The numerical analysis of higher-order nonlinear FE method for advection dominated problems

XX Cong eso de Ecuaciones Di e enciales y Aplicaciones X Cong eso de Ma em´ a ica Aplicada Se illa, 24-28 sep iemb e 2007 (pp. 1–6) The nume ical analysis o highe -o de nonlinea FE me hod o ad ec ion domina ed p oblems S. M. C. Mal a, R. C. C. de Almeida Labo a ´o io Nacional de Compu a¸c˜ao Cien ´ı ica (LNCC/MCT) A . Ge ´ulio Va gas,333 - Pe ´opolis, 22561-070, Rio de Janei o, B azil. E-mails: [email p o ec ed], [email p o ec ed]. Keywo ds: ad ec ion domina ed p oblems, ini e elemen me hod, nume ical analysis Resumen The nume ical analysis o nonlinea discon inui y-cap u ing me hods applied o ad ec ion domina ed p oblems has no been comple ely es ablished ye . Some pa - icula esul s gi e e y good con ibu ion owa ds he exis ence o disc e e solu ions al hough no uniqueness esul s a e demons a ed. This pape s udies he condi ions o he uniqueness o he solu ion when using o he CAU (Consis en App oxima e Upwind) Pe o -Gale kin o sol ing ad ec ion domina ed p oblems. The main issue in his analysis is ha i elies on he op imali y p ope y o he CAU solu ion which does no depend on any es ic ions o he app oxima ion spaces. 1. In oduc ion I is well known ha nume ical simula ions o ad ec ion-domina ed p oblems p esen nume ical di icul ies ela ed o he lack o s abili y: because ad ec ion domina es di usion, classical Gale kin ini e elemen (FE) me hods gene a e uns able app oxima ions, which usually exhibi spu ious oscilla ions. The SUPG (S eamline Upwind Pe o -Galekin) me hod, p oposed by B ooks and Hughes [1], was he i s a ia ionally consis en , s able and accu a e ini e elemen model o ad ec ion domina ed p oblems. Fo egula solu ions his me hod p esen s quasi-op imal a es o con e gence o he s eamline de i a i e and was i s analyzed by Johnson e al. [4]. Ne e heless, o non egula solu ions, localized oscilla ions a e s ill obse ed in he neighbo hood o s eep g adien s meaning ha he s eamline is no always he app op ia e upwind di ec ion. To o e come his lack o mono- onici y many discon inui y cap u ing e ms we e designed o enhance s abili y ei he in 1 S. M. C. Mal a, R. C. Almeida a linea [5] o nonlinea way [2, 3]. Qui e p omising esul s we e ob ained using he non- linea CAU (Consis en App oxima e Upwind) ini e elemen me hod p oposed in [2]. The sys ema ic way o upda ing he upwind di ec ion in he CAU me hod esul s in adding o he SUPG o mula ion a nonlinea discon inui y-cap u ing e m in a consis en way, en- gende ing an addi ional s abili y in he di ec ion o he app oxima e g adien . The heo y has been e ined o e he yea s in se e al di ec ions. The nume ical analysis o he CAU me hod, e en o highe -o de elemen s, has been discussed in a ecen pape [3]. The s abili y analysis was shown based on a linea ized i e a i e scheme, which uses he solu ion o he SUPG me hod as an ini ial guess in o de o sol e he CAU (nonlinea ) me hod. Howe e , his analysis has some open ends [6] conce ning he sol abili y o he i e a i e scheme. The e o e, ou main goal is o imp o e he de eloped analysis. In pa icula , we add ess he p oblem associa ed o he con e gence o he linea ized i e a i e scheme as well as he uniqueness o he solu ion o he nonlinea app oxima e me hod. 2. Ma hema ical Model We a e in e es ed in s eady-s a e solu ions o con ec ion-domina ed eac ion-di usion scala p oblems o he o m Lφ:= γφ +u·∇φ−²∆φ= in Ω φ= 0 on Γ,(1) whe e udeno es a gi en eloci y ield, ²is he gi en (small) posi i e di usion coe icien , γis he gi en non-nega i e eac ion coe icien and is a sou ce e m. The p oblem is de ined in he domain Ω ⊂ Rnwi h