ESAIM: MAN 43 (009) 1003 106 DOI: 10.1051/man/009015 ESAIM: Mathmatical Modlling and Numrical Analysis www.saim-man.org A POSERIORI ESIMAES FOR HE CAHN HILLIARD EQUAION WIH OBSACLE FREE ENERGY L ubomír Baňas 1 and Robrt Nürnbrg Abstract. W driv a postriori stimats for a discrtization in spac of th standard Cahn Hilliard quation with a doubl obstacl fr nrgy. h drivd stimats ar robust and fficint, and in practic ar combind with a huristic tim stp adaptation. W prsnt numrical xprimnts in two and thr spac dimnsions and compar our mthod with an xisting huristic spatial msh adaptation algorithm. Mathmatics Subjct Classification. 65M60, 65M15, 65M50, 35K55. Rcivd January 14, 008. Rvisd Novmbr 8, 008. Publishd onlin Jun 1, 009. 1. Introduction In this papr w driv spatial a postriori rrorstimats for a pic-wis linar finit lmnt approximation of th following Cahn Hilliard quation: γ u = Δw in Ω := Ω [0,], t w = γδu + 1 γ Ψ (u) in Ω, u ν = w ν =0 on Ω (0,], u(, 0) = u 0 in Ω, (1.1) whr Ω is a convx polyhdral domain in R d, d =, 3, and >0 is a fixd positiv tim. Morovr, Ψ is a givn nrgy potntial, and in this papr w will tak Ψ to b th so calld doubl obstacl potntial Ψ(s) := { 1 (1 s ) if s [ 1, 1], if s/ [ 1, 1]. (1.) Kywords and phrass. Cahn Hilliard quation, obstacl fr nrgy, linar finit lmnts, a postriori stimats, adaptiv numrical mthods. Supportd by th EPSRC grant EP/C548973/1. 1 Dpartmnt of Mathmatics and th Maxwll Institut for Mathmatical Scincs, Hriot-Watt Univrsity, Edinburgh, EH14 4AS, UK. L.Banas@hw.ac.uk Dpartmnt of Mathmatics, Imprial Collg London, London, SW7 AZ, UK. Articl publishd by EDP Scincs c EDP Scincs, SMAI 009
1004 L. BAŇAS AND R. NÜRNBERG W not that othr choics of Ψ ar also possibl, s.g. (1.4) blow. In addition, th paramtr γ>0isan intraction lngth, which is small compard to th dimnsions of Ω. Equation (1.1) was originally introducd by Cahn and Hilliard to modl spinodal dcomposition and coarsning phnomna in binary alloys, s [11,1]. Hr u is dfind to b th diffrnc of th local concntrations of th two componnts of an alloy and hnc u is rstrictd to li in th intrval [ 1, 1]. Mor rcntly, th Cahn Hilliard quation has bn usd.g. as a phas fild modl for sharp intrfac volutions and to study phas transitions and intrfac dynamics in multiphas fluids, s.g. [7,,3] and th rfrncs thrin. W not that in (1.1) w hav usd a tim scaling, so that in th limit γ 0, w rcovr th wll known sharp intrfac motions by Mullins Skrka. W rcall that this limit was first formally shown in [6], and latr provd rigorously in [1]. W not that as proprtis of commrcially producd matrials dpnd on microstructurs which ar gnratd using spcial procssing tchniqus, such as phas sparationand coarsningmchanisms, accurat prdictions of microstructur or th volution of pattrn formation during phas sparation and coarsning ar of considrabl intrst in matrials scinc. As it is difficult to obtain such information by ral-lif xprimnts, rliabl numrical computations ar vry important. It is th aim of this papr to prov suitabl a postriori stimats for th discrt approximation of th considrd problm that can b usd to construct robust and rliabl msh rfinmnt algorithms in two and thr spac dimnsions, which allow for fficint and rliabl numrical simulations. h thory of Cahn and Hilliard is basd on th following Ginzburg Landau fr nrgy ( E(u) := γ u + γ 1 Ψ(u) ) dx. (1.3) Ω h first trm in th fr nrgy pnalizs larg gradints and th scond trm is th homognous fr nrgy. hn (1.1) can b drivd from mass balanc considrations as a gradint flow for th fr nrgy E(u), with th chmical potntial w := δe δu bing th variational drivativ of th nrgy E with rspct to u. For notational convninc in (1.1) it was implicitly assumd that th fr nrgy Ψ is diffrntiabl. An xampl for such a potntial function is Ψ(s) = 1 4 (s 1), (1.4) which has th advantag of bing smooth but th disadvantag that physically non-admissibl valus with u > 1 can b attaind during th volution. Of cours, th obstacl fr nrgy (1.) forcs u to stay within th intrval [ 1, 1] of physically maningful valus. his is a clar advantag ovr a formulation involving (1.4). Hnc, in this papr w will from now on considr th obstacl fr nrgy (1.). hn th chmical potntial w nds to b computd with th hlp of a variational inquality, s (.1) blow. It is this variational inquality which rquirs spcial attntion in dvloping an a postriori rror stimat. ypical volutions of (1.1) starting from a wll mixd initial stat bgin with a rlativly short arly phas, calld spinodal dcomposition, in which th local concntrations u grow towards th minimizrs ± 1of(1.). his lads to a stup, whr larg parts of th domain ar occupid by rgions whr u = ± 1, which ar sparatd by intrfacial rgions whr u < 1, in which u smoothly varis from 1 to1. hnfollowsa much slowr volution phas, in which th total volum of ths intrfacial rgions is dcrasd. his phas is calld coarsning. h thicknss of th intrfacial rgions, i.., th rgion whr u < 1, is asymptotically of ordr O(γ). As mntiond arlir, it can b shown that in th sharp intrfac limit (i.., whnγ 0) th long tim dynamics of quations (1.1) corrspond to th Mullins Skrka quation. Finit lmnt mthods for quation (1.1) with(1.) hav bn proposd and analyzd in [9], s also [5,6]. In addition, xistnc and uniqunss of th solution u, w to (1.1), as wll as rgularity rsults, wr shown in [8]. In [7] a finit lmnt approximation for a rlatd, so calld dgnrat, Cahn Hilliard quation was considrd, and in addition a huristic adaptiv msh rfinmnt algorithm was usd for numrical simulations in two spac dimnsions, in ordr to incras th fficincy of th computations. his approximation and th corrsponding msh rfinmnt hav rcntly bn xtndd to thr spac dimnsions in [3], s also [4]. hr xist numrous works on finit lmnt approximations of (1.1) with smooth potntials such as (1.4).
