.2] that (4.4) L(v) 2 0 Clift 2 κ 0 h 1 [v n] 2 ds, v V(h), (4.5) M(v) 2 0 C2 lift E h E h κ 0 h 1 [v n] 2 ds, v V(h), for a constant C lift only dpnding on κ 0, th shap-rgularity of th msh and th polynomial dgr k. As a consqunc, th forms A h and B h ar wll-dfind and continuous on V(h) V(h) and V(h) L 2 ().

12 12 B. Cockburn, G. anschat and D. Schötzau Proposition 4.1. W hav that A h (u, v) ν C a u 1,h v 1,h u, v V(h), B h (v, q) C b v 1,h q 0 (v, q) V(h) L 2 (), for continuity constants C a and C b that ar indpndnt of th msh-siz. Proof. Th continuity proprtis of A h and B h follow from (4.4), (4.5) and th Cauchy-Schwarz inquality; s [16, Proposition 3.1] and [19, Lmma 7.5]. Proposition 4.2. Lt w 1, w 2 J(T h ), u V(h) and v V h. Thn w hav O h (w 1 ; u, v) O h (w 2 ; u, v) C o w 1 w 2 1,h u 1,h v 1,h, for a continuity constant C o that is indpndnt of th msh-siz. Th proof is givn in Sction 5.1. It is obtaind by using th mbdding and trac thorms of [13] and [10]. Corcivity and inf-sup condition. Nxt, w discuss th corcivity proprtis of th forms A h and O h. W hav th following rsult. Proposition 4.3. Lt κ b givn by (4.1). Thn, for any κ 0 > 0, thr xists a constant α > 0 indpndnt of th msh-siz such that A h (v, v) να v 2 1,h v V h. Furthrmor, for w J(T h ) and v V h, thr holds O h (w; v, v) = 1 w n [v n] 2 ds E I h Γ w n v 2 ds. In th intgrals ovr dgs in Eh I, w dnot by n any unit normal to th dg undr considration. For a proof of th corcivity of th form A h, w rfr to [2] or [16]. A similar corcivity rsult involving also th discrt vlocity gradint was usd in [9] and [8]. W furthr not that, for th similar symmtric intrior pnalty forms A h usd in th DG approach of [11], th paramtr κ 0 has to b chosn larg nough. Th proof of th scond assrtion is standard; s,.g., [8]. Finally, w hav th following inf-sup condition for th form B h. Proposition 4.4 ([11]). Thr xists a constant β > 0 indpndnt of th msh-siz such that B h (v, q) sup β q 0 q Q h. 0 v V h v 1,h Extnsions of this rsult to th hp-vrsion of th finit lmnt mthod and to quadrilatral mshs can b found in [23, 19]. Rmark 4.5. A carful inspction of th proof in [11] rvals that th discrt infsup condition in Proposition 4.4 also holds for th smallr spac Ṽh in (3.10). Consquntly, all th stability rsults of this sction hold for th particular LDG mthod in Rmark 3.1.

13 A locally consrvativ LDG mthod for th incomprssibl Navir-Stoks quations 13 Stability of th post-procssing oprator. Th nxt rsult stats that th oprator P is a boundd linar oprator from V h to Ṽh with rspct to th norm 1,h. It is on of th main tchnical rsults ndd to analyz th locally consrvativ LDG mthod (3.11) (3.13). Proposition 4.6. Lt v V h. Thn w hav Pv 1,h C stab v 1,h, with a stability constant C stab > 0 that is indpndnt of th msh-siz. Th proof of this proposition is givn in Sction 5.2 blow. Rmark that, for th LDG mthod in Rmark 3.1, w hav C stab = 1 sinc P is chosn as th idntity. W ar now rady to stat our main rsults Th rsults. Our first rsult stats that, undr a smallnss condition similar to th on for th xact solution, (1.7), th LDG mthod (3.11) (3.13) dfins a uniqu discrt approximation. Morovr, it actually givs an fficint way to comput it. Thorm 4.7 (Existnc and uniqunss of discrt solutions). Assum that (4.6) µ := C o C stab C p f 0 ν 2 α 2 < 1. Thn th LDG mthod (3.11) (3.13) dfins a uniqu solution (u h, p h ) V h Q h. It satisfis th bounds (4.7) (4.8) as wll as (4.9) E I h u h 1,h C p f 0 ν α, p h 0 β 1 ( Ca + 2α α ) C p f 0, Pu h n [u h n] 2 ds + Pu h n u h 2 ds C2 p f 2 0 Γ ν α. Morovr, if (u l+1 h, pl+1 h ) is th approximat solution givn by th LDG mthod in (3.11) (3.12) for th Osn quations with w = Pu l h, l 0, thn ( ) u h u l+1 h Cp f 0 µ l 1,h 2 να (1 µ), ( ) p h p l+1 h 0 2 β 1 Ca + 2α µ l C p f 0 α (1 µ), for any initial guss (u 0 h, p0 h ) V h Q h. This rsult, whos proof is givn in Sction 5.3, stats that th LDG mthod in (3.11) (3.13) is wll-dfind and that w can comput its approximat solution by solving a squnc of Osn problms. Sinc th paramtr µ is indpndnt of th msh-siz, th convrgnc of that squnc is always xponntial and so computationally fficint. Not that if w st (4.10) α = min{1, α}, C poinc = max{c P, C p }, C O = max{c, C o C stab },

