On the Stability of Continuous-Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems

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1 Nonam manuscript No. (will b insrtd by th ditor) On th Stability of Continuous-Discontinuous Galrkin Mthods for Advction-Diffusion-Raction Problms Andra Cangiani John Chapman Emmanuil Gorgoulis Max Jnsn th dat of rcipt and accptanc should b insrtd latr Abstract W considr a finit lmnt mthod which coupls th continuous Galrkin mthod away from intrnal and boundary layrs with a discontinuous Galrkin mthod in th vicinity of layrs. W prov that this consistnt mthod is stabl in th stramlin diffusion norm if th convction fild flows non-charactristically from th rgion of th continuous Galrkin to th rgion of th discontinuous Galrkin mthod. Th stability proprtis of th coupld mthod ar illustratd with a numrical xprimnt. Kywords stabilisd finit lmnt mthods, singularly-prturbd quations Mathmatics Subjct Classification (2000) 65M60, 76M0 Introduction It is wll known that th standard continuous Galrkin (cg) finit lmnt mthod xhibits poor stability proprtis for singularly prturbd problms. In prsnc of sharp boundary or intrior layrs, non physical oscillations pollut th numrical approximation throughout th solution domain, and thus stabilisation tchniqus nd to b mployd; s [6] for a survy. Th discontinuous Galrkin (dg) mthod offrs a framwork for th dsign of finit lmnt mthods with good stability proprtis; s,.g., [9] for a survy of thir dvlopmnt. This is achivd by rlaxing th continuity rquirmnts at th intr-lmnt boundaris, whr appropriat upwindd numrical fluxs can b mployd. On such dg mthod is th Intrior Pnalty (IP) mthod considrd hrin. Howvr dg mthods rquir mor dgrs of frdom (DOFs) compard to th standard cg mthod. In this work, w invstigat if som of ths additional DOFs can b rmovd without affcting th stability proprtis of th dg mthod. Th motivation of this work is twofold: firstly w wish to improv our undrstanding of th wll-stablishd dg mthods in rspons to th criticism in th incrasd numbr of DOFs; scondly w sk A. Cangiani E. Gorgoulis Dpartmnt of Mathmatics, Univrsity of Licstr, Licstr LE 7RH, UK J. Chapman M. Jnsn Dpartmnt of Mathmatical Scincs, Univrsity of Durham, Durham DH 3LE, UK

2 2 Andra Cangiani t al. insight into th dsign of mor fficint finit lmnt mthods, allowing to combin advantags th dg framwork with smallr approximation spacs. Concptually w nvisag in this work a finit lmnt spac that lis btwn th standard continuous and discontinuous Galrkin spacs. W construct such a spac by applying standard continuous lmnts away from any boundary or intrnal layrs (calld th cg rgion) and discontinuous lmnts in th rgion of such layrs (calld th dg rgion). Thrfor w call this mthod th continuous discontinuous Galrkin (cdg) finit lmnt mthod. In th broadr sns, mthods of this typ hav bn proposd bfor by Bckr, Burman, Hansbo and Larson [4], Prugia and Schötzau [5], and Dawson and Proft []. In [4] a globally rducd discontinuous Galrkin mthod is studid, whos approximation spac consists of th continuous picwis linar functions nrichd with picwis constant functions. In [5], [] th local discontinuous Galrkin mthod [0] has bn usd on th discontinuous rgion with transmission conditions on th subst of intrlmnt boundaris whr continuous and discontinuous lmnts mt. Our approach is diffrnt to [5], [] as w impos no such conditions byond thos alrady imposd by th dg mthod. W also mntion th hybridizabl dg approach of Cockburn and co-workrs which rsults into th rduction of dg dgrs of frdom via local hybridization [8,4]. Rmoving th transmission conditions maks th approach mor natural, but introducs som difficultis as possibl ovr- and undrshoots at th intrfac of th cg and dg rgions must b controlld. Indd, th analysis is limitd to th cas whr th cdg intrfac is non-charactristic, cf. Assumption 4. Thn th convctiv intrfac trms ar non-zro and prmit to control jumps across th intrfac. This allows us to driv a rigorous stability bound for th cdg approximation. On th cg rgion w split th numrical solution into an approximation to th hyprbolic problm which is wakly dpndnt on th diffusion cofficint ε and an approximation to th rmaining part, which dpnds mor strongly on ε but is small in siz. Stability of th approximation on th dg rgion is shown using th approach of Buffa, Hughs and Sangalli [5] with som xtnsions in th mannr of Ayuso and Marini [2]. To our knowldg this papr is th first to prsnt a stability rsult for th proposd cdg mthod. Comparisons of our cdg mthod with various dg and cg mthods hav bn undrtakn by Cangiani, Gorgoulis and Jnsn [7] and Dvloo, Forti and Goms [2]; s also [6] which studis th cdg approximation as th limit of a dg approximation as th jump pnalization on intrlmnt boundaris tnds to infinity. Th rmaindr of this work is organizd as follows. In Sction 2 w formulat th problm, introduc notation, and xplain th assumptions w mak. Control on th continuous rgion is considrd in svral stags in Sction 3 and Sction 4. Sction 5 prsnts an inf-sup bound on th discontinuous rgion. Th main rsult of th papr showing control indpndnt of th prturbation paramtr ε and msh siz h follows in Sction 6. Th thortical rsults ar illustratd in Sction 7 by a numrical xprimnt. 2 Th Modl Problm and Notation Lt Ω b a boundd Lipschitz domain in R d, d 2. W introduc th modl advctiondiffusion-raction (ADR) problm ε u + b(x) u + c(x)u = f (x) for x Ω R d, () u = 0 on Ω (2)