A POSERIORI ESIMAES FOR HE CAHN HILLIARD EQUAION 1005 Hr w rfr to.g. [16 18] and th rfrncs thrin. A postriori stimats for th Cahn Hilliard quation with th smooth potntial (1.4) hav vry rcntly bn obtaind in [19], whr th stimats for a continuous in tim smi-discrt approximation only dpnd on polynomial powrs of γ 1, a rsult which crucially dpnds on th spctral stimat from [13]. o our knowldg, so far thr is no work on a postriori stimats for th Cahn Hilliard quation with th obstacl potntial (1.4), apart from th numrical rsults in [], which ar basd on rsults rlatd to th work in this papr. It is th aim of this papr to prov a postriori stimats and xamin adaptiv finit lmnt mthods for (1.1) in two and thr spac dimnsions. Sinc thr is no spctral stimat corrsponding to that from [13] availabl for th non-smooth modl, w only xamin th rror du to th spatial discrtization. hrfor w rstrict our analysis to spatial a postriori rror stimats for a discrt in tim analogu of (1.1). In particular, w will driv stimats for a coupld systm that consists of an lliptic variational inquality involving two constant obstacls, and a linar lliptic problm; s (.1) blow. By using th idas of [7], whr rror stimats for linar finit lmnt approximations of lliptic obstacl problms ar introducd, w ar abl to obtain an stimat with localizd intrior rsidual, which nabls ffctiv and rliabl rror control by rfinmnt that is mainly concntratd in th intrfacial rgion, whr u < 1. h a postriori analysis of lliptic obstacl problms is a rlativly nw fild. A rsidual a postriori stimat with non-localizd intrior rsidual was obtaind in [14]. A sharpr stimat with localizd intrior rsidual was constructd in [8] for constant obstacls and in [7] for gnral obstacls. A short rviw on a postriori stimats for lliptic obstacl problms is givn in [10]. A postriori stimats for parabolic variational inqualitis wr drivd in [4] by xtnding th idas of [7]. W also rfr to work in optimal control thory, whr vry rcntly an rror stimator for a control problm with sid constraints involving PDEs and inquality constraints has bn introducd in [0,1]. Howvr, w strss that a crucial diffrnc btwn work on optimal control thory and work on obstacl problms involving variational inqualitis is that th formr only applis th inquality constraints on th right hand sid of th control PDE, and that th localization of th intrior rsidual is not ssntial to obtain a lowr bound for th rror,.g. s [1]. h papr is organizd as follows. In Sction, w introduc th continuous in spac and discrt in tim Cahn Hilliard quation and its finit lmnt approximation by conforming pic-wis linar lmnts. In Sction 3, w stablish an a postriori stimat with non-localizd rsidual, which can potntially lad to xtnsiv msh rfinmnt outsid of th intrfacial rgion, i.. in th rgion whr th solution u is constant. In Sction 4, w construct uppr and lowr bounds for th rror with localizd intrior rsidual. In Sction 5, w discuss a numbr of adaptiv algorithms for numrical computations. Finally, Sction 6 is dvotd to numrical xprimnts, whr w xamin th prformanc of th adaptiv algorithms in two and thr spac dimnsions.. Finit lmnt approximation W considr th following continuous in spac smi-discrt countrpart of th Cahn Hilliard quation obtaind by a backward-eulr tim discrtization of (1.1): Find u K:= {v H 1 (Ω) : v 1} and w H 1 (Ω) such that (u, φ)+ τ γ ( w, φ) = (f,φ) φ H1 (Ω), (.1) γ ( u, (ψ u)) (w, ψ u) (g, ψ u) ψ K, whr (φ, ψ) = Ω φψ is th L -innr product ovr Ω. hroughout this papr, w dnot th L -norm ovr D Ωby D, and similarly us 1,D for th H 1 -norm. For notational convninc, w drop th subscript in th cas D = Ω. In addition, w dnot th norm in th dual spac (H 1 (Ω)) by 1 and us, for th duality pairing btwn H 1 (Ω) and its dual. On introducing th linar finit lmnt spac V h := {φ C(Ω) : φ is linar h } H 1 (Ω),
1006 L. BAŇAS AND R. NÜRNBERG whr Ω= h, w considr th following finit lmnt approximation of (.1): Find u h K h and w h V h such that (u h,φ)+ τ γ ( w h, φ) = (f,φ) φ V h, (.) γ ( u h, (ψ u h )) (w h,ψ u h ) (g, ψ u h ) ψ K h, whr K h := K V h. Not that in viw of th drivation of (.1), w usually hav f = u old h, g = 1 γ uold h in (.), whr u old h is th solution from th prvious tim stp. hn (.) corrsponds to on tim stp of th ar picwis linar functions, whr Vold h is th finit lmnt spac corrsponding to th prvious tim stp. his cas will simplify som stps in th analysis blow, in particular whn Vold h V h. W dnot by u = u u h, (.3) w = w w h. unconditionally stabl, fully discrt approximation in [9]. Also, in that cas f, g V h old W rcall th following wll-known rsult concrning V h : (φ, χ) (φ, χ) h C h φ χ φ, χ V h ; (.4) whr (φ, χ) h = Ω Ih (φχ)forφ, χ C(Ω) is th usual mass lumpd innr product, and I h is th usual Lagrang intrpolation oprator onto V h. In addition to th triangulation h, w introduc th st of its nods P h and dgs E h. W dnot th nodal basis functions of V h as (χ p h ) p P h,whrχ p h (q) =1ifp = q and χp h (q) = 0 othrwis. Morovr, for ach h and E h w dnot thir diamtr by h and h, rspctivly. W also introduc th local msh siz function h :Ω R, which is picwis constant and such that h = h for all h. For any st D Ω, w dfin th discrt nighbourhood of D by D = { h ; D }. In addition, in a slight abus of notation, w also introduc th short hand notation h α [ u h ] := E h h α [ u h ],whrα R. 3. A POSERIORI stimat with positivity prsrving intrpolation In this sction w xtnd th idas of [14], in ordr to show how it is possibl to driv an uppr bound for th rror of th finit lmnt approximation in a rlativly simpl mannr. h obtaind stimat, howvr, dos not tak into account crtain spcial proprtis of th solution, and may lad to xcssiv msh rfinmnt in practic, in aras whr th solution u is constant. W rcall th dfinition of th positivity prsrving intrpolation oprator Π h 0 : L 1 (Ω) V h H0 1 (Ω) from [14], i.., whavthatu 0 Π h 0 u 0 for all u L1 (Ω). It is thn a straightforward mattr to xtnd this dfinition to th Numann boundary condition and doubl obstacl prsnt hr, to obtain an analogous oprator Π h : L 1 (Ω) V h such that u K Π h u K h. (3.1) In fact, w can choos Π h to b th oprator givn in [5], Exampl 1.1. W hav th following approximation proprtis of Π h for u H 1 (Ω) and u h V h : Π h u u, (3.a) u Π h u Ch u, (3.b) u Π h u Ch 1/ u ẽ, (3.c) u h Π h u h C h 3/ [ u h ], (3.d) u h Π h u h C h [ u h ] ẽ, (3.) cf. [14], whr w rcall that is th union of all th lmnts surrounding, and similarly for ẽ.
