A Family of Discontinuous Galrkin Finit Elmnts for th Rissnr Mindlin plat Douglas N. Arnold Franco Brzzi L. Donatlla Marini Sptmbr 28, 2003 Abstract W dvlop a family of locking-fr lmnts for th Rissnr Mindlin plat using Discontinuous Galrkin tchniqus, on for ach odd dgr, and prov optimal rror stimats. A scond family uss conforming lmnts for th rotations and nonconforming lmnts for th transvrs displacmnt, gnralizing th lmnt of Arnold and Falk to highr dgr. Introduction Rcntly thr has bn a considrabl intrst in th dvlopmnt of Discontinuous Galrkin mthods for lliptic problms (s, for instanc, [4] and th rfrncs thrin). Although thir practical intrst is still undr invstigation, it is clar that th DG approach oftn implis a diffrnt way of daling with th problm, that can somtims lad to nw conforming or nonconforming finit lmnts that would hav bn mor difficult to discovr starting with a classical approach. Exampls in this dirction ar, for instanc, th xtnsion of th Crouzix Raviart lmnt for Stoks problm or narly incomprssibl lasticity problms [23], and th rcnt papr by two of th prsnt authors using DG mthods to dvlop non-conforming lmnts for th Rissnr Mindlin plat [6]. Institut for Mathmatics and its Applications, Univrsity of Minnsota, Minnapolis, MN 55455 (arnold@ima.umn.du). Supportd by NSF grant DMS-007233. Dipartimnto di Matmatica, Univrsità di Pavia, Italy and IMATI-CNR, Via Frrata, 2700 Pavia, Italy (brzzi@imati.cnr.it). Partially supportd by Italian govrnmnt grant PRIN 200. Dipartimnto di Matmatica, Univrsità di Pavia, Italy and IMATI-CNR, Via Frrata, 2700 Pavia, Italy (marini@imati.cnr.it). Partially supportd by Italian govrnmnt grant PRIN 200.
Hr w prsnt a family of finit lmnt approximations for th Rissnr Mindlin plats. Ths ar mixd mthods, in which th rotation vctor, transvrs displacmnt, and transvrs shar ar all approximatd. Th starting point of th family is a totally discontinuous approach, in which both rotations θ and transvrsal displacmnts w ar locally polynomials of dgr k, whr k is an odd intgr, whil th transvrs shars ar approximatd by totally discontinuous polynomials of dgr k. Howvr, many variants ar possibl. For instanc, w could (i) kp θ discontinuous but us a nonconforming w (having momnts up to th ordr k continuous at th intrlmnt boundaris), or (ii) tak both θ and w nonconforming (by adding a suitabl st of bubbl functions to θ), or (iii) us a continuous θ and a nonconforming w, by adding a diffrnt st of bubbl functions to θ. This last option, for k =, will giv back th Arnold Falk (AF) lmnt [5], and thrfor, for k >, can b sn as a highr ordr vrsion of AF. On th othr hand, th othr options can b sn, for k =, as a discontinuous or nonconforming vrsions of AF. In particular hr w prsnt th analysis of th two xtrm cass, that is th fully discontinuous cas and th cas in which θ is continuous and w is nonconforming. Th analysis of th othr cass could b prformd along similar lins. It would b intrsting to compar ths nw lmnts with th mor classical MITC k familis (s [2] or [5]) and th lmnts in [5], as wll as with th mor rcnt mthods such as [6], [7], [8], [2], [22], [25], [26], and [27]. Evn mor intrsting would b th xtnsion of ths DG tchniqus to th tratmnt of shll problms. S for instanc [24], [9], [7], [8], [20], [22], [9] and th rfrncs thrin for a discussion of th difficultis in dsigning accurat and robust shll lmnts. W point out hr that our lmnts, at last in th totally discontinuous vrsion, us th sam dgrs of frdom for th rotations and th transvrs displacmnt, which is usually considrd as a vry favorabl fatur for th discrtization of shll problms in th Naghdi modl. Th papr is organizd as follows. Aftr a sction on notations and prliminaris, in Sction 3 w rcall th Rissnr Mindlin modl and driv our family of mthods in th fully discontinuous cas. Th corrsponding rror stimats ar provd in Sction 4. Finally, in Sction 5, w prsnt th cas of continuous θ and nonconforming w, togthr with its analysis. 2 Notations and prliminaris Lt Ω R 2 dnot th domain occupid by th middl surfac of th plat. W shall us th usual Sobolv spacs such as H s (T ), with th corrsponding smi-norm and norm dnotd by s,t and s,t, rspctivly. 2
Whn T = Ω w just writ s and s. By convntion, w us boldfac typ for th vctor-valud analogus: H s (Ω) = [H s (Ω)] 2. Occasionally w shall us calligraphic typ for symmtric-tnsor-valud analogus: H s (Ω) = [H s (Ω)] 2 sym. W us parnthss (, ) to dnot th innr product in any of th spacs L 2 (Ω), L 2 (Ω), or L 2 (Ω). W shall assum that th domain Ω is a polygon and dnot by T h a dcomposition of Ω into triangls T, by E h th st of all th dgs in T h, and by Eh 0 th st of intrior dgs. W us th notation for picwis polynomial spacs from [4], so L s k(t h ) = { v H s (Ω) : v T P k (T ), T T h }, () with P k (T ) th st of polynomials of dgr at most k on T. (Not that in this usag, calligraphic typ dos not rfr to tnsor-valud quantitis.) Our finit lmnts will b discontinuous and so not containd in th spac H (Ω), but rathr in a picwis Sobolv spac H (T h ) := { v L 2 (Ω) : v T H (T ), T T h }. Diffrntial oprators can b applid to this spac picwis. W indicat this by a subscript h on th oprator. Thus, for xampl, th picwis gradint oprator h maps H (T h ) into L 2 (Ω) and th picwis symmtric gradint, or infinitsimal strain, oprator ε h maps H (T h ) into L 2 (Ω). Th spac H (T h ) is quippd with th smi-norm v,h = h v 0 and th corrsponding norm v 2,h = v 2,h + v 2 0. As is usual in th DG approach, w dfin th jump and avrag of a function in H (T h ) as a function on th union of th dgs of th triangulation. Lt b an intrnal dg of T h, shard by two lmnts T + and T, and lt n + and n dnot th unit normals to, pointing outward from T + and T, rspctivly. If ϕ blongs to H (T h ) (or possibly th vctoror tnsor-valud analogu), w dfin th avrag {ϕ} on as usual: {ϕ} = ϕ+ + ϕ. 2 For a scalar function ϕ H (T h ) w dfin its jump on as [ϕ] = ϕ + n + + ϕ n, which is a vctor normal to. Th jump of a vctor ϕ H (T h ) is th symmtric matrix-valud function givn on by: [ϕ] = ϕ + n + + ϕ n, whr ϕ n = (ϕn T +nϕ T )/2 is th symmtric part of th tnsor product of ϕ and n. 3
