RELIABLE AND EFFICIENT A POSTERIORI ERROR ESTIMATES OF DG METHODS FOR A SIMPLIFIED FRICTIONAL CONTACT PROBLEM

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volum 16, Numbr 1, Pags c 2019 Institut for Scintific Computing and Information RELIABLE AND EFFICIENT A POSTERIORI ERROR ESTIMATES OF DG METHODS FOR A SIMPLIFIED FRICTIONAL CONTACT PROBLEM FEI WANG AND WEIMIN HAN Abstract. A postriori rror stimators ar studid for discontinuous Galrkin mthods for solving a rprsntativ lliptic variational inquality of th scond kind, known as a simplifid frictional contact problm. Th stimators ar drivd by rlating th rror of th variational inquality to that of a linar problm. Rliability and fficincy of th stimators ar thortically provd. y words. Elliptic variational inquality, discontinuous Galrkin mthod, a postriori rror stimators, rliability, fficincy. 1. Introduction For mor than thr dcads, adaptiv finit lmnt mthod (AFEM) has bn an activ rsarch fild in scintific computing. As an fficint numrical approach, it has bn widly usd for solving a varity of diffrntial quations. Each loop of AFEM consists of four stps: Solv Estimat Mark Rfin. That is, in ach loop, w first solv th problm on a msh, thn us a postriori rror stimators to mark thos lmnts to b rfind, and finally, rfin th markd lmnts and gt a nw msh. W can continu this procss until th stimatd rror satisfis crtain smallnss critrion. Th adaptiv finit lmnt mthod can achiv rquird accuracy with lowr mmory usag and lss computation tim. A postriori rror stimators ar computabl quantitis that provid th contribution of rror on ach lmnt to th global rror. Thy ar usd in adaptiv algorithms to indicat which lmnts nd to b rfind or coarsnd. To captur th tru rror as prcisly as possibl, thy should hav two proprtis: rliability and fficincy ([1, 4]). Hnc, obtaining rliabl and fficint rror stimators is th ky for succssful adaptiv algorithms. A varity of a postriori rror stimators hav bn proposd and analyzd in th litratur. Many rror stimators can b classifid as rsidual typ or rcovry typ ([1, 4]). Various rsidual quantitis ar usdtocapturlostinformationgoingfromutou h, suchasrsidualofthquation, rsidual from drivativ discontinuity and so on. Anothr typ of rror stimators is gradint rcovry, i.., G( u h ) u h is usd to approximat u u h, whr a rcovry oprator G is applid to th numrical solution u h to rbuild th gradint of th tru solution u. A postriori rror analysis has bn wll stablishd for standard finit lmnt mthods for solving linar partial diffrntial quations, and w rfr th radr to [1, 4, 30]. Du to th inquality fatur, it is mor difficult to dvlop a postriori rror stimators for variational inqualitis (VIs). Howvr, numrous articls can b found on a postriori rror analysis of finit lmnt mthods for th obstacl Rcivd by th ditors May 20, 2016 and, in rvisd form, August 17, Mathmatics Subjct Classification. 65N15, 65N30, 49J40. 49