14 14 B. Cockburn, G. anschat and D. Schötzau thn both th smallnss assumptions in (1.7) and (4.6) ar satisfid if w hav that C O C poinc f 0 ν 2 α 2 < 1. Hnc, both th Navir-Stoks quations and thir LDG approximation ar uniquly solvabl. Undr a smallnss condition that is slightly mor rstrictiv, w obtain th following stimats. Thorm 4.8 (Error stimats). Assum that C O C poinc f 0 (4.11) ν 2 α 2 1 2, and that th xact solution (u, p) of th Navir-Stoks quations (1.1) satisfis (4.12) u H s+1 () 2, p H s (), s 1. Thn u u h 1,h C u C app [ u s+1 + ν 1 p s ] h min{k,s}, u Pu h 1,h C w C app [ u s+1 + ν 1 p s ] h min{k,s}, p p h 0 C p C app [ ν u s+1 + p s ] h min{k,s}, σ σ h 0 C σ C app [ ν u s+1 + p s ] h min{k,s}, whr C app only dpnds on th rgularity of th msh and th polynomial dgr k, and { ( β + C b )(2 C a + 3 α ) 1 + C stab C u = max,, 2 C } b 2 (4.13),, β α C stab α α (4.14) C w = (1 + C stab ) + C stab C u, { ( C a + α ) C p = max Cu, α (1 + C stab ), β + C b, 1 } (4.15), β 2β C stab β β (4.16) C σ = (1 + C lift )C u. This rsult, whos proof is givn in Sction 5.4, stats that th LDG mthod undr considration convrgs with optimal ordr. Not also that, sinc th function w = Pu h is xactly divrgnc-fr, it provids an optimally convrgnt globally solnoidal approximation to th vlocity! Lt us brifly discuss som xtnsions of ths rsults: First, w point out that all th rsults of this sction ar valid vrbatim for th LDG mthod in Rmark 3.1 whr V h is rplacd by th div-conforming spac Ṽh in (3.10) and P is chosn to b th idntity. Although hr w only considrd th cas of triangular mshs, th rsults of this papr can b straightforwardly xtndd to simplicial mshs in thr dimnsions. Furthrmor, th LDG approach w propos hr can b asily xtndd to Q k Q k P k 1 lmnts on quadrilatral or hxahdral affin mshs, by using a post-procssing oprator P that is givn by a slight modification of th BDM projction on quadrilatrals or hxahdra; s [5]. Th rsults in this sction hold thn tru for this LDG mthod as wll. This fact is actually vrifid in our numrical xprimnts for which w hav usd squar mshs and Q 1 Q 1 P 0 lmnts. Howvr, th xtnsion of our rsults to Q k Q k Q k 1 lmnts on quadrilatral mshs is not straightforward. Although, by using a post-procssing oprator P