On a boundary dg, th avrag {ϕ} is dfind simply as th trac of ϕ, whil for a scalar-valud function w dfin [ϕ] to b ϕn (with n th outward unit normal), and for a vctor-valud function w dfin [ϕ] = ϕ n. It is asy to chck that (s,.g., [3]) ϕ n T v ds = {ϕ} [v] ds, ϕ H (Ω), v H (T h ). (2) T T h T Similarly, T T h T Sn T η ds = {S} : [η] ds, S H (Ω), η H (T h ). It is not difficult to s that both th abov rlations hold in mor gnral situations. For instanc, (2) also holds for ϕ H(div; Ω), whr H(div; Ω) is th spac of vctors ϕ L 2 (Ω) with div ϕ L 2 (Ω). In th squl w shall oftn us th following rsult (s [], [2]): lt T b a triangl, and lt b an dg of T. Thn thr xists a positiv constant C only dpnding on th minimum angl of T such that ϕ 2 0, C ( ϕ 2 0,T + ϕ 2,T ), ϕ H (T ). (3) Clarly, (3) also holds for vctor valud functions ϕ H (T h ). 3 Th problm and a DG discrtization In this sction w rcall th Rissnr Mindlin plat modl and driv a discontinuous Galrkin discrtization of it. Givn th load g in L 2 (Ω) and th tnsor of bnding moduli C, th Rissnr Mindlin quations with clampd boundary dtrmin th rotation θ, transvrs displacmnt w, and scald shar strss γ by th quations div C ε(θ) γ = 0 in Ω, (4) div γ = g in Ω, (5) w θ λ t 2 γ = 0 in Ω, (6) θ = 0, w = 0 on Ω. (7) Hr ε dnots th usual symmtric gradint oprator, λ th shar corrction factor, and t th plat thicknss. Hncforth w will incorporat λ in th thicknss (still dnoting it by t). To obtain a wak mixd formulation of th systm (4) (7) w multiply (4) by a tst function η H 0(Ω) and (5) by a tst function v H 0 (Ω), 4
intgrat by parts, and add th quations. Nxt, w multiply (6) by a tst function τ L 2 (Ω) and intgrat. W thus find that (θ, w) H 0(Ω) H 0 (Ω) and γ L 2 (Ω) satisfy (C ε(θ), ε(η)) + (γ, v η) = (g, v), (η, v) H 0(Ω) H 0 (Ω), (8) ( w θ, τ ) t 2 (γ, τ ) = 0, τ L 2 (Ω). (9) A natural way to discrtiz th Rissnr Mindlin systm is to rstrict th trial and tst functions in this wak formulation to finit dimnsional subspacs. That is, w choos finit dimnsions subspacs Θ h H 0(Ω), W h H 0 (Ω), and Γ h L 2 (Ω) and dfin (θ h, w h ) Θ h W h and γ h Γ h by th quations (C ε(θ h ), ε(η)) + (γ h, v η) = (g, v), (η, v) Θ h W h, ( w h θ h, τ ) t 2 (γ h, τ ) = 0, τ Γ h. In ordr to ovrcom th wll-known problm of locking th loss of accuracy for small plat thicknss this formulation is oftn gnralizd by th inclusion of a projction oprator P h : H (T h ) Γ h to obtain th systm (C ε(θ h ), ε(η)) + (γ h, P h ( v η)) = (g, v), (η, v) Θ h W h, (P h ( w h θ h ), τ ) t 2 (γ h, τ ) = 0, τ Γ h. (Th mthod without P h can b viwd as th spcial cas whr P h is takn to b th L 2 -projction onto Γ h.) A numbr of th most succssful finit lmnt mthods for th Rissnr Mindlin systm can b writtn in this form with appropriat choics for th spacs Θ h, W h, and Γ h and th projction oprator P h. Howvr, simpl choics of th finit lmnt spacs hav bn found to b unsuccssful vn with th us of a projction oprator. For xampl, th choic of continuous picwis linar functions for Θ h and W h and picwis constant functions for Γ h sms natural, but dos not giv a good mthod. In this papr w will show that vry simpl discontinuous finit lmnt spacs can b usd. To driv a finit lmnt mthod for th Rissnr Mindlin systm basd on discontinuous lmnts, w procd as bfor tsting (4) against a tst function η and (5) against a tst function v, intgrating by parts, and adding, with th diffrnc that now η and v may b discontinuous across lmnt boundaris, that is, thy blong to H (T h ) and H (T h ), rspctivly. Thus w obtain (C ε h (θ), ε h (η)) {C ε h (θ)} : [η] ds + (γ, h v η) {γ} [v] ds = (g, v), (η, v) H (T h ) H (T h ), (0) 5
