hp-bem for Contact Problems and Extended Ms-FEM in Linear Elasticity

Transcript

1 hp-bem for Coac Problems ad Exeded Ms-FEM i Liear Elasiciy Vo der Fakulä für Mahemaik ud Physik der Gofried Wilhelm Leibiz Uiversiä aover zur Erlagug des Grades Dokor der Naurwisseschafe Dr. rer. a. geehmige Disseraio vo Dipl.- Mah. Abderrahma Issaoui gebore am i Melaoui (Tuesie) 204

2 Refere: Prof. Dr. Ers P. Sepha, Leibiz Uiversiä aover Korrefere: Prof. Dr. Joachim Gwier, Uiversiä der Budeswehr, Müche Tag der Promoio:

3 Absrac We cosider a coac problem bewee a elasic body ad a rigid foudaio wih Tresca s fricio law, prese a hp-discreizaio echique of a mixed formulaio based o biorhogoal basis fucios ad solve he correspodig discree sysem wih he semi-smooh Newo mehod. A-poseriori error esimaes usig a error idicaor i a hp-adapive refieme algorihm are derived. We show covergece of he hpversio of BEM ad prese some umerical experimes. Furhermore we cosider a mixed boudary eleme formulaio, which is sabilized followig ideas of P.ild, Y. Reard ad V.Lleras for he FEM. A mesh-depede sabilizaio erm is added o he discree mixed formulaio, i order o avoid he discree if-sup codiio. Exisece ad uiqueess of he soluio of he discree problem are show ogeher wih a priori error ad a poseriori error esimaes. Numerical experimes are lised for he sabilized ad he o-sabilized cases. A sochasic coac problem wih a boudary iegral formulaio is aalyzed. We show ha he sochasic mixed formulaio is well-posed, sudy he deformaio of a elasic homogeeous maerial i which Youg s modulus (parameer ha characerizes he maerial properies) is a radom variable. A exeded muliscale fiie eleme mehod EMsFEM is derived for he aalysis of liear elasic heerogeeous maerials. The mai idea is o cosruc umerically fiie eleme basis fucios ha capures he small-scale iformaio (he fie mesh) wihi each coarse eleme. The cosrucio of he basis fucios is doe separaely for each coarse eleme wih piecewise liear fucios. The boudary codiios for he cosrucio of he muliscale basis fucios have a big ifluece o capurig he small-scale iformaio. We aalyse a correspodig FEM/BEM couplig ad derive a a priori error ad a-poseriori error esimae. Nex we prese fiie eleme implemeaios for operiodic case. Keywords. Tresca fricioal coac problem, biorhogoal basis fucios, sabilized hp-bem, semi-smooh Newo, Sochasic BEM, Muliscale-FEM. I

4 Zusammefassug Wir berache ei Koakproblem mi Trescareibug zwische eiem elasische Körper ud eiem sarre Uergrud. Für die esprechede gemische Formulierug präseiere wir eie hp-diskreisierugsechik basiered auf biorhogoale Basisfukioe ud löse das esprechede diskree Sysem mi eiem halbglae Newo-Verfahre. A poseriori Fehlerabschäzuge erlaube die Defiiio eies Fehleridikaors ud eie hp-adapive Verfeierugssraegie. Wir zeige die Kovergez eier hp-versio der BEM ud präseiere umerische Experimee. Darüber hiaus berache wir eie gemische Radelemeformulierug, die espreched de Idee vo P. ild, V. Lleras ad Y. Reard für FEM sabilisier wird. Ei gier-abhägiger Sabilisierugserm wird zur diskree gemische Formulierug addier, um die diskree if-sup-bedigug zu vermeide. Exisez ud Eideuigkei der Lösug des diskree Problems werde zusamme mi a priori ud a poseriori Fehlerabschäzuge gezeig. Numerische Experimee für de sabilisiere ud de ich sabilisiere Fall besäige die heoreische Ergebisse. Zudem wird ei sochasisches Koakproblem i eier Radiegralformulierug aalysier. Wir zeige, dass die sochasische gemische Formulierug wohlgesell is, ud sudiere die Verformug eies homogee elasische Maerials, i dem das Youg-Modul (das die Maerialeigeschafe charakerisier) eie Zufallsvariable is. Eie erweiere Muliskale-Fiie-Eleme-Mehode EMsFEM wird für die Aalyse vo heerogee lieare Maerialie hergeleie. Die aupidee beseh dari, umerisch Fiie-Eleme-Basisfukioe zu kosruiere, die die Mikrosrukur i jedem Grobeleme erfasse. Die Kosrukio der Basisfukioe wird für jedes Grobeleme mi sckweise lieare fukioe durchgeführ. Die Radbediguge für die Kosrukio der Mehrskale-Basisfukioe habe ämlich eie große Eifluss auf die Erfassug der Mikrosrukur. Wir aalysiere eie esprechede FEM/BEM- Kopplug sowie a priori ud a poseriori Fehlerabschzuge. Weier präseiere wir Fiie-Eleme-Implemeieruge für de ich periodische Fall. Schlagwore: Tresca Reibugskoak, biorhogoale Basisfukioe, sabilisieres hp-bem, halbglaes Newo-Verfahre, sochasische BEM, Mehrskale-FEM II

5 Ackowledgmes Firs of all, I would like o express my sicere graiude o my advisor Prof. Dr. Ers P. Sepha for he coiuous suppor of my Ph.D sudy ad research, for his paiece, moivaio, ehusiasm, ad immese kowledge. I have furhermore o hak my co-referee Prof. Dr. Joachim Gwier for his readiess o examie my hesis. My special haks go o Dr. Lohar Baz, Dr. eiko Gimperlei, ad my fried Zouhair Nezhi for heir suppor ad help i all sages of his hesis. I also hak all of he members of he Graduierekolleg for he various suppor. I wish o exed my haks o he whole saff a he Isiue for Applied Mahemaics. Paricularly o Mrs. Carme Gaze ad Mrs. Ulla Fleischhauer for heir kidess ad heir suppor o echical issues. I am also very graeful o my brohers Chokri, Imed, Taoufik, ad o my sisers Zohra ad ae, ad my mos hearfel haks o my broher Adel Aissaoui, my belle-soeur oussa. Mos especially, I would like o hak my wife Ime for her suppor ad paiece durig my work i his hesis. I hak my broher Adel for havig bee he hear of he family ad akig such good care of everybody. Las bu o leas, I would like o hak my pares for heir spiriual care ad proecio, ad for heir amazig love ad suppor. This hesis was suppored by IRTG 627 ( Ieraioal Research ad Traiig Group, graed by DFG). aover, de 09/04/204 Abderrahma Issaoui III

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7 Coes Iroducio 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy 3 2. Mixed Formulaio for Fricioal Coac Problem Exisece ad uiqueess of he soluio Discreizaio of he Lamé problem Semi-smooh Newo mehod ad algebraic represeaio A priori error esimae A poseriori error esimaes for coac wih fricio Numerical Experimes Sabilized mixed hp-bem i Liear Elasiciy 4 3. The mixed formulaio The sabilized mixed hp-bem formulaio Exisece ad uiqueess of he soluio A priori error aalysis for fricioal coac problem Reliable ad efficie a poseriori error esimaes for sabilized hp-bem for fricioal coac problems Reliabiliy of he BEM a poseriori error esimae Efficiecy of he BEM a poseriori error esimae Numerical Experimes hp-bem for Sochasic Coac Problems i Liear Elasiciy Mixed Formulaio for Sochasic Coac Problem Discreizaio for Sochasic Coac Problem A poseriori error esimaes for sochasic coac wih fricio Numerical Experimes Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy 0 5. The equaio of liear elasiciy The fiie eleme discreizaio Exeded muliscale fiie eleme mehod for he aalysis of liear elasic heerogeeous maerials The cosrucio of he basis fucios Siffess marix o he coarse mesh Covergece of he muliscale fiie eleme mehod V

8 Coes 5.5 A poseriori error for muliscale fiie eleme mehod Couplig FEM-BEM The weak formulaio of he Model Problem The discree problem for Ms-FEM-BEM A priori error esimae A poseriori error Numerical Experimes VI

9 Lis of Figures 2. Deformed geomery Esimaed errors Visualizaio of Lagrage muliplier Adapivey geeraed meshes for Lamé-BEM (bubble idicaor) Adapivey geeraed meshes for Lamé-BEM (residual idicaor) Iiial(lef) ad Deformed (righ) geomery Covergece for sabilized ad o-sabilized problems Deformed geomery wih he mea value of Youg modulus E Covergece of Sochasic Coac problem [54]The cosrucio of he umerical basis fucios Displaceme field of T for he basis fucio Φ x p x (lef) ad Φ x p y (righ) [54] Schemaic descripio of he EMsFEM The disribuios of he Mises sress obaied by he EMsFEM wih M c N c = 2 2, M f N f = 6 6, = 0.5, h = The disribuios of he Mises sress obaied by he EMsFEM wih M c N c = 24 4, M f N f = 8 8, = 0.25, h = The disribuios of he Mises sress obaied by he EMsFEM wih M c N c = 48 8, M f N f = 4 4, = 0.25, h = The disribuios of he Mises sress obaied by he EMsFEM wih M c N c = 96 6, M f N f = 2 2, = , h = The disribuios of he Mises sress obaied by FEM wih h = The Youg s modulus ad he Poisso s raio for he heerogeeous model Deformed geomery The disribuios of he Mises sress obaied by he EMsFEM wih M c N c = 96 6, = , L h = h = The disribuios of he Mises sress obaied by FEM wih L h = ad h = (L h = 8h) VII

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11 Iroducio Fricioal coac problems i liear elasiciy play a impora role i egieerig ad srucural mechaics. I mos cases, coac problems are reformulaed i erms of variaioal iequaliy problems, see Glowiski e al.[28], Kikuchi ad Ode [43]. The approximaio of coac problems ca be realized by he mixed fiie eleme mehod i which he o-peeraio codiio ad he fricio law are oly weakly eforced by a variaioal iequaliy. The heory of he mixed fiie eleme mehod is proposed by asliger e al.[33],[34] ad developed by Babuška [6], [7] ad Brezzi [4]. The key of his approach is he if-sup codiio. The fiie eleme spaces for he primal variables ad he mulipliers have o fullfil if-sup codiio, which is eeded i he covergece aalysis. owever, a discree if-sup codiio is difficul o obai for he hp-bem. Solvig fricioal coac problems wih he mixed hp-bem boudary eleme mehod is a challegig ask i mechaics. To circuve his difficuly a sabilizaio echique is used, by addig supplemeary erms i he weak formulaio, his mehod has bee iroduced ad aalyzed by Barbosa ad ughes i [] ad [2]. This approach was ake i [36, 37, 45] i he coex of h-fem ad is exeded o hp-bem i he prese work. The grea advaage of his approach is ha he sabilized scheme is sable for arbirary fiie eleme discreizaios, sice for he Lagrage muliplier space ay discreizaio ca be chose. The sabilized mehod for low-order fiie eleme discreizaios is based o liear - coform fucios for he displaceme ad piecewise liear (or cosa) fucios for he Lagrage mulipliers, see [36, 37, 45]. The use of higher-order polyomials leads o a cerai o-coformiy i he discreizaio which requires aeio i he covergece aalysis. May muliscale fiie eleme mehods MsFEM have bee developed ad sudied for he aalysis of liear elasic heerogeeous maerials. The MsFEM has bee origially idroduced by Babuška e al. [4, 5] ad was developed furher by ou e al. [38, 39] for solvig secod order ellipic boudary value problems. I his hesis, a exeded muliscale fiie eleme mehod EMsFEM is derived for he aalysis of liear elasic heerogeeous maerials. We aalyse a FEM/BEM couplig for EMsFEM ad derive a priori ad a poseriori error esimaes. The hesis is orgaized as follows. I Chaper, fricioal coac problems i liear elasiciy wih Tresca fricio i 2D are preseed ad aalyzed. A higher-order hp-bem discreizaio hechique of a mixed formulaio based a biorhogoal basis fucios

12 Iroducio is iroduced. The use of he biorhogoaliy allows a compoewise decouplig of he iequaliy cosrais. O he oher had he decoupled coac codiios ca be represeed by he problem of fidig he roo of a o-liear complemeariy fucio, ad herefore he so-called semi-smooh Newo mehods ca be applied. A a priori error aalysis is carried ou i he case, where we assume he discree if-sup codiio. A poseriori error esimaes usig a error idicaor i he hp-adapive refieme algorihm are derived. We show covergece of he hp-versio of BEM ad prese umerical experimes ha illusrae ad cofirm our heoreical resuls. I Chaper 2, we cosider a hp-bem sabilized mixed boudary eleme formulaio for fricioal coac problems i liear elasiciy i 2D. ere a mesh depede sabilizaio erm is added o he discree mixed formulaio, i order o avoid he if-sup codiio. The coac cosrais are imposed o he discree global se of affiely rasformed Gauss-Lobao pois o he elemes. Furhermore, a priori error esimaes are preseed ad a poseriori esimaes are derived for higher-order boudary eleme mehods. Numerical experimes are lised for he sabilized ad he osabilized cases. I Chaper 3, a sochasic coac problem wih boudary iegral formulaio is aalyzed. A sochasic mixed formulaio is show o be well-posed. This problem is rasformed io a equivale deermiisic oe by usig a Karhue-Loève specral decomposiio. The biorhogoaliy is adaped i he coex of a sochasic hp-bem. The compoewise decouplig of he weak cosrais allows o use a Uzawa algorihm o solve he discree problem. ere he Uzawa algorihm uses a poiwise projecio of he Lagrage muliplier. The deermiisic formulaio allows us o obai a residual a poseriori error esimaor. Fially i he las chaper, a exeded muliscale fiie eleme mehod EMsFEM is preseed for he aalysis of liear elasic heerogeeous maerials. The idea of he mehod is o cosruc umerically he muliscale basis fucios o capure he fie scale feaures of he coarse elemes i he muliscale fiie eleme aalysis. The cosrucio is doe separaely for each coarse eleme by solvig a subgrid problem ogeher wih suiable boudary codiios. We iroduce a FEM/BEM couplig for he EMsFEM wih sadard BEM ad derive a a poseriori error esimae. Fially, we give for he EMsFEM some umerical experimes. 2

13 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy I his chaper, we cosider a coac problem bewee a elasic body ad a rigid foudaio wih Tresca s fricio law ad prese a hp-discreizaio echique of a mixed formulaio based o biorhogoal basis fucios. For sadard h-versio BEM ad Sigorii codiios see [32]. We solve he correspodig discree sysem wih he semi-smooh Newo mehod, we derive a poseriori error esimaes ad use he error idicaor i a hp-adapive refieme algorihm. We show covergece for hp-versio of BEM ad prese some umerical experimes. Relaed a poseriori error esimaes for fricio problems have bee cosidered, for example, i [27, 30, 36, 42, 46, 50]. 2. Mixed Formulaio for Fricioal Coac Problem Le Ω R d, d = 2, 3, be a bouded Lipschiz domai wih he boudary Γ := Ω = Γ N Γ D be decomposed io o-iersecig Neuma Γ N, Dirichle Γ D ad coac boudaries, where ca come i coac wih he rigid foudaio. The problem he cosiss i fidig he displaceme fields u : Ω R such ha div σ(u) = 0 i Ω (2.a) σ(u) = C : ɛ(u) i Ω (2.b) u = 0 o Γ D (2.c) σ(u) = o Γ N (2.d) σ 0, u g, σ (u g) = 0 o (2.e) σ F, σ u + F u = 0 o (2.f) Le G(x,y) be he fudameal soluio of he Lamé equaio G(x, y) := λ+3µ 4πµ(λ+2µ) {log λ+3µ x y I + λ+µ λ+3µ 8πµ(λ+2µ) { x y I + λ+µ λ+3µ (x y) (x y) x y 2 }, for d = 2, (x y) (x y) x y 3 }, for d = 3, 3

14 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy We cosider a homogeous, isoropic, liear Sai Vea-Kirchhoff maerial, where he sress esor is give i erm of ooke s esor by σ(u) = C : ε(u) := λrε(u)i + 2µε(u) ε(u) = 2 ( u + ut ) where λ,µ are he Lamé cosas, moreover, r deoes he marix race operaor ad I he ideiy i R d, d = 2, 3 The scalar ormal ad ageial boudary sresses σ := σ(u) ad σ := σ(u) are well defied, where deoes he ui ormal exerior o he coac boudary. Le us deoe by F > 0 he give fricio hreshold o, we assume ha F is a cosa for he sake of simpliciy. The saic Tresca fricio codiio reads as follows σ F, a.e o if σ < F he u = 0 if σ = F he here exiss ν 0 such ha u = νσ We iroduice he sigle layer poeial V, he double layer poeial K, he adjoi double layer poeial K ad he hypersigular iegral operaor W for x Γ by V φ(x) := G(x, y)φ(y) ds y, Γ Ku(x) := (T y G(x, y) T )u(y) ds y, Γ K φ(x) := T x G(x, y)φ(y) ds y, Γ W u(x) := T x (T y G(x, y) T )u(y) ds y where T is he boudary racio operaor give by Γ T (u) := σ(u) Γ (2.2) Lemma 2.. [22, 23] Le Γ := Ω be he boudary of a Lipschiz domai Ω.The he iegral operaors V : 2 +s (Γ) 2 +s (Γ), K : 2 +s (Γ) 2 +s (Γ) K : 2 +s (Γ) 2 +s (Γ), W : 2 +s (Γ) 2 +s (Γ). 4

15 2. Mixed Formulaio for Fricioal Coac Problem are bouded for all s [ 2, 2 ],i.e. here exiss cosas C V,C K,C K,C W > 0 such ha V φ 2 +s (Γ) C V φ 2 +s (Γ), K φ 2 +s (Γ) C K φ 2 +s (Γ), Ku 2 +s (Γ) C K u 2 +s (Γ), W u 2 +s (Γ) C W u 2 +s (Γ). Lemma 2.2. [5, 9] Le Γ := Ω R d be boudary of a Lipschiz domai Ω.Le cap(ω) < i case d = 2. The he sigle layer poeial V is 2 (Γ)-ellipic,i.e. here exiss a cosa c V > 0, such ha V φ, φ Γ c V φ 2 (Γ), φ 2 (Γ) (2.3) Moreover is iverse operaor V : 2 (Γ) 2 (Γ) exiss ad V u 2 (Γ) c V u 2 (Γ), u 2 (Γ), (2.4) where c V is he ellipiciy cosa of V. The Seklov-Poicaré operaor S is defied by S := W + (K + 2 )V (K + 2 ). Lemma 2.3. [9] Le Γ 0 Γ. The he Seklov-Poicaré operaor S : 2 (Γ) 2 (Γ) is coiuous ad 2 (Γ 0 )-ellipic, i.e. here exiss c S, C S > 0 such ha Su 2 (Γ) C S u 2 (Γ) u 2 (Γ), (2.5) Su, u Γ c S u 2 2 (Γ 0 ) u 2 (Γ 0 ) (2.6) For ay displaceme u ad for ay boudary racio σ(u) defied o Ω he followig oaios are frequely used i his chaper. u = u + u ad σ(u) = σ (u) + σ (u) (2.7) We eed he followig fucio spaces V := [ 2 (Σ)] d = 2 (Σ) := {u 2 (Γ); supp(u) Σ} V := {u 2 (Σ) : u = 0 o Γ D } W := [ 2 (ΓC )] d = 2 (ΓC ) M := [ 2 (ΓC )] d = 2 (ΓC ) where Σ := Γ N, Γ N = 5

16 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy Le K be a closed covex ad oempy subse of V, o model he opeeraio codiio o he coac boudary, where g 0 is a give gap fucio. K := {u 2 (Σ) : u g o } We assume ha F L 2 ( ), 2 (Γ N ) ad g 2 ( ). We defie he D N (Dirichle-o-Neuma) mappig u Γ σ(u), Sice S is he D N mappig, here holds σ Su ΓC, σ Su ΓC (2.8) where u, σ, σ solve (2.) i he disribuioal sese. We defie he space of Lagrage mulipliers by M = M M (2.9) where M := {µ 2 (ΓC ) : µ, v ΓC 0, v 2 (ΓC ) wih v 0 a.e o } ad M (F) := {µ L 2 ( ) : µ F a.e o } are he ses of ormal ad ageial Lagrage mulipliers. The classical formulaio (2.) ca be rewrie i a weak sece as a saddle poi problem as follows: Fid (u, λ) V M such ha ere Su, v Σ + b(λ, v) =, v ΓN v V (2.0a) b(µ λ, u) g, µ λ ΓC µ M. (2.0b) b(µ, v) := µ, v ΓC + µ, v ΓC (2.) where he oaio, ΓC represes he dualiy pairig bewee 2 ( ) ad 2 ( ). Noe λ = σ(u) ΓC 0 sice σ(u) 0 o i (2.). Aoher classical weak formulaio of he problem (2.) is he primal variaioal iequaliy: Fid u K such ha Su, v u Σ + j(v) j(u), v u ΓN v K (2.2) 6

17 2.2 Exisece ad uiqueess of he soluio where j(u) = F u ds (2.3) is he fricio fucioal. As a miimizaio problem, i reads J(u) J(v) v K (2.4) where J(v) = 2 Sv, v Σ + j(v), v ΓN (2.5) 2.2 Exisece ad uiqueess of he soluio I his secio, we sudy he exisece ad uiqueess of he soluio of he mixed formulaio Theorem 2.. There exiss a uique soluio of problem (2.0). Proof. We kow from [9] ha he miimizaio problem (2.4) ad he variaioal problem (2.2) are equivale. Accordig o [24], K is a closed, covex se, ad J : K R is coiuous, covex ad coercive, i.e. J(v), for v K ad v 2. (Σ) This implies he exisece of a miimizer u of (2.4). Furhermore from [24] J is sricly covex. Therefore he miimizer u is uique. Noe ha j(v) = F v ds = sup µ, v ΓC. (2.6) µ M (F) We defie he Lagrage fucioal L(v, µ, µ ) := 2 Sv, v L(v) + µ, v g ΓC + µ, v ΓC (2.7) Noe ha by (2.6) we ge J(u) = if sup v V (µ,µ ) M M (F) L(v, µ, µ ) = L(v, λ, λ ). (2.8) Thus, for ay saddle poi (u, λ, λ ) V M M (F) of L, u is a miimizer of J. 7

18 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy Sice M (F) is bouded, he exisece of a uique saddle poi is guaraeed, if here exiss a cosa β > 0 such ha where β µ 2 ( ) sup µ, γ(v) c (2.9) v V, v = 2 (Σ) γ c (v) = v ΓC The above codiio (if-sup codiio ) follows from he closed rage heorem ad he surjeciviy of γ c, see [50]. We coclude ha here exiss a uique saddle poi (u, λ, λ ) V M M (F) of L, which is equivalely characerized by he mixed variaioal formulaio (2.0). 2.3 Discreizaio of he Lamé problem Le T hp deoe a pariio of Γ N io disjoi sraigh lie segmes I, such ha all corers of Γ N ad all ed pois Γ N, Γ D Γ N are odes of T hp. For simpliciy we assume meas( ) > 0 ad Γ D =. Furhermore we defie he se of Gauss-Lobao pois G I,hp o each eleme I T hp of correspodig polyomial degree p I ad se G hp := I Thp G I,hp, where p = (p I ) I Thp associaes o each eleme of T hp a polyomial degree p I. We iroduce he space of coiuous piecewise polyomials for he discreizaio of u: V hp := {u hp 2 (Σ) : I T hp, u hp I [P pi (I)] 2, u hp = 0 o Γ D } 2 (Γ) ad he space of piecewise polyomial discree boudary racios W hp := {φ 2 (Σ) : I Th, φ I [P pi (I)] 2 } 2 (Γ) The space V hp is spaed by he 2-d odal basis {φ i e k, i =,..., N V, k =, 2}, where e k deoes he k-h ui vecor, φ i he scalar Gauss-Lobao Lagrage basis fucio associaed wih he ode i ad N V he oal umber of he odes. We deoe by N C he se of all odes o he coac boudary. Furhermore we defie he space of discree vecorial Lagrage mulipliers by M hp, which is spaed by {ψ i e k, i =,..., N C, k =, 2}. The dual or biorhogoal basis fucios ψ i saisfy he orhogoaliy relaios ψ i φ j ds = δ ij φ j ds. (2.20) I order o obai a well-defied ormal we iroduce he averaged ormal a he ode i G hp as follows. 8

19 2.3 Discreizaio of he Lamé problem Le us defie by E i for i N C all surface elemes of he mesh coaiig ode i. E i = {e i T hp : e i, i N C } (2.2) The he ormal i ca be defied via i := ei, e i E i ei e i E i i N C (2.22) where ei is he ui ormal vecor of he surface edge e i. We ca he express he discree fucio u hp V hp for i dim V hp by u hp := i (u i i + u i )φ i. (2.23) The ormal ad ageial par of he discree fucio u hp are give by u hp := i u i φ i, u hp = i u i φ i, (2.24) where u i ad u i are he ormal ad ageial compoe of he odal value u i, give by u i := u i i, u i := u i u i i. (2.25) We defie i he same way he discree Lagrage muliplier for i dim M hp (F) as λ hp := i (λ i i + λ i )ψ i (2.26) For he fricioless coac problem he ageial compoe λ i equals 0. We iroduce he subse V hp := {vhp V hp : v hp 0}. (2.27) As i [4] we defie he space for he discree Lagrage muliplier by M hp (F) := {µ hp spa{ψ i } N C i= : µhp, v hp F, v hp h, v hp V hp }, (2.28) where he absolue value of he fucio v hp x by (see.[4]) v hp h := i We defie he weighed gap vecor for i dim M hp (F) by is give for i dim M hp (F) ad v i φ i (x). (2.29) g i := D i gψ i (x) ds, (2.30) where D i = φ i (x) ds. The spaces V hp ad M hp(f) ca be rewrie as follows; cf. [[4], Lemma 2.3] for he h-versio, i.e. p =. 9

20 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy Lemma 2.4 ([8], Lemma 6.). The space V hp i (2.27) ca be rewrie as V hp := {vhp = v i φ i V hp : v i 0, i N C } (2.3) i N C ad he space i (2.28) is equivale o M hp (F) := {µ hp = i N C µ i ψ i : µ i 0, µ i F, i N C } (2.32) Sice a explici represeaio of V is o kow, we eed o approximae he Seklov-Poicaré operaor. Le i hp : W hp 2 (Γ) ad j hp : V hp 2 (Γ) deoe he caoical imbeddig he dual i hp ad j hp The approximaio S hp of he Poicaré-Seklov operaor is give by S hp := W + (K + 2 )i hp(i hp V i hp) i hp (K + 2 ) (2.33) We ca rewrie he space M hp (F) as M hp (F) := M,hp M,hp (F), where M,hp ad M,hp are he ses of ormal ad ageial discree Lagrage muliplier, respecively. The discreized versio of (2.0) reads as: Fid u hp V hp ad λ hp = (λ hp, λ hp ) M hp (F) := M,hp M,hp (F) S hp u hp, v hp Σ + b(λ hp, v hp ) = L(v hp ) v hp V hp (2.34a) b(µ hp λ hp, u hp ) g, µ hp λ hp µ hp M hp (F). (2.34b) Lemma 2.5. [9] Le Γ := Ω R d be he boudary of a Lipschiz domai Ω ad arbirary Γ 0 Γ.The he approximaio of he Seklov-Poicaré operaor S hp : 2 (Γ) 2 (Γ) is coiuous ad 2 (Γ 0 )-ellipic, i.e. here exiss c Shp, C Shp > 0 such ha S hp u 2 (Γ) C S hp u 2 (Γ) u 2 (Γ), (2.35) S hp u, u Γ c Shp u 2 2 (Γ 0 ) u 2 (Γ 0 ). (2.36) We defie he operaor E hp = S S hp which represes he error i he approximaio of he Seklov-Poicaré operaor. Lemma 2.6. [9] The operaor E hp is bouded, i.e. here exiss C Ehp > 0 such ha E hp u 2 (Γ) C E hp u 2 (Γ) u 2 (Γ), (2.37) 0

21 2.3 Discreizaio of he Lamé problem ad here exiss a cosa C 0 > 0, such ha E hp u 2 C 0 if V (K + (Γ) Φ W hp 2 )u Φ 2 (Γ) u 2 (Γ). (2.38) Exedig he approach i [8] for he scalar case o he vecor case we obai he followig resul. Theorem 2.2. There exiss exacly oe soluio o he discree mixed formulaio (2.34). Proof. Uiqueess: Le (u hp, λhp ) ad (uhp formulaio (2.34). The we have S hp (u hp uhp 2 ), uhp uhp 2, λhp 2 2 Σ + b(λ hp Choosig µ hp = λhp 2 ad µ hp 2 = λhp i (2.34b) we ge Usig Lemma 2.5 we obai c Shp u hp uhp 2 2 S hp (u hp 2 (Σ) ) wo soluios of he discree mixed λhp 2, uhp uhp 2 ) = 0 (2.39) b(λ hp λhp 2, uhp uhp 2 ) 0 (2.40) uhp 2 ), uhp uhp Cosequely he firs argume u hp is uique. Sice u hp is uique we have for all v hp V hp 2 Σ + b(λ hp λhp 2, uhp uhp 2 ) = 0. (2.4) 0 = b(λ hp λhp 2, vhp ). (2.42) Usig he liear combiaios of λ hp,λhp 2 ad v hp ad he biorhogoaliy, we obai he relaio 0 = p G hp [(λ,p λ 2,p )v p + (λ,p λ 2,p )v p ]D p. (2.43) where D p = φ p (x) ds. Due o he arbirary choice of v p ad v p, choosig v p = 0 i (2.43) we ge (λ,p λ 2,p )v p = 0 ad λ,p = λ 2,p We choose ow v p = 0, we obai λ,p = λ 2,p, cosequely he secod argume λ hp is uique. Exisece: The iequaliy (2.34b) is obviously equivale o he followig codiios: λ hp M,hp, µ hp λ hp, u hp g 0, µ hp M,hp, (2.44) λ hp M,hp, µ hp λ hp, u hp 0, µ hp M,hp. (2.45)

