Another view for a posteriori error estimates for variational inequalities of the second kind

Transcription

1 Accptd by Applid Numrical Mathmatics in July 2013 Anothr viw for a postriori rror stimats for variational inqualitis of th scond kind Fi Wang 1 and Wimin Han 2 Abstract. In this papr, w giv anothr viw to undrstand a postriori rror analysis for finit lmnt solutions of lliptic variational inqualitis of th scond kind. This point of viw maks it simplr to driv rliabl rror stimators in solving variational inqualitis of th scond kind from th thory for rlatd linar variational quations. Rliabl rsidual-basd and gradint rcovry-basd stimators ar dducd. Efficincy of th stimators is also provd. ywords. Variational inquality; a postriori rror stimators; rliability; fficincy AMS Classification. 65N15, 65N30 1 Introduction Adaptiv finit lmnt mthods basd on a postriori rror stimats ar an activ rsarch fild. Many rror stimators can b classifid as rsidual typ or rcovry typ. Various rsidual quantitis ar usd to captur lost information going from u to u h, such as rsidual of th quation, rsidual from drivativ discontinuity and so on. In a gradint rcovry rror stimator, G h u h u h is usd to approximat u u h, whr a gradint rcovry oprator G h is applid to th numrical solution u h to rconstruct th gradint of th tru solution u. Th thory of a postriori rror stimation is wll stablishd for linar quations, and w rfr th radr to [1, 2, 16]. It is mor difficult to dvlop a postriori rror stimators for variational inqualitis (VIs) du to th inquality fatur. Nvrthlss, numrous paprs can b found on a postriori rror 1 School of Mathmatics and Statistics, Huazhong Univrsity of Scinc and Tchnology, Wuhan , China. Dpartmnt of Mathmatics, Univrsity of Iowa, Iowa City, Iowa 52242, USA. wangfitwo@163.com. Th work of this author was substantially supportd by th Chins National Scinc Foundation (Grant No ), and partially supportd by th Fundamntal Rsarch Funds for th Cntral Univrsity, HUST (Grant No. 2011QN169). 2 Dpartmnt of Mathmatics & Program in Applid Mathmatical and Computational Scincs, Univrsity of Iowa, Iowa City, Iowa 52242, USA. Th work of this author was supportd by th Simons Foundation. 1

2 stimation of finit lmnt mthods for obstacl problms, which is a rprsntativ lliptic variational inquality (EVI) of th first kind,.g., [3,10,13 15,17]. For VIs of th scond kind, in [4 7], th authors studid a postriori rror stimats and stablishd a framwork through th duality thory, but th sharpr stimation of on trm in th lowr bound is still an opn problm, i.., th fficincy was not compltly provd. In [8], Brass dmonstratd that a postriori rror stimators for finit lmnt solutions of th obstacl problm can b asily drivd by applying a postriori rror stimators for a rlatd linar lliptic problm. In this papr, w xtnd th idas thrin to giv anothr look at a postriori rror analysis for VIs of th scond kind. Morovr, w accomplish th proof for th fficincy of th rror stimators. W tak a stady stat frictional contact problm as an xampl to illustrat th drivation procss of a postriori rror stimators. Th idas and tchniqus prsntd hr for this modl problm can b xtndd to othr VIs of th scond kind. A frictional contact problm. Lt Ω R d (d 1) b a boundd domain with Lipschitz boundary Γ, Γ 1 a rlativly closd subst of Γ, and Γ 2 = Γ\Γ 1. Assum f L 2 (Ω), and g > 0 is a constant. Thn a frictional contact problm is to find u V = {v H 1 (Ω) : v = 0 a.. on Γ 1 } such that a(u, v u) + j(v) j(u) (f, v u) Ω v V, (1.1) whr (, ) Ω dnots th L 2 innr product in th domain Ω and a(u, v) = u v dx + u v dx, Ω Ω j(v) = g v ds. Γ 2 It was provd ( [12, Thorm 5.3], [11]) that this problm has a uniqu solution u V, and thr xists a uniqu Lagrang multiplir λ L (Γ 2 ) such that a(u, v) + g λ v ds = (f, v) Ω v V, (1.2) Γ 2 λ 1, λ u = u a.. on Γ 2. (1.3) It follows from (1.2) and (1.3) that th solution u of (1.1) is th wak solution of th boundary valu problm u + u = f in Ω, u = 0 on Γ 1, u n g, u n u + g u = 0 on Γ 2, whr n is th unit outward normal vctor. For any v V, st l(v) = (f, v) Ω g λ v ds. Γ 2 2