3 Stability of cdgfem for Singularly-Prturbd Problms 3 J Ω cg Ω dg Fig. An xampl of a cdg dcomposition. Right and abov th dashd lin is Ω cg ; lft and blow is Ω dg. Th rgion T dg is markd by gray lmnts, whil T cg consists of th whit lmnts. Th intrfac btwn T cg and T dg is J. with constant diffusion cofficint 0 < ε ε max, b [W, (Ω)] d, c L (Ω) and f L 2 (Ω). For 0 < ε th solution to this problm typically xhibits boundary or intrior layrs. Unlss othrwis statd, w dfin C throughout as a positiv constant, indpndnt of ε and th finit lmnt approximation spac and which may b rdfind from lin to lin. C may dpnd on ε max, b, c, f and Ω. By a b w man a Cb. W considr T h to b a subdivision of Ω into non-ovrlapping shap rgular simplics or hypr-cubs E, which w shall rfr to as th triangulation. Dnot by E h th union of dgs (or facs for d 3) of th msh and th union of intrnal dgs by E o h. Dfin Γ as th union of boundary dgs, i.., thos lying in Ω. Th diamtr of an lmnt E T h is dnotd by h E and h = max E Th h E. W also dnot th msh function by h E, thus for x E w lt h E (x) b qual to th diamtr of E. W only considr mshs whr h. Dfin h := min(h E,h E +) for E o h with = Ē+ Ē for E +, E T h. Th msh is shap rgular so thr xists C > 0 such that for all E w hav h 2 (h E + + h E ) Ch, and C such that C h E h E + Ch E. W dfin by Ω-dcomposition th splitting of Ω into two rgions Ω cg and Ω dg such that for th closur Ω = Ω cg Ω dg, and w dfin by T h -dcomposition th splitting of T h into two sub-mshs T cg and T dg such that T cg Ω cg and T dg := T h \ T cg. By abus of languag, w dnotd hr by T cg not just th sub-msh but also th rgion it occupis. Dfin Γ cg (rsp. Γ dg ) to b th intrsction of Γ with T cg (rsp. T dg ). Dfin J := T cg T dg and by convntion w say that th dgs lying in J ar only part of th discontinuous Galrkin sklton E dg, th union of facs in T dg, and not part of th continuous Galrkin sklton dfind by E cg := E h \ E dg. In Figur w illustrat a splitting for a problm whr Ω = (0,) 2 and th solution xhibits layrs at x = and y =. Th Ω-dcomposition is lablld, with th dmarcation btwn th Ω cg and Ω dg rgions givn by a dashd lin. A T h -dcomposition is shown with th T dg rgion shadd and th dgs in J markd with a havy lin. Givn a gnric scalar fild ν : Ω R, that may b discontinuous across an dg = Ē + Ē for E +, E T h, w st ν ± := ν E ±, th intrior trac on E ± and similarly dfin τ ± = τ E ± for a gnric vctor fild τ : Ω R d. Dfin th avrag and jump for a gnric scalar as {{ν}} := 2 (ν+ + ν ), ν := ν + n + + ν n, on E o h,

4 4 Andra Cangiani t al. and for a gnric vctor fild as {{τ}} := 2 (τ+ + τ ), τ := τ + n + + τ n, on E o h, whr n ± is th unit outward pointing normal from E ± on. For Γ th dfinitions bcom {{ν}} := ν, ν := νn, {{τ}} := τ, on Γ. W assum that th sign of b n is th sam for vry x. Givn a vctor b dnot th inflow and outflow boundaris of Ω by and for an lmnt Γ in := {x Ω : b n 0}, Γ out := {x Ω : b n > 0} in E := {x E : b n 0}, out E := {x E : b n > 0}. On an dg, w dnot by ν in th trac of a function ν, takn from th lmnt which contains in its inflow boundary. Dfinition Th cg spac is dfind by and th dg spac by V cg := {v H (Ω) : E T h,v E P k } V dg := {v L 2 (Ω) : E T h,v E P k } (3) whr P k is th spac of polynomials of ithr total (or on quadrilatral mshs partial) dgr at most k supportd on E. W sk a finit lmnt spac V cdg that lis btwn th cg and dg spacs in th sns that V cg V cdg V dg. (4) W construct this spac by using continuous shap functions away from any boundary or intrnal layrs prsnt in th solution to () and discontinuous functions in th rgion of such layrs. Dfinition 2 Th cdg spac is dfind by V cdg := {v L 2 (Ω) : E T h,v E P k ;v TcG C 0 (T cg );v ΓcG = 0}. (5) Throughout w us th sam polynomial dgr k for V cg, V dg and V cdg. Lt χ b th charactristic function on T dg, i.., that dfind by χ(x) := { : x TdG, 0 : x T cg. (6) Thn dfin V cdg (T dg ) := {χv : v V cdg } and V cdg (T cg ) := {( χ)v : v V cdg }. In ordr to nsur a uniqu solution w mak th following assumption.