22 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy The iequaliies ca be replaced by projecios P M,hp ad P M,hp respecively, where λ hp = P M,hp (λ hp + r(u hp g)), λ hp = P M,hp (λ hp + ru hp ) (2.46) ere he maps P M,hp ad P M,hp sad for he L 2 projecios oo M,hp ad M,hp, respecively, ad r > 0 is a arbirary parameer. The discree mixed formulaio leads o a fixed pois formulaio T : M,hp M,hp (F) M,hp M,hp (F) ( ) (λ hp, λ hp ) P M,hp (λ hp + r(u hp g)), P M,hp (λ hp + ru hp ) (2.47) where T is he fixed poi operaor,which defies he fixed poi ieraio ( (λ hp ) k+, (λ hp ) k+) ( = T (λ hp ) k, (λ hp ) k). (2.48) From Lemma 2.5 we have u hp uhp 2 2 L 2 ( ) uhp uhp 2 2 c 2 S (Σ) hp S hp (u hp Usig he oaio δu hp = u hp uhp 2 ad δλ hp = λ hp λhp 2 uhp 2 ), uhp uhp 2 Σ T (λ hp ) T (λhp 2 ) 2 L 2 ( ) δλhp + rδu hp ΓC 2 L 2 ( ) + δλhp + rδu hp 2 L 2 ( ) (2.49) δλ hp 2 L 2 ( ) +2rb(δλhp, δu hp ) + r 2 δu hp 2 L 2 ( ) δλ hp 2 L 2 ( ) 2r S hpδu hp, δu hp Σ + r 2 δu hp 2 L 2 ( ) δλ hp 2 L 2 ( ) 2rc S hp δu hp 2 L 2 ( ) +r2 δu hp 2 L 2 ( ) δλ hp 2 L 2 ( ) ( 2rc S hp β 2 + r 2 β 2 ) where β = δλhp 2 L 2 ( ) ad T is sric coracio for 0 < r < 2c δu hp 2 Shp. By he Baach L 2 ( ) fixed poi heorem exiss a λ hp = (λ hp, λ hp ) which saisfies (2.47). For ay give λ hp, problem (2.34a) reduces o a liear, fiie dimeioal problem. ece, he uiqueess resul of u hp implies he exisece of a u hp (λ hp ). 2.4 Semi-smooh Newo mehod ad algebraic represeaio For he soluio of he discree mixed formulaio (2.34) we describe a semi-smooh Newo approach which is equivale o a acive se sraegy. 2

23 2.4 Semi-smooh Newo mehod ad algebraic represeaio Lemma 2.7. The variaioal iequaliy cosrai (2.34b) is equivale o he decoupled poiwise o-peeraio codiios ad he fricio codiios u i g i, λ i 0, λ i (u i g i ) = 0 (2.50) λ i F i λ i < F i u i = 0 (2.5) λ i = F i α R : λ i = αu i for all i N C where F i is he fricio a he ode defied by F i := Fφ i ds (2.52) Proof. see [4] ad [8] The pealized Fischer-Burmeiser o liear complemeariy fucio preseed i [8] is defied by ( ) ϕ µ (u hp ΓC, λ hp ) = µ λ i + (g i u i ) λ 2i + (g i u i ) 2 wih µ (0, ] ad i N C. + ( µ)max {0, λ i }max {0, (g i u i )} (2.53) We defie he oliear complemeariy fucio (NCF) for Tresca fricio as C T (u hp, λ hp ) = max{f i, λ i + αu i }λ i F i (λ i + α u i ) (2.54) for ay posiive parameer α > 0. Theorem 2.3. The pair (u hp, λ hp ) saisfies he fricioal coac codiios (2.5) if ad oly if C T (u hp, λ hp ) = 0. (2.55) The decoupled poiwise o peeraio codiio is equivale o ϕ µ (u hp ΓC, λ hp ) = 0. (2.56) Proof. For he firs equivalece see[[4],heoerem5.]. For he secod equivalece we kow ha he NCF-fucio saisfies ϕ µ (u hp ΓC, λ hp ) = 0 u i g i, λ i 0, λ i (u i g i ) = 0. (2.57) 3

24 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy Lemma 2.8. The discree problem (2.34) is equivale o solvig he oliear sysem Au hp Dλ hp f 0 = F (u hp, λ hp ) = ϕ µ (u hp ΓC, λ hp ). (2.58) C T (u hp, λ hp ) Proof. Au hp Dλ hp f = 0 is he marix represeaio of he firs equaio of he discree problem. By Theorem (2.3) he codiios ϕ µ (u hp ΓC, λ hp ) = 0 ad C T (u hp, λ hp ) = 0 are equivale o he decoupled poiwise o-peeraio codiio (2.50) ad he fricioal codiio (2.5), respecively, which are equivale o (2.34b) by lemma 2.7. The geeralized Newo s mehod for he soluio of he oliear sysem (2.58) ca be defied as (u k, λ k ) = (u k, λ k ) F (u k, λ k ) F (u k, λ k ), k =, 2,... (2.59) Sice F is o differeiable everywhere, ad herefore he Jacobia marix F (u k, λ k ) does o exis everywhere, oe has o choose a suiable approximaio k F (u k, λ k ) ad solve he equaio (u k, λ k ) = (u k, λ k ) k F (uk, λ k ), k =, 2,... (2.60) which is equivale o k ( δu k δλ k ) = F (u k, λ k ). (2.6) ere k is he Clarke subdiffereial of F a (u k, λ k ) defied by A D ϕ k = µ(u k,λ k ) ϕ µ(u k,λ k ). (2.62) u C T (u k,λ k ) u We obai (δu k, δλ k ) by solvig he sysem λ C T (u k,λ k ) λ where A ϕ µ(u k,λ k ) u C T (u k,λ k ) u D ϕ µ(u k,λ k ) λ C T (u k,λ k ) λ ( δu k δλ k F (u, λ) = Au Dλ f F 2 (u, λ) = ϕ µ (u, λ) F 3 (u, λ) = C T (u, λ). ) F (uk, λ k ) = F 2 (u k, λ k ) (2.63) F 3 (u k, λ k ) 4

25 2.4 Semi-smooh Newo mehod ad algebraic represeaio Le S be he se of all odes o, ad N all remaiig odes. Now, he algebraic represeaio is give by he liear sysem A N N A N S 0 A SN A SS D S ϕ 0 µ(u k,λ k ) λ S 0 ϕ µ(u k,λ k ) u S C T (u k,λ k ) C T (u k,λ k ) u S λ S δu k N δu k S δλ k S F,N (u k, λ k ) F,S (u k, λ k ) = F 2 (u k, λ k ). (2.64) F 3 (u k, λ k ) We remark ha he zero block i he couplig marix D refers o he lies associaed wih he odes i se N. We have ϕ µ (u, λ) = ϕ µ (N.u ΓC, N.λ) = ϕ µ (u,s, λ,s ) (2.65) ad C T (u, λ) = C T (T.u ΓC, T.λ) = C T (u,s, λ,s ), (2.66) where N, T are he algebraic represeaios of he ormal,respecively, ageial vecor. Alogeher, A N N A N S 0 A SN A SS D S ϕ µ(u 0 k,s,λk,s ) u,s 0 C T (u k,s,λk,s ) u,s ϕ µ(u k,s,λk,s ) λ,s C T (u k,s,λk,s ) λ,s δuk N δu k S δλ k S F,N (u k, λ k ) F,S (u k, λ k ) = F 2 (u k, λ k ) (2.67) F 3 (u k, λ k ) 5

26 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy wih ϕ µ (u, λ) = u ϕ µ (u, λ) = λ C T u C T λ µ, if λ = u g = 0 ( u µ g )+( µ)λ, if λ >0 ad u >g λ 2 +(u g) 2 µ ( ) u g λ 2 +(u g) 2, oherwise µ, if λ = u g = 0 ( λ µ )+( µ) (u λ 2 g), if λ >0 ad u >g +(u g) 2 ( ) λ µ, oherwise λ 2 +(u g) 2 F α, if λ + αu F (u, λ) = α(λ F), if λ + αu > F α(λ F), oherwise 0, if λ + αu F (u, λ) = λ + λ + αu F, if λ + αu > F λ + λ + αu F, oherwise. If a fucio Ψ : R R is oegaive ad Ψ(x) = 0 if ad oly if x solves he NCF, he Ψ(x) is called a meri fucio of he NCF-fucio. ece fidig a soluio of he NCF is equivale o fidig a global miimum of he ucosraied miimizaio mi x R Ψ(x) wih opimal value zero. We defie he oegaive meri fucio Ψ(u, λ) = 2 F (u, λ)2 (2.68) Algorihm 2.. (Semi-smooh Newo algorihm). Iiialisaio: Choose iiial soluio u 0, λ 0 R, ρ > 0, β (0, ), σ (0, 2 ), p > 2 2. For k = 0,, 2,... do a) Termiaio Crierio If Ψ(u k, λ k ) < ol or Ψ(u k, λ k ) < ol he sop b) Search Direcio Calculaio s. Compue subdiffereial k F (u k, λ k ) ad fid d k = (d k u, d λ ) k R 2 k d k = F (u k, λ k ). (2.69) 6

27 2.4 Semi-smooh Newo mehod ad algebraic represeaio If (2.69) o solvable or if he desce codiio Ψ(u k, λ k )d k ρ d k p (2.70) is o saisfied, se d k := Ψ(u k, λ k ). (c) Lie Search Compue search legh k := maxβ l : l = 0,, 2,... s. Ψ(u k + k d k u, λ k + k d k λ ) Ψ(uk, λ k ) + σ k Ψ(u k, λ k )d k. (d) Updae Updae he soluio vecors ad goo sep 2. u k+ = u k + k d k u, λ k+ = λ k + k d k λ. Theorem 2.4. The semi-smooh Newo algorihm (2.7) coverges locally superliear, i.e. lim k e k+ e k = 0. (u k+, λ k+) T = ( u k, λ k) T k F (uk, λ k ) (2.7) wih k F (u k, λ k ) a subgradie of F a ( u k, λ k) T. I he fricioless case, i.e. F = 0 ad C T (u, λ) := λ, SSN coverges locally Q- e quadraic, i.e. lim k+ k = cos. e k 2 Proof. See[8],[0],[53] Lemma 2.9. (Galerki orhogoaliy) Le u V be a exac soluio of he coiuous problem (2.0) ad u hp V hp be he soluio of he discree problem (2.34). There holds Su S hp u hp, v hp Σ + b(λ λ hp, v hp ) = 0 v hp V hp (2.72) Proof. We choose v hp V hp V i (2.0) ad subrac (2.0) from he discree formulaio (2.34). Le u V ad u hp V hp. From [5], we defie he followig oaio ψ := V (K + 2 )u ψ hp := i hp V hp i hp (K + 2 )uhp (2.73) ψ hp := V (K + 2 )uhp. 7

28 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy Lemma 2.0. [5] Le u V be a exac soluio of he coiuous problem (2.0) ad u hp V hp be he soluio of he discree problem. There holds u u hp 2 W + ψ ψ hp 2 V = Su S hp u hp, u u hp + V (ψ h ψ hp), ψ ψ hp (2.74) where u u hp W := W (u u hp ), u u hp 2 ψ ψ hp V := V (ψ ψ hp ), ψ ψ hp 2 Lemma 2.. [3, 52, 9] Assume ha u +ν (Σ) wih ν [0, 2 ] ad ψ ν (Γ). There exiss a cosa C > 0 such ha he followig approximaio properies hold: if u v hp v hp C V hp 2 (Σ) if ψ φ φ W 2 C hp (Γ) ( h p ( h p ) 2 +ν u +ν (Σ) ) 2 +ν ψ ν (Γ) 2.5 A priori error esimae The a priori error esimae is based o he use of he discree if-sup codiio. I his secio we assume ha he discree if-sup codiio is valid. Assumpio 2.. There exiss a cosa β hp > 0 depedig o h ad p, such ha β hp if µ hp M hp (F) b(µ hp, v hp ) sup v hp V hp v hp 2 (Σ) µhp 2 ( ) (2.75) Lemma 2.2. Le (u, λ) V M be he soluio of he mixed formulaio (2.0)ad (u hp, λ hp ) V hp M hp (F) be he soluio of he discree problem (2.34). The here exiss a cosa C > 0 idepede of h ad p such ha for all µ hp M hp (F) here holds ( (CS λ λ hp 2 C + C Ehp ) u u hp ( ) β + C 0 hp 2 ψ φ (Σ) β 2 hp (Γ) +( + β hp ) λ µ hp 2 ( ) ) (2.76) Proof. I his proof we use he oaio i[9]. We use he followig ideiy Su S hp u hp = S(u u hp ) + E hp u hp = S(u u hp ) + E hp (u hp u) + E hp u 8

29 2.5 A priori error esimae where E hp = S S hp. From Lemma 2.5 ad Lemma 2.6 S hp ad E hp are coiuous. Therefore for all φ W hp ( ) Su S hp u hp, v hp (C S + C Ehp ) u u hp + E 2 hpu 2 v hp (Σ) 2 (Σ) 2 (Σ) ( ) (C S + C Ehp ) u u hp +C 2 0 ψ φ (Σ) 2 (Γ) v hp. 2 (Σ) (2.77) Usig he Galerki orhogoaliy lemma 2.9 ad (2.77) we ge λ hp µ hp, V hp = Su S hp u hp, v hp + λ µ hp, V hp ( ) (C S + C Ehp ) u u hp +C 2 0 ψ φ (Σ) 2 v hp (Γ) 2 (Σ) + λ µ hp 2 ( ) vhp 2 (Σ). (2.78) Usig he discree if-sup codiio (2.75) we obai ( λ hp µ hp 2 C(β hp) (C S + C Ehp ) u u hp +C ( ) 2 0 ψ φ (Σ) 2 (Γ) (2.79) ) + λ µ hp 2 ( ) The riagle iequaliy ad (2.79) yield he asserio. Theorem 2.5. Le (u, λ) V M be he soluio of mixed formulaio (2.0) such ha u +ν (Σ) wih ν [0, 2 ] ad (uhp, λ hp ) V hp M hp (F) be he soluio of he discree problem (2.34). We assume ha Su L 2 (Γ) ad λ ν ( )+ λ ν ( )+ F L2 ( ) u ν+ (Σ) (2.80) he here exiss C > 0 idepede of he polyomial degrees p ad of he mesh size h such ha wih (2.73) u u h 2 (Σ) + ψ ψhp 2 (Γ) + λ λhp 2 ( ) Cβ hp ( ) h mi{ 4,ν} u p +ν (Σ) (2.8) We have i paricular for u 3 2 (Σ) u u h 2 (Σ) + λ λhp 2 ( ) Cβ hp h 4 p 4 u 3 2 (Σ) (2.82) 9

30 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy Proof. We choose K hp := {u hp V hp : u hp (x) g(x), x G hp } which is a covex, closed subse of V hp, ad we defie he Lagrage ierpolaio operaor I hp o he se of Gauss-Lobao pois. Le v hp := I hp u K hp. Wih [[3],Theorems 4.2 ad 4.5] we have he followig approximaio properies: There exis cosas C, C > 0 idepede of u ad h, p such ha u v hp L2 (Γ) Ch ν p ν u ν (Γ), ν > 2 u v hp (Γ) C h ν p ν u ν (Γ), ν > (2.83) From Lemma 2.0 we obai u u hp 2 W + ψ ψ hp 2 V = Su S hp u hp, u u hp + V (ψ hp ψhp ), ψ ψ hp. Sice we have we obai Su S hp u hp = S(u u hp ) + E hp u hp = S(u u hp ) + E hp (u hp u) + E hp u, u u hp 2 W + ψ ψ hp 2 V = Su S hp u hp, u u hp + V (ψ hp ψhp ), ψ ψ hp Usig (2.34) ad he defiiio of S h we ge = S(u u hp ), u v hp + Su hp, u hp v hp + Su, v hp u hp + E hp u hp, u u hp + V (ψ hp ψhp ), ψ ψ hp. (2.84) Su hp, u hp v hp = E hp u hp, u hp v hp b(λ hp, u hp v hp ) +, u hp v hp (2.85) Usig (2.0), (2.84) ad (2.85), we obai C W u u hp 2 +C V ψ ψ hp 2 S(u u hp ), u v hp + Su, v hp u hp 2 (Σ) 2 (Γ) + E hp u hp, u v hp b(λ hp, u hp v hp ) +, u hp v hp Su, u v b(λ, u v) +, u v + V (ψ hp ψhp ), ψ ψ hp S(u u hp ), u v hp + E hp u hp, u v hp + Su, u hp v + Su, u v hp + b(λ, v u hp ) b(λ hp, u v hp ) + V (ψ hp ψhp ), ψ ψ hp + λ λ hp, u u hp. (2.86) 20

31 2.5 A priori error esimae For every v K, where C W ad C V are he ellipiciy cosas of V, W. As i [47], choosig v such ha v := g + if(u hp g hp, 0) v Γ N := u hp Γ N v Σ := u hp Σ (2.87) where g hp is he ierpolaio of g, we ca wrie v u hp = { 0 o ΓN (g g hp + δ u ) o wih δ u := if(0, g hp u hp ). The, we have v u hp L2 (Γ) g g hp L2 ( )+ δ u L2 ( ) (2.88) Due o u hp K hp we have I hp if(g u hp, 0) = 0 o, ad wih [[3],Theorems 4.2] we have By ierpolaio we have For he firs erm i (2.88) we have δ u 0 L2 ( ) g hp u hp L2 ( ) C h p δ u 0 ( ) C h p g hp u hp ( ). δ u 0 L2 ( ) C h 2 p 2 ghp u hp 2 (ΓC ). (2.89) g g hp L2 ( ) C h 2 p 2 g 2 (ΓC ). (2.90) From [47] we kow ha here exiss a cosa C 2 > 0, idepede of h,p, such ha u hp 2 (ΓC ) C 2 u 2 (ΓC ). (2.9) Therefore we have v u hp L2 (Γ) C 3 h 2 p 2 Fially, here exis a cosa C such ha ( ) g + u 2 (ΓC ) 2. (2.92) ( ) v u hp L2 (Γ) Ch 2 p 2 u +ν (Γ) (2.93) 2

32 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy wih ν [0, 2 ]. For he firs erm i (2.86), employig Cauchy-Schwarz ad Youg s iequaliy,we obai S(u u hp ), u v hp 2ɛ u vhp 2 + ɛ 2 (Σ) 2 u uhp 2 2 (Σ) h 2ν+ 2ɛ p 2ν+ u 2 +ν (Γ) + ɛ 2 u uhp 2. 2 (Σ) (2.94) Now we esimae he secod erm i (2.86). Accordig o Lemma2.ad Lemma2.6, we ge E hp u hp, u v hp = E hp (u hp u + u, u v hp 2ɛ u vhp 2 + ɛ 2 (Σ) 2 E hp(u hp u) 2 2 (Σ) + 2 E hp(u) (Σ) 2 u vhp 2 2 (Σ) ɛ u u hp 2 + h2ν+ +h2ν+ 2 (Σ) p 2ν+ u 2 +ν (Γ) p 2ν+ ψ 2 ν (Γ) (2.95) We ca easily esimae he hird erm i (2.86). From (2.83) ad (2.93), we obai Su, u hp v + Su, u v hp Su L2 (Σ) We esimae ow he erm ( ) u hp v L2 (Σ)+ u v hp L2 (Σ) Ch 2 p 2 u +ν (Γ). (2.96) b(λ, v u hp ) = λ, v u hp + λ, v u hp (2.97) We sar wih he firs erm i (2.97),usig (2.93), we ge λ, v u hp = λ (v u hp ) ds λ L2 ( ) v u hp L2 (Γ) Due o he approximaio (2.87) he secod erm i (2.97) is We esimae he erm Ch 2 p 2 u 2 +ν (Γ) (2.98) λ, v u hp = 0. (2.99) b(λ hp, u v hp ) = λ hp, u v hp + λ hp, u v hp (2.00) 22

33 2.5 A priori error esimae Employig Cauchy Schwarz ad Youg s iequaliy, we have λ hp, u v hp = (λ hp λ )(u v hp ) ds + λ (u v hp ) ds λ hp ɛ 2 λhp Similarly o (2.0), we ge λ hp, u v hp ɛ 2 λhp Combiig (2.0) ad (3.57), we obai λ 2 ( ) u v hp 2 (ΓC ) + λ L2 ( ) u v hp L2 ( ) λ 2 + h 2ν+ +hν+ 2 ( ) 2ɛ p 2ν+ u 2 +ν (Γ) p ν+ u 2 +ν (Γ) (2.0) λ 2 + h 2ν+ +hν+ 2 ( ) 2ɛ p 2ν+ u 2 +ν (Γ) p ν+ u 2 (2.02) +ν (Γ) b(λ hp, u v hp ) ɛ 2 λ λhp 2 + h 2ν+ +hν+ ( ) 2ɛ p 2ν+ u 2 +ν (Γ) p ν+ u 2 +ν (Γ) (2.03) ad we have, see[9] V (ψ hp ψhp ), ψ ψ hp (C K + 2 ) u uhp 2 (Σ) ψ φ 2 (Γ) + C V ψ ψ hp 2 (Γ) ψ φ 2 (Γ) 2ɛ (C K + 2 )2 ψ φ 2 + ɛ 2 (Γ) 2 u uhp 2 2 (Σ) + 2ɛ C2 V ψ φ 2 + ɛ 2 (Γ) 2 ψ ψhp 2 2 (Γ) φ W hp (2.04) Fially we esimae he erm λ λ hp, u u h = λ λ hp, u u hp + λ λ hp, u u hp. (2.05) Choosig µ = λ ad µ = 0, µ = 2λ i he iequaliy (2.0b) ad µ hp = λ hp ad = 0, µ hp = 2λ hp i he discree iequaliy (2.34b), we obai he complemeary µ hp codiios λ (u g) ds = λ hp (u hp g) ds = 0 (2.06) Le π Mhp be he L 2 -projecio operaor mappig M hp defied by π Mhp = { πm,hp : L 2( ) M,hp π M,hp : L 2 ( ) M,hp (2.07) which saisfies (λ π M,hp λ)µ ds = 0, (λ π M,hp λ)µ = 0 ds µ = (µ, µ ) M hp (2.08) 23

34 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy ad λ π M,hp λ L2 ( ) C ( ) h ν λ p ν(γc) (2.09) for ay real umber ν 0. We obai ad π M,hp λ λ 2 ( ) = λ π M,hp λ L2 ( ) C C sup 0 v 2 ( ) sup 0 v 2 ( ) ( ) h ν λ p ν(γc) (2.0) π M,hp λ λ, v π M,hp v v 2 (ΓC ) π M,hp λ λ L2 ( ) v π M,hp v L2 ( ) v 2 (ΓC ) ( ) h 2 πm,hp p λ λ L2 ( ) (2.) π M,hp λ λ 2 ( ) C ( h p ) 2 πm,hp λ λ L2 ( ) (2.2) Accordig o he coiuiy of he Direchle-o-Neuma operaor, we have λ 2 (ΓC ) + λ 2 (ΓC ) C u 3 2 (Σ). (2.3) We choose µ hp = π M,hp λ ad µ hp = π M,hp λ, as a cosequece we ge ( ) h if λ µ hp µ M 2 C 2 +ν u ( ) p ν+ (Σ) ( ) if λ µ hp h µ M (F) C 2 +ν u 2 ( ) p ν+ (Σ) (2.4) We sar wih he firs erm i (2.05). Accordig o (2.06), we have λ λ hp, u hp u = λ (u hp g) ds + λ hp (u g) ds (2.5) The ormal compoe ca be wrie as u hp := i N C α i φ i, µ hp := i N C β i ψ i, λ hp := i N C λ i ψ i. (2.6) 24

35 2.5 A priori error esimae For µ hp M,hp we have µ hp (u hp g) ds = i α i µ i D i µ hp, g = i = i α i µ i D i i α i µ i D i i µ i gψ i ds µ i g i D i ds = i (α i g i )µ i D i 0 (2.7) Choosig µ hp = π M,hp λ, from (2.7) we ge λ (u hp g) ds = (λ µ hp )(u hp g) ds + µ hp (u hp g) ds Γ C (λ µ hp )(u hp g) ds Γ C (λ π M,hp λ )(u hp g) ds Γ C = (λ π M,hp λ )((u hp g) (u g)) ds Γ C + (λ π M,hp λ )(u g π M,hp (u g)) ds Γ C = (λ π M,hp λ )(u hp u ) ds Γ C + (λ π M,hp λ )(u π M,hp u + π M,hp g g) ds (2.8) 25

36 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy ad we obai λ (u hp g) ds π M,hp λ λ 2 ( ) uhp u 2 (ΓC ) + π M,hp λ λ L2 ( ) u π M,hp u L2 ( ) + π M,hp λ λ L2 ( ) g π M,hp g L2 ( ) C + ( h p ( (h ) 2 +ν λ ν ( ) u u hp 2 (Σ) p ) 2ν ( ) h 2ν λ ν(γc) u +ν (Σ) + λ p ν ( ) g ν ( ) ɛ u u hp 2 2 (Σ)+C ( ) h 2ν u 2 p. +ν (Σ) (2.9) ) We cosider he hp-lagrage ierpolaio operaor I hp defied o he Gauss-Lobao pois mappig oo V hp. The liear combiaio of I hp u ca be wrie as I hp u := i a i φ i wih a i g i (2.20) where g i := gψ i ds, D i = φ i ds > 0 (2.2) D i Usig he liear combiaio i (2.6) ad he biorhogoaliy codiio, we ge λ hp (I hp u g) ds = i a i λ i D i λ hp, g = i = i a i λ i D i i a p λ i D i i λ i gψ i ds λ i g i D i ds = i (a i g i )λ i D i 0. (2.22) 26

37 2.5 A priori error esimae Accordig o (2.22) λ hp (u g) ds = = λ hp λ hp ((u g) (I hp u g)) ds + λ hp (I hp u g) ds (u I hp u ) ds (λ hp λ )(u I hp u ) ds + λ (u I hp u ) ds λ hp λ 2 ( ) u I hp u 2 (ΓC ) + λ L2 ( ) u I hp u L2 ( ) ɛ λ hp λ 2 hν+ +C 2 ( ) p ν+ u 2 ν+ (2.23) (Σ) We ow esimae he erm he secod erm i (2.05) From (2.34b), we have λ hp M,hp, µ hp λ hp, u hp 0, µ hp M,hp (2.24) Choosig µ hp = π M,hp λ, from (2.24) we ge λ λ hp, u hp u = λ µ hp, u hp u + µ hp λ µ hp, u hp u µ hp λ hp, u λ µ hp, u hp λ hp, u hp u u + λ µ hp, u + λ hp λ, u. (2.25) The esimae of he firs erm i (2.25) gives λ µ hp, u hp u = (λ µ hp )(u hp u ) ds 2ɛ λ µ hp 2 + ɛ 2 ( ) 2 uhp u 2 2 ( ) C h 2 +ν p 2 +ν u 2 ν+ (Σ) + ɛ 2 uhp u 2 2 ( ). (2.26) 27

38 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy We ow esimae he secod erm i (2.25) λ µ hp, u = (λ µ hp )u ds λ µ hp 2 ( ) u 2 (ΓC ) C h 2 +ν p 2 +ν u 2 ν+ (Σ). (2.27) The liear combiaio of I hp u ad λ hp I hp u := a i φ i, i N C ca be wrie as := λ hp i N C λ i ψ i. (2.28) Usig he liear combiaio i (2.28), he biorhogoaliy codiio ad he fac ha λ i F, we ge λ hp I hp u ds = i a i λ i D i F i a i D i F a i φ i ds i F I hp u ds. (2.29) We esimae ow he las erm i (2.25), usig he codio λ u F u = 0 ad (3.08): (λ hp λ )u ds = (λ hp λ )(u I hp u ) ds + (λ hp λ )I hp u ds Γ C + λ u ds F u ds Γ C (λ hp λ )(u I hp u ) ds + λ (u I hp u ) ds Γ C + F ( I hp u u ) ds Γ C (λ hp λ )(u I hp u ) ds + λ (u I hp u ) ds Γ C + F ( I hp u u ) ds λ hp λ 2 ( ) u I hp u 2 (ΓC ) + λ L2 ( ) u I hp u 2 (ΓC ) + F L2 ( ) u I hp u L2 ( ) ɛ λ hp λ 2 2 ( ) +hν+ 2 u 2 ν+ p ν+ 2 (Σ) (2.30) 28

39 2.6 A poseriori error esimaes for coac wih fricio Now combiig (2.26),(2.27),(2.30) we ge he esimae of he secod erm i(2.05) λ λ hp, u hp u ɛ λ hp λ 2 2 ( ) +ɛ uhp u 2 2 ( ) +hν+ 2 u 2 ν+ p ν+ 2 (Σ) (2.3) Fially, he esimae follows immediaely by usig Lemma2.2, Lemma(2.), (2.3), (2.94), (2.95), (2.96), (2.04), (2.98), (3.58), (2.9), (2.23). 2.6 A poseriori error esimaes for coac wih fricio Le (u, λ) be he exac soluio of he coiuous problem (2.0) ad le (u hp, λ hp ) be he soluio of he discree boudary eleme problem (2.34). We ow give a compuable upper boud for u u hp 2, where u u hp 2 := u u hp 2 + ψ ψ hp 2 + λ λ hp 2 (Σ) 2 2 (Γ) ( ) (2.32) Theorem 2.6. Le (u, λ) be he exac soluio of he boudary problem (2.0) ad (u hp, λ hp ) be he soluio of he discree boudary problem (2.34). The here holds he esimae : u u hp 2 + ψ ψ hp 2 η 2 (Σ) 2 hp 2 (I) + (λhp ) +, (g u hp ) + ΓC (Γ) I T hp where + (λ hp ) ( ) 4ɛ (uhp g) ( ) + ɛ λ λ hp 2 + ( λ hp 2 F) + 2 ( ) 2 ( ) ( + ( λ hp F) u hp + 2(λ hp u hp ) ) ds, η 2 hp (I) = h I p I S hp u hp 2 L 2 (I Γ N ) + h I p I ( λ hp ) S hp u hp 2 L 2 (I ) + h I s (V (ψ hp ψhp )) 2 L 2 (I) (2.33) Proof. Usig Lemma 2.0, sice W,, V, are posiive defiie, here exis cosas C W, C V > 0 such ha C W u u hp 2 2 (Σ)+C V ψ ψ hp 2 (Γ) Su S hpu hp, u u hp + V (ψ hp ψhp ), ψ ψ hp 29