15 A locally consrvativ LDG mthod for th incomprssibl Navir-Stoks quations 15 that is a slight modification of th standard Raviart-Thomas projction, it is asy to dfin a solnoidal vlocity fild w that blongs to th anisotropic polynomial spac Q k 1,k Q k,k 1, th approximation proprtis in this spac giv ris to only suboptimal convrgnc rats. If, on th othr hand, th polynomial dgr of th post-procssd vlocity is incrasd, th fild w cannot b shown to b solnoidal, as Ṽh Q h. Finally, lt us rmark that hr w hav usd th LDG approach to discrtiz th viscous trms. Howvr, our rsults rmain valid for any othr DG discrtization of ths trms whos primal bilinar form A h is both corciv and continuous, such as,.g., th intrior pnalty form. For dtails, w rfr th radr to th discussions in [2] and [19]. 5. Proofs In this sction, w provid th proofs of our main rsults Proof of Proposition 4.2. W bgin by proving Proposition 4.2. To do that, not that, if w insrt th dfinition of th upwinding numrical flux into th form O h, w hav O h (w; u, v) = u (v w) dx T h + T h [ w n {u } 1 ] 2 w n (u xt u) v ds. Hr, u xt dnots th xtrior trac of u takn ovr th dg undr considration and st to zro on th boundary. If w prform now a simpl intgration by parts, w gt O h (w; u, v) = u : (v w) dx This implis that whr T 1 = T h T 2 = T h T h + T h [ 1 2 w n (u xt u) 1 ] 2 w n (u xt u) v ds. O h (w 1 ; u, v) O h (w 2 ; u, v) = T 1 + T 2, T h u : ( v (w 1 w 2 ) ) dx, 1 2 (w 1 w 2 ) n (u xt u) v ds ( ) 1 w 1 n w 2 n (u xt u) v ds. 2 To bound th trm T 1, w rcall th following mbdding rsult provd in [13, Proposition 4.5] for smooth and convx domains and in [10, Lmma 6.2] for gnral polygons: v L 4 () C v 1,h, v {w L 2 () 2 : w H 1 () 2, T h },

16 16 B. Cockburn, G. anschat and D. Schötzau with a constant indpndnt of th msh-siz. (W point out that th brokn H 1 - norm usd in [13] is slightly diffrnt than th on w us hr. Howvr, a carful inspction of th proof of Proposition 4.5 thrin shows that th rsult holds in fact for our dfinition of 1,h.) It is thn clar that w can us Höldr s inquality to obtain T 1 w 1 w 2 L 4 () u 1,h v L 4 () C w 1 w 2 1,h u 1,h v 1,h. It rmains to stimat th trm T 2. Using th Lipschitz continuity of th function x x, w gt T 2 w 1 n w 2 n [u n] v ds, T h and, procding as in th proof of [13, Proposition 4.5], w obtain T 2 C w 1 w 2 1,h u 1,h v 1,h, with a constant indpndnt of th msh-siz. This complts th proof of Proposition Proof of Proposition 4.6. W prov Proposition 4.6 by first stablishing local stability bounds ovr patchs of lmnts and thn by summing up ths local rsults. Lt v V h b fixd. W procd in svral stps. Stp 1: Local bounds in th intrior. Lt = 1 2 b an intrior dg shard by two lmnts 1 and 2. W wish to stablish a local stability bound ovr th patch formd by 1 and 2. Namly, by dfining th local sminorm v 2 = v 2 0, 1 + v 2 0, 2 + h 1 [v n] 2 ds, w claim that (5.1) Pv 2 C [ v h 1 (v v p ) n 1 2 ds+ h 1 (v v p ) n 2 2 ds ], 2 with a constant C indpndnt of th msh-siz. To prov (5.1), it is nough to considr th cas whr 1 and 2 form a rfrnc patch of unit siz. Th gnral cas thn follows from a scaling argumnt and th shap-rgularity assumption on th msh, by mapping 1 2 onto th rfrnc patch using lmnt-wis Piola transforms. By th triangl inquality, w hav (5.2) Pv v Pv + v. It rmains to bound v Pv. To do so, w not that, for any lmnt T h, th rstriction (v Pv) blongs to P k () 2 and is uniquly dfind by th momnts (v Pv) n ϕ ds = (v v p ) n ϕ ds ϕ P k (),, (v Pv) ϕ dx = 0 ϕ P k 1 (), (v Pv) Ψ dx = 0 Ψ Ψ k ().