( h w θ, τ ) t 2 (γ, τ ) = 0, τ L 2 (Ω). Th two trms in th first quation involving intgrals ovr th dgs, which did not appar in (8), aris from th intgration by parts and ar ncssary to maintain consistncy. W now procd as is common for DG mthods. First, w add trms to symmtriz this formulation so that it is adjointconsistnt as wll. Scond, to stabiliz th mthod, w add intrior pnalty trms p Θ (θ, η) and p W (w, v) in which th functions p Θ and p W will dpnd only on th jumps of thir argumnts. Sinc [θ] = 0 and [w] = 0, w find that θ, w, and γ satisfy (C ε h (θ), ε h (η)) ({C ε h (θ)} : [η] ds + [θ] : {C ε h (η)}) ds + (γ, h v η) {γ} [v] ds + p Θ (θ, η) + p W (w, v) = (g, v), (η, v) H 2 (T h ) H 2 (T h ), ( h w θ, τ ) [w] {τ } ds t 2 (γ, τ ) = 0, τ H (T h ). () To obtain a DG discrtization, w choos finit dimnsional subspacs Θ h H 2 (T h ), W h H 2 (T h ), and Γ h H (T h ) and, in analogy with th continuous Galrkin cas, w incorporat a projction oprator P h : H (T h ) Γ h, so that th mthod taks th form: Find (θ h, w h ) Θ h W h and γ h Γ h such that (C ε h (θ h ), ε h (η)) ({C ε h (θ h )} : [η] ds + [θ h ] : {C ε h (η)}) ds + (γ h, P h ( h v η)) {γ h } [v] ds + p Θ (θ h, η) + p W (w h, v) = (g, v), (η, v) Θ h W h, (2) (P h ( h w h θ h ), τ ) [w h ] {τ } ds t 2 (γ h, τ ) = 0, τ Γ h. (3) For any choic of th finit lmnt spacs Θ h, W h, and Γ h, and any intrior pnalty functions p Θ and p W dpnding only on th jumps of thir argumnts, this givs a consistnt finit lmnt mthod whn th projction oprator P h is simply th L 2 -projction onto Γ h. Most othr choics of P h introduc a consistncy rror just as for continuous Galrkin mthods. 6
Th numrical mthod w will considr is of th form (2), (3). To complt th spcification of th mthod w nd to spcify thr things: th finit lmnt spacs Θ h, W h, and Γ h ; th intrior pnalty forms p Θ and p W ; and th projction oprator P h. For th finit lmnt spacs w mak a simpl choic, namly for an intgr k w us fully discontinuous picwis polynomials of dgr k to discrtiz θ and w, and of dgr k for γ. Using th notation introducd in (), Θ h = L 0 k(t h ), W h = L 0 k(t h ), Γ h = L 0 k (T h ). Not that this choic nsurs that h (W h ) Γ h, (4) an important rlation for this mthod as for many discrtizations of th Rissnr Mindlin systm. This, of cours, implis that, for any projction oprator P h : H (T h ) Γ h, P h h v = h v for all v W h. W mak a standard choic for th intrior pnalty trm p Θ : p Θ (θ, η) = κ Θ [θ] : [η] ds, (5) so that p Θ (η, η) can b viwd as a masur of th dviation of η from bing continuous. Th paramtr κ Θ is a positiv constant to b chosn; it must b sufficintly larg to nsur stability. For p W w us a wakr pnalization: p W (w, v) = κ W Q [w] Q [v] ds, whr Q is th L 2 -projction onto polynomials of dgr k on th dg and κ W is again a positiv constant to b chosn. Thus w pnaliz th dviation of w from th usual non-conforming dgr k finit lmnt spac rathr than th dviation from continuity. Finally, w nd to spcify th projction oprator P h. In th lowst ordr cas, k =, w simply choos th L 2 -projction onto th picwis constant spac L 0 0(T h ). For k > th dfinition of P h is mor complicatd and rquirs som notation and a lmma. For any odd intgr k > and any triangl T, dfin Γ (T ) = { τ + curl(b T v) τ P k (T ), div τ P k 3 (T ), v P k 2 (T ) }. Hr b T is th cubic bubbl givn by λ λ 2 λ 3 whr th λ i ar th barycntric coordinat functions on T, and curl v := ( v/ y, v/ x) (with formal adjoint rot δ := δ / y δ 2 / x). For k = w intrprt Γ (T ) = P 0 (T ). Not that dim Γ (T ) = dim P k (T ). 7