40 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy Usig Galerki orhogoaliy, we obai C W u u hp 2 2 (Σ) + C V ψ ψ hp 2 (Γ) Su S hpu hp, u u hp + Su S hp u hp, u hp v hp Σ + b(λ λ hp, u hp v hp ) + V (ψ hp ψhp ), ψ ψ hp Su S hp u hp, u v hp + V (ψ hp ψhp ), ψ ψ hp + b(λ λ hp, u hp v hp ), u v hp ΓN b(λ, u v hp ) S hp u hp, u v hp + V (ψ hp ψhp ), ψ ψ hp + b(λ λ hp, u hp v hp ) S hp u hp, u v hp ΓN + ( λ hp ) S hp u hp, u v hp ΓC + b(λ λ hp, u hp u) + V (ψ hp ψhp ), ψ ψ hp A + B + C + D (2.34) We esimae he erms A ad B, employig he Cauchy-Schwarz iequaliy. We obai: A + B ( λ hp ) S hp u hp L2 (I) u v hp L2 (I) I T hp + S hp u hp L2 (I) u v hp L2 (I) (2.35) I T hp Γ N Le Π hp be he hp-cléme ierpolaio operaor mappig oo V hp [48]. The here holds ( ) h 2 u Π hp u L2 (I) C u p, (2.36) 2 (ω(i)) where ω(i) cosiss of all eighborig elemes of I. We choose i (2.35) v hp = u hp + Π hp (u u hp ) we obai he followig approximaio: ( ) u v hp L 2 (I) C hi 2 u u hp p I 2 (ω(i)) (2.37) We apply a resul from [[5],Theorem 5.] for he erm D, o obai D := V (ψhp ψhp ), ψ ψ hp V (ψhp ψhp ) 2 ψ hp ψ (Γ) 2 (Γ) c h I 2 s (V (ψ hp ψhp )) 2 L 2 (I) ψ hp ψ (2.38) 2 (Γ) I T hp Fially we esimae he erm C. I order o obai a a poseriori error esimae, we have o esimae he erm: C = b(λ λ hp, u hp u) = λ λ hp, u hp u + λ λ hp, u hp u. (2.39) 30

41 2.6 A poseriori error esimaes for coac wih fricio Usig he codiio λ, u g = 0 ad (λ hp ) +, u g 0 where v + = max{0, v} ad v = mi{0, v}, i.e. v = v + + v. λ λ hp, u hp u = λ (λ hp ) +, u hp u (λ hp ), u hp u = λ (λ hp ) +, u hp g + g u (λ hp ), u hp u = λ (λ hp ) +, u hp g + λ, g u (λ hp ) +, g u (λ hp ), u hp u λ (λ hp ) +, u hp g (λ hp ), u hp u = (λ hp ) +, g u hp λ, (g u hp ) + + (g u hp ) (λ hp ), u hp u (λ hp ) +, g u hp + λ hp λ (λ hp ) + (λ hp ), (g u hp ) (λ hp ), u hp u = (λ hp ) +, (g u hp ) + + λ hp λ, (g u hp ) (λ hp ), (g u hp ) (λ hp ), u hp u = (λ hp ) +, (g u hp ) + + λ hp λ, (g u hp ) (λ hp ), u hp u (λ hp ) 2 ( ) uhp u 2 (Σ) + λ hp λ (g uhp 2 ( ) ) 2 (ΓC ) + (λhp ) +, (g u hp ) + ΓC. Usig he codio λ u + F u = 0 ad λ = ξf wih ξ : λ λ hp, u hp u = λ u + λ hp u + λ u hp = F u + λ hp u + ξfu hp F u + λ hp u + F u hp λ hp u hp λ hp u hp λ hp u hp ( λ hp F) + u + F u hp λ hp u hp ( λ hp F) + u u hp + ( λ hp F) + u hp + F u hp λ hp ( λ hp F) + u u hp 2 ( ) 2 (ΓC ) + [( λ hp F) + ( λ hp F)] u hp λ hp u hp ( λ hp F) + u u hp 2 ( ) 2 (ΓC ) + ( λ hp F) u hp + 2(λ hp u hp + λ hp u hp (2.40) u hp ) (2.4) Usig Youg s iequaliy we hrow he erm u u hp 2 (Σ) we obai he esimae of he heorem. o he lef had side, ad Now we look for a upper boud of he discreizaio error λ λ hp. 3

42 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy Lemma 2.3. Le (u, λ) solve he saddle poi problem (2.0), ad le (u hp, λ hp ) be he soluio of he discree problem (2.34). The here holds ( ) λ λ hp 2 C u u hp ψ ψ hp 2 + C ξ ( ) 2 (Σ) 2 hp 2 (I) (Γ) I T hp Where (2.42) ξ 2 hp (I) = h I p I S hp u hp 2 L 2 (I Γ N ) +h I p I ( λ hp ) S hp u hp 2 L 2 (I ) (2.43) Proof. For v V ad v hp V hp we have λ λ hp, v = λ λ hp, v v hp + λ λ hp, v hp Usig Galerki orhogoaliy ad he formulaio (2.0a), we obai λ λ hp, v = λ λ hp, v v hp Su S hp u hp, v hp Σ = L(v v hp ) Su, v v hp λ hp, v v hp Su S hp u hp, v Σ Su S hp u hp, v hp v Σ = S hp u hp, v v hp ΓN + ( λ hp ) S hp u hp, v v hp ΓC Su S hp u hp, v Σ = A + B + C (2.44) A = S hp u hp, v v hp ΓN S hp u hp L2 (I) u v hp L2 (I) (2.45) I T hp Γ N B = ( λ hp ) S hp u hp, v v hp ΓC ( λ hp ) S hp u hp L2 (I) u v hp L2 (I) I T hp (2.46) C = Su S hp u hp, v Σ (C s + C Ehp ) u u hp 2 (Σ) v 2 (Σ) + C 0 ψ ψ hp 2 (Γ) v 2 (Σ) (2.47) We choose v hp = Π hp v i A ad B, we ge ( ) λ λ hp, v (C s + C Ehp ) u u hp +C 2 0 ψ ψ hp (Σ) 2 (Γ) + C ( ) hi 2 Shp u hp p L2 (I) v I 2 (Σ) I T hp Γ N + C ( ) hi 2 ( λ hp ) S hp u hp L2 (I) v 2 (Σ) I T h Γ N p I v 2 (Σ) (2.48) 32

43 2.7 Numerical Experimes Usig he defiiio of he dual orm ad (a+b) 2 2a 2 +2b 2, he asserio immediaely follows. Theorem 2.7. Le (u, λ) be he exac soluio of he boudary problem (2.0) ad (u hp, λ hp ) be he soluio of he discree boudary problem (2.34), he here holds he a poseriori esimae : u u hp 2 + λ λ hp 2 η 2 (Σ) 2 hp 2 (I) + (λhp ) +, (g u hp ) + ΓC ( ) I T hp where + (λ hp ) 2 2 ( ) + (uhp g) ( ) + ( λ hp F) ( ) ( + ( λ hp F) u h + 2(λ hp η 2 hp (I) = h I p I S hp u hp 2 L 2 (I Γ N ) + h I p I ( λ hp ) S hp u hp 2 L 2 (I ) u hp ) ) ds (2.49) + h I s (V (ψ hp ψhp )) 2 L 2 (I) (2.50) Proof. Follows immediaely from Lemma 2.3 ad Theorem Numerical Experimes Numerical resuls are preseed wih he MATLAB package of L.Baz for he coac of he wo-dimeioal elasic body Ω = [ 0.5, 0.5] 2 wih a rigid sraigh lie. The coac boudary = [ 0.5, 0.5] 0.5 comes i coac wih rigid obsacle which occupies he half space y 0.5. The Youg s modulus ad he Poisso s raio are E = 200, ν = 0.3 respecively. The gap is assumed o be zero, i.e g = 0 ad he give fricio fucio F = 0.3. The applied Neuma boudary forces o he op, he lef ad he righ side of he domai are give by see [42] ( ) 400 sig(x)(y + side = 2 )( 2 y)exp( 0(y )2 ) 0(y + 2 )( ( ) 2 y) 0 op = 500( 2 x)2 ( 2 + x)2 I our umerical experimes we use he semi-smooh Newo Algorihm 2. o solve he discree problem. The iiial mesh is uiform ad cosiss of 6 elemes. We 33

44 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy iroduce a hp-adapive algorihm based o locally esig for smoohess as doe i [40] o decide of wheher o perform h-or-p-refieme. The deails of he procedure are described i Algorihm 2.2. Algorihm 2.2. (Mesh refieme sraegy). Geerae a iiial mesh T hp,0, discree spaces V hp,0, W hp,0, se l = 0 2. Choose a olerece T OL > 0 ad seerig parameer 0 γ 3. For l = 0,, 2,... a) Solve he discree problem, for (u hp l, λ hp l ) based o he hp-mesh T hp,l b) Compue idicaors η I for all segmes I T hp,l ad he global error esimaor η = ( I T hp,l η2 I c) Sop if ( I Thp,l η2 I ) 2 T OL d) Compue he local approximaio Ξ I of η I ad he global approximaio I T hp,l Ξ 2 I of η ) 2 e) For (θ (0, )), mark all elemes i N N = argmi ˆN T hp,l : Ξ 2 I θ I ˆN I T hp,l Ξ 2 I f) Esimae aalyiciy [40] (δ (0, )) i. Compue Legedre coefficies of u hp I wih p I u hp (x) I = a i L i (x), a i = 2i + 2 i=0 I u hp (x)l i (x) ii. Compue he slope m I R ad b I R, wih m I, b I R : i (i m I + b I log a i ) 2 mi iii. if exp( m I ) δ icrease p I by oe, else bisec I N ad keep he polyomial degree equal o p I o he resulig sub-elemes. g) Compue he ew hp-mesh T hp,l+ 34

45 2.7 Numerical Experimes h) Geerae he discree spaces, V hp,l+, W hp,l+ based o he mesh T hp,l+ i) Se l = l +, go o (a) I Figure2. we show he deformed cofiguraio. Figure 2.2 shows he esimaed errors for he h-uiform, h-adapive ad hp-adapive mehods wih (θ = 0.5, δ = 0.6). Figure2.3 shows he ormal ad he ageial compoe of he Lagrage muliplier. The refied meshes ad polyomial degrees obaied wih our approach are show i Figure 2.4 ad Figure 2.5. The firs figure shows he adapivey geeraed meshes ad polyomial degrees afer ad 3 refieme seps usig η bub (he bubble idicaor) as error idicaor see [47]. The secod figure shows adapivey refied meshes ad polyomial degrees obaied afer 9 ad seps usig he residual idicaor. 35

46 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy Figure 2.: Deformed geomery uf. h, p=, eergy diff. uf. h, p=, bubble uf. h, p=, residual residual h adap, p=, bubble residual h adap, p=, residual bubble hp adap, bubble residual hp adap, residual Error Degrees of Freedom Figure 2.2: Esimaed errors 36

47 2.7 Numerical Experimes (a) ormal compoe λ (b) ageial compoe λ Figure 2.3: Visualizaio of Lagrage muliplier 37

48 2 hp-bem For Fricioal Coac Problem i Liear Elasiciy (a) h-adapiviy, θ = 0.5, δ = (b) hp-adapiviy, θ = 0.5, δ = 0.6 Figure 2.4: Adapivey geeraed meshes for Lamé-BEM (bubble idicaor)

49 2.7 Numerical Experimes (a) h-adapiviy, θ = 0.5,δ = (b) hp-adapiviy, θ = 0.5,δ = 0.6 Figure 2.5: Adapivey geeraed meshes for Lamé-BEM (residual idicaor) 39

50

51 3 Sabilized mixed hp-bem i Liear Elasiciy I his chaper, we cosider a coac problem i 2D elasiciy wih Tresca fricio. We cosider a mixed boudary iegral formulaio, which is sabilized followig ideas of P.ild, Y. Reard ad V.Lleras [36],[37],[45] for he FEM. ere a mesh-depede sabilizaio erm is added o he discree mixed formulaio, i order o avoid he discree if-sup codiio.. Firs we sudy he exisece ad uiqueess of he soluio of he discree problem. A subsecio is devoed o a priori error ad a poseriori error esimaes. Fially, we prese some umerical experimes, which compare he sabilized ad he o-sabilized cases. 3. The mixed formulaio The oaios i Chaper 2 are used i his chaper. Le Ω R 2 be a bouded Lipschiz domai wih he boudary Γ := Ω = Γ N Γ D be decomposed io he o-iersecig Neuma codiioed segme Γ N, he Dirichle codiioed segme Γ D ad he coac codiioed segme which poeially ca come i coac wih he rigid foudaio wih Γ D = for simpliciy. The problem he cosiss i fidig he displaceme field u : Ω R such ha div σ(u) = 0 i Ω (3.a) σ(u) = C : ɛ(u) i Ω (3.b) u = 0 o Γ D (3.c) σ(u) = o Γ N (3.d) σ 0, u g, σ (u g) = 0 o (3.e) σ F, σ u + F u = 0 o (3.f) Recall ha he scalar ormal ad ageial boudary sresses are defied as σ := σ(u) ad σ := σ(u), ad for ay displaceme u ad for surface sresses defied o Ω we adop he followig oaio u = u + u ad σ(u) = σ (u) + σ (u) (3.2) 4

52 3 Sabilized mixed hp-bem i Liear Elasiciy As i Chaper2, we iroduce he fucio spaces V := [ 2 (Σ)] d = 2 (Σ) := {u 2 (Σ); supp(u) Σ} V := {u 2 (Σ) : u = 0 o Γ D } W := [ 2 (ΓC )] d = 2 (ΓC ) M := [ 2 (ΓC )] d = 2 (ΓC ) where Σ := Γ N ad we assume ha F L 2 ( ), 2 (Γ N ) ad g 2 ( ). We defie he D N (Dirichle-o-Neuma) mappig u Γ σ(u). There holds σ (u) Su ΓC, σ (u) Su ΓC. (3.3) The Seklov-Poicaré operaor S is defied by S := W + (K + 2 )V (K + 2 ). We defie he space for he Lagrage muliplier λ by where M = M M M := {µ 2 (ΓC ) : µ, v ΓC 0, v 2 (ΓC ) wih v 0 a.e o } ad M (F) := {µ L 2 ( ) : µ F a.e o }. are he ses of ormal ad ageial Lagrage mulipliers. The classical formulaio (3.) ca be rewrie i a weak sese as a saddle poi problem as follows: Fid (u, λ) V M such ha Su, v Σ + b(λ, v) =, v ΓN v V (3.4a) b(µ λ, u) g, µ λ ΓC µ M (3.4b) wih he fucioal b(µ, v) := µ, v ΓC + µ, v ΓC. (3.5) ere he oaio, represes he dualiy pairig bewee 2 ( ) ad 2 ( ). 42

53 3.2 The sabilized mixed hp-bem formulaio 3.2 The sabilized mixed hp-bem formulaio Le T hp be a subdivisio of Γ N io sraigh lie segmes I. We associae each eleme of T hp wih a polyomial degree p I ad se p = (p I ) I Thp. Furhermore we defie he se of Gauss-Lobao pois G I,hp o each eleme I T hp of correspodig polyomial degree p I as he affie mappig of he Gauss-Lobao pois o [, ] oo I ad se G hp := I Thp G I,hp.. We iroduce he space of coiuous piecewise polyomials for he discreizaio of u: V hp := {u hp C 0 (Σ) : I T hp, u hp I [P pi (I)] 2, u hp = 0 o Γ D } 2 (Γ) ad he space of piecewise polyomials for he discree racios W hp := {L 2 (Σ) : I T h, φ I [P pi (I)] 2 } 2 (Γ). A explici represeaio of V is o kow, which causes addiioal difficulies i he umerical reame. To resolve his problem we eed o approximae he Seklov- Poicaré operaors. Le i hp : W hp 2 (Γ) ad j hp : V hp 2 (Γ) deoe he caoical imbeddigs wih dual maps i hp ad j hp. The approximaio S hp of he Poicaré-Seklov operaors is give by S hp := W + (K + 2 )i hp(i hp V i hp) i hp (K + 2 ) (3.6) Recall ha he operaor E hp = S S hp represes he error i he approximaio of he Seklov-Poicaré operaor (see Chaper2). Le T q deoe a addiioal pariio of, which eeds o o coicide wih T hp ΓC. We defie he discree versio of he space M for he Lagrage muliplier as (cf [9]) where M q (F) := M,q M,q (F), M,q := {λ q W q : λ q (x) 0 x G q } ad M,q := {λ q W q : λ q (x) F x G q } W q := {λ q L 2 ( ) : J T q, λ q J P q (J)} 43

54 3 Sabilized mixed hp-bem i Liear Elasiciy Remark 3.. I follows from (3.4) ha λ = σ(u) i a weak sese. Therefore, λ has a ierpreaio as he egaive ormal coac racio. We cosider approximaios λ q ad σ(u h hp )(resp. λ q ad σ h (u hp )) of λ = σ (u) (resp.λ = σ (u)). where σ h (u hp ) := S hp u hp ΓC, σ h (u hp ) := S hp u hp ΓC. (3.7) The discreized versio of (3.4) wih sabilized Lagrage muliplier reads as: Fid u hp V hp ad λ q = (λ q, λ q ) M q (F) := M,q M,q (F): S hp u hp, v hp Σ + b(λ q, v hp ) + γ( λ q σ(u h hp ))σ(v h hp ) ds Γ C + γ( λ q σ h (u hp ))σ h (v hp ) ds = L(v hp ) v hp V hp b(µ q λ q, u hp ) + γ(µ q Γ C + γ(µ q λ q )( λ q σ(u h hp )) ds λ q )( λ q σ h (u hp )) ds g, µ q λ q (3.8a) µ q M (F). (3.8b) h ere γ is defied o each eleme I as he cosa γ = γ I 0, wih γ p 2 0 > 0 idepede I of h p. Noe ha he addiioal sabilizaio erm vaishes for he soluio of he coiuous problem as λ q λ ad S hp u hp Su Exisece ad uiqueess of he soluio I his secio we show he exisece ad uiqueess of a soluio o he sabilized formulaio. We follow ideas of V.Lleras [45] for he h-versio of sabilized FEM. Lemma 3.. [9][Coerciviy] For γ 0 sufficiely small, here exiss a cosa C > 0 idepede of h, p, ad q, such ha S hp v hp, v hp Σ γ(σ(v h hp )) 2 ds γ(σ h (v hp )) 2 ds C v hp 2, Γ 2 (Σ) vhp V hp C Lemma 3.2. For γ 0 small eough, Problem (3.8) admis a uique soluio. (3.9) Proof. Problem (3.8) is equivale o fidig a saddle-poi (u hp, λ q, λ q ) V hp M q (F) which saisfies L γ (u hp, ν q ) L γ (u hp, λ q ) L γ (v hp, λ q ) v hp V hp, ν q M (F), (3.0) 44

55 3.2 The sabilized mixed hp-bem formulaio wih L γ (v hp, ν q ) = 2 S hpv hp, v hp Σ L(v hp ) + b(ν q, v hp ) γ(ν q + σ 2 (v h hp )) 2 ds γ(ν q + σ h (v hp )) 2 ds. (3.) 2 Takig ν q = 0 i (3.) ad usig Lemma3. wih γ 0 small eough, we obai Tha yields L γ (v hp, 0) = 2 S hpv hp, v hp Σ L(v hp ) + b(0, v hp ) γ(σ 2 (v h hp )) 2 ds γ(σ h (v hp )) 2 ds 2 C 2 vhp 2 2 (Σ) L vhp 2 (Σ) (3.2) lim L γ (v hp, 0) = + (3.3) v hp 2 (Σ) Choosig v hp = 0 i (3.) we obai L γ (0, ν q ) = γ(ν q ) 2 ds γ(ν q ) 2 ds. 2 2 ad lim L γ (0, ν q ) = (3.4) ν q L2 ( ) The, due o (3.3) (3.4), L γ is sricly covex i v hp ad sricly cocave i ν q. The exisece of he soluio o problem (3.8) follows from he fac ha V hp ad M (F) are wo oempy closed covex ses, L γ (, ) is coiuous o V hp M (F), L γ (v hp, ) (resp. L γ (, ν )) is sricly cocave (resp. sricly covex) for ay v hp V hp (resp. for ay ν q M (F)) (see [35]). Le (λ q, uhp ) ad (λq 2, uhp 2 ) be wo soluio of (3.8). The, choosig µq = λ q 2 ad µ q 2 = λ q i (3.8b), we ge b(λ q 2 λ q, uhp ) + γ(λ q 2, λq, )( λq, σh (u hp )) ds Γ C + γ(λ q 2, λq, )( λq, σh (u hp )) ds g, λ q b(λ q λ q 2, uhp 2 ) + γ(λ q, λq 2, )( λq 2, σh (u hp 2 )) ds Γ C + γ(λ q, λq 2, )( λq 2, σh (u hp 2 )) ds g, λ q 2, λq,, λq 2, (3.5) (3.6) 45

56 3 Sabilized mixed hp-bem i Liear Elasiciy Addig he las wo iequaliies, we obai b(λ q λ q 2, uhp uhp 2 ) γ(λ q, λq 2, )(λq, λq 2, + σh (u hp Γ C γ(λ q, λq 2, )(λq, λq 2, + σh (u hp Fuhermore, subracig he wo (3.8a) wih v hp = u hp uhp 2 implies uhp 2 uhp 2 )) ds )) ds 0 0 = S hp (u hp uhp 2 ), uhp uhp 2 Σ + b(λ q λ q 2, uhp uhp 2 ) γ(λ q, λq 2, + σh (u hp uhp 2 ))σh (u hp uhp 2 ) ds Γ C γ(λ q, λq 2, + σh (u hp uhp 2 ))σh (u hp uhp 2 ) ds S hp (u hp uhp 2 ), uhp uhp 2 Σ γσ(u h hp uhp 2 )σh (u hp uhp 2 ) ds Γ C γσ h (u hp uhp 2 )σh (u hp uhp 2 ) ds + γ(λ q λ q 2 )2 ds (3.7) C u hp uhp γ 2 2 (λ q (Σ) λ q 2 ) 2 L 2 ( ) (3.8) The coformiy i he primal variable implies he uiqueess of he soluio of problem (3.8). Le u V ad u hp V hp. As i [5], we defie he followig vecors ψ := V (K + 2 )u ψ hp := i hp V hp i hp (K + 2 )uhp (3.9) ψ hp := V (K + 2 )uhp. Lemma 3.3. For ψ hp, ψhp defied i (3.9) here holds V (ψ hp ψhp ), φ = 0 φ W hp (3.20) Lemma 3.4. (Galerki orhogoaliy) Le u V be he soluio of he coiuous problem (3.4) ad u hp V hp he soluio of he discree problem. There holds Su S hp u hp, v hp Σ + b(λ λ q, v hp ) γ( λ q σ(u h hp ))σ(v h hp ) ds Γ C γ( λ q σ h (u hp ))σ h (v hp ) ds = 0 v hp V hp. (3.2) Proof. We choose v V hp V i (3.4a) ad subrac (3.4a) from he discree formulaio (3.8a). 46

57 3.3 A priori error aalysis for fricioal coac problem 3.3 A priori error aalysis for fricioal coac problem Lemma 3.5. Le u V, λ M solve he saddle poi problem (3.4) ad, le u hp V hp ad λ q M hp (F) be he soluio of he discree problem (3.8), we assume ha λ L 2 ( ). The here holds γ 2 (λ λ q ) 2 L 2 ( ) + γ 2 (λ λ q ) 2 L 2 ( ) µ )u ds where + + (λ µ q γ(λ λ q (λ q )(u hp µ )u ds + γ(λ λ q γ(µ q (λ q + γ( λ q σ (u hp )) ds )σ (u u hp ) ds λ λ q (λ µ q )(u hp )σ (u u hp ) ds λ λ q λ q )Eh (uhp ) ds γ(µ q, u hp u + γ( λ q, u hp u σ (u hp )) ds λ q )Eh (uhp ) ds (3.22) E h (uhp ) = (S S hp )u hp ΓC, E h (uhp ) = (S S hp )u hp ΓC (3.23) Proof. Recall ha λ = σ (u) ad λ = σ (u). The iequaliy i (3.8b) is equivale o he followig codiios: µ q λ q, u hp + γ(µ q Γ C µ q λ q, u hp + γ(µ q Noe ha γ 2 (λ λ q ) 2 L 2 ( ) Γ = γλ 2 ds 2 C γ 2 (λ λ q ) 2 L 2 ( ) = γλ 2 ds 2 Usig (3.24) ad (3.25) we ge γ(λ q ) 2 ds + γ(λ q ) 2 ds + γλ q γ(µ q γλ q γ(µ q λ q )( λ q σ(u h hp )) ds 0 (3.24) λ q )( λ q σ h (u hp )) ds 0. (3.25) µ q ds + γλ λ q Γ C γλ λ q ds + λ q µ q ds (µ q )σ(u h hp ) ds ds (µ q λ q )σ h (u hp ) ds. γ(λ q ) 2 ds (3.26) γ(λ q ) 2 ds (3.27) λ q )u hp ds λ q )u hp ds 47

58 3 Sabilized mixed hp-bem i Liear Elasiciy From he secod equaio of he coiuous problem (3.4b), we have for all (λ, λ ) M M (F): γ(λ ) 2 ds γλ µ ds (µ λ )u ds + γ(µ λ )σ (u) ds Γ C γ(λ ) 2 ds γλ µ ds (µ λ )u ds + γ(µ λ )σ (u) ds. This gives γ 2 (λ λ q ) 2 L 2 ( ) + γ 2 (λ λ q ) 2 L 2 ( ) γ(µ λ q )λ ds + γ(µ q λ )λ q ds + (λ µ )u ds Γ C + γ(µ λ )σ (u) ds + (λ q µ q )u hp ds + γ(µ q λ q )σ(u h hp ) ds Γ C + γ(µ λ q )λ ds + γ(µ q λ )λ q ds + (λ µ )u ds Γ C + γ(µ λ )σ (u) ds + (λ q µ q )u hp ds + γ(µ q λ q )σ h (u hp ) ds = γ(µ λ q )λ ds + γ(µ q Γ C + γ(µ λ )σ (u) ds + (λ q Γ C + γ(µ q λ q )σ (u hp ) ds Γ C + γ(µ λ q )λ ds + γ(µ q Γ C + γ(µ λ )σ (u) ds + (λ q Γ C + γ(µ q λ q )σ (u hp ) ds = (λ q µ )u ds + Γ C γ(λ λ q Γ C + (λ µ q )(u hp λ λ q, u hp u λ )λ q ds + (λ µ )u ds µ q )u hp ds γ(µ q λ q λ )λ q ds + µ q )u hp ds γ(µ q (λ µ q )σ (u u hp ) ds λ λ q + γ( λ q γ(µ q )Eh (uhp ) ds (λ µ )u ds λ q )Eh (uhp ) ds )(u hp + γ( λ q σ (u hp )) ds, u hp u + (λ q µ )u ds Γ C σ (u hp )) ds γ(λ λ q )σ (u u hp ) ds λ q )Eh (uhp ) ds γ(µ q λ q )Eh (uhp ) ds. 48

59 3.3 A priori error aalysis for fricioal coac problem Lemma 3.6. There exiss a cosa C > 0, idepede of h, p, such ha u hp 2 C (3.28) (Σ) I paricular, for u 0, here exiss a cosa C u such ha u hp 2 C u u (Σ) 2 (3.29) (Σ) Proof. Choosig v hp = u hp i (3.8a), we obai S hp u hp, u hp Σ + λ q u hp ds + Γ C + u hp ds + Choosig µ q λ q = 0 ad µ q λ q λ q γ( λ q σ h (u hp ))σ h (u hp ) ds γ( λ q σ h (u hp ))σ h (u hp ) ds = L(u hp ) (3.30) = λ q u hp i (3.8b), we ge ds λ q Now we choose µ q = 2λ q ad µ q = λ q i (3.8b), we obai u hp ds + Combiig (3.3) ad (3.32), we obai u hp ds + Similarly, we ge λ q λ q u hp ds + λ q γλ q γλ q ( λ q σ h (u hp )) 0 (3.3) ( λ q σ h (u hp )) 0 (3.32) ( λ q σ h (u hp )) = 0 (3.33) ( λ q σ h (u hp )) = 0 (3.34) Usig (3.30), (3.33), ad (3.34), we have L(u hp ) = S hp u hp, u hp Σ + λ q u hp ds + Γ C γ( λ q σ(u h hp ))σ(u h hp ) ds + λ q u hp ds + γ( λ q σ h (u hp ))σ h (u hp ) ds Γ C = S hp u hp, u hp Σ + γ(λ q ) 2 ds Γ C γ(σ(u h hp )) 2 ds + γ(λ q ) 2 ds γ(σ h (u hp )) 2 ds (3.35) From Lemma 3. ad (3.35), we obai C u hp 2 S hpu, u 2 Γ (Σ) γ(σ(u h hp )) 2 ds γ(σ h (u hp )) 2 ds L(u hp ) u hp 2 (Σ) (3.36) 49