3 Thn (1.2) bcoms a(u, v) = l(v) v V. (1.4) For a Lipschitz subdomain ω Ω, lt v 2 1,ω := a ω (v, v) = ( v 2 + v 2 ) dx. For a masurabl subst γ ω Γ 2, dfin { } λ,γ := sup g λ v ds : v H 1 (ω), v 1,ω = 1. (1.5) Th subscript γ and ω ar omittd if γ = Γ 2 and ω = Ω. W hav γ ω λ,γ = w 1,ω, (1.6) whr w H 1 (ω) is th solution of th auxiliary quation a ω (w, v) = g λ v ds v H 1 (ω). (1.7) Th rlation (1.6) is provd as follows. First, g λ v ds = a ω (w, v) w 1,ω v 1,ω. Thus, γ λ,γ = Ltting v = w in (1.7), w hav w 1,ω = γ sup g λ v ds/ v 1,ω w 1,ω. 0 v H 1 (ω) γ γ g λ w ds/ w 1,ω λ,γ. W introduc a family of finit lmnt spacs V h V corrsponding to partitions T h of Ω into triangular or ttrahdral lmnts (othr kinds of lmnts, such as quadrilatral lmnts, or hxahdral or pntahdral lmnts, can b considrd as wll). Th partitions T h ar compatibl with th dcomposition of Γ into Γ 1 and Γ 2. Thn th finit lmnt mthod for th VI (1.1) is: Find u h V h such that a(u h, v h u h ) + j(v h ) j(u h ) (f, v h u h ) Ω v h V h. (1.8) Similar to th continuous problm, th discrt problm has a uniqu solution u h V h and thr xists a uniqu Lagrang multiplir λ h L (Γ 2 ) such that ( [6, 12]) a(u h, v h ) + g λ h v h ds = (f, v h ) Ω v h V h, (1.9) Γ 2 λ h 1, λ h u h = u h a.. on Γ 2. (1.10) 3

4 For any v h V h, lt l h (v h ) = (f, v h ) Ω g λ h v h ds. Γ 2 From Hahn-Banach xtnsion thorm, th boundd linar functional l h, originally dfind on V h, can b xtndd to a boundd linar functional on V with th norm prsrvd. Thn (1.9) bcoms a(u h, v h ) = l h (v h ) v h V h. (1.11) Obviously, u h is also th finit lmnt approximation of th solution z V of th linar problm: a(z, v) = l h (v) v V, (1.12) which is th wak formulation of th boundary valu problm z + z = f in Ω, (1.13) z = 0 on Γ 1, z n = gλ h on Γ 2. Now w prsnt th procss to driv a postriori rror stimators for th finit lmnt mthod (1.8). From (1.4) and (1.12), for all v V, w hav a(u h u, v) = a(u h z, v) + a(z u, v) = a(u h z, v) + l h (v) l(v) = a(u h z, v) + g(λ λ h )v ds. Γ 2 Tak v = u h u in th abov rlation. Not that by (1.3) and (1.10), w hav g(λ λ h )v ds = g λ u h ds g λ u ds g λ h u h ds + g λ h u ds Γ 2 Γ 2 Γ 2 Γ 2 Γ 2 g u h ds g u ds g u h ds + g u ds Γ 2 Γ 2 Γ 2 Γ 2 Thn w obtain Thrfor, = 0. u h u 2 1 = a(u h u, u h u) a(u h z, u h u) u h z 1 u h u 1. (1.14) u h u 1 u h z 1. (1.15) 4