17 A locally consrvativ LDG mthod for th incomprssibl Navir-Stoks quations 17 Hnc, th rstriction of v Pv to th patch 1 2 blongs to th spac { V = v L 2 ( 1 2 ) 2 : v i P k ( i ) 2, (5.3) v i ϕ dx = 0, ϕ P k 1 ( i ), i } v i Ψ dx = 0, Ψ Ψ k ( i ), i = 1, 2. i Furthrmor, it can b asily sn that th mappings v v and v v, givn by v 2 = v 2 + h 1 v n 1 2 ds + h 1 v n 2 2 ds, v 2 = 1\ h 1 v n 1 2 ds \ h 1 v n 2 2 ds, dfin norms on V. By th quivalnc of all norms on a finit dimnsional spac, thr holds v C v v V, with a constant only dpnding on th th polynomial dgr. Thus, w obtain that On th othr hand, sinc, for i = 1, 2, w conclud that v Pv v Pv C v Pv. (v Pv) n i = (v v p ) n i on i, v Pv 2 C v Pv 2 = C h 1 (v v p ) n 1 2 ds + C 1 2 h 1 (v v p ) n 2 2 ds. This stimat and th inquality in (5.2) prov th local stability bound in (5.1). Stp 2: Local bounds on th boundary. Th analogous stability rsult holds on th boundary. Lt b an lmnt on th boundary and a boundary dg. By stting v 2 = v 2 0, + h 1 v n 2 ds, thr xists a constant C indpndnt of th msh-siz such that (5.4) Pv 2 C[ v 2 + h 1 (v v p ) n 2 ds ]. Stp 3: Summing up th local bounds. W complt th proof of Proposition 4.6 by summing up th local stability stimats stablishd in (5.1) and (5.4). To this nd, w first not that v v p = 1 2 (v vxt ) on intrior dgs and v v p = v on boundary dgs. Hr, w writ v xt to dnot th xtrior trac of v ovr th dg undr considration. Thrfor, w hav for any dg E h h 1 ( v p v) n 2 ds h 1 [v n] 2 ds.

18 18 B. Cockburn, G. anschat and D. Schötzau Using th local bounds in (5.1) and (5.4) and th abov stimat for v v p, w obtain Pv 2 1,h C Pv 2 E h C v 2 + C h 1 (v v p ) n 2 ds E h T h C v 2 + C h 1 [v n] 2 ds E h C v 2 1,h, E h with constants C indpndnt of th msh-siz. This complts th proof of Proposition Proof of Thorm 4.7. To prov Thorm 4.7, w procd as follows. First, w liminat th prssur from th problm by rstricting ourslvs to th wakly divrgnc-fr subspac of V h, (5.5) Z h = {v V h : B h (v, q) = 0 q Q h }. Th approximat vlocity is thus charactrizd as th only function u h Z h such that (5.6) A h (u h, v) + O h (Pu h ; u h, v) = f v dx v Z h. Thn, w construct a contractiv mapping S dfind on a ball of Z h whos only fixd point is prcisly th abov vlocity. Th proprtis for th corrsponding prssur p h follow thn from its charactrization, B h (v, p h ) = f v dx A h (u h, v) O h (Pu h ; u h, v) v V h /Z h, and from th inf-sup condition for th incomprssibility form B h. W procd in svral stps. Stp 1: Th oprator S. W bgin by introducing th oprator S. For u Z h, u = S(u) dnots th solution of th following problm: Find u Z h such that A h (u, v) + O h (Pu; u, v) = f v dx v Z h. Not that sinc u Z h w hav, by Proposition 3.2, that Pu J h (T h ). As a consqunc, this problm is uniquly solvabl. Furthrmor, by th corcivity of th form A h and O h in Proposition 4.3, να u 2 1,h A h(u, u) + O h (Pu; u, u) = f u dx f 0 u 0. By th Poincaré inquality in (4.3), w obtain να u 2 1,h C p f 0 u 1,h.

19 A locally consrvativ LDG mthod for th incomprssibl Navir-Stoks quations 19 Hnc, th solution u to th abov problm satisfis (5.7) u 1,h C p f 0 να. This implis that S maps h into h, whr h = {v Z h : v 1,h C p f 0 να }. Stp 2: Th oprator S is a contraction. Nxt, w show that S is a contraction on h undr th smallnss condition (4.6). To do so, lt u 1, u 2 b in h, and st u 1 = S(u 1 ), u 2 = S(u 2 ). Thn Sinc να u 1 u 2 2 1,h A h (u 1 u 2, u 1 u 2 ). A h (u 1 u 2, v) + O h (Pu 1 ; u 1, v) O h (Pu 2 ; u 2, v) = 0, for any v Z h, taking v = u 1 u 2 w gt να u 1 u 2 2 1,h O h(pu 2 ; u 1 u 2, u 1 u 2 ) + O h (Pu 2 ; u 1, u 1 u 2 ) O h (Pu 1 ; u 1, u 1 u 2 ) =: T 1 + T 2. By th corcivity proprty of th form O h in Proposition 4.3, T 1 0. Morovr, by th continuity proprty of O h in Proposition 4.2, th bound (5.7) and th continuity of th post-procssing oprator P in Proposition 4.6, This implis that T 2 C o Pu 1 Pu 2 1,h u 1 1,h u 1 u 2 1,h C oc p f 0 Pu 1 Pu 2 1,h u 1 u 2 1,h να C oc stab C p f 0 u 1 u 2 1,h u 1 u 2 1,h να = ναµ u 1 u 2 1,h u 1 u 2 1,h. u 1 u 2 1,h µ u 1 u 2 1,h, and so, if µ < 1, that is, if th smallnss condition (4.6) is satisfid, th mapping S is a contraction. Hnc, S has a uniqu fixd point u h h, which is th solution to th problm (5.6). Stp 3: Rcovring th prssur. Now that th vlocity u h has bn computd, th prssur is th solution p h Q h of (5.8) B h (v, p h ) = f v dx A h (u h, v) O h (Pu h ; u h, v) v V h /Z h. Du to Proposition 4.1, Proposition 4.2, and th Poincaré inquality in (4.3), th right-hand sid dfins a continuous linar functional on V h /Z h. Th inf-sup condition in Proposition 4.4 thn guarants th xistnc of a uniqu solution p h to th abov problm. It can thn b asily sn that th tupl (u h, p h ) is th uniqu solution to th LDG mthod in (3.11) and (3.12) with w = Pu h.