5 Stability of cdgfem for Singularly-Prturbd Problms 5 Assumption W assum for som ρ R. r(x) := c(x) b(x) ρ > 0 x Ω, (7) 2 Dfinition 3 W dfin th local msh Péclt numbr to b b L (E)h E /(2ε), s [6]. W considr mshs in th pr-asymptotic rgim by making th following assumption. Assumption 2 W assum that for ε = ε max and vry E T h th local msh Péclt numbr is gratr than / h E. Morovr, w rquir max{ h E /b L (T dg ),h E }. As a consqunc w hav ε ε max < 2 min h 3/2 E b L E T (Ω). (8) h This assumption, for a fixd b, rstricts th rfinmnt of th triangulation for a givn ε. If w allowd h 0 for fixd ε > 0 any layrs would b rsolvd by th msh and in th limit w would not s th non-physical oscillations associatd with th cg approximation. W rturn to this qustion in Rmark. To charactris admissibl Ω-dcompositions of th msh w introduc th rducd problm: b u 0 + cu 0 = f on Ω, u 0 = 0 on Γ in. (9) Furthr, w dfin u ε := u u 0, whr u is th solution to th ADR problm () (2). Th Ω-dcomposition is chosn such that u ε and u 0 hav additional rgularity on Ω cg. In gnral w do not xpct that u 0 H 2 (Ω), vn if w plac highr rgularity rquirmnts on f, s,.g., [3] and th rfrncs thrin. Assumption 3 Th st Ω cg Ω is chosn such that u 0 H 2 (Ω cg ) and u ε H 2 (Ω cg ) is boundd uniformly in ε, that is for vry 0 < ε ε max u ε H 2 (Ω cg ). (0) 2. Dcoupld and Standard Formulations W dfin by h th lmntwis gradint oprator. W discrtiz th advction trm by B a (w,ŵ) := (b h w)ŵdx b w ŵ in ds (b n)wŵds, () E T E h E o h Γ in and th raction trm by B r (w,ŵ) := cwŵdx. (2) E T E h Th advction and raction parts will frquntly occur togthr and so for brvity w also dfin B ar (w,ŵ) := B a (w,ŵ) + B r (w,ŵ). Th bilinar forms ar maningful for functions w,ŵ H (Ω) +V dg.

6 6 Andra Cangiani t al. For th diffusion trm, bsids th standard symmtric intrior pnalty mthod w also prsnt a modifid schm, which w call th dcoupld mthod. W rfr to [] for a comprhnsiv study of th intrior pnalty mthod and altrnativ discrtizations. For w,ŵ V dg th dcoupld mthod is dfind by B d (w,ŵ) := h w h ŵdx E T E h + E h \J σh w ŵ ({{ h w}} ŵ + {{ h ŵ}} w ) ds, (3) yilding B ε (w,ŵ) := ε B d (w,ŵ) + B a (w,ŵ) + B r (w,ŵ). (4) Hr σ > 0 is a discontinuity pnalization paramtr. With this formulation thr is no control on th fluxs across J (hnc th nam dcoupld). W rcall that th standard symmtric Intrior Pnalty (IP) mthod is givn by and B d (w,ŵ) := B d (w,ŵ) + J σh w ŵ ({{ h w}} ŵ + {{ h ŵ}} w ) ds, B ε (ŵ,w) := εb d (ŵ,w) + B a (ŵ,w) + B r (ŵ,w). Whn rstrictd to th cdg spac, th dcoupld and standard IP forms bcom th bilinar form for th standard cg mthod on th continuous rgion. W first analys th stability proprtis of th dcoupld formulation and infr stability for th standard IP mthod from a prturbation argumnt. W introduc th following msh dpndnt norm for w V dg : whr and w 2 := ε w 2 d + w 2 ar, (5) w 2 d := w 2 H (E) + σh w 2 L 2 (), E h w 2 ar = r /2 w 2 L 2 (Ω) + E 2 b n /2 w 2 L 2 (), h with r dfind in (7). Rcall that for th symmtric intrior pnalty mthod th paramtr σ is slctd indpndntly of T h such that B d is positiv dfinit with a corcivity constant which is also indpndnt of T h. W assum that σ is such that for all w V dg : {{ h w}} w L (E h ) 2 hw L 2 (Ω) σ/h w L 2 (E h ). (6) Thn, by Young s inquality, w hav B d (w,w) E h w h wdx + E h σh w w 2 {{ h w}} w L (E h ) 2 w 2 H (E) + E T 2 σh w 2 L 2 () = h E 2 w 2 d. h

7 Stability of cdgfem for Singularly-Prturbd Problms 7 W adopt for B d th sam σ as for B d. W introduc a projction oprator following th prsntation of [2] which allows us to considr non-constant b. For polynomial dgr k considr th L 2 -orthogonal projction Π D : L 2 (Ω) V cdg (T dg ) dfind by Ω Π D (v)wdx = vwdx w V cdg (T dg ). (7) Ω In particular Π D (v) TcG = 0. Furthrmor, for all lmnts E of th msh Π D (v) L 2 (E) v L 2 (E) v L 2 (E). (8) As Π D (v) V cdg (T dg ) w hav for all E T h th invrs inquality Π D (b h v) H (E) and using a trac inquality w hav h E Π D (b hv) L 2 (E), (9) Π D (b h v) 2 L 2 () h E Π D (b hv) 2 L 2 (E). (20) E h Dfin th stramlin norm by v 2 S := v 2 + τ E Π D (b h v) 2 L 2 (E), (2) whr τ E is dfind by τ E := τ min { h E b L (E), h2 E ε }, (22) and τ (0,] is a positiv numbr at our disposal. Dfinition 4 A dcoupld cdg approximation to () is dfind as ṽ h V cdg satisfying B ε (ṽ h,v) = f vdx v V cdg. (23) Ω Dfinition 5 A cdg approximation to () is dfind as v h V cdg satisfying B ε (v h,v) = f vdx v V cdg. (24) Ω Th nxt assumption imposs th rquirmnt that b points on J non-charactristically from T cg to T dg. Assumption 4 Th T h -dcomposition is such that for vry J 4 (b(x) nc ) > ε max σ h 3/2 x, (25) whr n C rprsnts th unit normal pointing from T cg to T dg. Obsrv that th scaling btwn ε and h mirrors that of Assumption 2.