2 50 F. WANG AND W. HAN problm, which is an lliptic variational inquality (EVI) of th first kind,.g., [5, 15, 24, 26, 29, 34]. In [11], Brass dmonstratd that a postriori rror stimators for finit lmnt solutions of th obstacl problm can b drivd by applying a postriori rror stimats for an associatd linar lliptic problm. For VIs of th scond kind, in [7, 8, 9, 10], th authors studid a postriori rror stimation and stablishd a framwork through th duality thory, but th fficincy was not compltly provd. In [31], th idas in [11] wr xtndd to giv a postriori rror analysis for VIs of th scond kind. Morovr, a proof was providd for th fficincy of th rror stimators. In rcnt yars, thanks to th flxibility in constructing fasibl local shap function spacs and th advantag to captur non-smooth or oscillatory solutions ffctivly, discontinuous Galrkin (DG) mthods hav bn widly usd for solving various typs of partial diffrntial quations. Whn applying h-adaptiv algorithm with standard finit lmnt mthods, on nds to choos th msh rfinmnt rul carfully to maintain msh conformity and shap rgularity. In particular, hanging nods ar not allowd without spcial tratmnt. For discontinuous Galrkin mthods, th approximat functions ar allowd to b discontinuous across th lmnt boundaris, so gnral mshs with hanging nods and lmnts of diffrnt shaps ar accptabl. Advantags of DG mthods includ th flxibility of mshrfinmnts and construction of local shap function spacs (hp-adaptivity), and th incras of locality in discrtization, which is of particular intrst for paralll computing. A historical account on th dvlopmnt of DG mthods can b found in [16]. In [2, 3], Arnold t al. stablishd a unifid rror analysis of nin DG mthods for lliptic problms and svral articls providd a postriori rror analysis of DG mthods for lliptic problms (.g. [6, 12, 14, 22, 25, 27]). Carstnsn t al. prsntd a unifid thory for a postriori rror analysis of DG mthods in [13]. In [32], th authors xtndd idas of th unifid framwork about DG mthods for lliptic problms prsntd in [3] to solv th obstacl problm and a simplifid frictional contact problm, and obtaind a priori rror stimats, which rach optimal ordr for linar lmnts. In [33], rliabl a postriori rror stimators of th rsidual typ wr drivd for DG mthods for solving th obstacl problm, and fficincy of th stimators is thortically xplord and numrically confirmd. A postriori rror analysis of DG mthods for th obstacl problm was also studid in [20]. A postriori rror analysis of DG mthods has not bn wll-studid for variational inqualitis of th scond kind. In this papr, w xplor this topic and study a postriori rror stimats of DG mthods for solving a rprsntativ lliptic variational inquality of th scond kind, namly, th simplifid frictional contact problm. Th stimators ar drivd by rlating th rror of th variational inquality to that of a corrsponding linar problm. Furthrmor, th rliability and fficincy of th stimators ar thortically provd. Evn though w only considr th rsidual typ rror stimators in this papr, an analysis for gradint rcovry typ rror stimation can b obtaind by th tchniqus usd in this papr and th standard argumnt of gradint rcovry typ rror analysis for th scond ordr lliptic quations. Th papr is organizd as follows. In Sction 2, w introduc th variational inquality problm and th DG schms for solving it. Thn w driv rliabl a postriori rror stimators of rsidual typ for th DG mthods of th simplifid frictional contact problm in Sction 3. Finally, w prov fficincy of th proposd rror stimators in Sction 4.

3 POSTERIORI ERROR ESTIMATES OF DGMS FOR A FRICTIONAL PROBLEM Th problm and DG formulations Contact mchanics is a rich sourc of variational inqualitis. Various frictional contact problms hav bn studid in th litratur, s,.g., [23, 21]. To simplify th notation, in this papr, w choos th simplifid frictional contact problm as an xampl to dvlop a postriori rror analysis of DG mthods in solving lliptic variational inqualitis of th scond kind. Similar a postriori rror analysis can b dvlopd for DG mthods in solving othr lliptic variational inqualitis of th scond kind, including thos arising in frictional contact problms Th problm. Lt R d (d = 2,3) b an opn boundd domain with Lipschitz boundary Γ that is dividd into two mutually disjoint parts, i.., Γ = Γ 1. Hr Γ 1 is a rlativly closd subst of Γ, and = Γ\Γ 1. Givn f L 2 () and a constant g > 0, th simplifid frictional contact problm is: find u V = H 1 Γ 1 () := {v H 1 () : v = 0 a.. on Γ 1 } such that (1) a(u,v u)+j(v) j(u) (f,v u) v V, whr (, ) dnots th L 2 innr product in th domain and a(u,v) = u vdx+ uvdx, j(v) = g v ds. Th frictional contact problm is an xampl of lliptic variational inqualitis of th scond kind and it has a uniqu solution u V ([18, 19]). Morovr, thr xists a uniqu Lagrang multiplir λ L ( ) such that (2) a(u,v)+ gλvds = (f,v) v V, (3) λ 1, λu = u a.. on. From (2) and (3), w know that th solution u of (1) is th wak solution of th following boundary valu problm u+u = f in, u = 0 on Γ 1, u n = gλ on, whr n is th unit outward normal vctor. For any v V, st l(v) = f vdx gλvds. Thn w hav by (2), (4) a(u,v) = l(v) v V. and Givn a triangulation T h of, for a Lipschitz subdomain ω, w dfin a ω,h (v,w) := ( v w +vw)dx Thn dfin { (5) λ,γ,h := sup ω v 1,ω,h := a ω,h (v,v) 1/2. γ } gλvds : v Hh(ω), 1 v 1,ω,h = 1,