60 3 Sabilized mixed hp-bem i Liear Elasiciy Theorem 3.. Le (u, λ) V M be a soluio of he problem (3.4) such ha u (Σ) ad λ L 2 ( ). Le L 2 (Σ). Le (u hp, λ q ) be he soluio of he discree problem (3.8). The here exiss a cosa C > 0 idepede of h,, p ad q, such ha u u hp 2 + ψ ψ hp 2 + γ 2 (Σ) 2 2 (λ λ q (Γ) [ ( C u v hp 2 2 (Σ)+ γ 2 σ (u v hp ) 2 L 2 ( ) µ q if v hp V hp M,q ) 2 L 2 ( ) + γ 2 (λ λ q ) 2 L 2 ( ) +b(λ q, u v hp ) + u v hp L2 (Σ) + γ(λ + σ (u hp ))σ (u hp v hp ) ds + γ(λ + σ (u hp ))σ (u hp v hp ) ds Γ C γ(λ q + σ (u hp ))E(u h hp v hp ) ds γ(λ q + σ (u hp ))E h (u hp v hp ) ds Γ C + γe(u h hp )σ (u hp v hp ) ds + γe h (u hp )σ (u hp v hp ) ds Γ C ) + γe(u h hp )E(u h hp v hp ) ds + γe h (u hp )E h (u hp v hp ) ds Γ ( ) C + if b(λ, v u hp ) + u hp v L2 (Σ) v V + if ψ φ 2 + if (λ q φ W 2 µ )u ds hp (Γ) µ M Γ C + if (µ q µ q λ )(u hp + γ( λ q σ (u hp )) ds M,q + if (λ q µ )u ds + if (µ q µ M (F) µ q λ )(u hp + γ( λ q σ (u hp )) ds M,q (F) ] + if γ(λ q µ q )Eh (uhp ) ds + if γ(λ q µ q )Eh (uhp ) ds M,q (F) Proof. From Lemma 2.0 we obai u u hp 2 W + ψ ψ hp 2 V = Su S hp u hp, u u hp + V (ψ hp ψhp ), ψ ψ hp. Sice we have wih E hp = S S hp Su S hp u hp = S(u u hp ) + E hp u hp = S(u u hp ) + E hp (u hp u) + E hp u, (3.37) µ q we obai u u hp 2 W + ψ ψ hp 2 V = Su S hp u hp, u u hp + V (ψhp ψhp ), ψ ψ hp = S(u u hp ), u v hp + Su hp, u hp v hp + Su, v hp u hp + E hp u hp, u u hp + V (ψhp ψhp ), ψ ψ hp. (3.38) 50

61 3.3 A priori error aalysis for fricioal coac problem Usig (3.8a) ad he defiiio of S hp, we ge Su hp, u hp v hp = E hp u hp, u hp v hp b(λ q, u hp v hp ) +, u hp v hp + γ(λ q + σ(u h hp ))σ(u h hp v hp ) ds + γ(λ q + σ h (u hp ))σ h (u hp v hp ) ds Usig (3.4), (3.38) ad (3.39), we obai C W u u hp 2 +C V ψ ψ hp 2 S(u u 2 (Σ) (Γ) hp ), u v hp + Su, v hp u hp 2 (3.39) + E hp u hp, u u hp + E hp u hp, u hp v hp b(λ q, u hp v hp ) +, u hp v hp Su, u v b(λ, u v) +, u v + V (ψ hp ψhp ), ψ ψ hp + γ(λ q + σ (u hp ))σ (u hp v hp ) ds + γ(λ q + σ (u hp ))σ (u hp v hp ) ds S(u u hp ), u v hp + E hp u hp, u v hp + Su, u hp v + Su, u v hp + b(λ, v u) b(λ q, u hp v hp ) + V (ψ hp ψhp ), ψ ψ hp + γ(λ q + σ(u h hp ))σ(u h hp v hp ) ds + γ(λ q + σ h (u hp ))σ h (u hp v hp ) ds We have (3.40) b(λ, v u) b(λ q, u hp v hp ) b(λ λ q, u hp u) = b(λ, v u hp ) b(λ q, u v hp ) (3.4) From (3.40) ad (3.4), we obai C W u u hp 2 +C V ψ ψ hp 2 λ 2 (Σ) (Γ) λ q, u hp u 2 S(u u hp ), u v hp + E hp u hp, u v hp + Su, u hp v + Su, u v hp + b(λ, v u hp ) b(λ q, u v hp ) + V (ψ hp ψhp ), ψ ψ hp + γ(λ q + σ(u h hp ))σ(u h hp v hp ) ds + γ(λ q + σ h (u hp ))σ h (u hp v hp ) ds (3.42) 5

62 3 Sabilized mixed hp-bem i Liear Elasiciy Usig Lemma 3.5 ad (3.42) we have C W u u hp 2 +C V ψ ψ hp 2 2 (Σ) 2 (Γ) + γ 2 (λ λ q ) 2 L 2 ( ) + γ 2 (λ λ q ) 2 L 2 ( ) S(u u hp ), u v hp + E hp u hp, u v hp + Su, u hp v + Su, u v hp + b(λ, v u hp ) b(λ q, u v hp ) + V (ψhp ψhp ), ψ ψ hp + γ(λ q + σ(u h hp ))σ(u h hp v hp ) ds + γ(λ q + σ h (u hp ))σ h (u hp v hp ) ds Γ C + (λ q µ )u ds + (λ µ q )(u hp + γ( λ q σ (u hp )) ds Γ C γ(λ λ q )σ (u u hp ) ds γ(λ λ q )σ (u u hp ) ds Γ C + (λ q µ )u ds + (λ µ q )(u hp + γ( λ q σ (u hp )) ds Γ C + γ(λ q µ q )Eh (uhp ) ds + γ(λ q µ q )Eh (uhp ) ds (3.43) Therefore C W u u hp 2 +C V ψ ψ hp 2 2 (Σ) 2 (Γ) + γ 2 (λ λ q ) 2 L 2 ( ) + γ 2 (λ λ q ) 2 L 2 ( ) S(u u hp ), u v hp + E hp u hp, u v hp + Su, u hp v + Su, u v hp + b(λ, v u hp ) b(λ q, u v hp ) + V (ψhp ψhp ), ψ ψ hp γ(λ + σ (u hp ))σ (u hp v hp ) ds + γ(λ + σ (u hp ))σ (u hp v hp ) ds Γ C + (λ q µ )u ds + (λ µ q )(u hp + γ( λ q σ (u hp )) ds Γ C γ(λ λ q )σ (u v hp ) ds γ(λ λ q )σ (u v hp ) ds Γ C + (λ q µ )u ds + (λ µ q )(u hp + γ( λ q σ (u hp )) ds Γ C γ(λ q + σ (u hp ))E(u h hp v hp ) ds γ(λ q + σ (u hp ))E h (u hp v hp ) ds Γ C γe(u h hp )σ (u hp v hp ) ds γe h (u hp )σ (u hp v hp ) ds Γ C + γe(u h hp )E(u h hp v hp ) ds + γe h (u hp )E h (u hp v hp ) ds Γ C + γ(λ q µ q )E(u h hp ) ds + γ(λ q µ q )Eh (uhp ) ds (3.44) Usig (3.37), Cauchy Schwarz iequaliy, ad he coiuiy of S hp ad E hp, here holds for all φ W hp A = S(u u hp ), u v hp C S u u hp 2 (Σ) u vhp 2 (Σ) (3.45) 52

63 3.3 A priori error aalysis for fricioal coac problem Usig Lemma2.6ad Cauchy Schwarz iequaliy, we have B = E hp u hp, u v hp = E hp (u hp + u u), u v hp E hp u 2 (Σ) u vhp 2 (Σ) +C E hp u u hp 2 (Σ) u vhp 2 (Σ) C 0 ψ φ hp 2 u (Γ) vhp +C 2 E hp u u hp u (Σ) 2 (Σ) vhp 2 (Σ) (3.46) for φ hp W hp. From Lemma3.3 ad (3.9) follows wih φ hp W hp ha C = V (ψh ψhp ), ψ ψ hp = V (ψh ψ), ψ φhp + V (ψ ψ hp ), ψ φ hp (C K + 2 ) u uh 2 (Σ) ψ φhp 2 (Γ) + C V ψ ψ hp 2 (Γ) ψ φhp 2 (Γ) (3.47) Employig Youg s iequaliy, we obai A 2ɛ C2 S u v hp 2 + ɛ 2 (Σ) 2 u uhp 2 2 (Σ) (3.48) B 2ɛ C2 E hp u v hp 2 + ɛ 2 (Σ) 2 u uhp 2 2 (Σ) + C 0 2 ψ φhp 2 + C 0 2 (Γ) 2 u vhp 2 2 (Σ) (3.49) C 2ɛ (C K + 2 )2 ψ φ 2 + ɛ 2 (Γ) 2 u uhp 2 2 (Σ) + 2ɛ C2 V ψ φ hp 2 + ɛ 2 (Γ) 2 ψ ψhp 2 2 (Γ) (3.50) Sice Su L 2 (Σ), we obai D = Su, u hp v + Su, u v hp Su L2 (Σ)( u hp v L2 (Σ)+ u v hp L2 (Σ)) (3.5) Employig Cauchy Schwarz ad Youg s iequaliy, we obai E = γ(λ λ q )σ (u v hp ) ds γ 2 (λ λ q ) L2 ( ) γ 2 σ (u v hp ) L2 ( ) ɛ 2 γ 2 (λ λ q ) 2 L 2 ( ) + 2ɛ γ 2 σ (u v hp ) 2 L 2 ( ) (3.52) 53

64 3 Sabilized mixed hp-bem i Liear Elasiciy E = γ(λ λ q )σ (u v hp ) ds γ 2 (λ λ q ) L2 ( ) γ 2 σ (u v hp ) L2 ( ) ɛ 2 γ 2 (λ λ q ) 2 L 2 ( ) + 2ɛ γ 2 σ (u v hp ) 2 L 2 ( ) (3.53) Usig (3.44), ad (3.48)-(3.53), we ge α u u hp 2 +α 2 ψ ψ hp 2 2 (Σ) 2 (Γ) + α 3 γ 2 (λ λ q ) 2 L 2 ( ) +α 4 γ 2 (λ λ q ) 2 L 2 ( ) α 5 u v hp 2 +α 6 ψ φ hp 2 2 (Σ) 2 (Γ) + ɛ γ 2 σ (u v hp ) 2 L 2 ( ) + ɛ γ 2 σ (u v hp ) 2 L 2 ( ) + Su L2 (Σ)( u hp v L2 (Σ)+ u v hp L2 (Σ)) + b(λ, v u hp ) b(λ q, u v hp ) + (λ q µ )u ds + (λ q µ )u ds (λ µ q (λ µ q )(u hp )(u hp + γ( λ q + γ( λ q σ (u hp )) ds σ (u hp )) ds + γ(λ + σ (u hp ))σ (u hp v hp ) ds + γ(λ + σ (u hp ))σ (u hp v hp ) ds γ(λ q + σ (u hp ))Ehp (uhp v hp ) ds γ(λ q + σ (u hp ))Ehp (uhp v hp ) ds + γehp (uhp )σ (u hp v hp ) ds + γehp (uhp )σ (u hp v hp ) ds + γehp (uhp )Ehp (uhp v hp ) ds + γehp (uhp )Ehp (uhp v hp ) ds + γ(λ q µ q )Ehp (uhp ) ds + γ(λ q µ q )Ehp (uhp ) ds, 54

65 3.3 A priori error aalysis for fricioal coac problem where he cosas α = 2C W 3ɛ α 2 = 2C V ɛ α 3 = 2 ɛ α 4 = 2 ɛ α 5 = C2 S ɛ + C2 E hp ɛ + C 0 α 6 = C 0 + ɛ (C K + 2 )2 + ɛ C2 V are idepede of h,, p ad q, α, α 2, α 3 ad α 4 are posiive if ɛ is small eough. The esimae of he heorem follows immediaely. Theorem 3.2. Le (u, λ) V M be he soluio of he problem (3.4) ad (u hp, λ q ) he soluio of he discree problem (3.8) wih L 2 (Γ N ) ad g = 0. Assume ha u +ν (Σ), λ ν ( ) for some ν [0, 2 ]. Suppose ha λ ν ( )+ λ ν ( )+ F L2 ( ) u +ν (Σ). The here exiss a cosa C > 0 idepede of h ad p, such ha u u hp + ψ 2 (Σ) ψhp 2 + γ 2 (λ λ q (Γ) ) C ( ν+ 2 + if µ M q ν+ 2 + h 2 ν pq ν u +ν (Σ) (λ q µ )u ds + if µ M (F) ) L2 ( )+ γ 2 (λ λ q ) L2 ( ) (λ q µ )u ds (3.54) Proof. The a priori esimae follows from he esimae i Theorem (3.) wih v = u hp. We esimae he erm b(λ q, u v hp ) = λ q, u v hp + λ q Employig Cauchy Schwarz ad Youg s iequaliy, we have λ q, u v hp = λ )(u v hp ) ds + λ q (λ q λ (u v hp ) ds λ L2 ( ) u v hp L2 ( )+ λ L2 ( ) u v hp L2 ( ), u v hp (3.55) ɛ 2 γ 2 (λ q λ ) 2 L 2 ( ) + 2ɛγ 0 p 2 h u v hp 2 L 2 ( ) + λ L2 ( ) u v hp L2 ( ) ɛ 2 γ 2 (λ q λ ) 2 L 2 ( ) + 2ɛ Similarly o (2.0), we ge λ q, u v hp ɛ 2 γ 2 (λ q λ ) 2 L 2 ( ) + 2ɛ h 2ν+ +hν+ p 2ν u 2 +ν (Γ) p ν+ u 2 (3.56) +ν (Γ) h 2ν+ +hν+ p 2ν u 2 +ν (Γ) p ν+ u 2 +ν (Γ) (3.57) 55

66 3 Sabilized mixed hp-bem i Liear Elasiciy Combiig (2.0) ad (3.57), we obai b(λ q, u v hp ) ɛ 2 γ 2 (λ q Now we esimae he erms + 2ɛ λ ) 2 L 2 ( ) + ɛ 2 γ 2 (λ q λ ) 2 L 2 ( ) h 2ν+ +hν+ p 2ν u 2 +ν (Γ) p ν+ u 2 (3.58) +ν (Γ) A 2 = u v hp 2 2 (Σ) ad A 3 = γ 2 σ (u v hp ) 2 L 2 ( ) We choose v hp = I hp u ad use he aproximaio propery of he Lagrage ierpolaio operaor: we have A 2 = u v hp 2 2 (Σ) = u I hp u 2 C h+2ν 2 (Σ) p +2ν u 2 +ν (3.59) (Σ) u v hp 2 (Γ) = u I hpu 2 (Γ) C ( h p ) 2ν u 2 +ν (Σ) (3.60) Employig he coiuiy codiio of he Dirichle-o-Neuma operaor ad (3.60) we obai We ow esimae he erm A 3 = γ 2 σ (u v hp ) 2 L 2 ( ) α γ 2 (u v hp ) 2 (Γ) = αγ 0 h p 2 u I hpu 2 (Γ) = Cαγ 0 h +2ν p 2+2ν u 2 +ν (Σ). (3.6) A 4, = γ(σ (u hp ) + λ )σ (u hp v hp ) ds. Recall ha λ = σ (u). Sice u hp v hp V hp, we ca apply he hp-iverse iequaliy h p 2 σ (u hp v hp ) 2 L 2 ( ) α h p 2 uhp v hp 2 (Γ) α u hp v hp 2 2 (Σ) (3.62) 56

67 3.3 A priori error aalysis for fricioal coac problem Usig Cauchy Schwarz, Youg s iequaliy, (3.59),(3.6) ad (3.62), we obai A 4, 2 γ h 0 p 2 σ (u hp v hp ) 2 L 2 ( ) + 2 γ h 0 p 2 σ (u u hp ) 2 L 2 ( ) h γ 0 p 2 σ (u hp v hp ) 2 L 2 ( ) + 2 γ h 0 p 2 σ (u v hp ) 2 L 2 ( ) Cαγ 0 h +2ν p 2+2ν u 2 +ν (Σ) +2αγ 0 u v hp 2 2 (Σ)+2αγ 0 u u hp 2 2 (Σ) ( h +2ν Č p 2+2ν u 2 +ν (Σ) + h+2ν p +2ν u 2 +ν (Σ) +γ 0 u u hp 2 2 (Σ) Similar o (3.63), we ge A 4, = γ(σ (u hp ) + λ )σ (u hp w hp ) ds Γ ( C Č h +2ν ( ) ) h +2ν p 2+2ν u 2 +ν + u 2 (Σ) p +γ 0 u u hp 2 +ν (Σ) 2 (Σ) We ow esimae he erm A 5, = µ q if As i Chaper2, le π Mhp M,q (µ q λ )(u hp ). (3.63) (3.64) + γ( λ q σ (u hp )) ds (3.65) be he L 2 -projecio operaor mappig M q defied by π Mq = { πm,q : L 2( ) M,q π M,q : L 2 ( ) M,q. (3.66) Choosig µ q = π M,q λ, we obai A 5, (π M,q λ λ )u hp ds + γ(π M,q λ λ )( λ q σ (u hp )) ds. (3.67) The esimae of he firs erm i (3.67) gives, usig (2.2): A 5,, = (π M,q λ λ )u hp ds Γ C = (π M,q λ λ )(u hp u ) ds + (π M,q λ λ )(u π M,q u ) ds π M,q λ λ 2 ( ) uhp u 2 (ΓC ) + π M,q λ λ L2 ( ) u π M,q u L2 ( ) ( ) 2 +ν C q λ 2 +ν ν ( ) u u hp ++2ν 2 (Σ) q +2ν λ ν ( ) u +ν (Σ) (3.68) 57

68 3 Sabilized mixed hp-bem i Liear Elasiciy We employ Youg s iequaliy o obai ( ) A 5,, C ɛ u u hp 2 + 2ν+ 2 (Σ) q 2ν+ u 2 +ν (Σ) (3.69) Now we esimae he secod iegral erm A 5,2, = if γ(µ q µ q M,q λ )( λ q σ (u hp )) ds. (3.70) We have A 5,2, γ(π M,q λ λ )( λ q + λ ) ds Γ C + γ(π M,q λ λ )(σ (u I hp u)) ds Γ C + γ(π M,q λ λ )(σ (I hp u u hp )) ds Cγ 2 0 ( h p 2 ) 2 ν + Cγ 2 0 ( h p 2 ) 2 ν + Cγ 2 0 ( h p 2 ) 2 ν q ν λ ν ( ) γ 2 (λ λ q ) L2 ( ) q ν λ ν ( ) γ 2 σ (u I hp u) L2 ( ) q ν λ ν ( ) γ 2 σ (I hp u u hp ) L2 ( ). (3.7) Usig Youg s iequaliy, (3.6), (3.60) ad (3.63), we obai ( h 2ν A 5,2, C p 2 q 2ν u 2 +ν (Σ) +ɛ γ 2 (λ λ q ) 2 L 2 ( ) + h+2ν p +2ν u 2 +ν (Σ) +αγ 0 u u hp 2 2 (Σ) ). (3.72) Fially we have ( A 5, C ɛ γ 2 (λ λ q ) 2 L 2 ( ) +γ 0 u u hp 2 2 (Σ) + 2ν+ q 2ν+ u 2 +ν (Σ) + h2ν p 2 q 2ν u 2 +ν (Σ) + h2ν+ p 2ν+ u 2 +ν (Σ) Similarly o (3.73), we obai A 5, = if (µ q µ q M,q (F) ( C λ )(u hp + γ( λ q ɛ γ 2 (λ λ q ) 2 L 2 ( ) +γ 0 u u hp 2 2 (Σ) σ (u hp )) ds + 2ν+ q 2ν+ u 2 +ν (Σ) + h2ν p 2 q 2ν u 2 +ν (Σ) + h2ν+ p 2ν+ u 2 +ν (Σ) ). (3.73) ). (3.74) 58

69 3.3 A priori error aalysis for fricioal coac problem We ow esimae he erm A 6, = if v hp V hp γ( λ q σ (u hp ))E h (u hp v hp ) ds, (3.75) which we wrie as ( ) A 6, = if v hp V hp γ(λ λ q )E(u h hp v hp ) ds + γσ (u u hp ))E(u h hp v hp ) ds (3.76) From a compuaio as i(3.63), we have ( ) h h γ 0 p 2 σ (u u hp ) 2 +2ν L 2 ( ) C p 2+2ν u 2 +ν + h+2ν (Σ) p +2ν u 2 +ν +γ 0 u u hp 2 (Σ) 2 (Σ) (3.77) ad wih coiuiy of E hp = S S hp ( ) h h γ 0 p 2 Eh (u hp v hp ) 2 +2ν L 2 ( ) C p +2ν u 2 +ν +γ 0 u u hp 2. (3.78) (Σ) 2 (Σ) We obai ( ) h +2ν A 6, C p +2ν u 2 +ν +γ 0 u u hp 2 +ɛ γ 2 (λ λ q (Σ) 2 ) 2 L 2 (Γ (Σ) C ) Similarly o (3.79), we ge A 6, = if γ( λ q v hp σ (u hp ))E h (u hp v hp ) ds V hp ( ) h +2ν C p +2ν u 2 +ν +γ 0 u u hp 2 +ɛ γ 2 (λ λ q (Σ) 2 ) 2 L 2 (Γ (Σ) C ) (3.79) (3.80) Cosider ow A 7, = if v hp V hp γe(u h hp )σ (u hp v hp ) ds (3.8) We have A 7, γe(u h hp u)σ (u hp v hp ) ds + γe(u)σ h (u hp v hp ) ds (3.82) Usig Cauchy Schwarz, Youg s iequaliy, we have A 7, ɛ 2 Eh (u hp u) 2 2 ( ) + 2ɛ γσ (u hp v hp ) 2 2 ( ) + 2 Eh (u) 2 2 ( ) + 2 γσ (u hp v hp ) 2 2 ( ) (3.83) 59

70 3 Sabilized mixed hp-bem i Liear Elasiciy Employig he coiuiy codiio of he Dirichle-o-Neuma operaor ad he iverse iequaliy, we ge γσ (u hp v hp ) 2 2 ( ) cγ2 u hp v hp ( ) cγ 2 0 u hp v hp 2 2 (Σ) ad wih he coiuiy of E hp (Lemma2.6) cγ0 2 h +2ν p +2ν u 2 +ν +cγ 2 (Σ) 0 u u hp 2 2 (Σ) ɛ 2 Ehp (u hp u) 2 2 ( ) ɛ 2 C E hp u hp u 2 2 (Σ) From Lemma2.6 ad Lemma2., we have (3.84) (3.85) Usig (3.84)-(3.86), we obai E h (u) 2 2 ( ) h+2ν p +2ν ψ 2 ν (Γ) C h+2ν p +2ν u 2 +ν (Σ) (3.86) A 7, C h+2ν p +2ν u 2 +ν (Σ) +[cγ 2 0( 2ɛ + 2 ) + ɛ 2 C E hp ] u hp u 2 2 (Σ) Similarly, we obai A 7, = if v hp V hp γe hp (uhp )σ (u hp v hp ) ds C h+2ν p +2ν u 2 +ν +[cγ 2 (Σ) 0( 2ɛ + 2 ) + ɛ 2 C E hp ] u hp u 2 2 (Σ) A 8, = if γe v hp hp (u hp )E(u h hp v hp ) ds V hp C h+2ν p +2ν u 2 +ν +[cγ 2 (Σ) 0( 2ɛ + 2 ) + ɛ 2 C E hp ] u hp u 2 2 (Σ) A 8, = if γe hp v hp (u hp )E h (u hp v hp ) ds V hp C h+2ν p +2ν u 2 +ν (Σ) +[cγ 2 0( 2ɛ + 2 ) + ɛ 2 C E hp ] u hp u 2 2 (Σ) (3.87) (3.88) We esimae ow he erm A 9, = µ q if M,q γ(λ q µ q )E hp (uhp ) ds (3.89) We have A 9, γ(λ q λ )Eh (uhp ) ds + γ(λ µ q )Eh (uhp ) ds (3.90) 60

71 3.3 A priori error aalysis for fricioal coac problem We have o esimae he firs erm i (3.90) A 9,, = γ(λ q λ )Eh (uhp ) ds (3.9) A 9,, = γ(λ q Γ C + γ(λ q λ )E h λ )Eh (uhp v hp ) ds γ(λ q λ )Eh (u vhp ) ds (u) ds (3.92) Employig Cauchy Schwarz ad Youg s iequaliy, we have A 9,, 3ɛ 2 γ 2 (λ λ q ) 2 L 2 ( ) +γ 0h ɛp 2 E h (uhp v hp ) 2 L 2 ( ) +γ 0h ɛp 2 E h (u vhp ) 2 L 2 ( ) + γ 0h ɛp 2 E h u 2 L 2 ( ) (3.93) wih he coiuiy of E hp, we have ad E h u 2 L 2 ( ) α u 2 +ν (Σ), for ν = 0, (3.94) γ 2 E h (u v hp ) 2 L 2 ( ) α γ 2 (u v hp ) 2 (Γ) = αγ 0 h p 2 u I hpu 2 (Γ) Usig (3.78), (3.94) ad (3.95), we obai A 9,, 3ɛ 2 γ 2 (λ λ q ) 2 L 2 ( ) +γ 0 ɛ + γ 0 ɛ uhp u 2 +αγ 0 2 (Σ) ɛ = Cαγ 0 h +2ν p 2+2ν u 2 +ν (Σ). (3.95) h +2ν p 2+2ν u 2 +ν + γ 0 h +2ν (Σ) ɛ p +2ν u 2 +ν (Σ) h p 2 u 2 +ν (Σ) (3.96) We esimae ow he secod erm i (3.90) A 9,2, = γ(λ µ q )Eh (uhp ) ds (3.97) The coiuiy of E h ad Lemma3.6, we ge h p 2 E h uhp 2 L 2 ( ) α h p 2 uhp 2 (Σ) α uhp 2 2 (Σ) α u 2 +ν (Σ) (3.98) Usig Cauchy Schwarz ad (3.98), we obai h 2 A 9,2, γ 0 p λ µ q h 2 L2 ( ) p E h uhp 2 L 2 ( ) γ 0 h 2 p γ 0 h 2 p ν q ν λ ν ( ) u +ν (Σ) ν q ν u 2 +ν (Σ) (3.99) 6

72 3 Sabilized mixed hp-bem i Liear Elasiciy Fially we obai A 9, 3ɛ 2 γ 2 (λ λ q ) 2 L 2 ( ) +γ 0 h +2ν ɛ p 2+2ν u 2 +ν + γ 0 h +2ν (Σ) ɛ p +2ν u 2 +ν (Σ) + γ 0 ɛ uhp u 2 +αγ 0 2 (Σ) ɛ Similarly o (3.00), we obai A 9, = if µ q M,q γ(λ q 2 h h p 2 u 2 +ν +γ 0 (Σ) p µ q )Eh (uhp ) ds ν q ν u 2 +ν (Σ) (3.00) 3ɛ 2 γ 2 (λ λ q ) 2 L 2 ( ) +γ 0 h +2ν ɛ p 2+2ν u 2 +ν + γ 0 h +2ν (Σ) ɛ p +2ν u 2 +ν (Σ) + γ 0 ɛ uhp u 2 +αγ 0 2 (Σ) ɛ 2 h h p 2 u 2 +ν +γ 0 (Σ) p ν q ν u 2 +ν (Σ) (3.0) Noe ha γ 0 is small eough. Morig he erms γ 0 u u h 2, ɛ γ 2 (λ λ q ) 2 2 L 2 (Γ (Σ) C ) ad ɛ γ 2 (λ λ q ) 2 L 2 ( ) o he lef had side, we obai he a priori error esimae of he heorem. We cosider he erms ad A, = A, = if µ M (λ q µ )u ds (3.02) if µ M (F) (λ q µ )u ds (3.03) Remark 3.2. The esimaio of he erms A, ad A, seems o be problemaic, due o he ocoformiy of our approach. ere he posiiviy codiio is eforced oly o he discree se of he Gauss Lobao pois. Remark 3.3. The esimae i Theorem3.3 is of order h 4 p 4 (whe = h ad p = q). I he osabilized case (see.chaper2), for vaishig gap fucio g = 0, we obai a covergece rae of order h 4 p 4 similar o he sabilized case, whe we assume he if-sup codicio o hold. For he compleeess of he covergece aalysis we also cosider he h-versio for p = ad q =. I order o esimae he erms A, ad A,, we have o defie he space of he discree Lagrage muliplier iroduced i [36, 37, 45] for he FEM. M, := {λ W : λ 0 o } 62

73 3.3 A priori error aalysis for fricioal coac problem M, := {λ W : λ F o } where W := {µ C( ) : J T, µ J P (J)} I is a coformig discreizaio o muliplier as M, M ad M, M. We obai he followig a priori error esimae which proposes a covergece rae of O(h 4 ) (whe = h) Corollary 3.3. Le (u, λ) V M be he soluio of he problem (3.4) ad (u h, λ ) he soluio of he discree problem (3.8). Assume ha u +ν (Σ), λ ν ( ) for some ν [0, 2 ]. Suppose ha λ ν ( )+ λ ν ( )+ F L2 ( ) u +ν (Σ). The here exiss a cosa C > 0 idepede of h ad, such ha u u h + ψ 2 (Σ) ψh 2 + γ 2 (λ λ ) L2 (Γ (Γ) C )+ γ 2 (λ λ ) L2 ( ) C (h ) 4 + ν+ 2 + h 2 ν u +ν (Σ) (3.04) Proof. I is sufficie o esimae he erms A, ad A,. We have o esimae A, = if µ M (λ µ )u ds (3.05) Seig µ = 0, sice u 0 ad λ 0 o, we have A, = We ow esimae he erm if µ M (λ µ )u ds λ u ds 0 (3.06) A, = if µ M (F) (λ µ )u ds (3.07) We cosider he Lagrage ierpolaio operaor I h defied o he Gauss-Lobao pois mappig oo V h, where V h is he space of coiuous piecewise polyomials for he discreizaio of u for p =. We have λ I h u ds F I h u ds (3.08) 63