8 8 Andra Cangiani t al. Thorm On V dg th bilinar forms B ε and B ε ar corciv with rspct to w : 4 w 2 B ε (w,w), 4 w 2 B ε (w,w), w V dg. (26) Proof For th advction and raction trms using intgration by parts w hav B ar (w,w) = r /2 w 2 L 2 (Ω) + b n w w ds. (27) E 2 h For th diffusion trm it follows from (6) and Young s inquality that σ B d (w,w) + w w ds J h 2 w 2 d, B d(w,w) 2 w 2 d. Combining th last inquality with (27), th rsult now follows with Assumption 4. It follows that ṽ h and v h xist and ar uniqu. Th following final assumption prmits th us of an invrs inquality on th continuous Galrkin rgion. Assumption 5 Th msh T cg is quasi-uniform. It is convnint to dnot th msh-siz on T cg by h TcG = h E L (T cg ). Th main rsult of this work is Thorm 4, which stats that th cdg approximation is stabl in th stramlin diffusion norm whnvr Assumptions, 2, 3, 4 and 5 ar satisfid. In ordr to prov this rsult w stablish first two sparat stability bounds: W dfin ṽ ε,ṽ 0 V cdg by th condition that for all v V cdg B ε (ṽ ε,v) = B ε (u ε,v), (28) B ε (ṽ 0,v) = B ε (u 0,v). (29) Obsrv that by linarity of th dcoupld cdg mthod w hav ṽ ε +ṽ 0 = ṽ h. In Sction 3 w driv a bound for th dcoupld cdg approximation ṽ ε to u ε on T cg ; in Sction 4 w obtain a bound for th dcoupld cdg approximation ṽ 0 to u 0 on T cg. In Sction 5 w stablish an inf-sup condition with stramlin control on T dg. Finally, in Sction 6 w combin ths rsults to show stability of th dcoupld and thn of th standard cdg approximation. 3 Bounds on th ṽ ε Componnt on T cg W introduc th projction oprator of Scott and Zhang, [7] [3, Sction.6.2]. Lmma (Scott-Zhang Projction) Th Scott-Zhang oprator SZ h : W l,p (Ω) V cg has th following proprtis: For l > 2 thr xists a C sz > 0 such that for all 0 m min(,l) SZ h (v) H m (T cg ) C sz v H l (T cg ) v H l (T cg ), (30) and providd l k + for all E T cg and 0 m l w hav th approximation v SZ h (v) H m (E) C sz h l m E v H l ( E ) v H l ( E ). (3) whr E is th nod patch of E, i.., th st of clls in T cg sharing at last on vrtx with E.

9 Stability of cdgfem for Singularly-Prturbd Problms 9 In th proof of th following thorm w initially aim to obtain an a priori rror stimat. Th particular choic of th projction v A into th approximation spac and Assumption 4 lad to a dcoupling of th convction trms, allowing to control v A ṽ ε in trms of th rstriction of u ε to Ω cg. As u ε has no layr in this rgion an invrs stimat may b applid to obtain stability. Thorm 2 Th dcoupld cdg approximation ṽ ε is stabl on th T cg rgion in th sns that ṽ ε H (T cg ). (32) Proof W pick th auxiliary solution v A V cdg as follows: On T cg, dfin v A to b th Scott-Zhang projction of u ε ; and on T dg to b th dg approximation with boundary conditions givn by SZ h (u ε ) on J and 0 on Γ dg, i.., v A = SZ h (u ε ) on T cg, B ε (v A,v) = B ε (u ε,v) v V cdg (T dg ). St η := u ε v A and ξ := v A ṽ ε, so η + ξ = u ε ṽ ε. Notic that ξ V cdg. Th Galrkin orthogonality xprssd by (28) and Thorm giv 4 ξ B ε (ξ,ξ ) = B ε (η,ξ ) = B ε (η,ξ χξ ), (33) whr χ is dfind in (6). Not that ξ χξ is continuous xcpt on J whr ξ χξ = ξ C n C and {{ξ χξ }} = 2 ξ C, whr th suprscript C indicats th trac takn from th continuous Galrkin sid of J. W xamin ach trm of B ε in turn. For th diffusion parts w us Young s inquality B d (η,ξ χξ ) 2 η 2 H (T cg ) + 8 ξ 2 H (T cg ). Rcall that by Assumption 4 th convction cofficint b points on J from T cg to T dg. Thn, according to (), in intgrals ovr J of B a (η,ξ χξ ) only th trac of (ξ χξ ) TdG appars, but not of (ξ χξ ) TcG. Bcaus (ξ χξ ) TdG = 0 all dg intgrals of B a (η,ξ χξ ) vanish. With Young s inquality w hav B a (η,ξ χξ ) 4 ρ b hη 2 L 2 (T cg ) + ρ 6 ξ 2 L 2 (T cg ), (34) whr ρ is dfind in (7). Finally for th raction trm B r (η,ξ χξ ) 4 ρ c 2 L (Ω) η 2 L 2 (T cg ) + ρ 6 ξ 2 L 2 (T cg ). Using th prvious thr rsults, (33), th dfinition of th norm (5), and Lmma w gathr ξ trms on th lft hand sid to show, with h TcG = h E L (T cg ), 8 ξ 2 2ε η 2 H (T cg ) + 4 ρ b hη 2 L 2 (T cg ) + 4 ρ c 2 L (Ω) η 2 L 2 (T cg ) (εh 2 T cg + h 2 T cg + h 4 T cg ) u ε 2 H 2 (Ω cg ) h2 T cg, (35) whr in th final stp w hav usd (0) and that th constant of may dpnd on c 2 L (Ω). As ρ > 0 w may us (35) and an invrs inquality to show ξ 2 H (T cg ) h 2 T cg ξ 2 L 2 (T cg ) h 2 T cg ξ 2. (36) As w alrady stablishd with Assumption 3 and (30) that η H (T cg ), w conclud that ṽ ε H (T cg ) u ε H (T cg ) + η H (T cg ) + ξ H (T cg ).