A POSERIORI ESIMAES FOR HE CAHN HILLIARD EQUAION 1007 Choosing φ = w in (.1) andφ =Π h w in (.), w obtain that ( u, w )+ τ γ w =(f, w Π h w ) (u h, w Π h w ) τ ( wh, ( w Π h w ) ). (3.3) γ Nxt, w tak ψ = u h in (.1), lading to and ψ =Π h u K h, rcall (3.1), in (.), which givs γ ( u, u ) (w, u ) (g, u ) 0, (3.4) γ ( u h, u ) (w h, u ) (g, u ) ( g, Π h u u ) γ ( u h, (Π h u u) ) + ( w h, Π h u u ). (3.5) W hav th simpl idntity Π h u u =(Π h u u )+(Π h u h u h ). hrfor, aftr w subtract (3.5) from(3.4), w obtain γ u ( w, u ) ( g, u Π h ) ( u γ uh, ( u Π h u ) ) + ( w h, u Π h ) u + ( g + w h, Π h ) ( u h u h γ uh, (Π h u h u h ) ) (3.6). Furthr, aftr mploying intgration by parts, sinc Δu h =0,wobsrvthat ( u h, ψ) = u h ψ = { } u h ν ψ Δu h ψ h h = [ u h ] ψ ψ H 1 (Ω), E h (3.7) whr ν dnots th outward unit vctor to h. Hnc it follows from (3.3), on applying th Cauchy Schwartz and Young inqualitis togthr with (3.b) and (3.7), that ( u, w )+ τ γ w C γ τ h(u h f) + τ 4γ w + C τ γ h1/ [ w h ] + τ 4γ w. (3.8) Similarly, it follows from (3.6) that γ u ( w, u ) C γ h(g + w h) + γ 4 u + γc h 1/ [ u h ] + γ 4 u + ( g + w h, Π h u h u h ) γ ( uh, (Π h u h u h ) ). (3.9) h last two trms on th right-had sid of (3.9) can b stimatd, on noting (3.7) and(3.d)-(3.), as ( g + wh, Π h ) ( u h u h γ uh, (Π h u h u h ) ) C (γ h 1/ [ u h ] + 1γ ) h(g + w h). By combining th prvious quation with (3.8), (3.9) w arriv at ( ) τ γ w + γ u γ C τ h(f u h) + τ γ h1/ [ w h ] + 1 γ h(g + w h) + γ h 1/ [ u h ]. (3.10) Upon subsquntly rscaling w obtain [ ] γ τ w + γ u C τ h(u h f) + τ h 1/ [ w h ] + h(g + w h ) + γ h 1/ [ u h ]. (3.11)
1008 L. BAŇAS AND R. NÜRNBERG Rmark 3.1. h disadvantag of th abov stimat is that th intrior rsidual h(g + w h ) corrsponding to th variational inquality in (.) is not localizd to th noncontact st (s dfinition in th nxt sction), which can caus xcssiv msh rfinmnt in th contact st, whr th solution u h is constant and whr w h usually attains larg valus. Howvr, as th variational inquality in (.) trivially holds in th contact st, idally thr should b no contribution from th intrior rsidual to th a postriori rror stimat. his problm will b addrssd in th nxt sction. 4. A POSERIORI stimat with localizd intrior rsidual In this sction w driv an a postriori stimat with an intrior rsidual localizd to th intrfac, i.. th intrior rsidual inducd by th variational inquality in (.) is zro in th rgion whr u h =1. his rsult givs ris to mor fficint a postriori rror basd msh rfinmnt stratgis, and it is furthrmor a thortical justification for th construction of huristical msh adaptiv algorithms, whr th msh rfinmnt is concntratd in th intrfacial ara, i.. whr u h < 1. W xtnd th idas of [7,8] to th smi-discrt formulation (.1) of th Cahn Hilliard quation. Hr w dfin th discrt functions f h := I h f, g h := I h g and not that by dfinition w hav (g h,φ) h =(g, φ) h,(f h,φ) h =(f,φ) h for all φ C(Ω). Instad of th discrt formulation (.), w considr th following discrt problm: Find u h K h and w h V h such that (u h,φ) h + τ γ ( w h, φ) = (f h,φ) h φ V h, γ ( u h, (ψ u h )) (w h,ψ u h ) h (g h,ψ u) h ψ K h. (4.1) h abov formulation only diffrs from (.) in th zro ordr trms, whr w us th rducd discrt innr product (, ) h. Givn th tru solution u, and following th tchniqu in [7] for a singl obstacl, w obtain th partition of th domain Ω=C(u) N(u) F(u), (4.) whr th contact st C(u) is th maximal opn st A Ω such that u 1onA; th noncontact st N (u) := ɛ>0 B ɛ ;whrb ɛ is th maximal opn st B Ω such that u < 1 ɛ; th fr boundary F(u) isthstω\ (C(u) N(u)). h contact st can b furthr dcomposd as C(u) =C + (u) C (u), whr u = ± 1onC ± (u). W dfin th continuous rsidual σ(u) (H 1 (Ω)) as σ(u),ψ =(g, ψ)+(w, ψ) γ( u, ψ) ψ H 1 (Ω). (4.3) h following proprtis can b obtaind from th dfinition of σ (not, u 1 in C(u)) h discrt rsidual is dfind as σ h V h such that σ 0 in C + (u), (4.4a) σ 0 in C (u), (4.4b) σ = g + w in C(u), (4.4c) σ =0 in N (u). (4.4d) (σ h,ψ) h =(g h,ψ) h +(w h,ψ) h γ( u h, ψ) ψ V h. (4.5)