74 3 Sabilized mixed hp-bem i Liear Elasiciy choosig µ = λ ad use he codio λ u F u = 0 we ge (λ λ )u ds = (λ λ )(u I h u ) ds + (λ λ )I h u ds + λ u ds F u ds (λ λ )(u I h u ) ds + λ (u I h u ) ds + F ( I h u u ) ds (λ λ )(u I h u ) ds + λ (u I h u ) ds + F ( I h u u ) ds ad we obai (λ λ )u ds γ 2 (λ λ ) L2 ( )γ 2 u I h u L2 ( )+ λ L2 ( ) u I h u L2 ( ) (3.09) + F L2 ( ) u I h u L2 ( ) ɛ γ 2 (λ λ ) 2 L 2 ( ) + C ɛγ 0 h 2ν+ u 2 ν+ (Σ) +Ch ν+ u 2 ν+ (Σ) (3.0) The corollary is esablished by employig he esimaio of he erms i Theorem Reliable ad efficie a poseriori error esimaes for sabilized hp-bem for fricioal coac problems 3.4. Reliabiliy of he BEM a poseriori error esimae Le (u, λ) be he soluio of he coiuous problem (3.4) ad (u hp, λ q ) he soluio of he discree problem (3.8). We ow derive a upper boud for u u hp 2, where u u hp 2 := u u hp 2 + ψ ψ hp 2 + λ 2 (Σ) 2 (Σ) λq 2 2 ( ) (3.) 64

75 3.4 Reliable ad efficie a poseriori error esimaes for sabilized hp-bem for fricioal coac problems Lemma 3.7. ([9].Lemma 3.2.9). There exiss a operaor Π hp : 2 (Σ) V hp, which is sable i he 2 -orm ad has he quasiopimal approximaio propriees i he L 2 - orm,ie. here exiss a cosa C, idepede of h ad p such ha for all u 2 (Σ) here holds Π hp u C u 2 (Σ) 2 (Σ) ( ) ( ɛ) h 2 u Π hp u L2 (Σ) C u p 2 (Σ) (3.2) (3.3) wih arbirary small ɛ (0; 2 ) Theorem 3.4. Le (u, λ) be he soluio of he problem (3.4) ad (u hp, λ q ) he soluio of he discree problem (3.8). The here holds he esimae for ɛ > 0 arbirary: u u hp 2 + ψ ψ hp 2 η 2 (Σ) 2 h 2 (I) + (λq ) +, (g u hp ) + ΓC (Σ) I T hp + (λ q ) 2 2 ( ) + 4ɛ (u hp g) ( ) + ɛ λ λ q 2 2 ( ) ( + ( λ q F) u hp + 2(λ q + ( λ q F) ( ) u hp ) ) ds where for a arbirary segme I ( ) ɛ ( ) ɛ ηh 2 (I) = hi S hpu hp 2 L p 2 (I Γ N ) + hi ( λ q ) S hpu hp 2 L I p 2 (I ) I + h I p 2 λ q I + σ(u h hp ) 2 L 2 (I ) +h I p 2 λ q + σ h (u hp ) 2 L 2 (I ) I + h I s (V (ψ hp ψhp )) 2 L 2 (I). Proof. Sice S,, V, are posiive defiie, here exis cosas c s, c v > 0 such ha c s u u hp 2 2 (Σ)+c v ψ ψ hp 2 (Σ) Su S hpu hp, u u hp + V (ψ hp ψhp ), ψ ψ hp. (3.4) 65

76 3 Sabilized mixed hp-bem i Liear Elasiciy The Galerki orhogoaliy propery (Lemma 3.4) shows ha c s u u hp 2 2 (Σ)+c v ψ ψ hp 2 (Σ) Su S hp u hp, u u hp Σ + V (ψp ψ hp ), ψ ψ hp Σ + Su S hp u hp, u hp v hp Σ + b(λ λ q, u hp v hp ) + γ(λ q + σ(u h hp ))σ(u h hp v hp ) ds + γ(λ q + σ h (u hp ))σ h (u hp v hp ) ds Su S hp u hp, u v hp + V (ψhp ψhp ), ψ ψ hp + b(λ λ q, u hp v hp ) + γ(λ q + σ(u h hp ))σ(u h hp v hp ) ds + γ(λ q + σ h (u hp ))σ h (u hp v hp ) ds, u v hp ΓN b(λ, u v hp ) S hp u hp, u v hp + V (ψhp ψhp ), ψ ψ hp + b(λ λ q, u hp v hp ) + γ(λ q + σ(u h hp ))σ(u h hp v hp ) ds + γ(λ q S hp u hp, u v hp ΓN + ( λ q ) S hp u hp, u v hp ΓC + b(λ λ q, u hp u) + V (ψhp ψhp ), ψ ψ hp + γ(λ q + σ(u h hp ))σ(u h hp v hp ) ds + γ(λ q + σ h (u hp ))σ h (u hp v hp ) ds + σ h (u hp ))σ h (u hp v hp ) ds (3.5) We esimae he firs ad he secod erms, employig he Cauchy-Schwarz iequaliy: A = S hp u hp, u v hp ΓN S hp u hp L2 (I) u v hp L2 (I) (3.6) I T hp Γ N B = ( λ q ) S hp u hp, u v hp ΓC ( λ q ) S hp u hp L2 (I) u v hp L2 (I) I T hp (3.7) We apply a resul from [5], we obai D := V (ψhp ψhp ), ψ ψ hp V (ψhp ψhp ) 2 ψ hp ψ (Σ) 2 (Σ) c h I 2 s (V (ψ hp ψhp )) 2 L 2 (I) ψ hp ψ. (3.8) 2 (Σ) I T hp Usig lemma 3.7, we choose v hp = u hp + Π hp (u u hp ) o obai he followig esimae: ere ω(i) is a eighbourhood of I. ( ) ( ɛ) u v hp L 2 (I) C hi 2 u u hp p I 2. (3.9) (ω(i)) 66

77 3.4 Reliable ad efficie a poseriori error esimaes for sabilized hp-bem for fricioal coac problems The coiuiy of S hp ad Youg s iequaliy imply p h 2 σ h (v hp u hp ) L2 (Γ) = p h 2 σ h (Π hp (u u hp )) L2 (Γ) p h 2 C Πhp (u u hp ) (Γ) C Π hp (u u hp ) 2 (Γ) CC u u hp 2 (3.20) (Γ) Sice v hp = u hp + Π hp (u u hp ), we ge E = γ(λ q + σ(u h hp ))σ(u h hp v hp ) ds = + σ(u h hp ))σ(π h hp (u hp u)) ds I I I γ(λ q γ 0 I γ 0 h I p 2 I I Ĉ h I p 2 I I h 2 I p I λ q λ q (λ q + σ(u h hp )) h 2 I p I σ h (Π hp (u u hp )) ds 2 + σ(u h hp ) 2 h I L 2 (I) p 2 σ(π h hp (u u hp )) 2 L 2 (I) I I + σ(u h hp ) 2 L 2 (I) 2 u u hp 2 (Γ). (3.2) 2 Similarly o (3.2), we obai E = γ(λ q + σ h (u hp ))σ h (u hp v hp ) ds Ĉ 2 λ q + σ h (u hp ) 2 L 2 (I) u u hp 2. (3.22) (Γ) h I p 2 I I Fially we esimae he erm C = b(λ λ q, u hp u) = λ λ q, u hp u + λ λ q, u hp u. (3.23) Usig he codiio λ, u g = 0 ad (λ q ) +, u g 0 where v + = max{0, v} 67

78 3 Sabilized mixed hp-bem i Liear Elasiciy ad v = mi{0, v}, i.e. v = v + + v. λ λ q, u hp u = λ (λ q ) +, u hp = λ (λ q ) +, u hp u (λ q ), u hp u g + g u (λ q ), u hp u = λ (λ q ) +, u hp g + λ, g u (λ q ) +, g u (λ q ), u hp u λ (λ q ) +, u hp g (λ q ), u hp u = (λ q ) +, g u hp λ, (g u hp ) + + (g u hp ) (λ q ), u hp u (λ q ) +, g u hp + λ q λ (λ q (λ q ), u hp u = (λ q ) +, (g u hp ) + + λ q λ, (g u hp ) (λ q ), (g u hp ) (λ q ), u hp u = (λ q ) +, (g u hp ) + + λ q λ, (g u hp ) (λ q ), u hp u (λ q ) 2 ( ) uhp u 2 (Σ) + λ q λ (g uhp 2 ( ) ) 2 (ΓC ) + (λq ) + (λ q ), (g u hp ) By he coac codiio, λ u + F u = 0. Seig λ = ξf wih ξ λ λ q, u hp u = λ u + λ q u + λ u hp = F u + λ q u + ξfu hp F u + λ q ( λ q ( λ q u + F u hp F) + u + F u hp F) + u u hp λ q u hp λ q u hp λ q λ q + ( λ q u hp u hp F) + u hp ( λ q F) + u u hp 2 ( ) 2 (ΓC ) + [( λ q F) + ( λ q F)] u hp λ q ( λ q F) + u u hp 2 ( ) 2 (ΓC ) + ( λ q F) u hp + 2(λ q u hp ) +, (g u hp ) + ΓC. + F u hp λ q + λ q u hp (3.24) u hp u hp ). (3.25) Usig Youg s iequaliy we move he erm u u hp 2 (Σ) esimae of he heorem follows. o he lef had side. The Now we fid a upper boud o he discreizaio error λ λ q. Lemma 3.8. Le λ solve he saddle poi problem (3.4), ad λ q he soluio of he 68

79 3.4 Reliable ad efficie a poseriori error esimaes for sabilized hp-bem for fricioal coac problems discree problem (3.8). The here holds ( ) λ λ q 2 C u u hp 2 + ψ ψ hp 2 + C ξ 2 ( ) 2 (Σ) 2 h 2 (I). (Γ) I T hp where ξ 2 h (I) = ( hi p I + h I p 2 λ q I ) ɛ ( ) ɛ S hpu hp 2 L 2 (I Γ N ) + hi ( λ q ) S hpu hp 2 L p 2 (I ) I + σ(u h hp ) 2 L 2 (I ) +h I p 2 λ q + σ h (u hp ) 2 L 2 (I ) I (3.26) (3.27) Proof. Le v V ad v hp := Π hp v V hp we have λ λ q, v = λ λ q, v v hp + λ λ q, v hp Usig Galerki orhogoaliy ad he formulaio (3.4) we obai λ λ q, v = λ λ q, v v hp Su S hp u hp, v hp Σ γ(λ q + σ(u h hp ))σ(v h hp ) ds γ(λ q =, v v hp Su, v v hp λ q, v v hp + σ h (u hp ))σ h (v hp ) ds Su S hp u hp, v Σ Su S hp u hp, v hp v Σ γ(λ q + σ(u h hp ))σ(v h hp ) ds γ(λ q + σ h (u hp ))σ h (v hp ) ds = S hp u hp, v v hp ΓN + ( λ ) S hp u hp, v v hp ΓC Su S hp u hp, v Σ γ(λ q + σ(u h hp ))σ(v h hp ) ds γ(λ q + σ h (u hp ))σ h (v hp ) ds We esimae he erms o he righ had side: A = S hp u hp, v v hp ΓN S hp u hp L2 (I) v v hp L2 (I) (3.28) I T hp Γ N B = ( λ q ) S hp u hp, v v hp ΓC ( λ q ) S hp u hp L2 (I) v v hp L2 (I) I T hp (3.29) C = Su S hp u hp, v Σ (C s + C Ehp ) u u hp 2 (Σ) v 2 (Σ) + C 0 ψ ψ hp 2 (Γ) v 2 (Σ). (3.30) 69

80 3 Sabilized mixed hp-bem i Liear Elasiciy Usig he iverse iequaliy as well as Lemma 3.7: D = γ(λ q + σ(u h hp ))σ(v h hp ) ds γ(λ q + σ(u h hp ))σ(π h hp v) I T I hp = γ 0 h 2 I (λ q + σ h p (u hp )) h 2 I σ I p (Π h hp v) I I T hp I Ĉ h I p 2 I I λ q + σ(u h hp ) 2 L 2 (I) 2 v 2 (Γ). (3.3) Similarly o (3.3), we ge D = γ(λ q + σ h (u hp ))σ h (v hp ) ds Ĉ h I p 2 I I λ q + σ h (u hp ) 2 L 2 (I) 2 v 2 (Γ) (3.32) Recallig v hp = Π hp v i A ad B ad Lemma 3.7 we obai ( ) λ λ q, v (C s + C Ehp ) u u hp +C 2 0 ψ ψ hp (Σ) 2 (Γ) + C ( ) ɛ hi 2 S hp u hp p L2 (I) v I 2 (Σ) I T hp Γ N + C ( ) ɛ hi 2 ( λ q ) S hp u hp L2 (I) v 2 (Σ) I T hp Γ N + Ĉ I h + Ĉ I h 2 I p I λ q + σ p (u h hp ) L2 (I) v I 2 (Γ) 2 I λ q + σ h (u hp ) p L2 (I) v I 2 (Γ) v 2 (Σ) (3.33) By defiiio of he dual orm ad (a+b) 2 2a 2 +2b 2, he asserio (3.26) follows. Theorem 3.5. Le (u, λ) be he soluio of problem (3.4) ad (u hp, λ q ) he soluio 70

81 3.4 Reliable ad efficie a poseriori error esimaes for sabilized hp-bem for fricioal coac problems of he discree problem (3.8). The he followig a poseriori esimae holds: u u hp 2 + ψ ψ hp 2 + λ 2 (Σ) 2 (Σ) λq 2 2 ( ) (I) + (λq ) +, (g u hp ) + ΓC where I T hp η 2 h + (λ q ) 2 2 ( ) + (uhp g) ( ) + ( λ q F) ( ) ( + ( λ q F) u hp + 2(λ q u hp ) ) ds ( ) ɛ ( ) ɛ ηh 2 (I) = hi S hpu hp 2 L p 2 (I Γ N ) + hi ( λ q ) S hpu hp 2 L I p 2 (I ) I + h I p 2 λ q I + σ(u h hp ) 2 L 2 (I ) +h I p 2 λ q + σ h (u hp ) 2 L 2 (I ) I + h I s (V (ψ hp ψhp )) 2 L 2 (I) (3.34) Proof. The esimae follows immediaely from Lemma 3.8 ad Theorem Efficiecy of he BEM a poseriori error esimae I his secio we derive a efficiecy of he a poseriori error esimaes. We follow ideas of [5], [6], [9]. Assumpio 3.. There exiss a cosa C > 0 such ha I, J T hp ad C is idepede of I,J. Le I J < C, p I p J < C (3.35) h max := max I T hp I, p max := max I T hp p I, h mi := mi I T hp I p mi := mi I T hp p I For he simpliciy of he he preseaio, we assume ha he gap fucio g = 0. Lemma 3.9. There exiss a cosa c > 0 such ha for ay eleme I T hp he local 7

82 3 Sabilized mixed hp-bem i Liear Elasiciy error idicaor η h (I) saisfies cγ 0 η 2 h (I) h I p I W (u u hp ) 2 L 2 (I Σ) +h I p I (K + 2 )(ψ ψhp ) 2 L 2 (I Σ) + h I s (V (ψ ψhp )) 2 L 2 (I) +h I s (K + 2 )(u uhp ) 2 L 2 (I) + p I γ 2 (λ λ q ) 2 L 2 (I ) + γ 2 σ (u u hp ) 2 L 2 (I ) + γ 2 σ (u u hp ) 2 L 2 (I ) (3.36) h he sabilizaio parameer γ is defied o each eleme I as γ = γ I 0, where γ p 2 0 > 0 is I idepede of h ad p ad γ 0 is chose small eough, where η 2 h = h I p I S hp u hp 2 L 2 (I Γ N ) +h I p I ( λ q ) S hp u hp 2 L 2 (I ) + h I p 2 λ q I + σ (u hp ) 2 L 2 (I ) +h I p 2 λ q + σ (u hp ) 2 L 2 (I ) I + h I s (V (ψ hp ψhp )) 2 L 2 (I) +h I p 2 I E(u h hp ) 2 L 2 (I ) +h I p 2 E h (u hp ) 2 L 2 (I ) I Proof. Noig ha = Su ΓN for he exac soluio u, we obai for I Γ N : γ 0 h I p I S hp u hp 2 L 2 (I Γ N ) = γp I Su S hp u h 2 L 2 (I Γ N ) = γp I W (u u hp ) + (K + 2 )(ψ ψhp ) 2 L 2 (I Γ N ) 2γp I W (u u hp ) 2 L 2 (I Γ N ) + 2γp I (K + 2 )(ψ ψhp ) 2 L 2 (I Γ N ). Noe ha λ = Su ΓC. The we obai for I h ( ) I γ 0 ( λ q ) S hp u hp 2 L p 2 (I) 2γp I λ λ q 2 L 2 (I) + Su S hpu hp 2 L 2 (I) I 4γp I W (u u hp ) 2 L 2 (I) + 4γp I (K + 2 )(ψ ψhp ) 2 L 2 (I) ad for ay I Γ we have + 2p I γ 2 (λ λ q ) 2 L 2 (I) γ 0 h I s (V (ψhp ψ hp )) 2 L 2 (I) = γ 0h I s (V (ψhp (K + 2 )uhp )) 2 L 2 (I) 2γ 0 h I s (V (ψ ψhp )) 2 L 2 (I) + 2γ 0 h I s (K + 2 )(u uhp ) 2 L 2 (I). 72

83 3.4 Reliable ad efficie a poseriori error esimaes for sabilized hp-bem for fricioal coac problems h I γ 0 p 2 λ q + σ (u hp ) 2 L 2 (I ) = γ 2 (λ λ q ) 2 L 2 (I) + γ 2 σ (u u hp ) 2 L 2 (I ) I Similarly o (3.37), we ge γ 2 (λ λ q ) 2 L 2 (I) + γ 2 σ (u u hp ) 2 L 2 (I ) (3.37) h I γ 0 p 2 λ q + σ (u hp ) 2 L 2 (I ) γ 2 (λ λ q ) 2 L 2 (I) + γ 2 σ (u u hp ) 2 L 2 (I ) I (3.38) γ 0 h I p 2 I E h (u hp ) 2 L 2 (I ) = γ 0 Fially we have γ 0 h I p 2 I h I p 2 I γ 0 h I p 2 I γ 0 h I p 2 I (S S hp )u hp 2 L 2 (I ) h I p 2 I Su S hp u hp 2 L 2 (I ) +γ 0 S(u u hp ) 2 L 2 (I ) ( W (u u hp ) 2 L 2 (I ) + (K + 2 )(ψ ψhp ) 2 L 2 (I ) + γ 2 σ (u u hp ) 2 L 2 (I ) (3.39) ( E h (u hp ) 2 L 2 (I ) γ h I 0 p 2 W (u u hp ) 2 L 2 (I ) + (K + ) I 2 )(ψ ψhp ) 2 L 2 (I ) + γ 2 σ (u u hp ) 2 L 2 (I ). (3.40) ) Lemma 3.0. Le P hp be he L 2 (Γ)-projecio operaor oo W hp, i.e P hp : L 2 (Γ) W hp such ha P hp ψ ψ, Φ = 0, Φ W hp (3.4) ad for ay real umbers µ 0 here exiss a cosa C such ha ψ µ (Γ) ψ P hp ψ L2 (Γ) C ( ) h µ ψ p µ(γ). (3.42) The here holds ψ P hp ψ 2 2 (Γ) C ( h p ) ψ 2 L 2 (Γ). (3.43) I paricular ψ P hp ψ 2 2 (Γ) C ( h p ) ψ P hp ψ 2 L 2 (Γ). (3.44) 73

84 3 Sabilized mixed hp-bem i Liear Elasiciy Proof. We observe ha ad ψ P hp ψ 2 (Γ) = ψ P hp ψ 2 (Γ) = sup φ 2 (Γ) = sup φ 2 (Γ) = sup φ 2 (Γ) ψ L2 (Γ) C sup φ 2 (Γ) = sup = C φ 2 (Γ) sup φ 2 (Γ) ψ P hp ψ, φ φ 2 (Γ) ψ P hp ψ, φ P hp φ φ 2 (Γ) ψ, φ P hp φ φ 2 (Γ) sup φ 2 (Γ) φ P hp φ L2 (Γ) φ 2 (Γ) ( ) h 2 ψ L2 (Γ) (3.45) p ψ P hp ψ, φ φ 2 (Γ) ψ P hp ψ, φ P hp φ φ 2 (Γ) ψ P hp ψ L2 (Γ) φ P hp φ L2 (Γ) φ 2 (Γ) ( ) h 2 ψ Php ψ p L2 (Γ) (3.46) Theorem 3.6. Le (u, λ) be he soluio of problem (3.4) ad (u hp, λ q ) he soluio of he discree problem (3.8). Assume ha λ L 2 ( ). The here exiss a cosa c(γ 0 ) idepede of h, p, ad q such ha c(γ 0 ) where I T hp η 2 h (I) Cp2 max + h 2 max ( ) u u hp 2 + ψ 2 (Σ) ψhp 2 + γ 2 (λ λ q ) 2 2 (Γ) L 2 ( ) ( ) u ψ 2 2 (Σ) 2 (Γ) η 2 h (I) = h I p I S hp u hp 2 L 2 (I Γ N ) +h I p I ( λ q ) S hp u hp 2 L 2 (I ) + h I p 2 λ q I + σ (u hp ) 2 L 2 (I ) +h I p 2 λ q + σ (u hp ) 2 L 2 (I ) I + h I s (V (ψ hp ψhp )) 2 L 2 (I) +h I p 2 I (3.47) E(u h hp ) 2 L 2 (I ) +h I p 2 E h (u hp ) 2 L 2 (I ) I 74

85 3.4 Reliable ad efficie a poseriori error esimaes for sabilized hp-bem for fricioal coac problems ad we have (λ q ) + (u hp ) + ds λ λ q L2 ( ) u u hp L2 ( ) + λ L2 ( ) u u hp L2 ( ) (λ q ) 2 2 ( ) λ λq 2 L 2 ( ) ( ( λ q F) u hp + 2(λ q u hp ) ) ds λ λ q L2 ( ) u u hp L2 ( ) + λ λ q L2 ( ) u L2 ( ) + λ L2 ( ) u u hp L2 ( ) ( λ q F) ( ) λ λq L2 ( ) (u hp ) ( ) p2 max h mi u u hp L2 ( ). (3.48) Proof. Usig he coiuiy of W ad K ad Lemma3.9 we obai h max γ 0 W (u u hp ) 2 L p 2 (Σ) Cγ h max 0 u u hp 2 (3.49) mi p (Γ) mi γ 0 h max s (K + 2 )(u uhp ) 2 L 2 (Γ) γ 0h max (K + 2 )(u uhp ) 2 (Γ) Cγ 0 h max u u hp 2 (Γ) (3.50) h max γ 0 (K + p mi 2 )(ψ ψhp ) 2 L 2 (Γ) γ h max 0 ψ ψ hp 2 L p 2 (Γ) (3.5) mi γ 0 h max s (V (ψ ψhp )) 2 L 2 (Γ) γ 0h max V (ψ ψ hp ) 2 (Γ) γ 0 h max ψ ψ hp 2 L 2 (Γ) (3.52) 75

86 3 Sabilized mixed hp-bem i Liear Elasiciy γ 2 σ (u u hp ) 2 L 2 ( ) γ 0 h max p 2 σ (u u hp ) 2 L 2 ( ) mi γ 0 h max p 2 mi ( ) σ (u v hp ) 2 L 2 ( ) + σ (v hp u hp ) 2 L 2 ( ) (3.53) The coiuiy codiio of he Dirichle-o-Neuma operaor, Youg s iequaliy ad we choose v hp = u hp + Π hp (u u hp ) we obai h max γ 0 p 2 σ (v hp u hp ) 2 L 2 (Γ) = γ h max 0 mi p 2 σ (Π hp (u u h )) 2 L 2 (Γ) mi γ 0 h max p 2 mi C Π hp (u u h ) 2 (Γ) Cγ 0 Π hp (u u h ) 2 2 (Γ) Cγ 0 u u h 2 2 (Γ) (3.54) We employ he coiuiy codiio of he Dirichle-o-Neuma operaor ad choosig v hp = I hp u we obai h max γ 0 p 2 σ (u v hp ) 2 L 2 ( ) Cγ h max 0 mi p 2 u v hp 2 (Γ) mi Cγ 0 h max p 2 mi u I hp u 2 (Γ) we obai h max γ 0 p 2 σ (u u hp ) 2 L 2 ( ) = Cγ 0 mi Similarly o (3.57), we ge h max γ 0 p 2 σ (u u hp ) 2 L 2 ( ) = Cγ 0 mi Cγ 0 h 2 max p 3 mi u (Σ) ( ) u u hp 2 2 (Γ) +h2 max p 3 u mi (Σ) ( ) u u hp 2 2 (Γ) +h2 max p 3 u mi (Σ) (3.55) (3.56) (3.57) h max γ 0 u u hp 2 p (Γ) γ h max 0 u I hp u 2 mi p (Γ) mi +γ 0 h max p mi I hp u u hp 2 (Γ) (3.58) We apply he iverse iequaliy for he secod erm we obai h max γ 0 I hp u u hp 2 p (Γ) γ h max p 2 max 0 I hp u u hp 2 mi p mi h mi 2 (Γ) ( ) Cγ 0 p max u u hp 2 + I hpu u 2 2 (Σ) 2 (Σ) (3.59) 76

87 3.4 Reliable ad efficie a poseriori error esimaes for sabilized hp-bem for fricioal coac problems we have h max γ 0 u I hp u 2 p (Γ) γ h 2 max 0 mi p 2 u mi (Σ) (3.60) ad here holds γ 0 h max p mi u u hp 2 (Γ) Cγ 0 γ 0 p max u I hp u 2 γ h 2 max 2 0 u 2 3 (Γ) p mi 2 (Σ) ( ) p max u u hp 2 2 (Σ) +h2 max u 2 3 p mi 2 (Σ) (3.6) (3.62) ad we have γ 0 h max u u hp 2 (Γ) γ 0p 2 max u u hp 2 2 (Σ) +γ 0h 2 max u (Σ) (3.63) h max ψ ψ hp 2 L 2 (Γ) h max ψ P h ψ 2 L 2 (Γ) +h max P h ψ ψ hp 2 L 2 (Γ) (3.64) usig he iverse iequaliy ad lemma(3.0) for he secod erm h max P h ψ ψ hp 2 L 2 (Γ) Cp2 max ψ ψ hp 2 2 (Γ) +Cp2 max P h ψ ψ 2 2 (Γ) Cp 2 max ψ ψ hp 2 2 (Γ) +Ch2 max ψ 2 2 (Γ) (3.65) Employig lemma(3.0) for he firs erm we obai ad γ 0 h max ψ ψ hp 2 L 2 (Γ) Cγ 0 h max ψ P h ψ 2 L 2 (Γ) C h2 max p mi ψ 2 2 (Γ) ( ) p 2 max ψ ψ hp 2 2 (Γ) +h2 max ψ 2 2 (Γ) ( ) h max γ 0 ψ ψ hp 2 L p 2 (Γ) Cγ 0 p max ψ ψ hp 2 mi 2 (Γ) +h2 max ψ 2 p mi 2 (Γ) (3.66) (3.67) (3.68) Fially we ge c(γ 0 ) I T hp η 2 h (I) Cp2 max + h 2 max ( ) u u hp 2 + ψ 2 (Σ) ψhp 2 + γ 2 2 (λ λ q ) 2 (Γ) L 2 ( ) ( ) u ψ 2 2 (Σ) 2 (Γ) (3.69) Sice u 0, we have 0 (u hp ) + u hp u o, (3.70) 77

88 3 Sabilized mixed hp-bem i Liear Elasiciy ad 0 (u hp ) + L2 ( I) u u hp L2 ( I). (3.7) If I T hp, le J I be he par of he edge where (λ q ) + = λ q I (λ q ) + (u hp ) + ds = J J λ q (u hp ) + ds λ q λ (u hp ) + ds + λ (u hp ) + ds J λ λ q L2 (I ) u u hp L2 (I ) + λ L2 (I ) u u hp L2 (I ) (3.72) Noig ha λ 0 o he we have 0 (λ q ) λ λ q o ad we obai (λ q ) 2 2 ( ) (λq ) 2 L 2 ( ) λ λ q 2 L 2 ( ) λ λq L2 ( ) (3.73) We ow esimae he erm ( ( λ q F) u hp + 2(λ q u hp ) ) ds (3.74) Sice λ u 0 o we have 0 (λ q we obai (λ q I u hp ) λ u λ q u hp λ (u u hp ) + (λ λ q u hp ) ds λ λ q L2 (I ) u u hp L2 (I ) )(u u hp ) + (λ λ q )u (3.75) + λ λ q L2 (I ) u L2 (I )+ λ L2 (I ) u u hp L2 (I ). (3.76) We esimae he secod erm i (3.74), If I T hp, le J I be he par of he edge where ( λ q F) = ( λ q F) 78

89 3.5 Numerical Experimes Sice λ F we obai ( λ q F) u hp ds = I = + J J J J ( λ q ( λ q ( λ q F) u hp ds λ F + λ ) u hp ds λ )( u hp u ) ds+ ( λ q λ ) u ds λ λ q L2 (I ) u u hp L2 (I ) We ow esimae he erm + λ λ q L2 (I ) u L2 (I ) (3.77) ( λ q F) ( ) (3.78) we have Sice λ F we ge Fialy we obai ( λ q F) ( ) ( λq F) + 2 L 2 ( ) (3.79) ( λ q F) + λ q λ F + F λ q λ. (3.80) ( λ q F) ( ) ( λq F) + 2 L 2 ( ) (λ λq ) L2 ( ) (3.8) Usig he iverse iequaliy, we obai From (3.7), we have (u hp ) ( I) p2 I h I (u hp ) + 2 L 2 ( I). (3.82) (u hp ) ( I) p2 I h I u u hp L2 ( I). (3.83) 3.5 Numerical Experimes Numerical resuls are preseed wih he MATLAB package of L.Baz for he coac of he wo-dimeioal elasic body Ω = [ 0.5, 0.5] 2 wih a rigid obsacle. = 79