4 52 F. WANG AND W. HAN whr γ is a masurabl subst of ω and H 1 h (ω) = {v L2 (ω) : v ω H 1 ( ω)}. If ω = and γ =, th subscript ω and γ ar omittd. W hav (6) λ,γ,h = w 1,ω,h, whr w Hh 1 (ω) is th solution of th following auxiliary quation (7) a ω,h (w,v) = g λ vds v Hh 1 (ω). γ Th formula (6) can b provd by an argumnt similar to that found in [31] Discontinuous Galrkin formulations. First, w introduc som notation. Lt {T h } b a family of triangulations of such that th minimal angl condition is satisfid. For a triangulation T h, lt E h b th st of all dgs, Eh i E h th st of all intrior dgs, Eh b := E h\eh i th st of all boundary dgs, E0 h E h th st of all dgs not lying on, Eh 1 := E0 h \Ei h, E2 h := E h\eh 0, and dfin E() as th st of sids of. Lt h = diam() for T h, h = lngth() for E h, and N h dnot th st of nods of T h. For any lmnt T h, dfin th patch st ω := {T T h, T Ø}, and for any dg shard by two lmnts + and, dfin ω := +. For a scalar-valud function v and a vctor-valud function q, lt v i = v i, q i = q i, and n i = n i b th unit normal vctor xtrnal to i with i = ±. Dfin th avrag { } and th jump [ ] on an intrior dg Eh i as follows: For a boundary dg Eh b, w lt {v} = 1 2 (v+ +v ), [v] = v + n + +v n, {q} = 1 2 (q+ +q ), [q] = q + n + +q n. [v] = vn, {q} = q, whr n is th outward unit normal. Lt us dfin th following linar finit lmnt spacs V h = {v h L 2 () : v h P 1 () T h }, W h = {w h [L 2 ()] 2 : w h [P 1 ()] 2 T h }. W dnot by h th brokn gradint whos rstriction on ach lmnt T h is qual to. Dfin som sminorms and norms by th following rlations: v 2 = v 2 dx, v 2 1, = v 2, v 2 = v 2 ds, v 2 0,h = v 2, v 2 1,h = v 2 1,, v 2 1,h = v 2 0,h + v 2 1,h. Throughout this papr, stands for C, whr C dnots a gnric positiv constant dpndnt on th minimal angl condition but not on th lmnt sizs, which may tak diffrnt valus at diffrnt occurrncs. Now, lt us introduc th discontinuous Galrkin mthods for solving th variational inquality (1). Hr, w tak th local DG mthod (LDG) as an xampl to show how to driv a postriori rror stimators of DG mthods for solving th frictional contact problm (1). Th drivation and analysis for th LDG mthod in this papr can b xtndd straightforward to othr DG mthods studid in [32].

5 POSTERIORI ERROR ESTIMATES OF DGMS FOR A FRICTIONAL PROBLEM 53 Th LDG mthod ([17]) for solving th frictional contact problm is to find u h V h such that (8) B h (u h,v h u h )+j(v h ) j(u h ) (f,v h u h ) v h V h, whr (9) B h (u,v) = ( h u h v +uv)dx E 0 h [u] { h v}ds { h u} [v]ds Eh 0 β [u][ h v]ds [ h u]β [v]ds Eh i Eh i +(r 0 ([u])+l(β [u]),r 0 ([v])+l(β [v]))+α j 0 (u,v). Hr β [L 2 (Eh i)]2 is a vctor-valud function which is constant on ach dg of Eh i, and αj 0 (u,v) = E η[u] [v]ds is th pnalty trm with th pnalty wighting h 0 function η : Eh 0 R givnbyη onach Eh 0, η bing apositiv numbron. For any w h W h, th lifting oprators r 0 : [L 2 (Eh 0)]2 W h and l : L 2 (Eh i) W h ar dfind by r 0 (q) w h dx = E 0 h q {w h }ds, l(v) w h dx = v[w h ]ds w h W h. Eh i Th bilinar form B h is continuous and lliptic with rspct to crtain DG-norm, and thrfor, in particular, th discrt problm has a uniqu solution u h V h (s [3, 32]). Similar to th continuous problm, thr xists a uniqu Lagrang multiplir λ h L ( ) such that ([19]) (10) B h (u h,v h )+ gλ h v h ds = (f,v h ) v h V h, (11) λ h 1, λ h u h = u h a.. on. Lt Thn (10) bcoms l h (v) = (f,v) gλ h vds. (12) B h (u h,v h ) = l h (v h ) v h V h. For any v V, w know that [u] = 0 and [v] = 0 on Eh 0. Thn w hav from (2) that (13) B h (u,v) = a(u,v) = l(v) v V Obviously, u h is also th finit lmnt approximation of th solution z V of th linar problm: (14) B h (z,v) = l h (v) v V, which is th wak formulation of th boundary valu problm (15) z +z = f in, z = 0 on Γ 1, z n = gλ h on.