10 0 Andra Cangiani t al. 4 Bounds on th ṽ 0 Componnt on T cg W now pick th auxiliary solution v A as follows: On T cg lt it b u 0 and on T dg b th dg approximation to u 0 with boundary conditions givn by u 0 on Γ dg J, i.., v A = u 0 on T cg, (37) B ε (v A,v) = B ε (u 0,v) v V cdg (T dg ). (38) Lmma 2 W hav for all v V cdg that B ε (v A,v) = B ε (ṽ 0,v). Proof Fix v V cdg. Thn using (29) B ε (ṽ 0,v) = B ε (u 0,v) = B ε (u 0,v χv) + B ε (u 0, χv) whr χ is dfind in (6). Obsrv that B ε (u 0, χv) = B ε (v A, χv) by (38). Notic that v χv and u 0 ar continuous on T cg. Rcall that no intgral ovr J appars in th dfinition of B d. For B a (ṽ 0,v χv), th intgral ovr J vanishs arguing as in th prvious proof for (34). Thrfor B ε (ṽ 0,v χv) = B ε (u 0,v χv) = B ε (v A,v χv). Th ovrall structur of th nxt proof is similar to that of Thorm 2. Howvr, hr th dcoupling of th convction trms is basd on th additional Galrkin orthogonality idntifid in Lmma 2, which is xploitd by th projction ṽ π. Lmma 3 W hav ṽ 0 H (T cg ). Proof Dfin ṽ π to b { SZh (u ṽ π := 0 ) on T cg, v A on T dg, and lt η := v A ṽ π, ξ := ṽ π ṽ 0. With ths dfinitions η + ξ = v A ṽ 0, η TdG = 0 and ξ and η ar continuous on T cg. Thn using Thorm and Lmma 2 w hav 4 ξ 2 B ε (ξ,ξ ) = B ε (η,ξ ) = ε h η h ξ + (b h η)ξ + cηξ dx + b η ξ in ds. T cg J Du to Assumption 4 w hav ξ in = ξ D, th trac from th dg sid of J, and η = η C n C, th trac and normal from th cg sid of J. W split ach of th trms using Young s inquality, giving 4 ξ 2 2ε h η 2 L 2 (T cg ) + ε 8 hξ 2 L 2 (T cg ) + 4 ρ b hη L 2 (T cg ) + ρ 6 ξ 2 L 2 (T cg ) + 4 ρ c 2 L (Ω) η 2 L 2 (T cg ) + ρ 6 ξ 2 L 2 (T cg ) + (b n C η C )ξ D ds. (39) For th final trm w not that ξ is a polynomial and so using Young s inquality and a trac and invrs inquality (with constant C ti ) givs J (b n C η C )ξ D ds 4C ti b 2 L (Ω) η C 2 h ρ L 2 (J) + ρ 6 ξ 2 L 2 (T dg ). (40) J

11 Stability of cdgfem for Singularly-Prturbd Problms W combin (39) and (40) to hid all trms of ξ undr th norm on th lft-hand sid of (39). Using (3) for th trms of η and a trac inquality givs ρ ξ 2 L 2 (T cg ) ξ 2 (εh 2 T cg + h 4 T cg + h 2 T cg ) u 0 2 H 2 (T cg ) h2 T cg u 0 2 H 2 (T cg ) and, by an invrs inquality, ξ 2 H. Now th rsult follows from th stability of (T cg ) th Scott-Zhang oprator. 5 Inf-Sup Condition Th following thorm is an adaptation of rlatd stability bounds in [5] and [2] to fit th abov assumptions. Although th vrification of th blow inf-sup condition follows th ovrall structur in [5], w stat it hr in dtail as th prsnt analysis xtnds th scop to non-constant advction cofficints via th incorporation of Π D as [5]. Morovr, it dals with th modification of th bilinar form on J and it only has stramlin control on th T dg sid. It is hlpful to rcall that Π D v TcG = 0 for any v. Thorm 3 Thr xists a positiv constant Λ is which is indpndnt of h and ε but may dpnd on th polynomial dgr, σ, and th constants in (9) and (20) such that: inf v V cdg sup ˆv V cdg Proof For an arbitrary v V cdg, w dfin Bε (v, ˆv) v S ˆv S Λ is. (4) ˆv := v + γ v S, v S := τ E Π D (b h v), (42) whr γ is a positiv paramtr at our disposal and τ E is dfind in (22). Not that through th dfinition of Π D w hav ˆv,v S V cdg. Thorm 3 follows from th following two bounds: ˆv S v S, (43) B ε (v, ˆv) v 2 S. (44) Considr first (43). W xamin ach trm of v S 2 S in turn. W hav ε v S 2 H (E) εh 2 E τ EΠ D (b h v) 2 L 2 (E) ττ E Π D (b h v) 2 L 2 (E) v 2 S. (45) Also r /2 v S 2 L 2 (Ω) r L (Ω) τe Π 2 D (b h v) 2 L 2 (E) v 2 S. (46) For th trms on th dgs w us (20) and (22). This givs b n /2 v S 2 L 2 () b L (Ω)τEh 2 E Π D (b hv) 2 L 2 (E) v 2 S. (47) E h