90 3 Sabilized mixed hp-bem i Liear Elasiciy [ 0.5, 0.5] { 0.5} ad Γ N = Ω \ ( Γ D ). The Youg s modulus ad he Poisso raio are E = 000, ν = 0.3 respecively. The Neuma force is = (0, 0) T, he gap g = x ,he sabilizaio parameer γ = 0 3 h ad he give fricio coefficie p 2 F = 0.3. I Figure3. we show he iiial ad he deformed cofiguraio. Figure 3.2 shows he esimaed errors for he h-uiform for he sabilized ad he o-sabilized problems, ad hp-adapive mehods for he o-sabilized problem. 80

91 3.5 Numerical Experimes Γ D Γ N Γ N Figure 3.: Iiial(lef) ad Deformed (righ) geomery 0 2 ui. h, p=, biorho hp adapive, biorho ui. h, p=, sabilized Residual Error Esimae Degrees of Freedom Figure 3.2: Covergece for sabilized ad o-sabilized problems 8

92

93 4 hp-bem for Sochasic Coac Problems i Liear Elasiciy Sochasic mehods are, as heir ame suggess, based o he use of probabiliy o model uceraiy. The objecive is o sudy he effecs o he oupu uceraiy o he ipu parameers, cosidered give, mechaical models. The kowledge abou he uceraiies of he ipu daa ca be modeled as: Radom variables, i.e fucios depedig oly o he hazard, so deermiisic i space ad ime. Sochasic fields, i.e fucios depedig o boh he hazard ad space. Sochasic process, i.e fucios depedig o he hazard ad ime. I his chaper we prese a sochasic mixed BEM formulaio for coac problems wih Tresca fricio.for he heoreical reame of radom variaioal iequaliies see [29], [3] We show ha he sochasic mixed formulaio is well-posed. We sudy he deformaio of a elasic homogeeous maerial i which Youg s modulus (parameer ha characerize he maerial properies) is a radom variable. Similary he surface force = (x, ω) ad he gap fucio g are assumed o be radom as well. 4. Mixed Formulaio for Sochasic Coac Problem Le D R 2 be a bouded Lipschiz domai wih boudary Γ := Ω = Γ N Γ D decomposed io he o-iersecig Neuma segme Γ N, he Dirichle segme Γ D ad he coac segme which poeially ca come i coac wih he rigid foudaio. Furher le (Ω, B, P ) be a probabiliy space wih he se of oucomes, B 2 Ω he algebra of eves ad P : B [0, ] a probabiliy measure. Le σ(u), ɛ(u) ad C deoe he sress esor, he liearized srai esor ad he elasiciy esor, respecively. Furher o u ad will deoe he displaceme field ad he ouward ui ormal. We defie κ(x) for each x D as a radom variable κ : D Ω R o he probabiliy space (Ω, B, P ). As a cosequece κ : D Ω R is a radom field ad oe obais he real umber κ(x, ω) for each realizaio ω Ω. 83

94 4 hp-bem for Sochasic Coac Problems i Liear Elasiciy Le he oucome of he elasiciy esor be a radom field C(x,.), x D, defied o a probabiliy space (Ω, B, P ). The problem he cosiss i fidig he displaceme fields u : D Ω R such ha P-a.e i Ω div σ(u(x, ω)) = f(x, ω) i D (4.a) σ(u(x, ω)) = C(x, ω) : ɛ(u(x, ω)) i D (4.b) u(x, ω) = 0 o Γ D (4.c) σ(u(x, ω)) = (ω, x) o Γ N (4.d) σ 0, u (x, ω) g(x, ω), σ (u (x, ω) g(x, ω)) = 0 i (4.e) σ F, σ u (x, ω) + F u (x, ω) = 0 o (4.f) Bocher spaces are a geeralizaio of he cocep of L p spaces o fucios whose values lie i a Baach space which is o ecessarily he space R or C of real or complex umbers. Le a measure space (T, Σ, µ) wih Σ a sigma algebra over T ad µ a measure o Σ be give, he for a Baach space (X, X ), he Bocher space L p (T ; X) is defied as he space of all measurable fucios u : T X such ha he orm u L p (T ;X) = ( u() p X dµ()) p < + for p (4.2) is fiie. T u L (T ;X) := ess sup T u() X < + if p = + (4.3) I he case where p = 2 ad X is a separable ilber space, we have by Fubii ha: L 2 µ(t ; X) = L 2 (T ) X (4.4) where deoes he esor produc bewee ilber spaces Remark 4.. I paricular L 2 µ(t ; X) is iself agai a ilber space Usig he defiiio of Bocher spaces, le L 2 P (Ω; 2 (Σ)) be he esor ilber space of he secod-order radom variables defied o he probabiliy space (Ω, B, P ) wih values i 2 (Σ). Le K P ω be he o empy closed covex subse of secod-order radom variables which saisfy he o-peeraio codiio u g P ω almos surely. (4.5) K P ω := {u L 2 P (Ω; 2 (Σ)) : u g P ω a.s o Ω} 84

95 4. Mixed Formulaio for Sochasic Coac Problem We iroduce some sochasic ermiology: The Karhue-Loève specral decomposiio cosiss i decomposig he radom fields κ wih eigevalues λ m i he form: κ(x, ω) = κ 0 (x) + λm b m (x)ξ m (ω), (4.6) m= where κ 0 is he mea of a radom field κ a he poi x D defied by κ 0 (x) := κ(x, ) := κ(x, ω)dp (ω). (4.7) We approximae he radom fields by a rucaed Karhue-Loève expasio: κ(x, ω) κ 0 (x) + Ω M λm b m (x)ξ m (ω), (4.8) m= This deermies a fiie umber M of idepede radom variable {ξ m } M m= wih mea zero ad ui variace.the umber of radom variables will be deoed M or he umber of he radom dimesios. These radom variables are deermied by ξ m (ω) = (κ(x, ω) κ 0 (x))b m (x) dx (4.9) λm D As a cosequece, he sochasic variaio of he radom fields is ow oly hrough is depedece o he radom variables ξ,, ξ M. A assumpio of fiie dimesioal oise mus be made for each radom ipu R(ω): R(ω) = R(x, ξ (ω), ; ξ M (ω)) =: R(x, ξ(ω)). We deoe he rage space of each variable by Θ m = ξ m (Ω) ad he produc rage space M Θ = Θ m Θ Θ Θ 2 Θ M. m= Sice we have assumed he {ξ m } M m= are idepede ad coiuous, hey have a joi desiy fucio ρ : Θ R ad ρ(y) = ρ (y )ρ 2 (y 2 ) ρ M (y M ) where he ρ m are he correspodig desiy fucios of he ξ m, y m Θ m ad y = (y,, y M ).. Wih his assumpio by he Doob-Dyki Lemma, he displaceme u ca be also described by a fiie umber of radom variables u(x, ω) = u(x, ξ (ω),, ξ M (ω)). (4.0) The goal of he umerical mehods is o seek he soluio u(x, ξ). We ca ow replace L 2 P (Ω; 2 (Σ)) by L 2 ρ(θ; 2 (Σ)) ad have K ρ := {u L 2 ρ(θ; 2 (Σ)) : u g (P ) a.s o Θ}. 85

96 4 hp-bem for Sochasic Coac Problems i Liear Elasiciy The variaioal iequaliy of he secod kid he cosiss i fidig a radom displaceme u K ρ such ha v K ρ A(u, v u) + j(v) j(u) L(v u) v K ρ (4.) wih he biliear form where A(u, v) := Θ S y u, v Σ ρ dy, (4.2) S y := W y + (K y + )Vy (K y ). wih V y, K y, K y ad W y as i i Chaper2 for where G(x, x 2 ) := E(y) { β log x x 2 I + β (x x 2 ) (x x 2 ) 2 x x 2 2 } β = (3 4ν)( + ν), β 2 = + ν 2π( ν) 4π( ν) wih Poisso s raio ν ad Youg s modulus E. The liear form L(v) := The fricio fucioal is give by j(v) = Defie he eergy fucioal J : L 2 ρ(θ; 2 (Σ)) R as Θ Θ, v ΓN ρ dy. (4.3) F u ρ ds dy (4.4) J(v) := A(u, v) + j(v) L(v). (4.5) 2 Remark 4.2. We deoe by {C(x, ), x D}, he radom elasiciy esor field. The biliear form is coiuous ad ellipic, if C(x, ) is uiformly bouded from below [2],[3],[] Lemma 4.. The biliear form A(, ) is a coiuous ellipic biliear form o L 2 ρ(θ; 2 (Σ)) L 2 ρ(θ; 2 (Σ)), i.e. here exis cosas C A > 0 ad c A > 0 such ha for all u, v L 2 ρ(θ; 2 (Σ)) A(u, v) C A u L 2 ρ (Θ; 2 (Σ)) v L 2 ρ(θ; 2 (Σ)) A(u, v) c A u 2 L 2 ρ(θ; 2 (Σ)) (4.6) (4.7) 86

97 4.2 Discreizaio for Sochasic Coac Problem Proof. Follows from he fac ha he Seklov-Poicaré operaor S is coiuous ad 2 (Σ)-coercive ad if he radom elasiciy esor field is uiformely bouded from below ad above [2],[3],[]. We ca formulae he sochasic classical formulaio as a saddle poi problem equivale o he sochasic variaioal iequaliy. The sochasic admissible space for Lagrage muliplier λ is give by { L(F) = µ L 2 ρ(θ; 2 ( )) : Θ µ, v ρ dy } Θ F, v ΓC ρ dy, v V where V = {v L 2 ρ(θ; 2 (ΓC )), v 0} (4.8) The mixed variaioal formulaio of he sochasic coac problem wih fricio is give equivalely i he deermiisic form by (cf.[8]): Fid (u, λ) L 2 ρ(θ; 2 (Σ)) L(F) such ha wih he fucioal A(u, v) + b(λ, v) = L(v) v L 2 ρ(θ; 2 (Σ)) (4.9a) b(µ λ, v) g, µ λ ΓC ρ dy µ L(F) (4.9b) Θ b(λ, u) = Θ u, λ ΓC ρ dy + u, λ ΓC ρ dy Θ We defie he srog poiwise o peeraio codiio o by u (x, y) g(x, y), λ (x, y) 0, λ (x, y)(u(x, y) g(x, y)) = 0 (4.20) ad he srog poiwise fricio codiio reads as λ (x, y) F λ (x, y) < F u (x, y) = 0 (4.2) λ (x, y) = F α R : λ (x, y) = α 2 u (x, y) 4.2 Discreizaio for Sochasic Coac Problem Le T hp deoe a pariio of Γ N, such ha all corers of Γ N ad all ed pois Γ N, Γ D Γ N are odes of T hp. For simpliciy we assume meas( ) > 0 ad 87

98 4 hp-bem for Sochasic Coac Problems i Liear Elasiciy Γ D =. T kq is he mesh of Θ. Furhermore we defie he se of Gauss-Lobao pois G I,hp o each eleme I T hp of correspodig polyomial degree p I ad se G hp := I Thp G I,hp.. Aalogously, le G kq be he affiely rasformed Gauss-Lobao pois i he sochasic domai, wih G kq := J Tkq G J,kq. For p = (p I ) I Thp associae each eleme of T hp wih a polyomial degree p I. Similary, for q = (q J ) J Tkq associae each eleme of T kq wih a polyomial degree q J 0. I paricular, he fiie eleme discreizaio of he associaed deermiisic problem ca be choose compleely idepedely of he sochasic discreizaio, The space V hp ca be spaed by he 2-d odal basis {φ i e k, i =,..., N V, k =, 2}, where e k deoes he i-h ui vecor, φ i he scalar Gauss-Lobao Lagrage basis fucio associaed wih he ode i ad N V he oal umber of he odes. We iroduce (cf.[8]) he coiuous piecewise polyomial space for he discreizaio of u, wih V hp := {u hp 2 (Σ) : I T hp, u hp I [P pi (I)] 2, u hp = 0 o Γ D }, V hp spa{φ i } dimv hp i= ad he piecewise polyomial space of discree racios, V T hp := {φ 2 (Σ) : I Th, φ I [P pi (I)] 2 } spa{φ T i } dimvt hp i=. We ex iroduce a fiie dimesioal subspace W kq := { v L 2 y(θ) : v kq J P pj (J) J T kq } spa{ζi } dimw kq i= of he sochasic parameer space ad approximae he esor produc space L 2 ρ(θ; 2 (Σ)) by he esor produc V hp W kq spa{φ i ζ j : φ i V hp, ζ j W kq } We defie he discree se of admissible displacemes as } K hp,kq := {v V hp W kq : (v hp ) ij i g ij, where g ij is a suiable approximaio of g. We oe i geeral ha K hp,kq is o a subse of K. The weighed gap vecor is give by g ij := D i Dj soch g(x, y)ψ i (x)ζ j (y) dx ρ dy, (4.22) Θ where D i = φ i dx, Dj soch = ρϑ j dy. Θ 88

99 4.2 Discreizaio for Sochasic Coac Problem Furhermore we defie he dual Lagrage space ad he sochasic dual Lagrage space M hp = (V hp ΓC ) = spa {ψ j } dim V hp ΓC j= T kq := spa {ζ j } dim W kq j=. The discree Lagrage muliplier space is give by { L hp,kq := µ hp M hp T kq : µ hp, v hp ρ dy where Θ V hp,kq := {v V hp W kq, v,ij 0}. Θ } F, v hp h ρ dy, v hp V hp,kq, (4.23) The dual or biorhogoal basis fucios ψ i, ζ j saisfy he orhogoaliy relaios ψ i φ j ds = δ ij φ j ds, ζ i ϑ j ρ dy = δ ij ϑ j ρ dy. (4.24) Θ Θ The discree fucio u hp V hp W kq he is of he form ( for i dimv hp ad j dimw hp ) u hp := ij := ij u ij φ i (x)ϑ j (y) (u,ij i + u,ij )φ i (x)ϑ j (y). (4.25) The ormal ad ageial par of he discree fucio u hp V hp W kq are give by u hp := ij u,ij φ i (x)ϑ j (y), (4.26) u hp := ij u,ij φ i (x)ϑ j (y). (4.27) The discree absolue value o of he ageial compoe is u hp h := ij u,ij φ i (x)ϑ j (y). Similary, he discree Lagrage muliplier is give by (for i dimm hp ad j dimt hp ) λ hp := ij := ij λ ij ψ i (x)ξ j (y) (4.28) (λ,ij i + λ,ij )ψ i (x)ξ j (y). (4.29) 89

100 4 hp-bem for Sochasic Coac Problems i Liear Elasiciy We ca express he ormal ad he ageial par of λ hp as follows: λ hp := ij λ,ij ψ i (x)ξ j (y), (4.30) λ hp := ij λ,ij ψ i (x)ξ j (y). (4.3) Lemma 4.2. The space L hp,kq (4.23) is equivale o L hp,kq := µhp := µ ij ψ i ξ j, µ,ij 0, µ,ij F, i dim M hp, j dim T hp ij (4.32) Proof. By meas of he biorhogoaliy relaios φ i ψ j ds = δ ij D i, D i = φ i ds > 0, (4.33) Γ C ϑ i ξ j ρ(y)dy = δ ij Di soch, Di soch = ϑ i ρ(y)dy > 0. (4.34) Θ µ hp L hp,kq ad v hp V hp W kq ca be wrie as Θ v hp := ij µ hp := ij (v,ij i + v,ij )φ i (x)ϑ j (y), (4.35) (µ,ij i + µ,ij )ψ i (x)ξ j (y). (4.36) Similary µ hp := ij µ,ij ψ i (x)ξ j (y), ad he orm of he ageial compoe of v hp is v hp h := ij v,ij φ i (x)ϑ j (y) Iserig µ hp ad v hp i he iequaliy µ hp, v hp ρ dy we ge ad Θ Θ µ hp, v hp ρ dy = ij Θ Θ F, v hp h ρ dy = F ij Sice v ij ad v,ij are arbirary ad D i > 0, D soch j F, v hp h ρ dy, (µ,ij v,ij + µ,ij v,ij )D i D soch j v,ij D i D soch j. > 0, we have µ,ij v,ij + µ,ij v,ij F v,ij. (4.37) 90

101 Sice v hp 0, we ge v,ij 0. Choosig v,ij = 0 i (4.37) 4.2 Discreizaio for Sochasic Coac Problem µ,ij v,ij 0, ad we ge µ,ij 0 Choosig ow v,ij = µ,ij ad v,ij = 0 i (4.37), we obai µ,ij F. We have Θ µ hp, v hp ρ dy = ij (µ,ij v,ij + µ,ij v,ij )D i D soch j (4.38) ad Θ F, v hp h ρ dy = F ij v,ij D i D soch j. (4.39) Sice v,ij 0, we ge for each µ hp L hp,kq µ,ij v,ij + µ,ij v,ij µ,ij v,ij F v,ij. Summig over all pois, we obai he asserio, (see[[4],lemma2.3]). The approximaio S hp of he Poicaré-Seklov operaor is give by S hp := W + (K + 2 )i hp(i hp V i hp) i hp (K + ). (4.40) 2 The discree variaioal iequaliy reads: Fid u hp K hp,kq such ha A h (u hp, v hp ) + j(v hp ) j(u hp ) L(v hp u hp ) v hp K hp,kq, (4.4a) where j(v hp ) = Θ F u hp (x, y) ρ ds dy. The discree biliear form is A h (u hp, v hp ) := S hp u hp, v hp Σ ρ dy = Θ Θ W u hp + (K + 2 )Ψ hp, v hp Σ ρ dy, (4.42) ad Ψ hp V T hp solves for P-a.e. y Θ he auxiliary problem V Ψ hp, v hp (K = + Σ 2 )uhp, v hp v hp V T hp. (4.43) Σ 9

102 4 hp-bem for Sochasic Coac Problems i Liear Elasiciy Lemma 4.3. The discree biliear form A h (u hp, v hp ) is coiuous ad L 2 ρ(θ; 2 (Σ))- coercive. The discree mixed sochasic problem i deermiisic formulaio reads : Fid u hp V hp W kq ad λ hp L hp,kq A h (u hp, v hp ) + b(λ hp, v hp ) = L(v hp ) v hp V hp W kq (4.44a) b(µ hp λ hp, u hp ) g, µ hp λ hp ρ dy µ hp L hp,kq (4.44b) wih he fucioal b(u hp, λ hp ) = Θ Θ u hp, λ hp ρ dy + Θ u hp, λ hp ρ dy. Theorem 4.. There exiss exacly oe soluio o he discree mixed formulaio (4.44). Proof. Uiquiess: We assume ha (u hp formulaio (4.44). The we have A h (u hp uhp 2 ), uhp uhp, λhp 2 ) + b(λhp ) ad (uhp 2, λhp 2 λhp 2, uhp uhp 2 ) solve he discree mixed ) = 0, (4.45) where A h (u hp, v hp ) := b(u hp, λ hp ) = Θ Θ S hp u hp, v hp Σ ρ dy u hp, λ hp ρ dy + Θ u hp, λ hp ρ dy. (4.46) Choosig µ hp = λhp 2 ad µ hp 2 = λhp i (4.44b) we ge Usig Lemma4.3 we obai c u hp uhp 2 2 A L 2 ρ(θ; 2 h (u hp (Σ)) b(λ hp λhp 2, uhp uhp 2 uhp 2 ), uhp uhp Cosequely he firs argume u hp is uique. Sice u hp is uique we have for all v hp V hp W kq ) 0. (4.47) 2 ) + b(λhp λhp 2, uhp uhp 2 ) = 0 (4.48) 0 = b(λ hp λhp 2, vhp ). (4.49) Usig λ hp, λhp 2 ad v hp ad he biorhogoaliy, we obai he relaio 0 = ij [(λ,,ij λ 2,,ij )v,ij + (λ,,ij λ 2,,ij )v,ij ]D i D soch j (4.50) 92

103 4.2 Discreizaio for Sochasic Coac Problem Choosig v,ij = 0 i (4.50) we ge because D i D soch j > 0. (λ,,ij λ 2,,ij )v,ij = 0 ad λ,,ij = λ 2,,ij We choose ow v,ij = 0, ad obai λ,,ij = λ 2,,ij. Cosequely he secod argume λ hp is uique. Exisece: We kow ha he iequaliy (4.44b) ca be wrie as a projecio equaio [44]: ( )) λ hp = P Lhp,kq (λ hp u hp g + r, (4.5) where he map P Lhp,kq sad for he orhogoal projecio oo L hp,kq ad where r > 0 is a arbirary parameer. The fixed poi operaor T is defied as follows [44] : u hp T : L hp,kq L hp,kq ( λ hp P Lhp,kq (λ hp + r From Lemma 4.3 we have u hp u hp g )). (4.52) u hp uhp 2 2 L 2 ρ (Θ;L 2( )) uhp uhp 2 2 L 2 ρ (Θ; 2 (Σ)) c S hp (u hp Θ uhp 2 ), uhp uhp 2 Σ ρ dy (4.53) Usig he oaio δu hp = u hp u hp 2, δλ hp = λ hp λ hp 2, = L 2 ρ(θ;l 2 ( )), we compue: ( ( T (λ hp ) T (λhp 2 ) 2 P Lhp,kq λ hp + r δλ hp + r ( δu hp δu hp δλ hp + rδu hp 2 ) u hp, g u hp, 2 )) P Lhp,kq ( δλ hp 2 +2rb(δλ hp, δu hp ) + r 2 δu hp 2 δλ hp 2 2r S hp δu hp, δu hp Σ ρ dy + r 2 δu hp 2 Θ δλ hp 2 2rc δu hp 2 +r 2 δu hp 2 δλ hp 2 ( 2rcβ 2 + r 2 β 2 ) λ hp 2 + r ( u hp,2 g u hp,2 where β = δλhp 2. I follows ha T is sric coracio for 0 < r < 2c. By he δu hp 2 Baach fixed poi heorem here exiss a λ hp = (λ hp, λ hp ) which saisfies (4.52). For )) 2 93

104 4 hp-bem for Sochasic Coac Problems i Liear Elasiciy ay give λ hp, problem (4.44a) reduces o a liear, fiie dimeioal problem. ece, he uiqueess resul of u hp implies he exisece of a u hp (λ hp ). Remark 4.3. Theorem 4. is he sysem case correspodig o he scalar problem cosidered i [8]. Bu Theorem 4. eeds a ew proof. Lemma 4.4. The coac cosrais (4.44b) are equivale o he poiwise codiios u,psp,p s g psp,p s, λ,psp,p s 0, λ,psp,p s (u,psp,p s g psp,p s ) = 0 (4.54) λ,psp,p s Fλ,ps,p s λ,psp,p s < Fλ,ps,p s u,psp,p s = 0 (4.55) λ,psp,p s = Fλ,ps,p s α R 2 : λ,psp,p s = α 2 u,psp,p s (p sp, p s ) G hp G kq. Proof. see [4], [0] ad [8] For Tresca s fricio law he fricio coefficie is give by a fucio F( ) : R. We defie F pspps associaed wih he odes (p sp, p s ) G hp G kq Θ by F pspps = F φ psp (x)ϑ ps (y) ρ dy ds, (4.56) If F is a cosa fucio we have Θ Γ s C F pspp s = F D psp D ps. (4.57) We will ow formulae he classical Uzawa algorihm for problem (4.44). As i [8],[44], we iroduce he followig equivale formulaio: Lemma 4.5. The pair (λ,psp,p s, u,psp,p s ) saisfies he poiwise o-peeraio codiio (4.54) if ad oly if i saisfies he poiwise codiio λ,psp,p s = max{0, λ,psp,p s + c(u,psp,p s g psp,p s )} (4.58) Proof. see [[4],Theorem 4.] Lemma 4.6. The pair (λ,psp,p s, u,psp,p s ) saisfies he fricio coac codiio (4.55) if ad oly if i saisfies he poiwise codiio λ,psp,p λ,psp,ps = F s + c u,psp,ps ij max{f ps,ps, λ,psp,ps + cu,ps,ps } (4.59) 94

105 4.3 A poseriori error esimaes for sochasic coac wih fricio Proof. see [[4],Theorem 5.] The Uzawa algorihm correspods o ieraios of he fixed poi operaor. I his algorihm, we use he poiwise codiio (4.58) ad (4.59). The Uzawa algorihm ca be wrie as follows: Algorihm 4.. (Uzawa algorihm). Iiialisaio: Choose iiial soluio λ 0, c > 0, ol > 0 2. For k = 0,, 2,... do a) For give λ k, fid u k by solvig (4.44a) ad (4.44b) b) Updae Lagrage muliplier by usig λ,psp,ps = max{0, λ,psp,ps + c(u,psp,ps g psp,ps )} λ,psp,p λ,psp,ps = F s + c u,psp,ps max{f ps,ps, λ,psp,ps + cu,ps,ps } c) Sop if λ k λ k < ol λ k or λ k λ k < ol, else go o sep A poseriori error esimaes for sochasic coac wih fricio Lemma 4.7. [[9], Lemma 3.2.9] There exiss a operaor Π hp : 2 (Σ) V hp, which is sable i he 2 -orm ad has quasiopimal approximaio properies i he L 2 - orm. More precisely, here exiss a cosa C, idepede of h ad p, such ha for all u 2 (Σ) here holds Π hp u C u 2 (Σ) 2 (Σ) ( ) ( ɛ) h 2 u Π hp u L2 (Σ) C u p 2 (Σ) (4.60) (4.6) wih arbirarily small ɛ (0; 2 ). Theorem 4.2. Le (u, λ) be he exac soluio of he boudary problem (4.9) ad (u hp, λ hp ) be he soluio of he discree boudary problem (4.4), he here holds he 95

106 4 hp-bem for Sochasic Coac Problems i Liear Elasiciy a poseriori esimae : u u hp 2 + λ λ hp L 2 ρ (Θ; 2 (Σ)) L 2 ρ (Θ; 2 ηhp 2 (I, J) ( )) I T hp J T kq + (λ hp ) +, (g u hp ) + ΓC ρ dy + (λ hp ) 2 Θ L 2 ρ(θ; 2 ( )) + (u hp g) + 2 L 2 ρ(θ; 2 ( )) + ( λhp F) + 2 L 2 ρ(θ; 2 ( )) ( + ( λ hp F) u h + 2(λ hp u hp ) ) ds ρ dy (4.62) where η 2 hp = ( + ( hi p I Θ ) ) ɛ ( ) S hpu hp 2 L 2 ρ (J;L 2(I Γ N )) + ( λhp ) S hp u hp 2 L 2 ρ (J;L 2(I )) + h I s (V (ψ hp ψhp )) 2 L 2 ρ(j;l 2 (I)), (4.63) wih arbirarily small ɛ (0; 2 ). Proof. Usig he same argumes as i Chaper 2. From (2.34) we have C W u u hp 2 + C L 2 ρ (Θ; 2 V ψ ψ hp 2 S (Σ)) L 2 ρ (Γ)) (Θ; 2 hp u hp, u v hp ΓN ρ dy Θ + ( λ hp ) S hp u hp, u v hp ΓC ρ dy Θ + λ λ hp, u hp u ρ dy + λ λ hp, u hp u ρ dy Θ Θ + V (ψhp ψhp ), ψ ψ hp ρ dy Θ + A + B + C (4.64) We esimae he firs ad he secod erms i (4.64), employig he Cauchy-Schwarz iequaliy. We obai A = S hp u hp, u v hp ΓN ρ dy + ( λ hp ) S hp u hp, u v hp ΓC ρ dy Θ Θ ( λ hp ) S hp u hp L 2 ρ (J;L 2 (I)) u v hp L 2 ρ (J;L 2 (I)) J T kq I T hp + S hp u hp L 2 ρ (J;L 2 (I)) u v hp L 2 ρ (J;L 2 (I)) (4.65) I T hp Γ N J T kq Le π kq he L 2 ρ-projecio oo W kq, which saisfies (π kq w w) v ρ dy. (4.66) Θ 96

107 4.3 A poseriori error esimaes for sochasic coac wih fricio Noe ha π kq is L 2 ρ-sable, usig Lemma4.7, we obai w π kq Π hp w 2 L 2 ρ (Θ;L2 (I)) w Π hpw 2 L 2 ρ (Θ;L2 (I)) + Π hpw π kq Π hp w 2 L 2 ρ (Θ;L2 (I)) ( ) ɛ hi C w 2 p I ( hi L 2 ρ (Θ; 2 (ω(i)))+ Π hp w 2 L 2 ρ(θ;l 2 (I)) ) ɛ C w 2 p I L 2 ρ (ω(i)))+ w 2 (Θ; 2 L 2 ρ (Θ; 2 (ω(i))) ( ( ) ) ɛ hi C + w 2 (4.67) L 2 ρ(θ; (ω(i))), 2 p I where we used he 2 -sabiliy of Π hp. We have w π kq Π hp w 2 L 2 ρ(θ;l 2 (I)) C ( + ( hi p I ) ɛ ) w 2 L 2 ρ (Θ; 2 (ω(i))), (4.68) Choosig v hp = u hp + π kq Π hp (u u hp ), we ge A = S hp u hp, u v hp ΓN ρ dy + ( λ hp ) S hp u hp, u v hp ΓC ρ dy Θ Θ ( ( ) ) ɛ 2 hi + ( λ hp ) S hp u hp L 2 ρ (J;L 2 (I)) u u hp L 2 ρ (J; 2 (Σ)) J T kq I T hp + J T kq I T hp Γ N ( + p I ( hi p I ) ɛ ) 2 S hp u hp L 2 ρ (J;L 2 (I)) u u hp L 2 ρ (J; 2 (Σ)) (4.69) As i (2.38) here holds for he las erm C (ψhp Θ V ψhp ), ψ ψ hp ρ dy V (ψhp ψhp ) L 2 ρ (Θ; 2 (Σ)) ψhp ψ L 2 ρ (Θ; 2 (Σ)) c h I 2 Θ s (V (ψ hp ψhp )) 2 L 2 (I) ψ hp ψ L 2 I T ρ (Θ; 2 (4.70) (Σ)) hp 97