20 20 B. Cockburn, G. anschat and D. Schötzau Stp 4: Th stability bounds. Nxt, lt us show th stability bounds for (u h, p h ) in Thorm 4.7. Th bound for u h 1,h in (4.7) follows in a straightforward way sinc u h h. To obtain th bound for th upwind trm in (4.9), not that να u h 2 1,h + O h (Pu h ; u h, u h ) C p f 0 u h 2 1,h 1 Cp f να 2 να u h 2 1,h. Similarly to th prvious argumnts, hr w hav usd th corcivity of A h, quation (5.6) with tst function v = u h, and th Poincaré inquality (4.3). Bringing th trm 1 2 να u h 2 1,h to th lft-hand sid and obsrving th corcivity of O h giv th stability bound in (4.9). Morovr, using th inf-sup condition in Proposition 4.4, th Poincaré inquality in (4.3), th continuity proprtis in Proposition 4.1 and Proposition 4.2, and th stability of P in Proposition 4.6, w hav from (5.8) β p h 0 B h (v, p h ) sup C p f 0 + ν C a u h 1,h + C o C stab u h 2 1,h 0 v V h v. 1,h Taking into account th stability bound for u h and assumption (4.6) givs ( p h 0 β 1 C p f 0 + C ac p f 0 + C ) oc stab C p f 0 α ν 2 α 2 C p f 0 β 1 C p f 0 ( 2 + C a α ). This givs th dsird bound (4.8) for p h. Stp 5: Th convrgnc stimats. It rmains to prov th rror stimats for th squnc {(u l h, pl h )} l 0. Lt us bgin with that of th vlocity. From Stp 3, w hav that u l+1 h = S(u l h ). As a consqunc, sinc S is a contraction with Lipschitz constant µ, w immdiatly gt ( ) µ u h u l+1 l h 1,h u 2 h u 1 (1 µ) h 1,h. Th rsult now follows from th fact that, by th stability bound (5.7), u m h 1,h C p f 0 να, for m 1. To obtain th stimat for th prssur, w procd as follows. First, w not that, from Stp 4, w hav B h (v, p l+1 h ) = f v dx A h (u l+1 h, v) O h(pu l h; u l+1 h, v) v V h. This implis that B h (v, p h p l+1 h ) = A h(u h u l+1 h, v) O h(pu h ; u h, v) + O h (Pu l h ; u h, v) O h (Pu l h ; u h, v) + O h (Pu l h ; ul+1 h, v) = A h (u h u l+1 h, v) O h(pu h ; u h, v) + O h (Pu l h ; u h, v) O h (Pu l h; u l+1 h u h, v). W insrt this xprssion in th inf-sup condition for B h, us th stability proprtis of A h, O h, and P, tak into account th bounds for u h 1,h, u l h 1,h, and th