12 2 Andra Cangiani t al. Similarly, σε v S 2 E h L 2 () τe 2 h Th final trm of th stramlin norm givs σε Π D (b h v) 2 L 2 (E) v 2 S. (48) τ E Π D (b h v S ) 2 L 2 (E) τ E b h (τ E Π D (b h v)) 2 L 2 (E) h 2 E τe b 3 2 L (E) h 2 E Π D (b hv) 2 L 2 (E) v 2 S. Combining th abov rsults w hav v S 2 S v 2 S. Using a triangl inquality w find ˆv S v S + γ v S S C(τ,σ,γ) v S, which concluds th proof of (43). To prov (44) first considr th advction and raction trms of th norm. Using th linarity of B ar w hav B ar (v, ˆv) = B ar (v,v) + γb ar (v,v S ). Th scond trm is givn by B ar (v,v S ) = E E o h cv(τ E Π D (b h v)) + (b h v)(τ E Π D (b h v))dx (49) b v (τ E Π D (b h v)) in ds (b n)v(τ E Π D (b h v))ds. Γ in (50) Using th proprtis of Π D givn in (7) th scond trm abov bcoms (b h v)(τ E Π D (b h v))dx = τ E Π D (b h v)π D (b h v)dx E E T E h = τ E Π D (b h v) 2 L 2 (E). (5) Using Young s inquality w hav cv(τ E Π D (b h v))dx c L (Ω) E T E h 2 v 2 L 2 (E) + 2 τ2 E Π D (b h v) 2 L 2 (E). W now com to th dg trms in (49). For ach dg w us a trac inquality with constant C ti and (22), so that for smooth f and polynomial g: f gds C ti f L 2 () g L 2 (E) C τ ti f he τ E b L 2 L () g L 2 (E). (E) Thus, with C E dpnding on C ti, b and th numbr of dgs pr lmnt, Young s inquality givs (b n)v(τ E Π D (b h v))ds b v (τ E Π D (b h v)) in ds E o h Γ in E h C E τ b n /2 v 2 L 2 () + τ E 4 Π D (b hv) 2 L 2 (E).

13 Stability of cdgfem for Singularly-Prturbd Problms 3 In conclusion, using Assumption and (27), w hav ( B ar (v, ˆv) ρ γ c ) ( ) L (Ω) 2 v 2 L 2 (E) + E T 2 γ C E τ b n /2 v 2 L 2 () h E h ( + γ τ E τ2 E E T 2 τ (52) E ) Π 4 D (b h v) 2L2(Ω). h Rcall that h E /b L (T dg ) by Assumption 2, which impls τ E for all E T h. For gnral v, all trms on th right-hand sid of (52) ar positiv, providd γ is small nough. Equation (6) nsurs th continuity of B d with rspct to d ; thus B d (v, ˆv) C v d ˆv d, (53) for som C > 0. Rcalling (45) and (48), it is clar that v S d C 2 v d, for som C 2 > 0. Hnc, with γ /(2C C 2 ), B d (v, ˆv) = B d (v,v) + γ B d (v,v S ) v 2 d σ v 2 ds γ C v d v S d J h 2 v 2 d σ v 2 ds. (54) h J Thus, if in addition γ min{ρ/ c L (Ω),/(8C E τ)}, thn (52) and (54) giv B ε (v, ˆv) ε 2 v 2 d + ρ 2 v 2 L 2 (Ω) b n /2 v 2 L 2 () ε σ v 2 ds E J h h + γ 4 τ E Π D (b h v) 2 L 2 (Ω). Rcalling Assumption 4 and h, it is clar that (44) is satisfid. 6 Stability of th Dcoupld and Standard Approximations W saw that, undr a st of suitabl assumptions, th dcoupld approximation satisfis th stability bounds: ṽ h H (T cg ), ṽ h S f L 2 (Ω). (55) Th first bound is a consqunc of Thorm 2 and Lmma 3, th scond of Thorm 3. So whil on has stramlin-diffusion stability on T dg, an vn strongr bound is availabl on T cg undr th aformntiond assumptions. W now driv a stability rsult for th cdg mthod. W rquir that th gomtry of th intrfac J dos not bcom significantly mor complicatd as th msh is rfind. Mor prcisly, w rquir th bounddnss of th trac oprator. Th proof is basd on th obsrvation that B B is of ordr ε and that thus th stability of th dcoupld mthod may in th convction-dominatd rgim b transfrrd to th classical dg bilinar form. Thorm 4 Suppos that th oprator norm of th trac oprator H (T cg ) L 2 (J) is boundd indpndntly of h. Thn, th cdg approximation v h is stabl in th sns that h TcG h v h 2 L 2 (T cg ) + v h 2 S + f 2 L 2 (Ω).

14 4 Andra Cangiani t al. Proof St ζ := v h ṽ h. Using th corcivity of B ε, Galrkin orthogonality and th norm of th trac oprator H (T cg ) L 2 (J), w hav 4 ζ 2 B ε (ζ,ζ ) = B ε (ṽ h,ζ ) B ε (ṽ h,ζ ) + B ε (v h,ζ ) B ε (ṽ h,ζ ) = B ε (ṽ h,ζ ) B ε (ṽ h,ζ ) = ε { h ṽ h } ζ + { h ζ } ṽ h σ ṽ h ζ ds J h ( ε h TcG h ṽ h L 2 (Ω) εσ ζ L 2 (J) + hζ L 2 (Ω) εσ ṽ h L 2 (J) σ + ṽ h ζ ds J h h 3/2 ) h /2 ( ) /2 ( ε h TcG h ṽ h 2 L 2 (Ω) + εσ ṽ h 2 L 2 (J) ε h ζ 2 L 2 (Ω) + εσ ζ 2 L 2 (J) and thus, using Assumption 4, h /2 ζ 2 εh TcG h ṽ h 2 L 2 (Ω) + εσ h 3/2 ) /2 h /2 T cg ṽ h 2 L 2 (J). (56) Dividing through by h TcG and using an invrs inquality on ρ ζ L 2 (E) givs h TcG h ζ 2 L 2 (T cg ) h T cg ζ 2 ε h ṽ h 2 L 2 (Ω) + εσ h 3/2 T cg ṽ h 2 L 2 (J). (57) Using Assumption 4 as wll as (55) w bound ach of th trms in (57). W conclud that h TcG h v h L 2 (T cg ) h T cg h ξ L 2 (T cg ) + h T cg h ṽ h L 2 (T cg ). To show that v h S is boundd w stablish an inf-sup condition for B ε. Indd, (43) may b usd without chang. It rmains to transfr (44) to B ε. Th inquality (52) is still availabl as th discrtsation of th lowr-ordr trms did not chang. Diffrnt is that w now us B d (v, ˆv) C v ˆv in plac of (53), justifid by Assumption 4. Appaling to (45) (48), on has v S C 2 v for som C 2 > 0. Hnc Bd(v, ˆv) = Bd(v,v) + γ Bd(v,v S ) v 2 d γ C v v S. (58) For γ C C 2 small nough and λ sufficintly larg, γ C v v S is boundd by 2 v 2 d + 2 B ar(v, ˆv), using again th positivity of th trms in (52). Rmark Du to Assumptions 2 and 4 th abov stability bound is valid for th rgim ε h 3/2 E b L (Ω). For compltnss w brifly outlin hr how h H (Ω) stability of th cdg mthod is stablishd if ε h 3/2 E b L (Ω). Th stability proof is in this cas asir bcaus th msh Péclt numbr is smallr. If Ω is smooth or convx and th cofficints hav sufficint rgularity thn u is in H 2 (Ω). Indd for ε 2 h3/2 E b L (Ω) th H 2 norm of u is uniformly boundd in ε. Suppos that th msh is quasi-uniform. By Cà s Lmma, with h := max E h E, a standard argumnt givs u v h H (Ω) ( + ε )h u H 2 (Ω) (h + h /2 ) u H 2 (Ω), and thus h u v h H (Ω).