bounda y Γ. The Gale kin o mula ion o p oblem (1) eads: ind φh∈Vp h⊂H1 0(Ω) such ha , B(φh, ηh) = ( , ηh),∀ηh∈Vp h, wi h B(φh, ηh) := ²(∇φh,∇ηh)+(u·∇φh, ηh) + γ(φh, ηh) and Vp h={ηh∈C0(Ω); ηh|Ωe∈Pp∀Ωe, ηh|Γ= 0}, whe e Ppis he se o in e pola ion polynomials o deg ee less o equal pde ined in each ini e elemen Ωewi h cha ac e is ic elemen leng h deno ed by h. Fo ad ec ion-domina ed p oblems, when ²is aken e y small in p oblem (1), i is well- known ha classical Gale kin and s abilized ini e elemen me hods gene a e global and local uns able app oxima ions, espec i ely. As a emedy, discon inui y-cap u ing a ian s a e conside ed as an addi ional (bu nonlinea ) s abiliza ion. Qui e p omising esul s ha e been ob ained using he CAU (Consis en -Upwind Pe o -Gale kin) [2, 3] o mula ion, which eads as: ind φh∈Vp hsuch ha D(φh, ηh) + Ne X e=1 (L(φh)− , τc[u− ]·∇ηh)|Ωe=F(ηh),∀ηh∈Vp h,(2) 2 The nume ical analysis o highe -o de ... wi h D(φh, ηh) := B(φ, η) + Ne X e=1 (Lφh, τsu·∇ηh)|Ωe; F(ηh) := ( , ηh) + Ne X e=1 ( , τsu·∇ηh)|Ωe, whe e τsand τca e he s abiliza ion pa ame e s om he SUPG [1, 4] and CAU [2] me hods, espec i ely. The e a e a a ie y o possible designs o hese pa ame e s in he li e a u e [1, 2, 3, 4, 6, 5]. The auxilia y ec o ield in equa ion (2) is a modi ied eloci y ield such ha i sa is ies he o iginal pa ial di e en ial equa ion (1) o he app oxima e solu ion φh∈Vp hin each elemen . This equi emen implies [u− (φh)] ·∇φh=L(φh)− =: R(φ).(3) Then, he ec o ield is de e mined such ha i is he eloci y ield closes o he eal eloci y ield uin he L2sense. Hence, i is ob ained by sol ing he ollowing local minimiza ion p oblem: ind u∈L∞(Ω)dsuch ha ku− kL2(Ωe)≤ ku−mkL2(Ωe),∀m∈Qm,(4) wi h Qm={m;m·∇φh−²∆φh+γφh− = 0 in Ωe, e = 1, . . . , Ne}. The solu ion o (4) oge he wi h (3) leads o ( −u= 0 ,i |∇φh|= 0 ; u− =R(φh) |∇φh|2∇φh,o he wise.(5) Thus, i |∇φh|= 0, he linea SUPG me hod is eco e ed. O he wise, |u− |=|R(φh)| |∇φh|=: β(φh), which ensu es ha , in each elemen Ωe, limh7→0 (φh) = u, when limh7→0φh=φ. Besides, as is selec ed minimizing ku− kL2(Ωe), he ollowing inequali y is e i ied β(φh) = |R(φh)| |∇φh|≤|R(ψh)| |∇ψh|=β(ψh),∀ψh∈Vp h.(6) Due o his p ope y, one may conclude ha he app oxima e solu ion o (2) using (5) is op imal. This is a ema kable ea u e o his me hod. In ac , he p ope y (6) is c ucial o yield he uniqueness esul p esen ed in nex sec ion. 3. Uniqueness o solu ion Le he CAU me hod (2) be e-w i en as: ind φh∈Vp hsuch ha a(φh;φh, ηh) = F(ηh),∀ηh∈Up h,(7) whe e a(φh;φh, ηh) := D(φh, ηh) + Ne X e=1 c(φh;φh, ηh)|Ωe,(8) 3 S. M. C. Mal a, R. C. Almeida wi h c(φh;φh, ηh)|Ωe:= µτc|R(φh)|2 |∇φh|2∇φh,∇ηh¶|Ωe. The non-symme ic bilinea o m D(·,·) is coe ci e (see, o example, [3]), which implies he ollowing SUPG-s abili y p ope y D(wh, wh)≥C(θ)|||wh|||2,∀wh∈Vp h,(9) wi h C(θ)=1−1/2√1 + θ, whe e θis de ined in [5]. The bilinea o m D(·,·) is also con inuous in he ollowing s abilized mesh dependen no m: |||ηh|||2:≡²k∇ηhk2+γkηhk2+ Ne X e=1 τsku·∇ηhk2 Ωe.(10) yielding uniqueness and exis ence solu ion o he SUPG me hod [4], aking τc= 0 in p oblem (7). Besides, p o iding ha τc≥0, c(ηh;ηh, ηh)|Ωe≥0 o all ηh∈Vp h, he s abili y o he CAU me hod ollows immedia ely om he p e ious SUPG s abili y esul , a(ηh;ηh, ηh)≥C0(θ)|||ηh|||2.