108 4 hp-bem for Sochasic Coac Problems i Liear Elasiciy From (2.40) ad (2.4), we obai B = λ λ hp, u hp u ρ dy + Θ (λ hp ) L 2 ρ (Θ; 2 ( )) uhp + λ hp + Θ Θ λ λ hp, u hp u ρ dy u L 2 ρ (Θ; 2 (Σ)) λ L 2 ρ (Θ; 2 ( )) (uhp g) + L 2 ρ (Θ; 2 ( )) (λ hp ) +, (g u hp ) + ΓC ρ dy + ( λ hp F) + L 2 ρ (Θ; 2 ( )) uhp u L 2 ρ (Θ; 2 (Σ)) ( + ( λ hp F) u h + 2(λ hp u hp ) ) ds ρ dy. (4.7) Θ Now we combie esimaes (4.69), (4.70), ad (4.7), use he Cauchy-Schwarz iequaliy, ad Youg s iequaliy, we obai u u hp 2 + ψ ψ hp 2 η L 2 ρ(θ; 2 (Σ)) L 2 ρ(θ; 2 hp 2 (I, J) (Γ)) I T hp J T kq + (λ hp ) +, (g u hp ) + ΓC ρ dy + (λ hp ) 2 Θ L 2 ρ (Θ; 2 ( )) + (u hp g) + 2 L 2 ρ(θ; 2 ( )) + ( λhp F) + 2 L 2 ρ(θ; 2 ( )) ( + ( λ hp F) u h + 2(λ hp u hp ) ) ds ρ dy where η 2 hp = ( + ( hi p I Θ + ɛ λ hp λ L 2 ρ (Θ; 2 ( )), (4.72) ) ) ɛ ( ) S hpu hp 2 L 2 ρ(j;l 2 (I Γ N )) + ( λhp ) S hp u hp 2 L 2 ρ(j;l 2 (I )) + h I s (V (ψ hp ψhp )) 2 L 2 ρ(j;l 2 (I)) (4.73) As i Lemma2.3, usig esimae (4.68), we obai ( ) λ hp λ L 2 ρ (Θ; 2 C u u hp 2 + ψ ψ hp 2 ( )) L 2 ρ(θ; 2 (Σ)) L 2 ρ(θ; 2 (Γ)) + C (I, J), (4.74) where ξ 2 hp = ( + ( hi p I ξhp 2 I T hp J T kq ) ) ɛ ( ) S hpu hp 2 L 2 ρ (J;L 2(I Γ N )) + ( λhp ) S hp u hp 2 L 2 ρ (J;L 2(I )). Combiig (4.72) ad (4.74) yields he aposeriori error esimae. (4.75) 98

109 4.4 Numerical Experimes 4.4 Numerical Experimes I his secio, we cosider a radom posiio-idepede Youg s modulus E ad a deermiisic posiio-idepede Poisso raio ν = 0.3. This eables he esor produc of marices, he global marix ca be wrie as esor produc of he marix represeaio of he Seklov-Poicaré operaor S ad he sochasic mass marix. Where S is he Seklov-Poicré operaor for E = ad S := W + (K + 2 )V (K + 2 ) = E S Numerical resuls are preseed for he coac of he wo-dimesioal elasic body D = [ 0.5, 0.5] 2, wih a rigid obsacle.the rigid obsacle occupies he half space x 2 < 0.5, = [ 0.5, 0.5] { 0.5}. We use he uiform disribuio, which correspods o he desiy ρ 2 ad he sochasic domai Θ = [ 3, 3 ]. The Neuma force is 3 = (0, ) T, he gap g = 0.2 ad he give fricio coefficie F = 0.3. Mea values of he Youg modulus E was adoped as 2000P a. The radom Youg s modulus is modeled by a uifom radom variable wih values i he ierval bewee 800 ad I Figure 4.2 we prese he error i he eergy J(u) := 2 u, Su, u ad i he eergy orm J(u) := u, Su wih respec o he umber of he degrees of freedom, for h- uiform ad p-versio. The exac value of he poeial J(u) , J(u) are obaied from exrapolaio of he poeial values of he lowes order h-versio for p = ad q = 0. I Figure4. we show he deformed cofiguraio for he mea value of Youg modulus E. To solve he oliear equaio, we require more ha 300 Uzawa ieraios o achieve a olerace of

110 4 hp-bem for Sochasic Coac Problems i Liear Elasiciy Figure 4.: Deformed geomery wih he mea value of Youg modulus E Figure 4.2: Covergece of Sochasic Coac problem 00

111 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy I his chaper, a exeded muliscale fiie eleme mehod EMsFEM is derived for he aalysis of liear elasic heerogeous maerials. The mai idea is o cosruc umerically a fiie eleme basis fucios ha capures he small-scale iformaio ( he fie mesh ) wihi each coarse eleme [26], [54]. The cosrucio of he basis fucios is doe separaely for each coarse eleme wih a liear boudary codiio. The boudary codiios for he cosrucio of he muliscale basis fucios have a big ifluece o capurig he smale-scale iformaio. We aalyse a correspodig FEM/BEM couplig ad derive a a priori error ad a-poseriori error esimae. Nex we prese fiie eleme implemeaios for operiodic case. 5. The equaio of liear elasiciy I his secio we cosider a wo-dimesioal pla srai deermiisic problem Le Ω be a bouded domai i R 2 wih polygoal boudary Ω i which every poi is represeed by caresia coordiae x = (x, x 2 ) T. We cosider a solid body i Ω deformed uder he ifluece of a volume force f ad a esio force. The displaceme field u of he body is govered by he liear elasiciy sysem: div σ(u) = f i Ω (5.a) σ(u) = C : ε(u) i Ω, (5.b) where σ is he sress esor, he srai esor ε is give by he symmeric par of he deformaio gradie ε(u) = 2 ( u + ut ) (5.2) C = C(x), x Ω is he 4-h order elasiciy esor, i describes he elasic siffess of he maerial uder load. The sysem give i equaio (5.) follows he boudary codiios u = g o Γ D (5.3a) σ(u) = o Γ N, (5.3b) 0

112 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy where is he ui ouer ormal vecor o Ω. The boudary Γ = Ω be decomposed io wo disjoi subses Γ D ad Γ N, such ha Γ = Γ D Γ N ad meas(γ D ) > 0. Defiiio 5.. Defie he Sobolev space for vecor fields i R 2 by V = (Ω) = [ (Ω)] 2 where u(x) (Ω) meas ha u i (x) (Ω) for ay i =, 2, ad equiped wih he orm (Ω) = {f L 2 (D) : 2 f L 2 (Ω)} u (Ω)= s Ω 2 u dx 2 where s = (s, s 2 ) N 2 is a muliidex wih s = s + s 2. We defie he followig coiuous fucios spaces V 0 = {u (Ω); u = (u, u 2 ) : u = 0 o Γ D } V we defie he biliear form a(u, v) := Ω ε(u) : C : ε(v) dx. (5.4) This form is symmeric, coiuous ad coercive, he coerciviy i.e C 0 > 0 : a(u, v) C 0 u (Ω) u V The variaioal formulaio is give by where a(u, v) = F (v) v V 0, (5.5) F (v) = Ω f v dx + v ds Γ N (5.6) 5.2 The fiie eleme discreizaio Le T h be a quasi-uiform riagulaio of Ω R 2, wih he mesh parameer h ad le N h be he se of verices of T h coaied i Ω, we deoe he umber of grid pois i N h by p. 02

113 5.2 The fiie eleme discreizaio Le {ϕ j } p j= be he sysem of piecewise liear basis fucios o he riagulaio T h of Ω such ha ϕ i (x j ) = δ ij x j N h. Space V 0 is he replaced by heir discree approximaio V h 0, he discree soluio u h is give by: u h = (u h, u 2h ) V h 0 where p u αh = u j α ϕ j ad u h = u j α ϕ j e α α =, 2 (5.7) j= The we defie he basis fucio p j= φ j(α) = ϕ j e α : Ω R 2 (5.8) of V h as a vecor field wih a scalar odal fucio i oe of heir compoes ad zero i he ohers. We se U = (u, u 2, u 2, u 2 2,..., u p, up 2 ) where u j α are odal values of u h i.e u αh (x j ) = u j α. I he wo-dimeioal problem d = 2 p deoes he oal umber of degrees of freedom of V h 0. The discreizaio form is : Fid u h V h such ha where a(u h, v h ) = F (v h ) v h V h (5.9) ad a(u h, v h ) = F (v h ) = Ω Ω ε(u h ) : C : ε(v h ) dx (5.0) f v h dx + v h ds Γ N (5.) Iegrals i (5.9) are compued as sums of iegrals over all eleme T usig he fac ha Ω = T Th T ε(u h ) : C : ε(v h ) dx = f v h dx v h ds T T E Γ N We wrie T T h u = T T h u x u 2 x 2 u x + u 2 x 2 03

114 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy The equaio (5.9) ca be wrie as C Ω u x u 2 x 2 u x + u 2 x 2 v x v 2 x 2 v x + v 2 x 2 = Ω ( ) v (f, f 2 ) v 2 ( + (, 2 ) Γ N The correspodig liear algebra equaio serie is give by AU = b. We iroduice he marix B i relaed o a ode x i by v v 2 ) B i = ϕ i x 0 ϕ 0 i x 2 ϕ i ϕ i x x 2 The biliear form i equaio (5.9) applied o he basis fucio of V h reads a(φ i(α), φ j(β) ) = ε(ϕ i e α ) Cε(ϕ j e β ) dx (5.2) = Ω Ω r,l= 2 C αrβl l ϕ j r ϕ i dx (5.3) ad we defie he siffess marix A = (a ij ) d=2 p i,j= R d d I holds where ε(φ i(α) ) = ε(ϕ i e α ) = B i e α φ i(α) = B i e α We ca wrie A i(α)j(β) = (e α ) Ω B i C B j dx e β α, β =, 2 (5.4) For each eleme T T h we defie he siffess marix A T = BT C B T dx (5.5) where he marix B T coais he odal marices B Ti, he 4 verices of T Elemeary calculaios provide o eleme T B T = 0 ϕ x 0 T i =, 2, 3, 4 correspodig o B T = [B T, B T2, B T3, B T4 ] (5.6) ϕ 2 ϕ x 0 3 ϕ x 0 4 x 0 ϕ ϕ x ϕ x ϕ x x 2 ϕ ϕ ϕ 2 ϕ 2 ϕ 3 ϕ 3 ϕ 4 ϕ 4 x 2 x x 2 x x 2 x x 2 x R

115 5.3 Exeded muliscale fiie eleme mehod for he aalysis of liear elasic heerogeeous maerials 5.3 Exeded muliscale fiie eleme mehod for he aalysis of liear elasic heerogeeous maerials We prese a approach where, isead of approximaig i V, we use a beer space of muliscale fucios V Ms 0(Ω). The muliscale fiie eleme soluio foud from solvig he fiie eleme problem usig V Ms The space V Ms is he spa of he se of muliscale basis focio {ΦMs i } which are defied for each ode of a coarse mesh T (Ω).The idea of he mehod is o cosruc umerically he muliscale basis fucios o capure he fie scale feaures of he coarse elemes i he muliscale fiie eleme aalysis.. I his secio, we also give he defiiios of he muliscale basis ad he muliscale coarse space. We cosider a wo dimeioal pla sai problem, le Ω be a bouded domaie i R 2 wih polygoal boudary Ω = Γ where Γ D ad Γ N are he segmes o which we prescribe homogeeous Dirichle ad Neuma boudary codiios ad u = [u x, u y ] T is he displaceme field, f ad are he body forces ad racios respecively ad C is he symmeric elasiciy esor ad is he ouward ormal o Ω, i is assumed ha f L 2 (Ω) ad L 2 (Γ N ). The weak formulaio he cosiss i fidig u V, where such ha V := {v (Ω) = [ (Ω)] 2 : v ΓD = 0} a(u, v) = F (v) v V (5.7) where a : V V R ad F : V R are biliear ad liear form o V defied as a(u, v) := σ(u) : ε(v) dx ad F (v) := f v dx + v ds (5.8) Ω Ω Γ N The soluio u V is called he weak soluio o he boudary value problem ad Lax-Milgra lemma esures is exisece ad uiqueess. Le T be he coarse riagulaio of Ω, here we assume agai ha each coarse eleme T cosiss of uio of fie elemes τ T h of he fie riagulaio. The coarse elemes T T are cosruced by agglomeraio of he fie elemes, we cosruc a se of agglomeraed elemes {T } = T such ha each T = τ Th τ is a simply coeced uio of fie grid elemes, le Σ he coarse grid pois i Ω. For each coarse ode x p Σ we deoe he k-h coarse degree of freedom for k {, 2} relaed o his ode by p(k), we deoe w p = {T T : x p T } he uio of quadrilaerals coeced o he ode x p ad wp = diam(w p ), we eummerae is four verices by p =, 2, 3, 4 ad for every ode x p o he Dirichle boudary, we 05

116 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy also deoe Γ p,d = w P Γ D ad for every eleme T i T we deoe by w T he pach of quadrilaerals coaiig T ha is w T = { T : T T } T ad se wt = diam(w T ), i holds ha T 2 T, w p 2 wp ad w T 2 wt. The shape regulariy of he mesh T imposed, E = E T for every edge E of T, whereas he local quasi-uiformiy implies i pariculariy ha T wp wt The cosrucio of he basis fucios We cosruc a vecor valued muliscale basis fucio Φ Ms p(k) : w p R 2 where Φ Ms p(k) is he basis associed wih he ode p ad suppored o w p. The cosrucio is doe separaly for each eleme T T, Φ Ms p(k) is used for he displaceme i he direcio k. We deoe he scalar coarse odal basis fucio correspodig o x p by φ li p : w p R, φ li p is liear i T ad φ li p (x q ) = δ pq, wih x q Σ he basis fucios Φ Ms p(k) whose resricio Φ Ms p(k),t o T mus solve he followig subgrid problem: Fid Φ Ms p(k),t (T ) such ha a T (Φ Ms p(k), v) = 0 for all v (T ), T w p, (5.9) subjec o a suiable boudary codiio Φ Ms p(k),t = φli p,t e k o T ; T w p, (5.20) where x p Σ (T ) Φ Ms p(k) = Ω ek o T. (5.2) O T, liear boudary codiios are imposed i he k-h compoe of he vecor-field ad zero boudary codiios i he {, 2} \ {k}. The local boudary codiios will be cosruced so ha hey are coiuous across eleme edges, ha is Φ Ms p(k),t (x) = ΦMs (x) = φli p(k),t p,t e k for x T T. For he wo-dimesioal problem wo kids of basis fucios are cosruced oe is used for he x-axis direcio ad he oher is used for he y-axis direcio. Firsly, le us cosider he cosrucio of he basis fucio Φ p (x) = Φ x p x o ode of he coarse eleme Figure 5. (lef). The displaceme a all boudary odes are o cosrai i y-axis direcio excep ode 3, for which he displaceme are fixed o zero i boh coordiae direcios i order o avoid rigid displaceme, a ui displaceme is applied o ode i he posiive x-direcio Figure 5. (lef). The displaceme a odes 2, 3 ad 4 are fixed o zero i x-direcio, he values vary liearly 06

117 5.3 Exeded muliscale fiie eleme mehod for he aalysis of liear elasic heerogeeous maerials Figure 5.: [54]The cosrucio of he umerical basis fucios Figure 5.2: Displaceme field of T for he basis fucio Φ x p x (lef) ad Φ x p y (righ) alog each side. We igore he displaceme values i y-axis direcio ad remaiig displaceme values i x-axis direcio o esablish he basis fucio Φ p (x) = Φ x p x The cosrucio of Φ p (y) = Φ y p y is similar o ha of Φ p (x), we have obaied he basis fucios Φ p (x) ad Φ p (y) Figure 5.2, he res of he basis fucios of he coarse eleme ca be cosruced i he similar way. A modificaio of he cosrucio of he basis fucios is developed i [55] ad used for small deformaio elaso-plasic aalysis of periodic russ maerials. So ha he muliscale basis fucios for he displaceme fields ca be cosruced i a more accurae way. The mos key poi is ha he addiioal couplig erms of he basis fucios are iroduced i he improved cosrucio mehod. For he eleme T Figure 5. (righ), he displacemes a all boudary odes are cosrai i y direcio, a he same ime he odes o boudary 34 ad boudary 23 are cosrai i x- direcio, a ui displaceme is applied o ode i he posiive x-direcio ad he values vary liearly alog boudaries 2 ad 4,he ieral displaceme field of he eleme ca be obaied direcly by sadard fiie eleme aalysis i fie scale mesh ad he basis fucios Φ x p (x) ad Φ x p (y) ca be obaied, hus he basis fucios Φ x p = {Φ x p x, Φ x p y}. ere Φ x p i y is a coupled addiioal erm ad meas ha he displaceme field i y- direcio wihi he eleme iduced by ui displaceme of ode i i he x-direcio. The res of he basis fucios ca be cosruced i he similar way ad he fial basis fucios ie.φ x p i x, Φ x p i y, Φ y p i y ad Φ y p i x for i =, 2, 3, 4 obaied by his way 07

118 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy Figure 5.3: [54] Schemaic descripio of he EMsFEM Siffess marix o he coarse mesh The displaceme vecor of he odes i he fie scale omis he represeaio u h = Ru (5.22) where u is he displaceme vecor of odes i he coarse mesh ad R is he basis fucio marix which coais he coefficie vecors, represeig a coarse basis fucio i erms of he fie scale basis. R = [R T R T 2 R T ] T (5.23) [ ] Φ x p R i = x Φ x p y Φ x p 2 x Φ x p 2 y Φ x p 3 x Φ x p 3 y Φ x p i y Φ x p i y Φ y p x Φ y p y Φ y p 2 x Φ y p 2 y Φ y p 3 x Φ y p 3 y Φ y p i y Φ y p i y i =, 2..., (5.24) where is he oal umber of he he fie scale mesh wihi he sub-grids, usig (5.22) ad (5.24), we have u e h = G eu E (5.25) where G e = R e R e2 R e3 R e4 (5.26) (5.27) is he rasformaio marix bewee he displaceme vecors of micro-scale odes ad macro-scale odes ad e is a arbirary fie-scale eleme wihi he coarse eleme show Figure

119 5.4 Covergece of he muliscale fiie eleme mehod The siffess marix of he coarse eleme is give by K E = N e= K e, Ke = G T e K e G e (5.28) where K e is he eleme siffess marix. The global siffess marix o he coarse mesh is obaied as follows K = A M i=k E (i) (5.29) where A M i= is a marix assembled operaor ad M is he oal umber of he coarse eleme. The correspodig muliscale coarse space is coformig, we defie he coarse space V Ms := {Φ Ms p(k), xp Σ, k =, 2} 0(Ω) (5.30) Usig he defied space of muliscale fucios V Ms problem gives a muliscale fiie eleme approximaio u Ms v Ms VMs a Ω (u Ms, v Ms ) = = Ω Ω σ(u Ms 0(Ω) i he fiie eleme which saisfies for all ) : ε(v Ms ) dx F v Ms dx (5.3) I holds ha Ω σ(u Ms ) : ε(v Ms ) dx = T T T σ(u Ms ) : ε(v Ms ) dx 5.4 Covergece of he muliscale fiie eleme mehod I his secio, he covergece of he MsFEM is preseed. The MsFEM is defied for o-periodic problem bu he covergece aalysis is made i he periodic case. As i [26],[39], we resric ourselves o a periodic case. We cosider he followig elasiciy problem L ɛ u ɛ = f i Ω u ɛ = 0 o Ω (5.32) 09

120 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy L ɛ is he elasiciy operaor, L ɛ = div(c( x ɛ ) : ε( )), ad C(x) = (C ijkl( x ɛ )) is he fourh order elasiciy esor, which saisfies symmery ad posiiv defiieess. There exiss α > 0 such ha C ijkl (y)ε ij ε kl αε ij ε ij y = x ɛ, ε ij symmeric The muliscale basis fucios saisfy Recall ha is he coarse mesh size. L ɛ Φ Ms j(k) = 0 i T T. We refer o [39] for he covergece aalysis of MsFEM, where he covergece for < ɛ ad > ɛ cases are preseed for a Dirichle problem. The covergece rae of MsFEM coais a erm ɛ (see [39]), he error becomes large whe he wo scale are close, whe ɛ he muliscale mehod aais a large error i ad L 2 orms (see [26]). The muliscale basis fucios are smooh if < ɛ ad ca be well approximaed by he sadard coiuous liear (biliear) basis fucios, we apply he radiioal fiie eleme mehod aalysis o he muliscale mehod. Whe > ɛ he muliscale basis fucios coais a smooh par ad a oscillaory par, which cao be approximaed by liear ( biliear ) fucios, he MsFEM gives a covergece resul uiform i ɛ as ɛ eds o zero, while he radiioal FEM wih piecewise polyomial basis fucios does o. Theorem 5.. [39](Covergece for < ɛ) Le u ad u Ms be he soluios of (5.32) ad (5.3), respecively. The here exiss a cosa C, idepede of ad ɛ, such ha u u Ms (Ω) C( ɛ ) f L 2 (Ω) (5.33) u u Ms L 2 (Ω) C( ɛ )2 f L 2 (Ω) (5.34) Remark 5.. Deails of he proof for a Dirichle problem ca be foud i [26],[39]. The proof for he rasmissio problem similarly follows from a covergece aalysis of he sadard fiie eleme / boudary eleme couplig, usig from [26],[39] ha he muliscale basis fucios do o differ sigificaly from piecewise liear ha fucios. Remark 5.2. For < ɛ he muliscale mehod gives he same rae of covergece as he liear fiie eleme mehod, bu his esimae is isufficie for pracical applicaios, he esimaes (5.33) ad (5.34) blows up as ɛ 0. Le I be he sadard ierpolaio operaor, ad I Ms : C(Ω) V Ms be he odal ierpolaio operaor defied i [20] by I Ms u = j u(x j )Φ Ms j (5.35) 0

121 5.4 Covergece of he muliscale fiie eleme mehod From he defiiio of he muliscale basis fucios, we have L ɛ (I Ms u) = 0 i T, I Ms u = I u o T (5.36) From he approximaio heory, we have u I u L 2 (T ) C u (T ) (5.37) u I u (T ) C 2 u (T ) (5.38) Lemma 5.. Le u (Ω) be he soluio of (5.32). There exis cosas C ad C 2 > 0 idepede of, such ha u I Ms u L 2 (T ) = C ( u (T ) +2 f L 2 (T ) ) (5.39) u I Ms u (T ) = C 2( u (T ) + f L 2 (T ) ) (5.40) Proof. We refer o [26],[39], sice I Msu = I u o T, we have I Msu I u 0(T ), ad by he Poicaré-Friedrichs iequaliy we obai From (5.36), we have I Ms u I u L 2 (T ) C IMs u I u (T ) T C( x ɛ )ε(ims u MS ) : ε(i Ms u I u) dx = 0, (5.4) ad similar o he proof of Lemma 6.3 i [26] we obai I Ms u I u 2 (T ) = C( x T ɛ )ε(ims u I u) : ε(i Ms u I u) dx = C( x T ɛ )ε(u I u) : ε(i Ms u I u) dx C( x )ε(u) : ε(ims u I u) dx T ɛ = C( x ɛ )ε(u I u) : ε(i Ms u I u) dx C( x ɛ )f(ims u I u) dx T C u I u (T ) IMs T u I u (T ) + IMs u I u L 2 (T ) f L 2 (T ) C u I u (T ) IMs u I u (T ) + IMs u I u (T ) f L 2 (T ) C I Ms u I u (T ) ( u I u (T ) + f L 2 (T ) ) (5.42) We obai I Ms u I u (T ) C( u (T ) + f L 2 (T ) ) (5.43) I Ms u I u L 2 (T ) C( u (T ) +2 f L 2 (T ) ) (5.44)

122 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy Usig (5.37), (5.38),(5.43) ad (5.44), we ge u I Ms u (T ) u I u (T ) + IMs u I u (T ) C 2 ( u (T ) + f L 2 (T ) ) u I Ms u L 2 (T ) u I u L 2 (T ) + IMs u I u L 2 (T ) C ( u (T ) +2 f L 2 (T ) Now we discuss he covergece aalysis for > ɛ by explorig he asympoic behavior of boh u ad Φ Ms j. We cosider he expasio of u: u = u 0 + ɛu +... (5.45) where u 0 is he soluio of he homogeized equaio For more deails (see [26],[39]). Le u I be he ierpola of u 0, usig he muliscale basis fucios Φ Ms j, oe ha u I is differe from he defiiio of I Ms u. I he lieraure he followig homogeizaio esimaes are oly available for he Dirichle [39] ad Neuma [49] problems, o he rasmissio problem. Lemma 5.2. Le u be he soluio of (5.32) ad u I he ierpola of he homogeized soluio u 0, usig he muliscale basis fucios Φ Ms j. The here exis cosas C ad C 2, idepede of ɛ ad, such ha u u I (Ω) C f L 2 (Ω) +C 2( ɛ ) 2 (5.46) u u I L 2 (Ω) C 2 f L 2 (Ω) +C 2( ɛ ) 2 (5.47) Theorem 5.2. (Covergece for > ɛ) Le u ad u Ms be he soluios of (5.32) ad (5.3), respecively. The here exis cosas C ad C 2, idepede of ɛ ad, such ha I is show i [39] ha u u Ms (Ω) C f L 2 (Ω) +C 2( ɛ ) 2 (5.48) (5.49) u u Ms L 2 (Ω) = O(2 + ɛ ) (5.50) Remark 5.3. While we expec hese resuls o hold for he rasmissio problem, a aalysis of he homogeized rasmissio problem has o bee repored. We leave he ecessary aalysis as a ope problem. Remark 5.4. For > ɛ he muliscale mehod coverges o he homogeized soluio u 0 i he limi as ɛ 0. 2

123 5.5 A poseriori error for muliscale fiie eleme mehod 5.5 A poseriori error for muliscale fiie eleme mehod We cosider he problem L ɛ u ɛ = f i Ω (5.5) Lemma 5.3. Le u be he soluio of (5.5) ad u Ms problem (5.3). The here holds he esimae: where e Ms a(e Ms, v) = T T T + 4 = u ums. T T l= f(v v Ms ) dx E l T be he soluio of he discree [σ(u Ms ) ] El (v v Ms ) ds (5.52) Proof. We cosider he quaiy e Ms = u ums, called muliscale discreizaio error, i he followig we derive ad aalyze a upper boud for he quaiy measured i he eergy orm. e Ms Ω := Give he muliscale soluio u Ms he so called error residual equaio VMs a(e Ms, ems ) (5.53) of he discree problem (5.3), we obai Recallig o a(e Ms, v) = a(u u Ms, v) = F (v) a(u Ms, v) (5.54) a T (Φ Ms p(k), v) = 0 for all v 0(T ), T w p (5.55) ad we obai V Ms := {Φ Ms p(k), xp Σ, k =, 2} 0(Ω) Le v V ad v Ms a T (u Ms, v) = 0 for all v 0(T ) (5.56) VMs, usig his we rewrie he discree problem (5.3) as follows 0 = F (v Ms ) a(u Ms, v Ms ) (5.57) 3

124 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy wih he discreizaio error e Ms = u ums he Galerki-mehod for fiie sub-domais yields he impora Galerki orhogoaliy Subracig (5.54) from (5.57) we obai a(e Ms, v) = ( T T T T a(e Ms, v Ms ) = 0 v Ms V Ms (5.58) T T f(v v Ms ) dx + Γ N T ) (v v Ms ) ds σ(u Ms ) : ε(v v Ms ) dx v V (5.59) iegraio by pars of he erm T σ(ums ) : ε(v vms ) dx i (5.59) we obai ( a(e Ms, v) = T T + T T ( T T f(v v Ms ) dx + div σ(u Ms Γ N T )(v v Ms ) dx ) (v v Ms ) ds T σ(u Ms ) ) (v v Ms ) ds (5.60) For fixed T T le E l be a edge of T wih l =, 2, 3, 4, we iroduce he jump of racios a he eleme boudaries ( ) 2 σ + (u Ms ) + E l + σ (u Ms ) E l if E l Γ T = T [σ(u Ms ) ] El := σ(u Ms ) E l if E l Γ T,N = T Γ N 0 if E l Γ T,D = T Γ D (5.6) here El is he ormal o E l Recall ha by corucio of he muliscale basis fucios, we have divσ(u Ms ) = 0 i T, we ca ow regroup he boudary erms i(5.60) for every fixed T so ha (5.60) becomes a(e Ms, v) = T T T + 4 T T l= f(v v Ms ) dx E l T [σ(u Ms ) ] El (v v Ms ) ds (5.62) I he followig, we derive a-poseriori error esimaes for < ɛ ad > ɛ. 4

125 5.5 A poseriori error for muliscale fiie eleme mehod Theorem 5.3. (A poseriori error esimae for < ɛ) Le u be he soluio of (5.32) ad u Ms The here holds he esimae: where a(e Ms T h Ω T Ω E h E Ω E Ω, e Ms ) ηt 2 h + T h Ω T Ω η 2 T h = T h Ω η 2 T = η 2 E h = T Ω η 2 E = be he soluio of he discree problem (5.3). η 2 T + 2 (T h ) f f 2 dx T h 4 f 2 E h E Ω E Ω f(x) = T (E h ) ([σ(u Ms E h 2 ([σ(u Ms ) ])2 T f dx, E h E Ω η 2 E h + E Ω ) ] [σ(u Ms ) ])2 η 2 E (5.63) [σ(u Ms ) ](x) = [σ(u Ms ) ] dx E E Proof. From Lemma 5.3, we have a(e Ms, v) = T T T + 4 Firs, we esimae he erm T T l= A := f(v v Ms ) dx T T Le f(x) = T T f, if x T he Ω f 2 = Ω owever T (f f) = 0 (f f + f) 2 = Ω Ω f 2 = E l T T [σ(u Ms ) ] El (v v Ms ) ds (5.64) f(v v Ms ) dx (5.65) (f f) Ω T T f (f f) + T Ω (f) 2 (f f) 2 + (f) 2 (5.66) Ω 5