Lmma 3. Lt k b a positiv odd intgr and T a triangl. P k (T ) satisfis T δ ρ dx = 0 for all ρ Γ (T ), thn δ = 0. If δ Proof. This is obvious for k = so w assum k 3. By intgration by parts, w hav T (rot δ)b T v dx = 0 for all v P k 2 (T ). In particular, w can tak v = rot δ and conclud that rot δ = 0. Thrfor δ = ψ for som ψ P k (T ) which w can normaliz to hav man valu 0 on T. Now, givn an arbitrary q P k 2 (T ) and an arbitrary picwis polynomial µ of dgr k on T (that is, µ rstricts to a polynomial of dgr k on ach dg of T ), w hav that th quation div τ = q in T, τ n = µ on T (6) has a solution τ P k (T ) if and only if T q dx = µ ds. (This can b T chckd by counting dimnsions and noting that τ satisfis (6) for q = 0, µ = 0 if and only if τ = curl(b T p) for som p P k 3 (T )). Taking q = 0 and µ an arbitrary picwis polynomial of man valu 0 on T, w can solv (6) to find τ Γ (T ). Thn intgration by parts givs 0 = ψ τ dx = ψµ ds. T This, togthr with our normalization T ψ ds = 0 shows that ψ T is orthogonal to all picwis polynomials of dgr k. Thrfor on ach dg ψ must b a multipl of th Lgndr polynomial of dgr k and hnc it mush chang sign xactly k tims on ach dg (unlss it is idntically 0). Th global continuity of ψ, howvr, ruls out an odd numbr (3k) of sign changs, so w conclud that ψ = 0 on T, i.., ψ = b T φ for som φ P k 3 (T ). Now tak q = φ, µ = constant on T in (6). Th rsulting τ blongs to Γ (T ) and so is orthogonal to δ = (b T φ), and now intgration by parts immdiatly implis that φ = 0. Lt Γ h = { τ L 2 (Ω) τ T Γ (T ), T T h }. (7) In viw of th lmma, w may dfin P h : L 2 (Ω) Γ h by 4 Error analysis T (δ P h δ, τ ) = 0, τ Γ h. (8) Having compltd th spcification of our family of DG mthods (on for ach positiv odd intgr k), in this sction w stat and prov th basic 8
rror stimats for th mthods. For this purpos w first dfin norms η 2 Θ := η 2,h + ( ) [η] 2 0, + {C ε h (η)} 2 0,, (9) v 2 W := v 2,h + [v] 2 0,, (20) τ 2 Γ := τ 2 0 + {τ } 2 0,, (2) for η H 2 (T h ), v H (T h ), and τ H (T h ). Rmark 4. If w rplac [v] in (20) with its projction into som polynomial spac on th dg, w obtain an quivalnt norm. That is, v 2 W v 2,h + Q ([v]) 2 0, v 2,h + Q0 ([v]) 2 0,, v H (T h ), (22) whr Q is, as abov, th L 2 -projction onto polynomials of dgr k on th dg, and Q 0 th L 2 -projction onto constants on, and th constants of quivalnc dpnd only on minimum angl of th triangulation. Obviously v 2,h + Q0 ([v]) 2 0, v 2,h + Q ([v]) 2 0, v 2 W, so, to stablish (22), w nd only show that [v] 2 0, C v 2,h + Q0 ([v]) 2 0,. Now if v H (T ) for som triangl T with dg and Q 0 v dnots th avrag of v on (i.., th L 2 -projction into constants of its trac on ), w hav v Q0 v 2 0, C( 2 v Q0 v 2 0,T + v 2 0,T ) C v 2 0,T, whr w hav usd (3) and a simpl approximation rsult. It follows that, for v H (T h ), [v] Q0 ([v]) 2 0, C v 2,h, 9