5 Rcalling (1.6), w hav λ λ h = u z 1 u u h 1 + u h z 1 2 u h z 1. W summariz th abov rsults in th following thorm. Thorm 1.1 Lt u and z b th solutions of th problm (1.1) and (1.12), and lt u h b th finit lmnt solution of u. Thn, u h u 1 + λ λ h 3 u h z 1. This rsult is th starting point for drivation of a postriori rror stimators, whn combind with th standard rsults ( [1]) on rror stimators for th trm u h z 1. Basd on this obsrvation, w discuss rsidual typ rror stimators and gradint rcovry typ rror stimators in th nxt two sctions. Rliability and fficincy ar provd for both typs of rror stimators. 2 Rsidual typ rror stimators First, w introduc som notations. Givn a boundd st D R d and a positiv intgr m, H m (D) is th usual Sobolv spac with th corrsponding norm m,d and smi-norm m,d, which ar abbrviatd by m and m, rspctivly, whn D coincids with Ω. For convninc, w rwrit 0,D as D. W assum Ω is a polyhdral domain and dnot by {T h } h a family of partitions of Ω. For a partition T h, dnot all th dgs of T h by E h, and Eh i = E h\γ, E h,γ2 = E h Γ 2. Lt h = diam() for T h and h = diam() for E h. For any lmnt T h, dfin th patch st ω := {T T h, T }, and for any dg shard by two lmnts and, dfin ω :=. For a givn lmnt T h, N () and E() dnot th sts of th nods of and sids of, rspctivly; n dnots th unit outward normal vctor to th boundary of and n a unit vctor on. Throughout th papr, C dnots a gnric positiv constant indpndnt of th lmnt siz, which may tak diffrnt valus at diffrnt occurrncs. Dfin th intrior rsiduals and dg-basd jumps R := u h + u h f for ach T h, { [ u h R := ] if E i n h, u h + g λ n h if E h,γ2. 5

6 Hr [ u h ] = u n h n + u h n rprsnts th discontinuity of th gradint of u h across th dg shard by th nighboring lmnts and. Thy lad to th local stimators 1/2 η R, = h 2 k R h R 2 + h R 2 for any T h. (2.1) 2 E() E h,γ2 E() E i h It follows from [1] that th rsidual-typ a postriori rror stimator for th lliptic quation (1.13) satisfis ( ) 1/2 u h z 1 C. Hnc, w hav th following thorm. T h η 2 R, Thorm 2.1 Lt u V and u h V h b th solutions of th problms (1.1) and (1.8). Thn, u h u 1 + λ λ h Cη R, η 2 R := T h η 2 R,. Now w turn to considr lowr bounds with rsidual rror stimators. This can b achivd by following th standard argumnt for lowr bounds with rsidual rror stimators for linar lliptic problms, s [1, pp ]. Dfin ( ) a (u, v) = u v + uv dx, so that for u, v H 1 (Ω), a(u, v) = T h a (u, v). For any v V, by intgration by parts, w hav a (u h u, v) = a (u h z, v) + a(z u, v) T h T h = R v dx + R v ds + T h E i h E h,γ 2 E h,γ2 g(λ λ h )v ds. (2.2) W will us th bubbl functions. For ach T h, lt λ 1, λ 2 and λ 3 b th barycntric coordinats on. Thn th intrior bubbl function ϕ is dfind by ϕ = 27λ 1 λ 2 λ 3, and th thr dg bubbl functions ar givn by τ 1 = 4λ 2 λ 3, τ 2 = 4λ 1 λ 3, τ 3 = 4λ 1 λ 2. W rcall som proprtis of th bubbl functions ( [1, Thorms 2.2 and 2.3]). 6