21 A locally consrvativ LDG mthod for th incomprssibl Navir-Stoks quations 21 contraction proprty of S to obtain ( p h p l+1 h 0 β 1 νc a µ + C ) oc stab C p f 0 (1 + µ) u h u l να h 1,h β 1 µ (νc a + 2να) u h u l h 1,h. Th dsird bound for p h p l+1 h 0 thn follows from th bound for u h u l h 1,h. This complts th proof of Thorm Proof of Thorm 4.8. Hr, w driv th rror stimats in Thorm 4.8. To do that, w modify th approach usd in [8] to gt rror stimats for th LDG mthod for th Osn problm in two ways. First, w us th non-conforming approach introducd in [16] and latr usd in [19], and considr th xprssion R h (u, p) := R h (u, p; v) sup, 0 v V h v 1,h whr R h (u, p; v) := A h (u, v) + O h (u; u, v) + B h (v, p) f v dx, v V h. Th scond modification is, of cours, du to th prsnc of th convctiv nonlinarity. Proof. W procd in svral stps. Stp 1: Th abstract stimat for u u 1,h. Lt us bgin with th stimat for th rror u u 1,h. W claim that w hav [ (5.9) u u h 1,h C u inf u v 1,h + inf u ṽ 1,h v V h v V h 1 + inf q Q h ν p q ν R h(u, p) whr C u is givn by (4.13). To prov this rsult, w procd as in th rror analysis of standard mixd mthods, s,.g., [5], and considr first an lmnt v Z h whr Z h is th krnl in (5.5). W thn hav th following trivial inquality (5.10) u u h 1,h u v 1,h + v u h 1,h. Nxt, w obtain an stimat of v u h. By th corcivity proprty of th form A h in Proposition 4.3, w hav (5.11) ν α v u h 2 1,h ν α v u h 2 1,h A h (v u h, v u h ), and sinc A h (u u h, v) + O h (u; u, v) O h (Pu h ; u h, v) + B h (v, p p h ) = R h (u, p; v) for any v V h, for v = v u h, w gt that A h (v u h, v u h ) = A h (v u, v u h ) ], O h (u; u, v u h ) + O h (Pu h ; u h, v u h ) B h (v u h, p p h ) + R h (u, p; v u h ) =: T 1 + T 2 + T 3 + T 4.

22 22 B. Cockburn, G. anschat and D. Schötzau Hnc α ν v u h 2 1,h T 1 + T 2 + T 3 + T 4. Th trms T 1, T 3 and T 4 can b asily stimatd as follows. First, by th continuity of th form A h, Thn, by dfinition of R h and R h, Finally, sinc v u h Z h, w hav T 1 ν C a u v 1,h v u h 1,h. T 4 R h (u, p) v u h 1,h. B h (v u h, p p h ) = B h (v u h, p) = B h (v u h, p q), for any q Q h. From Proposition 4.1, it follows that T 3 C b v u h 1,h p q 0 q Q h. It rmains to stimat th trm T 2. To do that, considr th idntity T 2 = O h (Pv; u, v u h ) + O h (Pu h ; u, v u h ) O h (Pu h ; v u h, v u h ) + O h (Pv; u, v u h ) O h (u; u, v u h ) + O h (Pu h ; v u, v u h ) =: T 21 + T 22 + T 23 + T 24. Not that du to Proposition 4.2, Proposition 4.6, th stability bound for u in (1.5), and th dfinitions of th paramtrs in (4.10), w hav T 21 C o C stab u 1 v u h 2 1,h C o C stab C P f 0 v u h 2 1,h ν C O C poinc f 0 να v u h 2 1,h 1 2 να v u h 2 1,h, by th smallnss condition (4.11). Nxt, by th corcivity proprty of O h in Proposition 4.3, T 22 0, and, by Proposition 4.2, Proposition 4.6, and th bound for u h in Thorm 4.7, by th smallnss condition (4.11). T 24 C oc stab C p f 0 u v 1,h v u h 1,h να C OC poinc f 0 να u v 1,h v u h 1,h 1 2 να u v 1,h v u h 1,h,