and so as dsird. [v] 2 0, = ( [v] Q0 ([v]) 2 0, + Q 0 ([v]) 2 0,) C v 2,h + Q0 ([v]) 2 0,, Th following thorm is th principal rsult of th papr. Thorm 4.2 Lt θ, w, γ solv th Rissnr Mindlin systm (8), (9). Lt k b a positiv odd intgr and suppos that th pnalty paramtr κ Θ is sufficintly larg and th pnalty paramtr κ W is positiv. Thn thr xists a uniqu solution θ h, w h, γ h to th discontinuous Galrkin mthod (2) (3). Morovr, thr xists a constant C, indpndnt of h and t, such that θ θ h Θ + w w h W + t γ γ h Γ C h k ( θ k+ + w k+ + γ k ). Rmark 4.3 This stimat is clarly optimal with rspct to th powr of h and with rspct to th rgularity of θ and w. With rspct to th rgularity of γ on might hop to rplac γ k with t γ k + γ k + div γ k on th right-hand sid. Howvr, such an stimat dos not follow from th currnt analysis. W will howvr b abl to prov it, in th last sction, for th continuous-nonconforming cas. W now turn to th proof Thorm 4.2, bginning by introducing som notation. Lt a h (θ, η) = (C ε h (θ), ε h (η)) ({C ε h (θ)} : [η] + [θ] : {C ε h (η)}) ds + p Θ (θ, η), j(τ, v) = {τ } [v] ds. (23) Clarly w hav a h (θ, η) C θ Θ η Θ, θ, η H 2 (T h ), (24) j(τ, v) C τ Γ v W, v H (T h ), τ H (T h ). (25) 0
In this notation w may rwrit () as a h (θ, η) + (γ, h v η) j(γ, v) + p W (w, v) = (g, v), and (2) (3) as (η, v) H 2 (T h ) H 2 (T h ), (26) ( h w θ, τ ) j(τ, w) t 2 (γ, τ ) = 0, τ H (T h ), (27) a h (θ h, η) + (γ h, h v P h η) j(γ h, v) + p W (w h, v) = (g, v), (η, v) Θ h W h, (28) ( h w h P h θ h, τ ) j(τ, w h ) t 2 (γ h, τ ) = 0, τ Γ h. (29) Dfining a lifting oprator J : H (T h ) Γ h by th quation w can liminat γ h in (29): (J(v), τ ) = j(τ, v), τ Γ h, (30) γ h = t 2 ( h w h J(w h ) P h θ h ). (3) Substituting in (28), w obtain an altrnat formulation of th mthod: a h (θ h, η) + t 2 ( h w h J(w h ) P h θ h, h v J(v) P h η) + p W (w h, v) = (g, v), η Θ h, v W h. Th following stimat for J will play an important rol in th analysis. (Hr and throughout th squl w continu to dnot by C a gnric constant which may dpnd on th msh through its shap rgularity but not othrwis and which is indpndnt of t.) Proposition 4.4 J(v) 2 Γ C Q [v] 2 0,, v W h. Proof. First w not that, by a local invrs inquality, τ Γ C τ 0, τ Γ h. (32) Now J(v) 2 0 = (J(v), J(v)) = j(j(v), v) = {J(v)} [v] ds = {J(v)} Q [v] ds.
Thrfor ( ) /2 ( ) /2 J(v) 2 0 {J(v)} 2 0, Q [v] 2 0, ( ) /2 J(v) Γ Q [v] 2 0,, and so th proposition follows using (32). Th nxt two propositions ar analogus of Poincaré s inquality and Korn s inquality for picwis smooth functions. Proposition 4.5 v 0 C v W, v H (T h ). (33) Proof. This sort of rsult is wll-known. S for instanc [2] or th mor gnral rsults of [0]. For th convninc of th radr w includ th proof. As a first stp, w considr a smooth domain Ω such that Ω Ω, and w xtnd v by zro outsid Ω. W dnot again by v th xtnsion. Thn w dfin th function ψ as th solution of ψ = v in Ω, with ψ H 0 ( Ω). W obviously hav ψ 2, Ω C v 0. Thn w hav, using th dfinition of ψ, intgrating by parts in ach triangl and using (2), and finally using (25): v 2 0 = ( ψ, v) = ( ψ, h v) + j( ψ, v) Using (3) it is not difficult to s that ψ Γ C ψ 2, Ω C v 0, ψ v,h + C ψ Γ v W. and (33) follows. Lmma 4.6 η 2,h C( ε(η) 2 0,T + [η] 2 0,), T T h η H (T h ). (34) Proof. This is ssntially a spcialization of th rsults in []. From Thorm 3. of that papr, with Φ chosn as in Exampl 2.3, w gt η 2,h C( ε(η) 2 0,T + η 2 0 + [η] 2 0,), η H (T h ). T T h 2