126 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy Usig (5.39), choosig v = e Ms A ad vms f L 2 (T ) ems T Ω = IMs ems, we obai I Ms e Ms L 2 (T ) C f L 2 (T ) ( ems (T ) + 2 f L 2 (T ) ) T Ω C( C T Ω T Ω 2 f 2 L 2 (T ) + T Ω f L 2 (T ) ems (T ) 2 f 2 L 2 (T ) +ɛ 2 ems 2 (T ) (5.67) Now we esimae he erm T Ω 2 f 2, employig (5.66), we ge L 2 (T ) 2 f 2 L 2 (T ) = 2 ( f 2 dx) T Ω T Ω T h T T h = 2 { f f 2 dx + f 2 dx} T Ω T h T T h T h T T h = 2 { f f 2 dx + T h f 2 } T Ω T h T T h T h T = 2 { f f 2 dx+ T f 2 } T Ω T h T T h = 2 (T h ) f f 2 dx + 4 f 2 T h Ω T h T Ω = ηt 2 h + ηt 2. (5.68) T h Ω T Ω ere f T is a cosa o T, ad T 2. We obai A T h Ω η 2 T h + T Ω η 2 T + ɛ 2 ems 2 (T ) (5.69) Now we esimae he erm A 2 := E Ω where E is a ierior edge of T. Choosig v = e Ms ad vms v v Ms L 2 (E ) = ems = IMs [σ(u Ms E ems I Ms ) ] E (v v Ms ) ds. (5.70), sice IMs ems = I e Ms o T, we ge e Ms L 2 (E ) = ems I e Ms L 2 (E ). (5.7) From [[2],Lemma 4], for every edge E of T ad u (T ), we have u 2 L 2 (E ) C( u 2 L 2 (T ) +2 u 2 (T ) ) (5.72) 6

127 5.5 A poseriori error for muliscale fiie eleme mehod Usig (5.72), we ge v v Ms 2 L 2 (E ) = ems C ( C ( I Ms e Ms 2 L 2 (E ) e Ms 2 e Ms I e Ms 2 L 2 (T ) +2 e Ms 2 (T ) + 2 e Ms 2 (T ) ) I e Ms 2 (T ) ) C e Ms 2 (T ). (5.73) Usig (5.73), we obai wih Youg s iequaliy A 2 [σ(u Ms ) ] L 2 (E ) v vms L 2 (E ) E Ω e Ms (Ω) C E Ω E Ω Now we esimae he firs erm i (5.74), wih 2 [σ(u Ms ) ] L 2 (E ) ([σ(u Ms ) ]) 2 ds + ɛ E 2 ems 2 (Ω). (5.74) we have A 22 := E Ω = E Ω = E Ω = E Ω = E h E Ω = E h E Ω where E. E h E [σ(u Ms ) ](x) = [σ(u Ms ) ] (5.75) E E ([σ(u Ms ) ]) 2 = E ([σ(u Ms E h E h E E h E (E h ) η 2 E h + ([σ(u Ms E h ([σ(u Ms E h ([σ(u Ms E h E Ω E Ω E h E E h ([σ(u Ms ) ]) 2 ) ] [σ(u Ms ) ])2 + E h E ([σ(u Ms E h ) ] [σ(u Ms ) ])2 + E h ([σ(u Ms E h E ) ] [σ(u Ms ) ])2 + E ([σ(u Ms ) ] [σ(u Ms ) ])2 + E Ω ) ])2 ) ])2 ) ])2 2 ([σ(u Ms ) ])2 η 2 E. (5.76) We obai A 2 E h E Ω η 2 E h + E Ω η 2 E + ɛ 2 ems 2 (Ω). (5.77) 7

128 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy We hrow he erm ɛ 2 e Ms 2 (Ω) follows immediaely. Theorem 5.4. (A poseriori error esimae for > ɛ) Le u be he soluio of (5.32) ad u Ms The here holds he esimae: a(e Ms, e Ms ) ηt 2 h + T h Ω T Ω o he lef had side, he esimae of he Theorem be he soluio of he discree problem (5.3). η 2 T + + f 2 L 2 (T ) ( ɛ ) T Ω E h E Ω 2 η 2 E h + E Ω η 2 E (5.78) Proof. To prove he heorem, we firs deoe u I (x) = I Ms u 0 (x) = j u 0 (x j )Φ Ms j, (5.79) where u 0 is he soluio of he homogeized equaio ad u I is he ierpola of u 0 usig he muliscale basis fucios Φ Ms j, oe ha u I is differe from he defiiio of I Msu. We kow ha L ɛ (u I ) = 0 i T, u I = I u 0 o T (5.80) Firs, we esimae he erm A = T T T f(v v Ms ) dx (5.8) Choosig v = e Ms we obai ad vms = u I u Ms, usig (5.47) ad Cauchy Schwarz iequaliy, A f L 2 (T ) u u I L 2 (T ) T Ω f L 2 (T ) (C 2 f L 2 (Ω) +C 2( ɛ ) 2 ) T Ω T Ω C 2 f 2 L 2 (T ) + T Ω C 2 f L 2 (T ) ( ɛ ) 2 (5.82) From (5.68), we obai A ηt 2 h + ηt 2 + f 2 L 2 (T ) ( ɛ ) T h Ω T Ω T Ω 2 (5.83) 8

129 5.6 Couplig FEM-BEM Now we esimae he erm Choosig v = e Ms A 2 = ad vms A 2 E Ω [σ(u Ms E ) ] E (v v Ms ) ds. (5.84) = I (u u Ms ) o T, we obai (see (5.72)-(5.77)) ηe 2 h + E h E Ω We hrow he erm ɛ 2 e Ms 2 (Ω) follows immedialy. E Ω η 2 E + ɛ 2 ems 2 (Ω). (5.85) o he lef had side, he esimae of he heorem 5.6 Couplig FEM-BEM Le Ω be a bouded Lipschiz domai wih boudary Γ := Ω ad exerior domai Ω c := R 2 \ Ω. The displaceme field u of he body saisfies he elasiciy maerial behaviour σ(u) = C : ε(u), (5.86) where σ is he sress esor, he srai esor ε is give by he symmeric par of he deformaio gradie. Give a volume force f he equilibrium equaio reads ε(u) = 2 ( u + ut ). (5.87) div σ(u) + f = 0 i Ω. (5.88) The exerior problem cosiss of he Navier-Lamé equaio 0 = u c := µ 2 u c (λ 2 + µ 2 )grad div u i Ω c, (5.89) ad a radiaio codiio of he form D α (u c a)(x) = O( x α ), α = 0,, ( x ), (5.90) where D = / x j ad a R 2 is a cosa vecor. The rasmissio problem has he daa f L 2 (Ω), u 0 2 (Γ), ad 0 2 (Γ) ad cosiss i fidig (u, u c ) (Ω) loc (Ω c) saisfyig (5.88), (5.89), (5.90), ad he ierface codiios: u u c = u 0 o Γ (5.9) σ(u) = T (u c ) + 0 o Γ (5.92) 9

130 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy where T is he coormal derivaive relaed o he Lamé operaor. T (u c ) := 2µ 2 u c + λ 2 div u c + µ 2 curlu c (5.93) ere, is he ormal derivaive ad is he ui ormal vecor o Γ poiig from Ω io Ω c. The Cauchy daa of u c loc (Ω c) wih u c = 0 saisfy (ξ, φ) := (u c Γ, T (u c ) Γ ) 2 (Γ) 2 (Γ) (5.94) The Seklov-Poicaré operaor for he exerior Lamé problem is give as S := (W + (K )V (K )) 2, (5.95) where W = 2W, K = 2K, V = 2V, K = 2K i Chaper2 which saisfies T u c Γ = Su c Γ (5.96) 5.6. The weak formulaio of he Model Problem The weak form of he ierface problem (5.88)-(5.92) reads as: Fid u (Ω), such ha v (Ω) Le σ(u), ε(v) + Su Γ, v Γ = f, v Su 0, v Γ (5.97) 2 (Γ)/R := {φ 2 (Γ),, φ = 0} (5.98) We cosider he followig weak formulaio for a muliscale problem ( cf. [7]): Fid (u, ξ, φ) (Ω) 2 (Γ) 2 (Γ)/R, such ha C ɛ ε(u), ε(v) φ, v = f, v + 0, v v (Ω) (5.99a) u, ψ 2 Vφ, ψ + 2 (K + )ξ, ψ = u 0, ψ ψ 2 (Γ) (5.99b) 2 (K + )φ, θ + Wξ, θ = 0 θ 2 (Γ)/R (5.99c) Wih he soluio u (Ω), we defie φ := S(u 0 u Γ ) ad ξ := u Γ u 0. We rewrie (5.99) as follows: Fid (u, ξ, φ) (Ω) 2 (Γ) 2 (Γ)/R, such ha B(u, ξ, φ; v, ψ, θ) = L(v, ψ) (v, ψ, θ) (Ω) 2 (Γ) 2 (Γ)/R (5.00) 20

131 5.6 Couplig FEM-BEM The discree problem for Ms-FEM-BEM Le V Ms 2 2 be a family of fiie dimesioal subspaces of V 2 2 (Γ)/R, where V Ms is he defied space of muliscale fucios. The discreized versio of (5.00) reads as: B(u Ms, φ, ξ ; v Ms, ψ, θ ) = L(v Ms As i [7], we cosider he relaio:, ψ ) (v Ms, ψ, θ ) V Ms 2 2 (5.0) φ φ, u u Ms Γ = 2 W(ξ ξ ), ξ ξ 2 V(φ φ ), φ φ + u u Ms, φ φ + 2 V(φ φ ), φ φ 2 (K + )(ξ ξ ), φ φ + 2 (K + )(φ φ ), ξ ξ + 2 W(ξ ξ ), ξ ξ (5.02) for all φ 2, ξ A priori error esimae Usig he relaio (5.02) ad Cauchy-Schwarz iequaliy, we obai B(u u Ms, φ φ, ξ ξ ; u u Ms, φ φ, ξ ξ ) = C ɛ ε(u u Ms ), ε(u u Ms ) φ φ, u u Ms Γ u u Ms, φ φ 2 V(φ φ ), φ φ + 2 (K + )(ξ ξ ), φ φ 2 (K + )(φ φ ), ξ ξ 2 W(ξ ξ ), ξ ξ = C ɛ ε(u u Ms ), ε(u u Ms ) + 2 W(ξ ξ ), ξ ξ + 2 V(φ φ ), φ φ 2A C 2 ɛ ε(u u Ms ) 2 L 2 (Ω) + ξ ξ 2 + φ φ 2 2A (5.03) 2 (Γ)/R 2 (Γ) where A := 2 W(ξ ξ ), ξ ξ 2 V(φ φ ), φ φ + u u Ms, φ φ + 2 V(φ φ ), φ φ 2 (K + )(ξ ξ ), φ φ + 2 (K + )(φ φ ), ξ ξ + 2 W(ξ ξ ), ξ ξ (5.04) 2

132 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy O he oher had, for all (v Ms, ψ, θ ) V Ms 2 2, we have B(u u Ms, φ φ, ξ ξ ; u u Ms, φ φ, ξ ξ ) = B(u u Ms, φ φ, ξ ξ ; u v Ms, φ ψ, ξ θ ) + B(u u Ms, φ φ, ξ ξ ; v Ms u Ms, ψ φ, θ ξ ) (5.05) Sice B(u u Ms, φ φ, ξ ξ ; v Ms C 2 ɛ ε(u u Ms ) 2 L 2 (Ω) + ξ ξ 2 + φ φ 2 2 (Γ)/R 2 (Γ) B(u u Ms = C ɛ ε(u u Ms ums, ψ φ, θ ξ ) = 0, we obai, φ φ, ξ ξ ; u v Ms, φ ψ, ξ θ ) + 2A ), ε(u v Ms ) φ φ, u v Ms Γ u u Ms, φ ψ 2 V(φ φ ), φ ψ + 2 (K + )(ξ ξ ), φ ψ 2 (K + )(φ φ ), ξ θ 2 W(ξ ξ ), ξ θ + 2 u u Ms, φ φ + V(φ φ ), φ φ (K + )(ξ ξ ), φ φ + (K + )(φ φ ), ξ ξ + W(ξ ξ ), ξ ξ = C ɛ ε(u u Ms ), ε(u v Ms ) φ φ, u v Ms Γ + u u Ms, φ ψ 2 φ + V(φ φ ), 2 φ 2 ψ φ + (K + )(ξ ξ ), φ 2 ψ 2 φ + (K + )(φ φ ), 2 ξ 2 θ ξ + W(ξ ξ ), 2 ξ + 2 θ ξ ( ) C 2 ɛ ε(u u Ms ) L2 (Ω)+ ξ ξ 2 + φ φ (Γ)/R 2 (Γ) ( ) C 2 ɛ ε(u v Ms ) L2 (Ω)+ ξ θ 2 + φ ψ (Γ)/R 2 (Γ) Afer choosig θ = ξ, ψ = φ, we ge he followig heorem. Theorem 5.5. C 2 (5.06) ɛ ε(u u Ms ) 2 L 2 (Ω) + ξ ξ 2 + φ φ 2 2 (Γ)/R 2 (Γ) { } if C 2 ɛ ε(u v Ms ) L2 (Ω)+ ξ θ 2 + φ ψ (Γ)/R 2 (Γ) (v Ms,ψ,θ ) V Ms 2 2 (5.07) A poseriori error I his secio, we derive a-poseriori error esimaes for < ɛ ad > ɛ. 22

133 5.6 Couplig FEM-BEM Theorem 5.6. (A poseriori error esimae for < ɛ) Le (u, φ, ξ) be he soluio of (5.00) ad (u Ms, φ, ξ ) be he soluio of he discree problem (5.0). The here holds he esimae: T h Ω E h E Ω E Ω ɛ ε(u u Ms ) 2 L 2 (Ω) + ξ ξ 2 + φ φ 2 2 (Γ)/R 2 (Γ) σ(u Ms ) 0 φ 2 L 2 (E ) C 2 E Γ + E Γ + E Γ + T h Ω η 2 T h = η 2 E h = s (u 0 u Ms + 2 V φ 2 (K + )ξ ) 2 L 2 (E ) (K + )φ + W ξ 2 L 2 (E ) η 2 T h + T h Ω η 2 E = E h E Ω E Ω f(x) = T T Ω η 2 T + E h E Ω 2 (T h ) f f 2 dx, T h (E h ) ([σ(u Ms E h 2 ([σ(u Ms ) ])2 T f dx, η 2 E h + T Ω E Ω η 2 E η 2 T = T Ω ) ] [σ(u Ms ) ])2 4 f 2 [σ(u Ms ) ](x) = [σ(u Ms ) ] dx E E Proof. Recall ha C 2 ɛ ε(u u Ms ) 2 L 2 (Ω) + ξ ξ 2 + φ φ 2 2 (Γ)/R 2 (Γ) B(u u Ms, φ φ, ξ ξ ; u v Ms, φ ψ, ξ θ ) + 2A = L(u v Ms, φ ψ ) B(u Ms, φ, ξ ; u v Ms, φ ψ, ξ θ ) + 2 u 0 u Ms + 2 V(φ φ ) 2 (K + )(ξ ξ ), φ φ + (K + )(φ φ ) + W(ξ ξ ), ξ ξ = f(u v Ms ) dx + 0, u v Ms u 0, φ ψ Ω C ɛ ε(u Ms ), ε(u v Ms ) + φ, u v Ms + u Ms, φ ψ + 2 Vφ, φ ψ 2 (K + )ξ, φ ψ + 2 (K + )φ, ξ θ + 2 Wξ, ξ θ + 2 u 0 u Ms + 2 V(φ φ ) 2 (K + )(ξ ξ ), φ φ + (K + )(φ φ ) + W(ξ ξ ), ξ ξ (5.08) 23

134 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy We esimae he las erm i (5.08) A := (K + )(φ φ ) + W(ξ ξ ), ξ ξ (5.09) Usig (5.99c), Cauchy Schwarz ad he iverse iequaliy, we ge Similarly we have A (K + )φ + Wξ 2 (Γ) ξ ξ 2 (Γ) 2 (K + )φ + Wξ L 2 (Γ) if ξ ξ ξ 2 (Γ) (5.0) A 2 := 2 (K + )φ, ξ θ + 2 Wξ, ξ θ 2 (K + )φ + Wξ L 2 (Γ) if θ ξ θ 2 (Γ), (5.) Usig (5.99b), Cauchy Schwarz ad he iverse iequaliy, we ge A 3 := u 0 u Ms + 2 V(φ φ ) 2 (K + )(ξ ξ ), φ φ u 0 u Ms Similarly we have 2 Vφ + 2 (K + )ξ 2 (Γ) if φ φ φ 2 (Γ) 2 s (u 0 u Ms + 2 Vφ 2 (K + )ξ ) L 2 (Γ) if φ φ φ 2 (Γ) (5.2) A 4 := u Ms, φ ψ u 0, φ ψ + 2 Vφ, φ ψ 2 (K + )ξ, φ ψ 2 s (u 0 u Ms 2 Vφ + 2 (K + )ξ ) L 2 (Γ) if φ ψ ψ 2 (Γ) (5.3) We esimae he ierior erms, emploig he iegraio by par, we obai A 5 = Ω = T Ω E Γ f(u v Ms ) dx C ɛ ε(u Ms ), ε(u v Ms ) {f + divσ(u Ms T σ(u Ms E h )}(u v Ms ) E Ω [σ(u Ms E ) ] E (u v Ms ) ds ) E (u v Ms ) ds. (5.4) Recall ha by corucio of he muliscale basis fucios, we have divσ(u Ms ) = 0 i T. 24

135 5.6 Couplig FEM-BEM Usig (5.09)-(5.4), we obai C 2 ɛ ε(u u Ms ) 2 L 2 (Ω) + ξ ξ 2 + φ φ 2 2 (Γ)/R 2 (Γ) f(u v Ms ) dx + [σ(u Ms ) ] E (u v Ms ) ds E Ω E Ω σ(u Ms ) 0 φ, u v Ms + 2 (K + )φ + Wξ L 2 (Γ) if ξ ξ ξ 2 (Γ) + 2 s (u 0 u Ms + 2 Vφ 2 (K + )ξ ) L 2 (Γ) if φ φ φ 2 (Γ) (5.5) Firs, we esimae he erm B := Le f(x) = T T f, if x T he Ω f 2 = Ω (f f + f) 2 = Ω Ω f(u v Ms ) dx (5.6) (f f) T T f (f f) + T Ω (f) 2 owever T (f f) = 0 Ω f 2 = Ω (f f) 2 + (f) 2 (5.7) Ω Usig (5.39), choosig v Ms B T Ω = ums IMs f L 2 (T ) u vms (u ums), wih e := u ums, we obai L 2 (T ) C f L 2 (T ) ( e (T ) + 2 f L 2 (T ) ) T Ω C( C T Ω T Ω 2 f 2 L 2 (T ) + T Ω f L 2 (T ) e (T ) 2 f 2 L 2 (T ) + ɛ 2 e 2 (T ) (5.8) 25

136 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy Now we esimae he erm T Ω 2 f 2, employig (5.7), we ge L 2 (T ) T Ω 2 f 2 L 2 (T ) = 2 ( f 2 dx) T Ω T h T T h = 2 { f f 2 dx + f 2 dx} T Ω T h T T h T h T T h = 2 { f f 2 dx + T h f 2 } T Ω T h T T h T h T = 2 { f f 2 dx+ T f 2 } T Ω T h T T h = 2 (T h ) f f 2 dx + 4 f 2 T h Ω T h T Ω = ηt 2 h + ηt 2. (5.9) T h Ω T Ω ier f T is a cosa o T, ad T 2. We obai B ηt 2 h + ηt 2 + ɛ 2 e 2 (T (5.20) ) T h Ω T Ω Now we esimae he erm Choosig v Ms = ums B 2 = IMs E Ω [σ(u Ms E ) ] E (u v Ms ) ds. (5.2) (u ums), sice IMs u = I u o T, we ge u v Ms L 2 (E ) = e IMs e L 2 (E ) = e I e L 2 (E ). (5.22) From [[2],Lemma 4], for every edge E of T ad u (T ), we have u 2 L 2 (E ) C( u 2 L 2 (T ) +2 u 2 (T ) ) (5.23) Usig (5.23), we ge u v Ms 2 L 2 (E = e IMs ) e 2 L 2 (E ) C ( ) e I e 2 L 2 (T ) +2 e I e 2 (T ) C ( ) 2 e 2 (T ) + 2 e 2 (T ) C e 2 (T ). (5.24) 26

137 5.6 Couplig FEM-BEM Usig (5.24), we obai B 2 E Ω e (Ω) C E Ω [σ(u Ms ) ] L 2 (E ) u vms L 2 (E ) E Ω Now we esimae he firs erm i (5.25), wih 2 [σ(u Ms ) ] L 2 (E ) ([σ(u Ms ) ]) 2 ds + ɛ E 2 e 2 (Ω). (5.25) we have B 22 = E Ω = E Ω = E Ω = E Ω = E h E Ω = E h E Ω where E. E h E [σ(u Ms ) ](x) = [σ(u Ms ) ] (5.26) E E ([σ(u Ms ) ]) 2 = E ([σ(u Ms E h E h E E h E (E h ) η 2 E h + ([σ(u Ms E h ([σ(u Ms E h ([σ(u Ms E h E Ω E Ω E h E E h ([σ(u Ms ) ]) 2 ) ] [σ(u Ms ) ])2 + E h E ([σ(u Ms E h ) ] [σ(u Ms ) ])2 + E h ([σ(u Ms E h E ) ] [σ(u Ms ) ])2 + E ([σ(u Ms ) ] [σ(u Ms ) ])2 + E Ω ) ])2 ) ])2 ) ])2 2 ([σ(u Ms ) ])2 η 2 E + ɛ 2 e 2 (Ω). (5.27) We obai B 2 ηe 2 h + ηe 2 + ɛ 2 e 2 (Ω). (5.28) E h E Ω E Ω Now we esimae he erm B 3 := σ(u Ms ) 0 φ, u v Ms (5.29) 27

138 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy Choosig v Ms = ums IMs (u ums ), B 3 σ(u Ms ) 0 φ L 2 (Γ) u vms L 2 (Γ) C e (Ω) 2 σ(u Ms ) 0 φ L 2 (Γ) σ(u Ms ) 0 φ 2 L 2 (Γ) + ɛ 2 e 2 (5.30) (Ω) E Γ Usig Youg s iequaliy, we ge ad B 4 := 2 (K + )φ + Wξ L 2 (Γ) if ξ ξ ξ 2 (Γ) (K + )φ + Wξ 2 L 2 (E ) + ɛ 2 ξ ξ 2, (5.3) 2 (Γ) E Γ B 5 := 2 s (u 0 u Ms + 2 Vφ 2 (K + )ξ ) L 2 (Γ) if φ φ φ 2 (Γ) s (u 0 u Ms + 2 Vφ 2 (K + )ξ ) 2 L 2 (E ) + ɛ 2 φ φ 2 2 (Γ) E Γ (5.32) We hrow he erms e 2 (Ω), ξ ξ 2 ad φ φ 2 o he lef had side, 2 (Γ) 2 (Γ) ad we obai he esimae of he heorem. Theorem 5.7. (A poseriori error esimae for > ɛ) Assume ha Lemma 5.2 ad Theorem 5.2 hold for he rasmissio problem. Le (u, φ, ξ) be he soluio of (5.00) ad (u Ms, φ, ξ ) be he soluio of he discree problem (5.0). The here holds he esimae: C 2 ɛ ε(u u Ms ) 2 L 2 (Ω) + ξ ξ 2 + φ φ 2 2 (Γ)/R 2 (Γ) σ(u Ms ) 0 φ 2 L 2 (E ) E Γ + E Γ + E Γ + T h Ω s (u 0 u Ms + 2 Vφ 2 (K + )ξ ) 2 L 2 (E ) (K + )φ + Wξ 2 L 2 (E ) η 2 T h + T Ω η 2 T + E h E Ω η 2 E h + E Ω ηe 2 + f 2 L 2 (T ) ( ɛ ) T Ω 2 Proof. To prove he heorem, we firs deoe u I (x) = I Ms u 0 (x) = j u 0 (x j )Φ Ms j, (5.33) 28

139 5.6 Couplig FEM-BEM where u 0 is he soluio of he homogeized equaio ad u I is he ierpola of u 0,usig he muliscale basis fucios Φ Ms j, oe ha u I is differe from he defiiio of I Msu. We kow ha From (5.5), we have L ɛ (u I ) = 0 i T, u I = I u 0 o T (5.34) ɛ ε(u u Ms ) 2 L 2 (Ω) + ξ ξ 2 + φ φ h 2 2 (Γ)/R 2 (Γ) f(u v Ms ) dx + [σ(u Ms ) ] E (u v Ms ) ds E C 2 Ω E Ω σ(u Ms ) 0 φ, u v Ms + 2 (K + )φ + Wξ L 2 (Γ) if ξ ξ ξ 2 (Γ) + 2 s (u 0 u Ms + 2 Vφ 2 (K + )ξ ) L 2 (Γ) if φ φ φ 2 (Γ) (5.35) Firs, we esimae he erm B := Ω f(u v Ms ) dx (5.36) We choose v Ms = u I, usig (5.47) ad Cauchy Schwarz iequaliy, we obai B f L 2 (T ) u u I L 2 (T ) T Ω f L 2 (T ) (C 2 f L 2 (Ω) +C 2( ɛ ) 2 ) T Ω T Ω C 2 f 2 L 2 (T ) + T Ω C 2 f L 2 (T ) ( ɛ ) 2 (5.37) From (5.9), we obai B ηt 2 h + ηt 2 + f 2 L 2 (T ) ( ɛ ) T h Ω T Ω T Ω 2 (5.38) Now we esimae he erm B 2 := E Ω [σ(u Ms E ) ] E (u v Ms ) ds. (5.39) 29

140 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy Usig v Ms = I u o T, we obai (see (5.22)-(5.28)) B 2 ηe 2 h + ηe 2 + ɛ 2 e 2 (Ω). (5.40) E h E Ω Similarly o Theorem 5.6, we obai E Ω B 3 := σ(u Ms ) 0 φ, u v Ms σ(u Ms ) 0 φ 2 L 2 (Γ) + ɛ 2 e 2 (5.4) (Ω) E Γ ad B 4 := 2 (K + )φ + Wξ L 2 (Γ) if ξ ξ ξ 2 (Γ) (K + )φ + Wξ 2 L 2 (E ) + ɛ 2 ξ ξ 2, (5.42) 2 (Γ) E Γ B 5 := 2 s (u 0 u Ms + 2 Vφ 2 (K + )ξ ) L 2 (Γ) if φ φ φ 2 (Γ) s (u 0 u Ms + 2 Vφ 2 (K + )ξ ) 2 L 2 (E ) + ɛ 2 φ φ 2 2 (Γ) E Γ (5.43) We hrow he erms e 2 (Ω), ξ ξ 2 ad φ φ 2 o he lef had side, 2 (Γ) 2 (Γ) he esimae of he Theorem follows immedialy. 5.7 Numerical Experimes Example : I his example we show he covergece of he EMsFEM for a homogeeous problem. The resuls obaied by he radiioal FEM ca be regarded as referece values. We choose Youg s modulus E = 2000 ad Poisso s raio ν = 0.3. The umber of coarse elemes is M c N c, where M c ad N c deoe he umbers of he elemes i he x ad y direcios, respecively. We cosider a umber of coarse elemes M c N c bewee 2 2 ad There are M f N f fie scale elemes wihi each coarse eleme. I Figure we show he disribuios of he microscopic vo-mises sress obaied by he exeded muliscale fiie eleme mehod EMsFEM ad he radiioal fiie eleme mehod FEM. The differece bewee he EMsFEM ad he radiioal FEM ca be observed for M c N c = 2 2. The resuls obaied by EMsFEM coverge o he referece values as he umber of elemes M c N c icreases as show i Figure Example 2. I his example we assume ha he Poisso s raio is a cosa, ν = 0.3, ad Youg s modulus varies i he rage of accordig o a uiform 30

141 5.7 Numerical Experimes disribuio. Youg s modulus ad he Poisso raio are give as show i Figure 5.9. The legh ad he widh of he model are give by L x, L y = L respecively where L x = N x L y ad N x N. ere we choose L = ad N x = 6. The small scale of he heerogeeiy is deoed by L h, ad we assume ha Youg s modulus is cosa i each eleme L h L h. The umber of coarse elemes is M c N c, where M c ad N c deoe he umbers of coarse elemes i he x ad y direcios, respecively. I Figure 5.0 we show he deformed cofiguraio for = I his example we use a discreizaio of he size of he heerogeeiy, L h = h. For he FEM, good resuls are obaied oly if he mesh size is smaller ha he size of he heerogeeiies L h. The resuls obaied by he FEM ca be regarded as referece values for L h = 8h. The resuls obaied by he radiioal FEM have relaive large errors if L h h [54]. The EMsFEM has higher accuracy ha he radiioal FEM. The umerically cosruced basis fucios capure he micro scale heerogeeiies wihi each coarse eleme as show i Figure 5.. 3

142 5 Exeded Muliscale Fiie Eleme Mehod i Liear Elasiciy Figure 5.4: The disribuios of he Mises sress obaied by he EMsFEM wih M c N c = 2 2, M f N f = 6 6, = 0.5, h = Figure 5.5: The disribuios of he Mises sress obaied by he EMsFEM wih M c N c = 24 4, M f N f = 8 8, = 0.25, h = Figure 5.6: The disribuios of he Mises sress obaied by he EMsFEM wih M c N c = 48 8, M f N f = 4 4, = 0.25, h = Figure 5.7: The disribuios of he Mises sress obaied by he EMsFEM wih M c N c = 96 6, M f N f = 2 2, = , h = Figure 5.8: The disribuios of he Mises sress obaied by FEM wih h =

143 5.7 Numerical Experimes Figure 5.9: The Youg s modulus ad he Poisso s raio for he heerogeeous model Figure 5.0: Deformed geomery Figure 5.: The disribuios of he Mises sress obaied by he EMsFEM wih M c N c = 96 6, = , L h = h = Figure 5.2: The disribuios of he Mises sress obaied by FEM wih L h = ad h = (L h = 8h) 33