A POSERIORI ESIMAES FOR HE CAHN HILLIARD EQUAION 1009 Altrnativly w can writ (σ h,ψ) h =(g h,ψ) h +(w h,ψ) h + γ(δ h u h,ψ) h, (4.6) whr Δ h : V h V h is th usual discrt Laplacian on V h. W dfin th jump across an innr lmnt dg/fac = 1 E h as [ u h ] = 1 ( u h 1 u h ) ν, whr ν is a unit normal vctor of pointing from 1 to. For a Numann boundary dg E h Ω w dfin [ u h ] = u h ν, whr ν is h outward unit vctor to th boundary Ω. Similarly to (4.), th domain Ω can b dcomposd into Ω=C h (u h ) F h (u h ) N h (u h ), (4.7) whr C h (u h ) := C h (u h ) + C h (u h ), C h (u h ) ± := { h ; u h = ± 1on }, N h (u h ) := { h ; u h < 1on }, F h (u h ) := Ω\ [C h (u h ) N h (u h )]. In our contxt, C h dnots th subdomains with pur matrials, N h dnots th diffus intrfac and F h is th so-calld discrt fr boundary btwn C h and N h. Similarly as in (4.4), on can stablish for all nods p P h that σ h (p) 0 if p C + h, (4.8a) σ h (p) 0 if p C h, (4.8b) σ h (p) =g h (p)+w h (p) if u h = 1 on supp χ p h, (4.8c) σ h (p) =0 if u h (p) < 1. (4.8d) Not that Δ h u h =0inC h (u h ). Following [7], w dfin th Galrkin functional G h (H 1 (Ω)) as G h,ψ = γ( (u h u), ψ) (w h,ψ)+(w, ψ)+(σ h σ, ψ) ψ H 1 (Ω). (4.9) W dirctly hav from (4.3) that G h,ψ = γ( u h, ψ) (w h + g, ψ)+(σ h,ψ) ψ H 1 (Ω). (4.10) Lmma 4.1 (prturbd Galrkin orthogonality). hr xists a constant C dpnding only on th msh rgularity, such that G h 1,h := sup G h,ψ h = C ( γ h ) Δ h u h + g h g 1,h. ψ h V h, ψ h =1 Proof. On rcalling th dfinitions of (, ) h, σ h and G h,whavforanyψ h V h that G h,ψ h = (g, ψ h ) h +[γ( u h, ψ h ) (g + w h,ψ h ) h ]+(w h,ψ h ) h (w h + g, ψ h )+(σ h,ψ h ) = (g, ψ h ) h (g, ψ h )+(w h,ψ h ) h (w h,ψ h )+(σ h,ψ h ) (σ h,ψ h ) h = (g + w h σ h,ψ h ) h (g h + w h σ h,ψ h )+(g h g, ψ h ) = γ(δ h u h,ψ h ) h + γ(δ h u h,ψ h )+(g h g, ψ h ). (4.11)
1010 L. BAŇAS AND R. NÜRNBERG Furthrmor, it follows from (.4) that γ(δ h u h,ψ h ) h + γ(δ h u h,ψ h ) γ h Δ h u h ψ h, which yilds th dsird rsult. Also th following is just a gnralisation of [7], Lmma 3.4, xcpt that th trm σ h σ 1 appar on th lft hand sid of (4.1). Lmma 4.. h following inquality holds dos not Proof. It follows from (4.9) that γ (u h u) (w h w, u h u) C 1 γ G h 1 C (σ h σ, u h u). (4.1) γ (u h u) (w h w, u h u) = G h,u h u (σ h σ, u h u) G h 1 (u h u) (σ h σ, u h u). Hnc by Young s inquality w gt γ (u h u) (w h w, u h u) 1 γ G h 1 + γ (u h u) (σ h σ, u h u). (4.13) h assrtion of th lmma thn asily follows from th last inquality. 4.1. Global uppr bound In th following lmma w stimat th Galrkin functional. Lmma 4.3. hr xists a constant C dpnding only on th msh rgularity, such that ( ) 1/ G h 1 C γ h 1/ [ u h ] + h(g + w h σ h ) + γ h Δ h u h + g gh 1,h. E h Proof. For ϕ H 1 (Ω) w writ G h,ϕ = G h,ϕ I h ϕ + G h,i h ϕ, whr I h ϕ dnots th Clémnt intrpolant for ϕ, s[15]. h scond trm in th abov quation can b stimatd using Lmma 4.1 (th prturbd Galrkin orthogonality) and th proprtis of I h as G h,i h ϕ C G h 1,h I h ϕ C(γ h Δ h u h + g g h 1,h ) ϕ. Similarly, on rcalling (4.10), w can stimat th first trm using standard argumnts of a postriori stimation as G h,ϕ I h ϕ = γ( u h, (ϕ I h ϕ)) (w h + g σ h,ϕ I h ϕ) = γ [ u h ] (ϕ I h ϕ) (w h + g σ h )(ϕ I h ϕ) E h ( ) h 1/ C γ h 1/ [ u h ] + h(g + w h σ h ) ϕ, E h L () which concluds th proof. h following lmma is an adaption of [7], Proposition 3.7.