W can now rpat, ssntially, th proof of (33) in ordr to bound th η 0 in trms of ε(η) 0,T and th jumps, and thn w asily dduc (34). Using Lmma 4.6, (3), and an invrs inquality, it is straightforward to vrify th following proposition. Proposition 4.7 Thr xist positiv constants κ 0 and α dpnding only on th polynomial dgr k and th shap rgularity of th partition T h, such that: if th constant κ Θ κ 0 (whr κ Θ is th pnalty paramtr apparing in (5)), thn a h (η, η) α η 2 Θ, η Θ h. (35) To procd with th analysis w dfin, for θ H (Ω), w H (Ω), and γ L 2 (Ω), approximations θ I Θ h, w I W h, and γ I Γ h. For θ I w simply tak th L 2 -projction of θ onto Θ h. Sinc Γ h Θ h, an obvious (but important) consqunc is that Of cours w hav P h θ = P h θ I, θ H (Ω). (36) θ θ I Θ C h k θ k+. (37) For w I w us a standard non-conforming P k intrpolant. Namly on ach triangl T w dfin w I T P k (T ) by (w w I )µ ds = 0, µ P k () for ach dg of T, (38) (w w I )v dx = 0, v P k 3 (T ). (39) T Not that (w w I ) τ dx = T T (w w I ) div τ dx + (w w I )τ n ds, T which vanishs if τ P k (T ) with div τ P k 3 (T ) and crtainly if τ = curl(b T v) for som v. Thus P h ( w) = P h ( h w I ) = h w I, w H (Ω), (40) with th last quality coming from (4). Standard approximation thory givs w w I W C h k w k+. (4) W also not that (38) implis that Q [w w I ] = 0 on vry dg. Hnc, p W (w w I, v h ) = 0, v h W h. (42) 3
Finally w dfin γ I = P h γ. Standard approximation argumnts stablish that γ γ I Γ C h k γ k. (43) Most importantly, (36) and (40) togthr imply that if γ = t 2 ( w θ), thn γ I = t 2 ( h w I P h θ I ). (44) Following idas from Duran and Librmann [2], our analysis will rly on this last rlation. Also important, but spcific to th cas of discontinuous lmnts, is th rlation or, quivalntly, j(τ, w I ) = 0, w H (Ω), τ Γ h, J(w I ) = 0, (45) which follows dirctly from (38). W will bound th rror btwn th xact solution θ, w, γ, dtrmind by (26) and (27), and th Galrkin solution θ h, w h, γ h, dtrmind by (28) and (29), in trms of th rrors in θ I, w I, and γ I which can in turn b boundd as in (37), (4), and (43). Lt θ δ = θ h θ I, w δ = w h w I, γ δ = γ h γ I. (46) From (3), (44), and (45) w hav P h θ δ = t 2 γ δ + h w δ J(w δ ). (47) Using (35), thn adding and subtracting θ w obtain α θ δ 2 Θ a h (θ δ, θ δ ) = a h (θ h θ, θ δ ) + a h (θ θ I, θ δ ) =: a h (θ h θ, θ δ ) + T. (48) Thn w tak η = θ δ, v = 0 in (26) and (28), and w add and subtract P h θ δ, to obtain a h (θ h θ, θ δ ) = (γ h, P h θ δ ) (γ, θ δ ) By (47), = (γ h γ, P h θ δ ) + (γ, P h θ δ θ δ ) =: (γ h γ, P h θ δ ) + T 2. (49) (γ h γ, P h θ δ ) = t 2 (γ h γ, γ δ ) + (γ h γ, h w δ J(w δ )) = t 2 γ δ 2 0 t 2 (γ I γ, γ δ ) + (γ h γ, h w δ J(w δ )) =: t 2 γ δ 2 0 + T 3 + (γ h γ, h w δ J(w δ )). 4