15 Stability of cdgfem for Singularly-Prturbd Problms 5 (a) Solution u givn by (60). (b) Solution u ε givn by (6) Fig. 2 Solution u and u ε for ε = Numrical Exprimnt Lt Ω = (0,) 2. W sk to solv ε u + ( x, y) u = x y (59) with Dirichlt boundary conditions chosn such that th solution is givn by u(x,y) = x + y Erf( x/ 2ε ) + Erf ( y/ 2ε ) Erf ( / 2ε ), (60) whr Erf is th rror function dfind by Erf(x) = 2 π x 0 t2 dt. For 0 < ε this problm xhibits an xponntial boundary layr along th outflow boundaris x = 0 and y = 0 of width O( ε) δ = δ =0.90 δ = δ = ε Fig. 3 u ε H 2 (Ω cg ) for diffrnt valus of δ.

16 6 6 Andra Cangiani t t al. al. 6 Andra Cangiani t al wh vh wh vh L 2 L 2 εh εh u vh u vh L 2 L 2 εh εh δ h δ h (a) (a) Diffrnc btwn cdg v h and and dg w h. mations. (b) (b) Error Error of th in cdg th cdg approximation. (a) Diffrnc btwn cdg and dg approximations. (b) Error in th cdg approximation. Fig. 4 Comparing th cdg approximation v h with th dg approximation w h and th xact solution u. Fig. Fig. 2 Comparing 2 approximations for for ε = ε = ThlossofaccuracyasT cg cg covrs covrs th th layr layr is is apparnt. Away from th layrs th boundary conditions on th inflow boundaris x = and y = arw wll W finally finally approximatd rmark rmark that, by that, y at last last andfor xfor th th rspctivly. xampl considrd Th hyprbolic hr, hr, th solution thchoic with of of Tths cg T cg laving boundary ing on on layr layr conditions of of lmnts is givn at at th byth uoutflow 0 (x,y) = boundary x + y 2. isthis optimal. givs Indd, adding vn vna asingl lmnt lmnt to to th th T cg lav- T cg rgion rgion rsults rsults in in oscillations polluting th th solution. For For xampl, for for u ε (x,y) T T cg =[2 5,] ] [0,2 5 cg =[2 5 = 2 Erf( x/ 2ε ) + Erf ( y/ 2ε ),] 2 ([0.5,0.5 Erf ( / ε ] ) [0,2 5. ]), ]), (6) i.., i.., adding W adding plot a singl a(60) singl and lmnt lmnt (6) for to to Tε cg T= cg 0halfway 3 in along Figur alongth 2; th x-axis, not x-axis, that rsults rsults awayinfrom th layrs th solution u ε is clos to zro. w h v h L 2 (Ω) = , Rcovring th gomtry of w Figur h v, h w L 2 (Ω) lt = Ω cg b a st of th, typ ( δ,) 2, 0 < δ <. In accordanc to Assumption ε h (w h v h ) L 2 (Ω) = ε h (w 0 3 h 3, vfor h ) ach δ, L 2 (Ω) = th suprmum 0 3, asignificantincrasonthnormsfort sup cg u =[2 5,] 2 cg =[2 ε 5 H,] 2 (Ω 2 cg ).NoticthatthischoicofT cg cg violats violats ε (0,ε max ] Assumptions Assumptions 3 and 3 and is finit. Th dpndnc of u ε H 2 (Ω cg ) with rspct to δ and to ε is illustratd in Figur 3. For this xampl c 2 b =, so Assumption is satisfid. Furthr, w fix ε = 0 6 8Acknowldgmnts: 8Acknowldgmnts: and considr uniform squar mshs of dg lngth 2 5 so that Assumption 2 is also satisfid as th smallst local msh Péclt numbr is W W gratfully W gratfully dfinthank T thank th th Archimds Archimds Cntr Cntr for for Modling, Modling, Analysis Analysisand andcomputation Computationin in cg = [ δ h,] 2, whr δ h = m2 5, m {0,...,32}. Not that, having fixd Crt th Crt msh, for for δhosting th th authors authors during during th th prparation prparation of of this this manuscript. manuscript. h is a discrt paramtr. Th intrfac J is composd of th dgs lying on th lins y = δ h for x δ h and x = δ h for y δ h. Th smallst valu of b n is δ h occuring on th dgs containing th point (δ h,δ h ). Thus, in this cas, Assumption 4 rads 4 Rfrncs Rfrncs mh > ε σ h 3/2 and is satisfid for all m {,...,32} (not that this assumption is trivially satisfid whn m = 0) for this choic of ε and h if σ < 200; in th shown computations σ = 0.. D. N. ARNOLD, F. BREZZI, B. COCKBURN, AND L. D. MARINI, Unifid analysis of discontinuous. D. InN. ARNOLD, F. BREZZI, B. COCKBURN, AND L. D. MARINI, Unifid analysis of discontinuous Galrkin Figurmthods 4 w plot forth lliptic L 2 (Ω) problms, norm, SIAMJournalonNumricalAnalysis,39(200),pp.749 ε wightd H (T h ) smi-norm, and L 2 norm of th Galrkin jumps mthods for lliptic problms, SIAMJournalonNumricalAnalysis,39(200),pp on E 779. h, rprsntd by, for both th diffrnc in th dg and cdg approximations B. and th rror 2. B. AYUSO AND L. D. MARINI, Discontinuous Galrkin mthods for advction-diffusion-raction problms, SIAMJournalonNumricalAnalysis,47(2009),pp AYUSO AND L. in D. th MARINI, cdg approximation. Discontinuous Galrkin Not from mthods Figur for advction-diffusion-raction 4(a) that th diffrnc in problms, approximations th 3. C. SIAMJournalonNumricalAnalysis,47(2009),pp incrass only vry slowly until th final data points (whr T cg T h ). BARDOS AND J. RAUCH, Maximal positiv boundary valu problms as limits of singular prturbation 3. Whn C. BARDOS AND J. RAUCH, Maximal positiv boundary valu problms as limits of singular prturba- th problms, continuous TransactionsofthAmricanMathmatical rgion covrs th layr, non-physical Socity, oscillations 270 (982), pollut pp. pp. th approximation 4. R. problms, in BECKER, function TransactionsofthAmricanMathmatical E.BURMAN, of δ h, s Figur P.HANSBO, 4(b). AND M. LARSON, Socity, ArducdP-discontinuousGalrkin 270 (982), pp. pp R. BECKER, E.BURMAN, P.HANSBO, AND In M. LARSON, ArducdP-discontinuousGalrkin mthod, Tabltch.rp.,EPFL,2004.EPFL-IACSrport w show th numbr of dgrs of frdom (dofs) as th continuous rgion 5. mthod, A. BUFFA, tch.rp.,epfl,2004.epfl-iacsrport T.J.R.HUGHES, AND G. SANGALLI, Analysis of a multiscal discontinuous Galrkin 5. is A. incrasd. mthod BUFFA, for T.J.R.HUGHES, Rducing th dgrs AND convction-diffusion problms, G. of SANGALLI, frdom to SIAMJournalonNumricalAnalysis,44(2006),pp.420 Analysis approximatly of a multiscal 30% of discontinuous th dg mthod Galrkin mthod 440. for convction-diffusion problms, SIAMJournalonNumricalAnalysis,44(2006),pp δ h δ h