7 Lmma 2.2 For ach T h, E(), lt ϕ and τ b th corrsponding intrior and dg bubbl functions. Lt P () H 1 () and P () H 1 () b finit-dimnsional spacs of functions dfind on and. Thn thr xists a constant C, indpndnt of h, such that for all v P (), C 1 v 2 ϕ v 2 dx C v 2, (2.3) C 1 v ϕ v + h ϕ v 1, C v, (2.4) C 1 v 2 τ v 2 ds C v 2, (2.5) h 1/2 τ v + h 1/2 τ v 1, C v. (2.6) For ach T h, ϕ and τ ar rspctivly th intrior and dg bubbl functions on or E i h E h,γ 2, and R is an approximation to th intrior rsidual R from a suitabl finit lmnt spac containing u h and u h. In (2.2), choosing v = R ϕ on lmnt and using an argumnt similar to that in [1, pp ], w obtain R C ( R R + h 1 u h u 1, ). For E i h, lt R b an approximation to th jump R from a suitabl finit-dimnsional spac and lt v = R τ in (2.2). W hav R C ( h 1/2 u h u 1,ω + h 1/2 R R ω + R R ). For E h,γ2, w obtain a ω (u h u, R τ ) = R R τ dx + R R τ ds + g(λ λ h )R τ ds. ω Thrfor, R 2 τ ds = R (R R )τ ds + a ω (u h u, R τ ) R R τ dx g(λ λ h )R τ ds. ω Applying Lmma 2.2, w bound th trms in th abov rlation as follows: R 2 τ ds C 1 R 2, R (R R )τ ds R τ R R C R R R, a ω (u h u, R τ ) u h u 1,ω R τ 1,ω Ch 1/2 u h u 1,ω R, R R τ dx R ω R τ ω Ch 1/2 R ω R, ω g(λ λ h )R τ ds λ λ h, R τ 1,ω Ch 1/2 λ λ h, R. 7

8 Hnc, R R + R R C ( h 1/2 u h u 1,ω + h 1/2 λ λ h + h 1/2 R R ω + R R ). (2.7) Not that u h +u h in and u h / n on ar polynomials. Hnc, th trms R R and R R can b rplacd by f f and λ h λ h, with discontinuous picwis polynomial approximations f and λ h. Thn w obtain th fficincy bound of th local rror indicator η R, (s also [5, 6]). Thorm 2.3 Lt u and u h b th solutions of (1.1) and (1.8), rspctivly, and lt η R, b th stimator (2.1). Thn ηr, 2 C u u h 2 1,ω + λ λ h 2, + h 2 f f 2 ω + h λ h λ h 2. (2.8) E() E h,γ2 Du to th inquality natur of th variational inqualitis, in th fficincy bound (2.8) of η R,, thr is a trm involving λ and λ h. In [5, 6], bcaus of th prsnc of this trm, th fficincy of th stimators was not provd compltly. From Thorm 2.1, w s that th involvmnt of th trm λ λ h, in th bound (2.8) is vry natural. This commnt is also valid for th cas of gradint rcovry typ rror stimators. 3 Gradint rcovry typ rror stimators In this sction, w study a gradint rcovry typ rror stimator for th linar finit lmnt solution of th frictional contact problm (1.1). Som additional notations ar ndd in this sction. W dnot by N h th st of nods of T h, and N h,0 is th st of fr nods, i.., thos nods that do not li on Γ 1. Lt N v N h b th st of th lmnt vrtics of th partition T h, N v,γ1 N v th subst of th lmnt vrtics lying on Γ 1, N v,i N v th st of th intrior vrtics, and N v,0 = N v N h,0. Lt {ϕ a : a N v } dnot th nodal basis functions of th linar lmnts for all th vrtics. Dfin an quivalnc rlation { a, if a Nv,0 ; ξ(a) := b, b N v,i and T h, s.t. a, b if a N v,γ1. Thn w can classify th st of vrtics N v into card(n v,0 ) classs of quivalnc, that is, I(a) = {ã N v : ξ(ã) = a} for ach nod a N v,0. W st ψ a = ϕã for vry nod a N v,0. ã I(a) 8