23 A locally consrvativ LDG mthod for th incomprssibl Navir-Stoks quations 23 Finally, by th Lipschitz proprty of th form O h in Proposition 4.2, w hav T 23 C o u Pv 1,h u 1 v u h 1,h C oc P f 0 u Pv 1,h v u h 1,h ν να u Pv 1,h v u h 1,h, 2 C stab by th bound for u in (1.5) and th smallnss condition (4.11). Now, tak an arbitrary function ṽ in Ṽh. Sinc P rproducs functions in Ṽh, w hav Pṽ = ṽ, and so u Pv 1,h u ṽ 1,h + Pṽ Pv 1,h u ṽ 1,h + C stab ṽ v 1,h (1 + C stab ) u ṽ 1,h + C stab u v 1,h, by Proposition 4.6. This implis that T 23 να (( ) ) 1 + Cstab u ṽ 1,h + u v 1,h v u h 1,h. 2 C stab Thus, gathring all th stimats abov and insrting thm in th right-hand sid of inquality (5.11), and bringing T 21 to th lft-hand sid, w obtain ( ) ( ) Ca + α 1 + Cstab v u h 1,h 2 u v 1,h + u ṽ 1,h α C stab ( ) ( ) 2 Cb 2 + p q 0 + R h (u, p). να να Insrting this stimat in th right-hand sid of inquality (5.10), w gt ( ) ( ) 2Ca + 3α 1 + Cstab u u h 1,h u v 1,h + u ṽ 1,h α C stab (5.12) ( ) ( ) 2 Cb 2 + p q 0 + R h (u, p), να να for any v Z h, ṽ Ṽh, and q Q h. It rmains to rplac v Z h by an arbitrary function in v V h. To this nd, fix v V h and considr th problm: Find r V h such that B h (r, q) = B h (u v, q) q Q h. Th inf-sup condition in Proposition 4.4 guarants that such a solution r xists. Furthrmor, it can b asily sn that w hav r 1,h β 1 C b u v 1,h, in viw of th inf-sup condition for B h and th continuity of th form B h. By construction and sinc B h (u, q) = 0 for any q Q h, w furthr hav that r+v Z h. Insrting r + v in (5.12), mploying th triangl inquality, and taking into account th abov bound for r yild th abstract rror stimat (5.9) for th vlocity. Stp 2: Th abstract stimat for u Pu h 1,h. As a consqunc of th approximation rsult (5.9), w obtain th following stimat of th rror btwn u and its globally solnoidal approximation Pu h (5.13) u Pu h 1,h (1 + C stab ) inf v V h u ṽ 1,h + C stab u u h 1,h.

24 24 B. Cockburn, G. anschat and D. Schötzau To s this, not that u Pu h 1,h u ṽ 1,h + ṽ Pu h 1,h u ṽ 1,h + C stab ṽ u h 1,h (1 + C stab ) u ṽ 1,h + C stab u u h 1,h, whr ṽ is any lmnt of Ṽh. Hr, w hav usd th stability bound in Proposition 4.6 and th fact that P rproducs polynomials in Ṽh. This shows th inquality (5.13). Stp 3: Th abstract stimat for th prssur. Now, lt us obtain th stimat for th prssur. W claim that th rror in th prssur satisfis [ (5.14) p p h 0 C p inf ν u v 1,h + inf ν u ṽ 1,h v V h v V h ] + inf p q 0 + R h (u, p), q Q h whr C p is givn by (4.15). To s this, w procd in a way similar to th on usd to dal with th vlocity. Thus, w bgin by noting that for q Q h p p h 0 q p h 0 + p q 0 β 1 B h (v, q p h ) sup + p q 0 v V h v 1,h β 1 B h (v, p p h ) sup + β 1 B h (v, q p) sup + p q 0, v V h v 1,h v V h v 1,h whr w hav us th inf-sup condition in Proposition 4.4. Thrfor, (5.15) p p h 0 β 1 sup v V h B h (v, p p h ) v 1,h + (1 + β 1 C b ) p q 0, by th continuity of th form B h. To bound th first trm on th right-hand sid of (5.15), w not that B h (v, p p h ) = A h (u u h, v) O h (u; u, v) + O h (Pu h ; u h, v) + R h (u, p; v), for any v V h, and procd as in th prvious stp to obtain and sinc from Stp 2, w gt B h (v, p p h ) = A h (u u h, v) + O h (Pu h ; u h u, v) + O h (Pu h ; u, v) O h (u; u, v) + R h (u, p; v) [ (ν C a ν α ) u u h 1,h + ν α ] u Pu h 1,h + R h (u, p) v 1,h, 2 C stab u Pu h 1,h (1 + C stab ) u ṽ 1,h + C stab u u h 1,h, B h (v, p p h ) (C a + α ) ν u u h 1,h + α ( ) 1 + Cstab ν u ṽ 1,h + R h (u, p). 2 C stab