6 54 F. WANG AND W. HAN 2.3. A bridg btwn u h u and u h z. In this subsction, w rlat th rror := u h u to u h z through th inquality (16) 1,h + λ λ h,h u h z 1,h + E 0 h [u h ] 2 Thn w us this rlation to driv a postriori rror stimators for DG solutions of th simplifid frictional contact problm by utilizing a postriori rror stimators of th rlatd linar lliptic problm (15). Not that a similar approach can b applid to othr lliptic variational inqualitis of th scond kind. To driv th inquality (16), w first dfin a continuous picwis linar function in V h H 1 Γ 1 (), whos valu is clos to th numrical solution. For any givn v h V h, xprssd as v h = 3 j=1 α(j) φ(j), whr φ(j), 1 j 3, ar th linar basis functions corrsponding to th thr vrtics of, w construct a function χ V h HΓ 1 1 () as follows: At vry intrior nod and th nods on of th conforming msh T h, th valu of χ is st to b th avrag of th valus of v h computd from all th lmnts sharing that nod, and χ = 0 at th boundary nods on Γ 1. For ach ν N h, lt ω ν = { T h : ν } and dnot its cardinality by ω ν, which is boundd by a constant dpnding only on th minimal angl condition of th msh. To ach nod ν, th associatd basis function φ (ν) is givn by suppφ (ν) =, φ (ν) = φ (j) for x(j) = ν. ω ν Thn w dfin χ V h H 1 Γ 1 () by χ = β (ν) φ (ν), whr β (ν) = 1 ω ν ν N h (j) α x (j) =ν 1/2 if ν N h and ν Γ 1. For nonconforming mshs, lt Nh 0 b th st of all hanging nods. Thn w construct χ from v h th sam way as in th conforming msh cas on all th nods ν N h \Nh 0. For an uppr bound of th rror v h χ, w quot a rsult from [22] (which is Thorm 2.2 thr for conforming mshs; th sam rsult also holds for nonconforming mshs, which is Thorm 2.3 in [22]). Lmma 2.1. Lt T h b a conforming triangulation. Thn for any v h V h, w can construct a continuous function χ V h H 1 Γ 1 () from v h, such that (17) v h χ 2 i, C E 0 h h 1 2i [v h ] 2, i = 0,1, whr th constant C is indpndnt of msh siz and v h, but it may dpnd on th lowr bound of th minimal angl of th lmnts in T h. Now, lt us driv th inquality (16). From (13) and (14), for all v V, w hav B h (u h u,v) = B h (u h z,v)+b h (z u,v) = B h (u h z,v)+ g(λ λ h )vds..

7 POSTERIORI ERROR ESTIMATES OF DGMS FOR A FRICTIONAL PROBLEM 55 By th dfinition (9) and noticing [v] = 0 on ach Eh 0, th abov quation bcoms ã(,v) [] { h v}ds β [][ h v]ds Eh 0 Eh i =ã(u h z,v) [u h z] { h v}ds whr Thn, E 0 h β [u h z][ h v]ds+ Eh i ã(u,v) = ã(,v) =ã(u h z,v) E 0 h + g(λ λ h )vds. g(λ λ h )vds, ( h u h v +uv)dx. [u z] { h v}ds β [u z][ h v]ds Eh i Not that [u z] = 0 on ach Eh 0. W hav (18) ã(,v) = ã(u h z,v)+ g(λ λ h )vds. Lt χ V h H 1 Γ 1 () b th function constructd from u h, satisfying (17) for v h = u h. Taking v := χ u = χ u h +u h u in (18) and using Cauchy-Schwarz inquality, w hav 2 1,h u h z 1,h ( χ u h 1,h + 1,h )+ 1,h χ u h 1,h + g(λ λ h )(χ u)ds = 1,h ( u h z 1,h + χ u h 1,h )+ u h z 1,h χ u h 1,h + g(λ λ h )(χ u)ds ,h ( u h z 1,h + χ u h 1,h ) 2 + u h z 1,h χ u h 1,h + g(λ λ h )(χ u)ds. Writ g(λ λ h )(χ u)ds = g(λ λ h )(u h u)ds+ g(λ λ h )(χ u h )ds. Not that by (3) and (11), w hav g(λ λ h )(u h u)ds Γ 2 = gλu h ds gλuds gλ h u h ds+ gλ h uds Γ 2 g u h ds g u ds g u h ds+ g u ds = 0.