9 Not that {ψ a : a N v,0 } is a partition of unity. Lt a = supp(ψ a ) and h a = diam( a ). For a givn v L 1 (Ω), lt v a = vψ a dx ϕ a dx, a N v,0. Thn a Clémnt typ intrpolation oprator Π h : V V h is dfind as follows: Π h v = a N v,0 v a ϕ a. Th nxt thorm summarizs som basic stimats for Π h. Its proof can b found in [9]. Thorm 3.1 Thr xists an h-indpndnt positiv constant C such that for all v V and f L 2 (Ω), v Π h v 2 1,Ω C v 2 1,Ω, f(v Π h v) dx C v 1,Ω h 2 a min f f a 2 Ω f a R 0, a a N v,0 h 1 (v Π hv) 2 C v 2 1,Ω, T h h 1/2 (v Π h v) 2 C v 2 1,Ω. E h Thr ar many typs of gradint rcovry oprators G h. For G h u h to b a good approximation of th tru gradint u, a st of sufficint conditions can b found in [1, Lmma 4.5]. Considr a gradint rcovry oprator G h : V h (V h ) d dfind as follows: G h v h (x) = G h v h (a)ϕ a (x), G h v h (a) = 1 a N v v h dx. a a From (1.11) and (1.12), w gt th Galrkin orthogonality a(u h z, v h ) = 0 v h V h. Using th abov quation, for any v V, w gt 1/2, whr a(u h z, v) = a(u h z, v Π h v) = I 0 + G h u h (v Π h v) dx + u h (v Π h v) dx a(z, v Π h v) Ω Ω I 0 = ( u h G h u h ) (v Π h v) dx C u h G h u h Ω v 1,Ω. Ω 9

10 Prform lmnt-wis intgration by parts, G h u h (v Π h v) dx = G h u h (v Π h v) dx T h Ω = div(g h u h ) (v Π h v) dx + (G h u h n ) (v Π h v) ds. T h T h E() Th first summation is rwrittn as div( u h G h u h ) (v Π h v) dx + u h (v Π h v) dx. T h T h Sinc G h u h is continuous across th lmnt boundaris, (G h u h n ) (v Π h v) ds = (G h u h n )(v Π h v) ds. T E() h E h,γ2 Applying th abov rlations and using th quation (1.12), w obtain that a(u h z, v) = I 0 + I 1 + I 2 + I 3, (3.1) whr I 1 = div( u h G h u h )(v Π h v) dx, T h I 2 = ( u h + u h f)(v Π h v) dx = R (v Π h v) dx, T h T h I 3 = E h,γ2 (G h u h n + gλ h )(v Π h v) ds. It is shown in [6] (s also [4]) that I 1 C v 1,Ω ( T h u h G h u h 2 ) 1/2, I 2 C v 1,Ω a N v,0 ( ) h 4 a u h 2 a + h 2 a min f f a 2 a, f a R I 3 C v 1,Ω h G h u h n + gλ h 2 E h,γ2 Taking v = u h z in (3.1) and rcalling Thorm 1.1, w obtain th nxt rsult. 10 1/2.