25 A locally consrvativ LDG mthod for th incomprssibl Navir-Stoks quations 25 Insrting this inquality in (5.15) and using th bound (5.9) for u u h 1,h from Stp 1, w immdiatly obtain th abstract stimat (5.14) for th prssur. Stp 4: Th abstract stimat for th vlocity gradint. To driv th rror stimat for σ σ h, w not that, from (3.14), w hav σ σ h = ν [ u h u h + L(u h ) ]. Hnc, σ σ h 0 ν u u h 1,h + ν L(u h ) 0. Using th stability bound (4.4) for th lifting oprator L yilds L(u h ) 2 0 Clift 2 κ 0 h 1 [u h n] 2 ds Clift u 2 u h 2 1,h. E h Th last inquality follows as th jumps of th xact solution vanish. This shows that (5.16) σ σ h 0 ν (1 + C lift ) u u h 1,h. Stp 5: Approximation stimats. Undr th rgularity assumption (4.12), th following standard approximation proprty holds inf ν u v 1,h + inf p q 0 C app h min{k,s}[ ] ν u s+1 + p s. v V h q Q h Morovr, from th rsults in [5], s also [11, Sction 3], w hav Finally, w hav that inf u ṽ 1,h C app h min{k,s} u s+1. v V h R h (u, p) C app h min{k,s}[ ν u s+1 + p s ], with a constant C app indpndnt of th msh-siz. To s th stimat of th rsidual, w procd as follows. For v V h, it is asy to s that R h (u, p; v) is givn by R h (u, p; v) = {ν u P h (ν u) } : [v n] ds {p P h p }[v n] ds, E h E h with P h : L 2 () 2 2 Σ h and P h : L 2 ()/R Q h dnoting th L 2 -projctions onto Σ h and Q h, rspctivly. Th dsird stimat follows thn by procding as th proof of [19, Proposition 8.1] and using standard approximation rsults for P h and P h. Stp 6: Conclusion. It is now a simpl mattr to s that th rror stimats of Thorm 4.8 follow by insrting th approximation stimats obtaind in th prvious stp into th abstract bounds for th vlocity, (5.9), for its globally solnoidal post-procssing (5.13), th prssur, (5.14) and th vlocity gradint, (5.16). This complts th proof of Thorm 4.8.

26 26 B. Cockburn, G. anschat and D. Schötzau 6. Numrical rsults In this sction, w prsnt numrical xprimnts that show that th thortical rats of convrgnc ar sharp. W also display th bhavior of th itrativ mthod as a function of th Rynolds numbr. As a rfrnc solution in our tsts, w tak th analytical solution (u, p) of th incomprssibl Navir-Stoks quations that was obtaind by ovasznay in [14]. For a givn viscosity ν, this solution is givn by whr u 1 (x, y) = 1 λx cos(2πy), u 2 (x, y) = λ 2π λx sin(2πy), p(x, y) = 1 2 2λx + p, λ = 8π 2 ν 1 + ν π 2. It solvs (1.1) with a suitably chosn right-hand sid f. Hr, th constant p is such that p dx = 0. W furthr us th valus of th xact solution u = (u 1, u 2 ) to prscrib inhomognous Dirichlt boundary data g for th vlocity on th whol boundary of th computational domain, which w tak to b = ( 1 2, 3 2 ) (0, 2). Not that in this cas, th numrical fluxs (on dgs lying on th boundary) must b modifid as follows: û w h (x) = { lim ɛ 0 u h (x ɛw(x)), x \ Γ, g(x), x Γ, σ h = σ h κ(u h g) n, û σ h = g, û p h = g, p h = p h. W considr squar mshs that ar gnratd by rfining th singl grid cll ( 1 2, 3 2 ) (0, 2) uniformly. Thrfor, a msh on lvl L consists of 2L by 2 L clls. All computations ar prformd with bilinar shap functions for σ h and u h and picwis constants for p h, according to th rmarks in Sction 4.3. Th stabilization function κ is chosn as in (4.1) with κ 0 = 4. Our analysis thn prdicts first ordr convrgnc for u h in th norm. 1,h and for th prssur in th L 2 -norm. In Tabl 1 w show th rrors and convrgnc rats in p, u and σ obtaind for ν = 0.1. Th rrors in p and σ ar masurd in th L 2 -norm whil u u h and u Pu h ar valuatd in th norm. 1,h. W obsrv th prdictd first ordr convrgnc for all th rror componnts, in full agrmnt with th rsults of Thorm 4.8. Notic that w hav scald th L 2 -rror in σ by ν 1 so that this rror can b dirctly compard to th H 1 -rrors in u u h and u Pu h. Ths thr rrors ar all of th sam magnitud, with a slight advantag for th post-procssd solution. In Tabl 2 w show th sminorm of th rrors u u h and u Pu h which masurs thir jumps. Notic that it suprconvrgs with ordr 3/2. This mans that th rlativ contribution of this sminorm to th 1,h norm diminishs as h dcrass. An analysis of this phnomnon rmains to b carrid out. In Tabl 3, w show th L 2 -rrors in th vlocitis and thir corrsponding convrgnc ordrs. In th first column, w obsrv that th vlocitis convrg with