A POSERIORI ESIMAES FOR HE CAHN HILLIARD EQUAION 1011 Lmma 4.4. h following inquality holds for th solutions u and u h of (.1) and (4.1), rspctivly. whr (σ h σ, u h u) C γ h 4 Δ h u h + 1 h w h + g h + γ h [ u h ], γ h h E h h = { h ; p 1,p P h, u h (p 1 ) =1and u h (p ) < 1}, E h = { E h ; P h } with P h = {p P h; u h (p) =1and u h 1 on supp χ p h } Proof. W rwrit (σ h σ, u h u) =(σ h,u h u)+(σ, u u h ). Sinc u h K, w can stimat th scond trm using (.1) (σ, u u h )=γ( u, (u h u)) (g, u h u) (w, u h u) 0. Nxt, on noting that Ω = C h N h h, w rwrit th first trm on th right-hand sid as (σ h,u h u) = σ h ( 1 u)+ σ h (1 u)+ σ h (u h u)+ σ h (u h u). C h C + h N h h Using (4.8a)-(4.8b) w gt Rcalling (4.8d) w hav C h σ h ( 1 u) 0, C + h σ h (1 u) 0. σ h (u h u) =0. N h h rmaining trm is stimatd as follows. Considr h u h (p ) < 1andσ h 0wgt and p 1,p P h, with u h (p 1 ) = ± 1, σ h (u h u) = σ h (u h 1) + σ h (1 u) σ h (u h 1) σ h u h 1, if σ h 0wgt σ h (u h u) = From [7], Lmma 3.6, w obtain (E h (p) :={ E h ; p }) u h 1 Ch E h (p 1) σ h (u h +1)+ σ h ( 1 u) σ h (u h +1) σ h u h +1. h [u h 1] 1/ Ch E h (p 1) h [u h ] 1/. W hav from σ h = σ h w h g h + w h + g h σ h w h g h + w h + g h,
101 L. BAŇAS AND R. NÜRNBERG that σ h σ h w h g h + w h + g h γh Δ h u h + w h + g h. Finally w gt σ h (u h u) C γh 4 Δ hu h + 1 γ h w h + g h + γ h [u h ], E h (p 1) which, on noting that E h = p P h E h(p), concluds th proof. h following lmma is a simpl consqunc of (4.1) and Lmmas 4.3 and 4.4. Lmma 4.5. [ γ (u h u) (w h w, u h u) C 1 γ h 1/ [ u h ] γ + h(g + w h σ h ) E h + γ h Δ h u h + g gh 1,h + ] h (w h + g h ). h nxt lmma givs an stimat for th first quation in (4.1). Lmma 4.6. [ τ γ (w h w) +(u h u, w h w) C γ τ τ γ h 1/ [ w h ] + h(u h f) E h + h (u h f h ) ] + f f h 1,h. h Proof. W start with th idntity τ γ ( w, φ)+( u,φ)= τ γ ( w, (φ I h φ)) + ( u,φ I h φ)+ τ γ ( w, I h φ)+( u,i h φ) for any φ H 1 (Ω). Nxt, according to (.1), (4.1), w can rwrit th abov quation as τ γ ( w, φ)+( u,φ)= τ γ ( w h, (φ I h φ)) + (f u h,φ I h φ)+ τ γ ( w, I h φ)+( u,i h φ). (4.14) Similarly as in Lmma 4.1, onnoting(.1), (4.1) and(.4), w obtain that τ γ ( w, I h φ)+( u,i h φ) h (u h f h ) φ + f f h 1,h φ, (4.15) and similarly to Lmma 4.3, whavthat τ γ ( w h, (φ I h φ)) + (f u h, (φ I h φ)) C [ ( ) τ 1/ ] h 1/ [ w h ] + h(u h f) φ. (4.16) γ E h h proof can b concludd by combining (4.15), (4.16) and(4.14), and by subsquntly applying a Young s inquality for φ = w.
A POSERIORI ESIMAES FOR HE CAHN HILLIARD EQUAION 1013 h following corollary is a simpl consqunc of Lmmas 4.5 and 4.6. Corollary 4.1. h following stimat is valid for u h, w h : γ (u h u) + τ (w h w) [ C τ h 1/ [ w h ] + γ τ h(u h f) + γ τ E h h (u h f) + γ h 1/ [ u h ] + h(g + w h σ h ) + h (w h + g h ) E h h + γ ] h Δ h u h + g gh 1,h + γ τ f f h 1,h. (4.17) Rmark 4.1. h stimat (4.17) diffrsfrom(3.11) in th following: th intrior rsidual, which is now h(g + w h σ h ) + h (w h +g h ), is localizd to th discrt noncontact st Ω \C h, on rcalling (4.8c); for simplicity, w did not considr coarsning in th drivation of (3.11), i.. th trms g g h 1,h, γ τ f f h 1,h ar not includd in (3.11); th trms γ τ h (u h f), γ h Δ h u h in (4.17) ar du to th us of th discrt innr product (, ) h and ar thrfor not prsnt in (3.11). Finally, w not that th quantity h Δ h u h is 0 within th discrt contact st, cf. [7], Rmark 3.7, and so it will not contribut to th a postriori rror stimat in that rgion. 4.. Local lowr bounds h o ach function f L (Ω) w assign a picwis constant function f dfind as f = 1 f h. Furthr, th so-calld local data oscillation is dfind as osc h (f,) = h (f f). Lmma 4.7. h following local stimat holds for all h [ γ h 1/ [ u h ] + h (g + w h σ h ) + γ h Δ h u h ] 1/ { C γ (u h u) + h (g g h ) + h (σ h σ) + h (w h w) +osc h (g + w h σ h,) Proof. h proof is basd on th local argumnt of Vrfürth [9]. With vry h, E h w rspctivly associat th standard canonical bubbl functions ψ, ψ. For tchnical rasons, w introduc th auxiliary function z h := γu h. hn, following a similar argumnt in [14], }
1014 L. BAŇAS AND R. NÜRNBERG for any h, w can construct a function φ := α ψ + β ψ,whrα, β ar chosn such that ([ z h ],φ ) = h [ z h ], (g + w h σ h,φ ) = h g + w h σ h, and [ ] 1/ φ C h 1/ [ z h ] + h (g + w h σ h ), [ ] 1/ φ Ch h 1/ [ z h ] + h (g + w h σ h ). W hav, on rcalling (3.7) and(4.3), that γ h 1/ [ u h ] + h (g + w h σ h ) = = ([ z h ],φ ) +(g + w h σ h,φ ) h 1/ [ z h ] + h (g + w h σ h ) = γ ([ u h ],φ ) +(g + w h σ h,φ ) = γ( u h, φ ) +(g + w h σ h,φ ) +(g + w h σ h (g + w h σ h ),φ ) = γ( (u u h ), φ ) (w w h,φ ) +(σ σ h,φ ) +(g + w h σ h (g + w h σ h ),φ ) γ (u u h ) φ + w w h φ + σ σ h φ + g + w h σ h (g + w h σ h ) φ [ ] C γ (u u h ) + h w w h + h σ σ h +osc h (g + w h σ h,) [ γ ] 1/ h 1/ [ u h ] + h (g + w h σ h ). (4.18) Nxt, w hav from(4.6) and an invrs inquality that γ h Δ h u h = h (g h + w h σ h ) Ch g h + w h σ h C [h g + w h σ h + h g g h +osc h (g + w h σ h,)]. (4.19) Finally, th assrtion of th lmma follows on combining (4.18) and(4.19) and on noting that h (g + w h σ h ) h (g + w h σ h ) +osc h(g + w h σ h,).