(11) Fo a qui e simila me hod, i was assumed in [7] ha he quan i y |β(·)|sa is ies q0≤ |β(φh)| ≤ q1,(12) o q0, q1>0, implying coe ci i y and con inui y o he bilinea o m a(ηh;·,·) and a unique solu ion ha comes om he Lax-Milg am heo em [5, 6, 7]. Howe e , a p oo o he lowe bounded is no exhibi ed in [7]. La e on, i was applied a a ian o B ouwe ’s ixed poin heo em o p o e he exis ence o a disc e e solu ion o he CAU nonlinea p oblem (7) bu a uniqueness esul emained s ill open [5]. The e o e, using only he op imal-CAU p ope y (6) we will show his ac in he ollowing heo em. Theo em 1: Assuming ha he minimiza ion p ope y (6) holds, he CAU me hod (7) has a unique solu ion φh∈Vp h. P oo : Conside wo solu ions φhand ˜ φhbelong o Vp ho p oblem (7). Using he de ini ion o unc ion β(·), we ha e D(φh−˜ φh, ηh) + Ne X e=1 c(φh;φh, ηh)− Ne X e=1 c(˜ φh,˜ φh, ηh) = D(φh−˜ φh, ηh) + Ne X e=1 h(τc[β(φh)]2∇φh,∇ηh)|Ωe−(τc[β(˜ φh)]2∇˜ φh,∇ηh)|Ωei= 0.(13) F om he op imal p ope y o he CAU solu ion, β(φh)≤β(ψh),∀ψh∈Vp h, ha is, β(φh)≤β(˜ φh), since ˜ φh∈Vp h. Simila ly, β(˜ φh)≤β(φh)as well. Hence, β(φh) = β(˜ φh) = α, (14) 4 The nume ical analysis o highe -o de ... whe e αis a non-nega i e cons an . No ice ha α= 0 only when R(φh) = 0, ha is, φh sa is ies he esidual equa ion and he non-linea ope a o anishes. Now, subs i u ing (14) in o (13) yields D(φh−˜ φh, ηh) + Ne X e=1 ³τcα2(∇φh−∇ ˜ φh),∇ηh´|Ωe= 0,∀ηh∈Vp h.(15) Nex , aking ηh=φh−˜ φh∈Vp hand using he SUPG-s abili y p ope y (9), he equa ion (15) is e-w i en as C(θ)|||φh−˜ φh|||2+α2 Ne X e=1 ³τc(∇φh−∇ ˜ φh),∇φh−∇ ˜ φh´|Ωe≤ C1(|||φh−˜ φh|||2+ Ne X e=1 τc|∇φh−∇ ˜ φh|2 Ωe)≤0, whe e C1= max{C(θ), α2}is a posi i e cons an . The e o e, since τc≥0, he abo e inequali y yields φh=˜ φh.¤ 4. Linea ized p oblem To sol e he nonlinea p oblem (7) i is necessa y o conside i e a i e me hods which a e in ended o keep he main p ope ies o he CAU me hod a each i e a ion. This can be achie ed by using he ollowing simple me hod: om he p e ious compu ed i e a e solu ion φn h,n∈ N, we ge c(φn h;φn+1 h, ηh)|Ωe= (R(φn+1 h), τn c[u− (φn h)]·∇ηh)|Ωe=¡τc|β(φn h)|2∇φn+1 h,∇ηh¢|Ωe,(16) whe e τn c=τc(u− (φn h)) and (u− (φn h)) = R(φn h) |∇φn h|2∇φn h. Hence, he i e a i e algo i hm consis s in: gi en φn hand [u− (φn h)], ind {φn+1 h} ∈ Vp h such ha a(φn h;φn+1 h, ηh) = F(ηh),∀ηh∈Vp h,(17) whe e a(φn h;φn+1 h, ηh) = D(φn+1 h, ηh) + Ne X e=1 c(φn h;φn+1 h, ηh)|Ωe is a non-symme ic bilinea o m de ined on Vp h×Vp h, o a gi en φn h∈Vp h. I is coe ci e and con inuous [3, 5, 6] in he no m (10). Consequen ly, he linea ized p oblem (17) is uniquely sol able ia he Lax-Milg am heo em. The ze o- h i e a i e solu ion φ0 his he SUPG solu ion, which can be seen as he ze o o de esidual co ec ion o each τ0 c= 0. Due o he p ope y (6), any i e a i e solu ion {φn+1 h}o (17) sa is ies β(φh) = |R(φh)| |∇φh|≤|R(φn h)| |∇φn h|=β(φn h).(18) 5 S. M. C. Mal a, R. C. Almeida In addi ion, he sequence o solu ions {φn+1}is also uppe bounded [3], ha is, he e is a posi i e cons an C, independen o n, such ha , |||φn+1 h||| ≤ Ck k,∀n. (19) F om his ac and assuming ha {φn+1 h}, solu ion o he linea ized p oblem (17), con e ges o φh, solu ion o he nonlinea CAU me hod (7), when n7→ ∞, we may p o e ha he addi ional e m c(φh, φh, ηh) does no deg ade he a es o con e gence as long as egula solu ions a e conce ned [3]. This esul is p esen ed in he ollowing heo em. Theo em 2: Conside ing ²=O(h) and τs=O(h/p) we a i e a he ollowing a p io i e o es ima e: |||φ−φh|||2≤C Ne X e=1 µh p¶2s+1 |φ|2 s+1,Ωe,(20) whe e φ|Ωe∈Hk+1(Ωe) o some k≥1, o any 0 ≤s≤m´ın(p, k), p≥1. The p oo o he p e ious esul is based on an impo an open p oblem: o p o e ha he sequence {φn+1 h}con e ges o φh, solu ion o he nonlinea CAU me hod (7), when n7→ ∞. Wo k in his subjec is ongoing and will be add essed in a o hcoming pape . Acknowledgmen s The au ho s would like o hank he B azilian Go e nmen , h ough he Agency CNPq, o he inancial suppo p o ided. Re e ences [1] A.N. B ooks and T.J.R. 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