8 56 F. WANG AND W. HAN In addition, Hnc, Rcalling (6), w hav g(λ λ h )(χ u h )ds λ λ h,h χ u h 1,h ǫ λ λ h 2,h + 1 4ǫ χ u h 2 1,h. 2 1,h u h z 2 1,h + χ u h 2 1,h +ǫ λ λ h 2,h. λ λ h,h = u z 1,h 1,h + u h z 1,h. Thn, w obtain th following rsult 1,h + λ λ h,h u h z 1,h + χ u h 1,h. Using (17) with i = 1 to bound χ u h 1,h, th abov inquality can b rwrittn as 1,h + λ λ h,h u h z 1,h + 1/2 (19) [u h] 2. Th rlation (19) srvs as a starting point for drivation of rliabl and fficint rror stimators of DG mthods for a frictional contact problm. In this papr, w focus on th drivation and analysis of rsidual typ rror stimators drivd from th inquality (19). A similar approach can also b applid to rcovry typ rror stimators. 3. Rliabl rsidual-typ stimators Now w driv a postriori rror stimators of DG mthods for solving th simplifid frictional contact problm. Th dtaild drivation and analysis of a postriori rror stimators is givn for th LDG mthod [17]. For othr DG mthods discussd in [32], similar rsults can b obtaind by similar argumnts. Givn intrior rsiduals and dg-basd jumps as follows, w dfin local stimators as (20) E 0 h R := u h u h +f for ach T h, { [ h u R := h ] if Eh i, h u h n+gλ h if Eh 2, η := h 2 R h R 2 + h R 2 1/2, (21) η := 1 2 E i h [u h ] 2 + E 2 h [u h ] 2 1/2. E i h E 1 h Applying Corollary 3.3, provd at th nd of this sction, to bound u h z 1,h in (19), w hav th following thorm. Thorm 3.1. Lt u H 2 () and u h solv (1) and (8) rspctivly. Thn w hav ( ) 1/2 u u h 1,h + λ λ h,h η 2 + (22) η 2.

9 POSTERIORI ERROR ESTIMATES OF DGMS FOR A FRICTIONAL PROBLEM 57 Now, lt us show how to driv th rsult in Corollary 3.3 to bound th trm u h z 1,h. W rcall on rsult in [13]. Not that th a postriori rror analysis in [13] was only for th Poisson problm with homognous Dirichlt boundary condition, but it is asy to xtnd th rsult to gnral lliptic problms with Numann boundary conditions. For th scond-ordr lliptic problm u+u = f in, u = 0 on Γ 1, rwrit it as a first ordr systm (23) p = u, p+u = f in, u = 0 on Γ 1, Thn th DG formulation for th problm is p h τ h dx = u h h τ h dx+ (24) (25) (p h h v h +u h v h )dx = f v h dx+ u n = g on, u n = g on. uˆ h n τ h ds τ h W h, pˆ h n v h ds v h V h, whr uˆ h and pˆ h ar numrical fluxs. Diffrnt choic of th numrical fluxs lads to diffrnt DG mthod. Th following rsult (s [13]) holds for th LDG mthod and othr mthods discussd in [3]. Thorm 3.2. Assum u H 1 Γ 1 () and p W := [L 2 ()] 2 ar th solution of th problm (23), and u h V h and p h W h ar th solution of th problm (24) (25). Thn, p p h C(η +ζ ), whr η 2 := h 2 divp h u h +f 2 + h [p h ] 2 + h p h n g 2, ζ 2 := E 0 h [u h] 2 E i h and C is a msh-siz indpndnt constant which dpnds only on th domain and th minimal angl condition. E 2 h From th rlation btwn p h and u h ([3, 13]), w dduc th following rsult. Corollary 3.3. With th sam notation as in Thorm 3.2, w hav u h u h C(η +ζ ), whr η 2 := h 2 u h u h +f 2 + E i hh [ h u h ] 2 + E 2 h Proof. By [28, Lmma 7.2], for any v h V h, r 0 ([v h ]) 2 C [v h] 2, l(β [v h]) 2 C E 0 h From [3, (3.9)], w know that h h u h n g 2. E i h p h = h u h r 0 ([ uˆ h u h ]) l({û h u h }). [v h] 2.