A POSERIORI ESIMAES FOR HE CAHN HILLIARD EQUAION 1015 Lmma 4.8. h following stimat holds for all h [ τ γ [ w h ] + h (u h f) + h (u h f) ] 1/ h 1/ C ( ) τ γ (w h w) + h (u h u) + h (f f h ) +osc h (u h f,). (4.0) Proof. h proof is similar to th proof of th prvious lmma. Similarly to bfor, w can construct a function φ := α ψ + β ψ,whrα, β ar chosn such that ([ w h ],φ ) = τ γ h [ w h ], (f u h,φ ) = h u h f, and φ C [ τ γ φ Ch [ τ h 1/ [ w h ] + h (u h f) ] 1/, γ h 1/ [ w h ] + h (u h f) ] 1/. W can writ, on rcalling (.1), that τ γ h 1/ [ w h ] + h (u h f) = τ ([ w h ],φ ) (u h f,φ ) γ = τ γ ( w h, φ ) (u h f,φ ) (u h f (u h f),φ ) = τ γ ( (w w h), φ ) +(u u h,φ ) (u h f (u h f),φ ) τ γ (w w h) φ + u u h φ + u h f (u h f) φ τ ] C[ γ (w w h) + h u u h +osc h (u h f,) [ τ γ h 1/ [ w h ] + h (u h f) ] 1/. (4.1) Finally, similarly to (4.19), w hav from an invrs inquality that h (u h f) C h (u h f) C [ h (u h f) +osc h (u h f,) ]. (4.) Combining (4.1) and(4.) concluds th proof. Rmark 4.. h rror quantitis in Lmmas 4.7 and 4.8 contain additional trms,.g. h (w h w) and h (u h u), which ar not prsnt in th rror xprssion for th uppr bound, cf. Corollary 4.1. hrfor, w ar not abl to combin ths two lmmas in ordr to obtain a lowr bound that corrsponds prcisly to th uppr rror stimat in Corollary 4.1, i.. a lowr bound for th rror γ (u h u) + τ (w h w).
1016 L. BAŇAS AND R. NÜRNBERG Naturally, such a lowr bound would b dsirabl, as it would giv a thortical proof of th fficincy of th drivd a postriori stimator. 5. Adaptiv algorithms In this sction w introduc svral msh adaption stratgis, that ar basd on th a postriori rror stimator drivd in Sction 4. hroughout this sction, w assum that f = u old h, g = 1 γ uold h aris from a fully discrt approximation of (1.1), whr u old h is th discrt solution from th prvious tim lvl. Hnc f, g ar picwis linar functions on Vold h, th finit lmnt spac from th prvious tim lvl, and thy will only diffr from f h = I h f and g h = I h g, rspctivly, if Vold h V h, i.., whn msh coarsning is mployd. W dfin th following local rror indicators: η u, = 1 h 1/ [ u h ] + 1 γ h (g + w h σ h ) + 1 γ h (g + w h ) ( S S) ; h η w, = τ γ h 1/ [ w h ] + 1 τ h (u h f) ; h (u h f). η c, = h Δ h u h + 1 τ h global rror indicators ar thn dfind as a corrsponding sum of local rror indicators, i.., η u = η w = h η u,, η w,, η c = η c,. h h By using th abov dfinition of th rror indicators, Corollary 4.1 can b rformulatd as (u h u) + τ [ γ (w h w) C η u + η w + η c + 1 γ g g h 1,h + 1 ] τ f f h 1,h. (5.1) Furthr, in th numrical xprimnts w masurd th rlativ rror by th indicator dfind as: η rl = η u + η w + η c u h 1 Rmark 5.1. h rror contributions in (5.1) can b classifid as follows η u corrsponds to th discrtization rror of u; η w corrsponds to th discrtization rror of w; η c corrsponds to th consistncy rror causd by th us of th mass lumpd product (, ) h ; th trms 1 γ g g h 1,h, 1 τ f f h 1,h corrspond to th rror in th approximation of th solution from th prvious tim-lvl causd by msh coarsning, i.., thy ar zro if no lmnts ar coarsnd; w introduc a huristic indicator η τ for tim stp control as follows η τ = 1 γ u h g h 1. Rmark 5.. h discrt dual norm 1,h is difficult to comput in practic, cf. [4], Rmark 5.. Instad of using th dual norm w dfin a simpl coarsning indicator using th L norm as follows: η h, = 1 γ g Ih g 1 γ g Ih g 1, 1 γ g g h 1,h,. Also not that for our choic f = u old h, g = 1 γ uold h,whavthat 1 γ g g h 1,h = τ γ 4 ( 1 τ f f h 1,h ). Hnc th trm 1 τ f f h 1,h can b nglctd, whn τ = O(γ ), which is gnrally th cas in our xprimnts.