11 Thorm 3.2 Lt u and u h b th solutions of (1.1) and (1.8), rspctivly. Thn u u h λ λ h 2 CηG 2 + C ( ) h 4 a u h 2 a + h 2 a min f f a 2 a, (3.2) f a R whr a N h,0 η 2 G = T h η 2 G,, η 2 G, = u h G h u h 2 + E() E h,γ2 h G h u h n + gλ h 2. (3.3) Th trm 1/2 h 4 a u h 2 a is boundd by O(h 2 ), and 1/2 h 2 a min f f a 2 a f a R a N h,0 a N h,0 is boundd by o(h) if f L 2 (Ω) or boundd by O(h 2 ) if f H 1 (Ω) (s [6]), which guarants th rliability of stimator η G. For th fficincy of th stimator, it is shown in Lmma 3.1 in [6] that ηg, 2 C h R 2 + h R 2, E() E h,γ2 E ω whr E ω dnots th st of innr sids of th patch ω corrsponding to th lmnt. Using th rlation (2.7), w obtain th following rsults. Thorm 3.3 Lt u and u h b th solutions of (1.1) and (1.8), rspctivly, and lt η G, b th stimator (3.3). Thn ηg, 2 C u u h 2 1,ω + λ λ h 2, + h 2 f f 2 ω + h λ h λ h 2. (3.4) E() E h,γ2 This thorm shows th fficincy of gradint rcovry typ stimators η G,. Th inquality (3.4) is comparabl to (2.8). 4 Summary In this papr, w study a postriori rror stimation of finit lmnt mthods for a frictional contact problm, and stablish a compact framwork to driv rliabl rsidual typ and gradint rcovry typ rror stimators by applying a postriori rror analysis for a rlatd linar lliptic problm. Furthrmor, w prov th fficincy of th rror stimators, which was an opn problm statd in [6]. This framwork can also b usd to driv rliabl and fficint a postriori rror stimators for othr variational inqualitis of th scond kind. Acknowldgmnt. W thank th anonymous rfrs for thir valuabl commnts that lad to an improvmnt of th papr. 11

12 Rfrncs [1] M. Ainsworth and T. J. Odn, A Postriori Error Estimation in Finit Elmnt Analysis, Wily, Chichstr, [2] I. Babuška and T. Strouboulis, Th Finit Elmnt Mthod and its Rliability, Oxford Univrsity Prss, [3] S. Bartls and C. Carstnsn, Avraging tchniqus yild rliabl a postriori finit lmnt rror control for obstacl problms, Numr. Math. 99 (2004), [4] V. Bostan and W. Han, Rcovry-basd rror stimation and adaptiv solution of lliptic variational inqualitis of th scond kind, Communications in Mathmatical Scincs 2 (2004), [5] V. Bostan and W. Han, A postriori rror analysis for a contact problm with friction, Computr Mthods in Applid Mchanics and Enginring 195 (2006), [6] V. Bostan and W. Han, Adaptiv finit lmnt solution of variational inqualitis with application in contact problm, in Advancs in Applid Mathmatics and Global Optimization, ds. D. Y. Gao and H. D. Shrali, Springr, Nw York, 2009, [7] V. Bostan, W. Han and B. D. Rddy, A postriori rror stimation and adaptiv solution of lliptic variational inqualitis of th scond kind, Applid Numrical Mathmatics 52 (2004), [8] D. Brass, A postriori rror stimators for obstacl problms anothr look, Numr. Math. 101 (2005), [9] C. Carstnsn, Quasi-intrpolation and a postriori rror analysis in finit lmnt mthods, RAIRO Math. Modl. Num. 33 (1999), [10] Z. Chn and R. H. Nochtto, Rsidual typ a postriori rror stimats for lliptic obstacl problms, Numr. Math. 84 (2000), [11] G. Duvaut and J.-L. Lions, Inqualitis in Mchanics and Physics, Springr-Vrlag, Brlin, [12] R. Glowinski, Numrical Mthods for Nonlinar Variational Problms, Springr-Vrlag, Nw York, 1984 [13] R. ornhubr, A postriori rror stimats for lliptic variational inqualitis, Comput. Math. Appl. 31 (1996),

13 [14] R. H. Nochtto,. G. Sibrt and A. Vsr, Pointwis a postriori rror control for lliptic obstacl problm, Numr. Math. 95 (2003), [15] A. Vsr, Efficint and rliabl a postriori rror stimators for lliptic obstacl problms, SIAM J. Numr. Anal. 39 (2001), [16] R. Vrfürth, A Rviw of a Postriori Error Estimation and Adaptiv Msh-rfinmnt Tchniqus, Wily-Tubur, [17] N. Yan, A postriori rror stimators of gradint rcovry typ for lliptic obstacl problms, Adv. Comput. Math. 15 (2001),