10 58 F. WANG AND W. HAN Thn u h u h u p h + p h h u h C(η +ζ )+ r 0 ([ uˆ h u h ]) + l({ uˆ h u h }). From th choics of numrical fluxs uˆ h in Tabl 3.1 of [3], w hav So [ uˆ h u h ] = [u h ] or 0, r 0 ([ uˆ h u h ]) C which implis E 0 h [u h ] 2, { uˆ h u h } = β [u h ] or 0. l({û h u h }) C E i h p h h u h ζ and u h u h C(η +ζ ). [u h ] 2, Finally, by th invrs inquality and trac inquality, w gt η 2 = h 2 divp h u h +f 2 + [p h ] Ehh 2 + h p h n g 2 i Eh 2 2 (η 2 + h 2 div(p h u h ) 2 + h [p h h u h ] 2 Eh i + ) h (p h h u h ) n 2 Eh 2 2η 2 +2 ( h 2 div(p h u h ) 2 +C p h u h 2 + ) h 2 p h u h 2 1, 2η 2 +C p h u h 2 = 2η2 +C p h h u h 2 2η 2 +Cζ 2. Thrfor, η C(η +ζ ) and th rsult is provd. 4. Efficincy of th stimators Now w prsnt lowr bounds of th stimators. W follow th standard argumnt to driv lowr bounds of rsidual rror stimators for lliptic problms, s [1, pp ]. First, w introduc th bubbl functions. Lt T h, and 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 list proprtis of bubbl functions statd in Thorms 2.2 and 2.3 of [1] in th form of a lmma. Lmma 4.1. For ach T h,, lt ϕ and τ b th corrsponding intrior and dg bubbl functions. Lt P() H 1 () and P() H 1 () b finitdimnsional spacs of functions dfind on or. Thn thr xists a constant C

11 POSTERIORI ERROR ESTIMATES OF DGMS FOR A FRICTIONAL PROBLEM 59 indpndnt of h such that for all v P(), C 1 v 2 ϕ v 2 dx C v 2, Dnot Thn for u,v H 1 (), C 1 v ϕ v +h ϕ v 1, C v, C 1 v 2 τ v 2 ds C v 2, /2 τ v +h 1/2 τ v 1, C v. a (u,v) = a(u,v) = ( u v +uv)dx. a (u,v). For all v HΓ 1 1 (), noting that [v] = 0 and [u z] = 0 on Eh 0, w hav a (,v) = a (u h z,v)+a(z u,v) = a (u h z,v)+ g(λ λ h )vds. T h Now, ( a (u h z,v) = (uh z) v +(u h z)v ) dx T h = ( (uh z)+u h z ) vdx+ (u h z) n vds. Hnc, (26) a (,v) = ( u h +u h f)vdx+ [ u h ] vds + ( u h n+gλ h )vds+ E 2 h E i h E 2 h g(λ λ h )vds. For ach T h, ϕ and τ ar rspctivly th intrior and dg bubbl functions on or Eh i E2 h. R is an approximation to th intrior rsidual R from a suitabl finit-dimnsional subspac. In (26), choos v = R ϕ on lmnt. W know ϕ vanishs on th boundary of by its dfinition, so v can b xtndd to b zro on th rst of domain as a continuous function. Thrfor, w gt a (, R ϕ ) = R R ϕ dx. Thn R 2 ϕ dx = R ( R R )ϕ dx+a (, R ϕ ). Applying th Cauchy-Schwarz inquality and Lmma 4.1, w obtain R ( R R )ϕ dx R ϕ R R R R R, a (, R ϕ ) 1, R ϕ 1, 1, R.