A POSERIORI ESIMAES FOR HE CAHN HILLIARD EQUAION 1017 Blow w outlin th dtaild dfinitions of th adaptiv algorithms that w usd for our numrical xprimnts. h huristic adaptiv algorithm (VOL1) wasusdin[3,7] for computations for th dgnrat Cahn Hilliard quation. h ida of th algorithm (VOL1) is to locally rfin th msh in such a way, that on has uniformly small lmnts of a prscribd volum vol( ) vol f for h \C h (u h ). h lmnts C h (u h ) ar coarsnd if vol( ) vol c / and rfind if vol( ) >vol c. Not, that in our implmntation w st vol f = h min /, vol c = h max/ indandvol f = h 3 min /6, vol c = h 3 max/6 in3d,whrh min and h max ar givn dsird minimum and maximum msh sizs, rspctivly. h scond adaptiv algorithm (VOL) is basd on th obsrvation that th stimator attains maximum valus at th lmnts from th discrt boundary F h (u h ). h (VOL) algorithm is similar to th (VOL1) algorithm with th addition of an adaptiv control of th constants vol f <vol c to kp th valu of η u blow a prscribd tolranc. Algorithm (VOL) (1) comput u h ; () for all h ; if F h and vol( ) >vol f mark for rfinmnt; if N h and vol( ) > vol f mark for rfinmnt, ls if vol( ) vol f mark for coarsning; if C h and vol( ) vol c /mark for coarsning, ls if vol( ) >vol c mark for rfinmnt; (3) rfin/coarsn msh; if no lmnts wr rfind/coarsnd procd with stp 4 ls procd with stp ; (4) comput η u,ifη u >OLst vol f := vol f / and procd with stp 1, ls procd with stp 5; (5) if η τ >OL τ dcras tim stp τ := τ/; if η τ < 0.01 OL τ incras tim stp τ := min{ τ,τ max }; (6) procd to th nxt tim lvl. h adaptiv algorithm (MAX) is similar to th maximum rror adaptiv stratgy from [] and is dscribd blow. For givn tolrancs OL and OL τ, and coarsning/rfinmnt paramtrs ɛ c, ε c, ε r, vol f, vol c w start with th msh from th prvious tim stp, i.., h = old h, and improv th msh for th nxt tim lvl with th following stps, whr w us th notation η max := max h η u,. Algorithm (MAX) (1) comput u h and η u,, η h,, h ; () for all h,ifη u >OLand η u, >ε r η max mark for rfinmnt; if η u, + η h, <ε c η max mark for coarsning; (3) if η τ >OL τ dcras tim stp τ := τ/; if η τ < 0.01 OL τ incras tim stp τ := min{ τ,τ max }; (4) procd to th nxt tim lvl. h constants ε r, ε c wr chosn as 0.6 and0.05, rspctivly. Not, that th algorithm (MAX) rally only uss th indicator η u for th msh rfinmnt. As confirmd by th numrical xprimnts blow, this also guarants th control ovr th rmaining rror contributions in practic. Rmark 5.3. W not that th coarsning stimat η h was not mployd in th adaptiv stratgy in []. h coarsning stimat is critically important whn computing spinodal dcomposition, whr msh coarsning may lad to an xcssiv loss of information and an unphysical ris of th discrt analogu of th fr nrgy (1.3). W usd a Uzawa-multigrid algorithm for th solution of th discrt systm of nonlinar algbraic quations arising from (.). For mor dtails on this itrativ solvr s [3,4]. 6.1. Failur of th non-localizd stimator 6. Numrical rsults W dmonstrat that a localizd stimator is ssntial for fficint numrical computations. W comput an volution of a squar to a circl for γ = 1 8π on a tim intrval (0, 10 4 ). W mploy th adaptiv stratgy (VOL1) with h min =1/3, h max =1. InFigur1 w display for t =10 4 th computd solution u h,thmsh,
1018 L. BAŇAS AND R. NÜRNBERG Figur 1. Solution u h, th msh and th indicators η u, η u at t =10 4. Figur. Solution u h and adaptiv mshs VOL1, VOL, MAX. th localizd stimator η u and non-localizd stimator from Sction 3 dfind as: η u = h ( 1 h 1/ [ u h ] ) + 1 γ h (g + w h ). Clarly, th indicator ηu dos not rflct th charactr of th solution proprly and lads to a substantial ovrstimation of th rror in th aras whr th solution is constant. On th othr hand, th localizd indicator η u is non-zro only in th intrfacial rgion. 6.. Comparison of diffrnt adaptiv stratgis, discrt convrgnc W compar th adaptiv algorithm (VOL1) with th adaptiv algorithm with rfinmnt along th fr boundary (VOL), th maximum stratgy (MAX), and th uniform global msh rfinmnt. In ordr to highlight th diffrncs btwn th adaptiv stratgis (VOL1), (VOL) and (MAX), w display in Figur an xampl of mshs gnratd by th rspctiv adaptiv stratgis. W xamin th convrgnc of η u, η w, η c with rspct to th numbr of dgrs of frdom, with th hlp of an xampl computation for an stablishd intrfac in th form of an llips and γ = 1 8π. W computd with uniform tim stps τ = ( h min ) 64 10 7,whrh min is th minimum msh siz in th rspctiv computations. h bhaviour was similar at all tim lvls, and w thrfor only prsnt th rsults at tim t = ˆt := 10 5. h profil of u h at tims t =0andt = ˆt for a uniform msh computation can b sn in Figur 3. h graphs of th dpndnc of η u, η w, η c on th numbr of vrtics at tim ˆt ardpictdinfigurs4 6, rspctivly. A logarithmic scaling is usd in th figurs, which allows us to intrprt th slop α as an xprimntal convrgnc rat of α, sinch #N h in D. h abov rsults support th assumption that th control of η u in th prsntd adaptiv algorithms (or in othr words th rfinmnt in th intrfacial ara only) is sufficint to guarant th control ovr th rmaining indicators η w, η c. h only qualitativ diffrnc btwn th uniform msh rfinmnt and adaptiv stratgis is in th convrgnc rats of η w, which appars to b O(h ) for uniform msh rfinmnt and O(h) for adaptiv msh rfinmnt. h diffrnc can b
A POSERIORI ESIMAES FOR HE CAHN HILLIARD EQUAION 1019 Figur 3. Solution u h at t =0andt = ˆt on a uniform msh with h =1/64. 10 MAX VOL1 VOL uniform O(h) 1 0.1 100 1000 10000 100000 1+06 Figur 4. Convrgnc of η u at t =10 5. Plot of stimator against numbr of dgrs of frdom (dof). accountd to th fact that η w is not localizd to th intrfacial rgion, th rgion that is mainly rfind by th adaptiv mthods. Not, that in th prsnt cas th wors convrgnc rats do not influnc th ovrall convrgnc rat, which is O(h). W conclud, that apart from th abov disadvantag of th (VOL1) algorithm thr is no significant qualitativ diffrnc in th prformanc of th thr adaptiv algorithms. h algorithm (MAX) is prhaps th most flxibl and ffctiv of all thr algorithms; howvr, its prformanc dpnds on th choic of th rfinmnt/coarsning constants. h algorithm (VOL1) is th simplst to implmnt. 6.3. Dpndnc of th stimator on γ W study th fficincy of th adaptiv algorithms with rspct to th paramtr γ. In ordr to obtain rliabl rsults it is dsird that th adaptiv algorithm producs mshs for which th stimat η rl (γ) OL,
100 L. BAŇAS AND R. NÜRNBERG 1 MAX VOL1 VOL uniform O(h) O(h^) 0.1 0.01 0.001 0.0001 100 1000 10000 100000 1+06 Figur 5. Convrgnc of η w at t =10 5. Plot of stimator against dof. 10 MAX VOL1 VOL uniform O(h) 1 0.1 0.01 100 1000 10000 100000 1+06 Figur 6. Convrgnc of η c at t =10 5. Plot of stimator against dof.