17 Stability of cdgfem for Singularly-Prturbd Problms 7 δ dofs % of dg dofs Tabl Dgrs of frdom with ε = 0 6. ε h (w h v h ) L 2 (Ω) dg cg dgrs of frdom rsults in only a vry slight diffrnc in th norm, thus showing that a considrabl saving can b mad without compromising stability. W finally rmark that, at last for th xampl considrd hr, th choic of T cg laving on layr of lmnts at th outflow boundary is optimal. Indd, adding vn a singl lmnt to th T cg rgion rsults in oscillations polluting th solution. For xampl, for T cg = [2 5,] 2 ([0.5, ] [0,2 5 ]), i.., adding a singl lmnt to T cg halfway along th x-axis, rsults in w h v h L 2 (Ω) = , ε h (w h v h ) L 2 (Ω) = , a significant incras on th norms for T cg = [2 5,] 2. Notic that this choic of T cg violats Assumptions 3 and 4. 8 Acknowldgmnts: W gratfully thank th Archimds Cntr for Modling, Analysis and Computation in Crt for hosting th authors during th prparation of this manuscript. Rfrncs. D. N. ARNOLD, F. BREZZI, B. COCKBURN, AND L. D. MARINI, Unifid analysis of discontinuous Galrkin mthods for lliptic problms, SIAM Journal on Numrical Analysis, 39 (200), pp B. AYUSO AND L. D. MARINI, Discontinuous Galrkin mthods for advction-diffusion-raction problms, SIAM Journal on Numrical Analysis, 47 (2009), pp C. BARDOS AND J. RAUCH, Maximal positiv boundary valu problms as limits of singular prturbation problms, Transactions of th Amrican Mathmatical Socity, 270 (982), pp. pp R. BECKER, E. BURMAN, P. HANSBO, AND M. LARSON, A rducd P-discontinuous Galrkin mthod, tch. rp., EPFL, EPFL-IACS rport A. BUFFA, T. J. R. HUGHES, AND G. SANGALLI, Analysis of a multiscal discontinuous Galrkin mthod for convction-diffusion problms, SIAM Journal on Numrical Analysis, 44 (2006), pp A. CANGIANI, J. CHAPMAN, E. H. GEORGOULIS, AND M. JENSEN, On local supr-pnalization of intrior pnalty Galrkin mthods, submittd jounral articl, (202). 7. A. CANGIANI, E. H. GEORGOULIS, AND M. JENSEN, Continuous and discontinuous finit lmnt mthods for convction-diffusion problms: A comparison, in Intrnational Confrnc on Boundary and Intrior Layrs, Göttingn, July 2006.

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