12 60 F. WANG AND W. HAN Us Lmma 4.1 again, R 2 Combining th abov rlations, w obtain R 2 ϕ dx. R R R + 1,. Finally, by th triangl inquality R R R + R, w gt R R R + 1,. Nowchoosthfinit-dimnsionalsubspacfromwhichth R comasthfunction spac spannd by th local nodal basis φ (i) with i = 1,2,3. Thn, R R rducs to f f whr w tak (27) f = 3 i=1 f i φ (i) For Eh 2, w obtain a ω (u h u,r τ ) = R R τ dx+ ω with f i = (f,φ (i) ) /(1,φ (i) ). R R τ ds+ g(λ λ h )R τ ds and 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 From Lmma 4.1, w stimat th trms in abov rlation as C 1 R 2 R 2 τ ds, 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 C/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 C/2 λ λ h, R. Hnc, w obtain R R + R R C ( /2 u h u 1,ω +/2 λ λ h, +h 1/2 R R ω + R R ). For E i h, lt R b an approximation to th jump R from a suitabl finitdimnsional spac and lt v = R τ in (26). By a similar argumnt, w hav R C ( /2 u h u 1,ω +h 1/2 R R ω + R R ). 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 λ h. Thn w obtain th fficincy bound of th local rror indicator η.

13 POSTERIORI ERROR ESTIMATES OF DGMS FOR A FRICTIONAL PROBLEM 61 Thorm 4.2. Lt u and u h b th solutions of (1) and (8), rspctivly, and η b th stimator (20). Thn ( (28) η C u u h ω + λ λ h, +h f f h ω E() E 2 ) + h λ h λ h, E() E 2 whr th constant C is dpndnt on th angl condition and indpndnt of h. 5. Summary In this papr, w provid a postriori rror analysis of DG mthods for a rprsntativ lliptic variational inquality of th scond kind, th simplifid frictional contact problm. By rlating th rror of th variational inquality to th rror of a corrsponding linar problm, w driv rsidual-typ rror stimators, which ar thortically provd to b rliabl and fficint. Not that w considr only th rsidual typ rror stimators in this papr, but th analysis for gradint rcovry typ rror stimation can b obtaind by th tchniqus usd in this papr and th standard argumnt of gradint rcovry typ rror analysis for scond ordr lliptic boundary valu problms. Acknowldgmnts This rsarch was supportd by th Chins National Scinc Foundation (Grant No ), and NSF undr grant DMS Rfrncs [1] M. Ainsworth and T. J. Odn, A Postriori Error Estimation in Finit Elmnt Analysis, Wily, Chichstr, [2] D. N. Arnold, F. Brzzi, B. Cockburn, and L. D. Marini, Discontinuous Galrkin mthods for lliptic problms, in Discontinuous Galrkin Mthods. Thory, Computation and Applications, B. Cockburn, G. E. arniadakis, and C.-W. Shu, ds., Lctur Nots in Comput. Sci. Engrg. 11, Springr-Vrlag, Nw York, 2000, [3] D. N. Arnold, F. Brzzi, B. Cockburn, and L. D. Marini, Unifid analysis of discontinuous Galrkin mthods for lliptic problms, SIAM J. Numr. Anal. 39 (2002), [4] I. Babuška and T. Strouboulis, Th Finit Elmnt Mthod and its Rliability, Oxford Univrsity Prss, [5] S. Bartls and C. Carstnsn, Avraging tchniqus yild rliabl a postriori finit lmnt rror control for obstacl problms, Numr. Math. 99 (2004), [6] R. Bckr, P. Hansbo, and M. G. Larson, Enrgy norm a postriori rror stimation for discontinuous Galrkin mthods, Comput. Mthods Appl. Mch. Engrg. 192 (2003), [7] V. Bostan, and W. Han, Rcovry-basd rror stimation and adaptiv solution of lliptic variational inqualitis of th scond kind, Commun. Math. Sci. 2 (2004), [8] V. Bostan, and W. Han, A postriori rror analysis for a contact problm with friction, Comput. Mthods Appl. Mch. Engrg. 195 (2006), [9] 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, [10] V. Bostan, W. Han, and B. D. Rddy, A postriori rror stimation and adaptiv solution of lliptic variational inqualitis of th scond kind, Appl. Numr. Math. 52 (2004), [11] D. Brass, A postriori rror stimators for obstacl problms anothr look, Numr. Math. 101 (2005), [12] Z. Cai, X. Y, and S. Zhang, Discontinuous Galrkin finit lmnt mthods for intrfac problms: a priori and a postriori rror stimations, SIAM J. Numr. Anal., 49 5 (2011),

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