A POSERIORI ESIMAES FOR HE CAHN HILLIARD EQUAION 101 0.46 0.45 gamma=1/3pi gamma=1/16pi gamma=1/8pi 0.44 0.43 0.4 0.41 0.4 0.39 0.38 0.37 0.36 0 1-05 -05 3-05 4-05 5-05 6-05 7-05 8-05 9-05 0.0001 Figur 7. Evolution of η rl for γ = 1 8π, 1 16π, 1 3π for th (MAX) algorithm. whr OL is a tolranc indpndnt of γ. On th othr hand, in ordr to obtain an fficint adaptiv msh rfinmnt, th minimum msh siz h min (γ), ndd in ordr to kp th rror blow a givn tolranc, should hav a linar dpndnc on γ. W computd an volution of a squar using th adaptiv algorithms (VOL1) and (MAX). In Figurs 7 and 8 w display th tim volution of η rl for γ i = 1, i =0, 1, for th two algorithms. h paramtrs for i 8π th adaptiv msh rfinmnt in th (VOL1) algorithm wr chosn as h min = γi γ, h 018 max = γi γ 0 for i =0, 1,. h tolranc in th (MAX) algorithm was OL = 0.45, which rsultd in similar maximum and minimum msh sizs in both algorithms. In ordr to xclud th influncs of th adaptiv tim stpping on th rror w usd a fixd tim stp τ = (γi ) 10 6 (γ 0 ). h rsults show that th volution of η rl is similar for diffrnt valus of γ if th numbr of msh points in th intrfac is kpt constant (i.. for th abov choics of h min/max ). his is a natural rquirmnt, which undrlins th fficincy of th adaptiv msh rfinmnt. In Figur 9 w display th computd solution u h and th undrlying adaptiv msh obtaind by th (MAX) algorithm for γ = 1 8π. 6.4. Spinodal dcomposition In th nxt xprimnt w prform an xampl computation of spinodal dcomposition. h initial data is obtaind by dfining a coars solution ũ 0 as a random prturbation around 0 on a uniform msh with h =1/0. A smooth initial condition u 0 is thn obtaind as u 0 (x) = ũ 0 (y) 1000 y x dy. Ω Not, that th abov intgral is computd approximatly. W computd th xampl for γ = 1 8π using a uniform msh with h =1/3 and using th adaptiv msh rfinmnt stratgy (MAX). W usd adaptiv tim-stpping basd on th indicator η τ, giving a tim stp siz 10 1 τ 1.1 10 9. h solution and adaptiv msh at diffrnt tim lvls is displayd in Figur 10.
10 L. BAŇAS AND R. NÜRNBERG 0.6 0.59 gamma=1/3pi gamma=1/16pi gamma=1/8pi 0.58 0.57 0.56 0.55 0.54 0.53 0.5 0 1-05 -05 3-05 4-05 5-05 6-05 7-05 8-05 9-05 0.0001 Figur 8. Evolution of η rl for γ = 1 8π, 1 16π, 1 3π for th (VOL1) algorithm. Figur 9. u h at tims t =0,10 5,10 4 for γ = 1 8π.
A POSERIORI ESIMAES FOR HE CAHN HILLIARD EQUAION 103 Figur 10. Spinodal dcomposition: u h at tims t =0,10 6,3 10 6,10 5. h volution of th indicator η u is displayd in Figur 11. Clarly, th rror on th uniform msh is almost two tims largr than th tolranc in th initial part of th computation, whil th rror on adaptiv mshs is always blow th tolranc. W found that too much coarsning in th computations of spinodal dcomposition could lad to an unphysical ris in th discrt nrgy, which undrlins th importanc of th coarsning stimat. 6.5. Coarsning in 3D h last xprimnt is to dmonstrat th prformanc of adaptiv msh rfinmnt in 3D computations. h zro lvl st of th initial condition consistd of two cubs of slightly diffrnt sizs. W computd th xampl using th adaptiv stratgy (VOL1) with fixd tim stp τ =10 6. h volution of th zro lvl st of th computd solution and a cut through th adaptiv msh at x 3 = 0 ar displayd in Figur 1. h volution of η rl in Figur 13 indicats a good control of th approximation rror.
104 L. BAŇAS AND R. NÜRNBERG 0.7 MAX uniform 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0-08 4-08 6-08 8-08 1-07 Figur 11. Evolution of th stimat η u for a uniform msh of fixd siz h =1/3 and th algorithm (MAX). Figur 1. 3D coarsning: zro lvl st of u h and cut through th msh at x 3 =0attims t =0,10 5,10 4,1. 10 4.
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