Th first trm in th right-hand sid is ngativ, and will go to th lft in th final stimat. To dal with th last trm, w not that (28) with η = 0, v = w δ, and (30) giv (γ h, h w δ J(w δ )) = (g, w δ ) p W (w h, w δ ) = (g, w δ ) + p W (w w h, w δ ), and (26) givs so (γ, h w δ ) = (g, w δ ) + j(γ, w δ ), (γ h γ, h w δ J(w δ )) = p W (w w h, w δ ) + (γ, J(w δ )) j(γ, w δ ) =: p W (w w h, w δ ) + T 4. Finally, adding and subtracting w I, and using (42) w dduc p W (w w h, w δ ) = p W (w w I, w δ ) p W (w δ, w δ ) = p W (w δ, w δ ). Th last trm in th right-hand sid is ngativ, and gos to th lft-hand sid. Collcting th abov quations w hav whr α θ δ 2 Θ + t 2 γ δ 2 Γ + p W (w δ, w δ ) T + T 2 + T 3 + T 4, (50) T = a h (θ θ I, θ δ ) C θ θ I Θ θ δ Θ, (5) T 2 = (γ, P h θ δ θ δ ), (52) T 3 = t 2 (γ γ I, γ δ ) t 2 γ γ I 0 γ δ 0, (53) T 4 = (γ, J(w δ )) j(γ, w δ ). (54) To stimat T 4 w add and subtract γ I (25) and Proposition 4.4, obtaining using (30), and thn w us T 4 = (γ, J(w δ )) j(γ, w δ ) = (γ γ I, J(w δ )) j(γ γ I, w δ ) C γ γ I Γ w δ W. This stimat is not, howvr, satisfactory, sinc w do not hav a trm lik w δ W in th lft-hand sid of (50). Hnc, w hav to bound h w δ 0 as wll. For this, w apply (47), Proposition 4.4, and th L 2 -bounddnss of P h to obtain h w δ 0 = t 2 γ δ + J(w δ ) + P h θ δ 0 C(t 2 γ δ Γ + θ δ Θ + (p W (w δ, w δ )) /2 ), 5
and thrfor, thanks to (22), w δ W C(t 2 γ δ Γ + θ δ Θ + (p W (w δ, w δ )) /2 ). (55) It rmains to bound T 2. From th dfinition of P h, w hav T 2 = (γ δ, P h θ δ θ δ ) γ δ 0 P h θ δ θ δ 0 Ch γ δ 0 θ δ Θ, whr δ is an arbitrary lmnt of Γ h. W may choos, for xampl, δ to b th L 2 -projction of γ onto L 0 k 2(T h ) and gt γ δ 0 Ch k γ k. Thus T 2 Ch k γ k θ δ Θ. Combining th prcding stimats and invoking th arithmtic-gomtric man inquality w obtain θ δ 2 Θ + t 2 γ δ Γ + p W (w δ, w δ ) C ( θ θ I 2 Θ + ( + t 2 ) γ γ I 2 Γ + w w I 2 W + h 2k γ 2 k ). In viw of (22), this bcoms θ δ 2 Θ + t 2 γ δ Γ + w δ 2 W C ( θ θ I 2 Θ + ( + t 2 ) γ γ I 2 Γ + w w I 2 W + h 2k γ 2 k ). Finally, combining with th triangl inquality and th intrpolation rror bounds (37), (4), (43), and assuming as natural that t is boundd from abov, w complt th proof of th Thorm 4.2. 5 Continuous θ and nonconforming w In this final sction w considr a mthod in which θ is discrtizd by mans of continuous lmnts, and w by mans of nonconforming ons. Our mthod is still of th form (2) (3), and again w must spcify th finit lmnt spacs Θ h, W h, and Γ h, th pnalty functions p Θ and p W, and th projction oprator P h. Th pnalty functions ar not ndd for this mthod, and can b takn to vanish. W kp Γ h = L 0 k (T h ) as bfor, and w kp th dfinition (8) of P h whr Γ h is still givn by (7). For th choic of W h w tak th spac of nonconforming picwis polynomials of dgr at most k, that is W h = { v h L 0 k Q [v h ] = 0, E h } (56) whr Q is as bfor th L 2 -projction on th spac of polynomials of dgr k on. Obviously w still hav h (W h ) Γ h. 6
Th abov dfinitions allow us to again tak γ I := P h γ Γ h and to again dfin w I W h by (38) (39). Thn (40) still holds. In ordr to hav th fundamntal proprty (44), on which th rror analysis is basd, w nd to dfin th spac Θ h so it admits an intrpolation oprator θ θ I Θ h satisfying (36) (which, togthr with (40), implis (44)). Th continuity w ar rquiring for θ I prcluds th choic θ I = P h θ mad formrly, and lads to a mor complicatd construction of Θ h. In particular, th somwhat natural choic Θ = L k dos not work (vn for k = ) as w would not hav nough dgrs of frdom to forc (36) in ach lmnt. Instad, w start from L k and add a sufficint numbr of bubbl functions to nsur (36). Dfin Θ(T ) = P k (T ) + b T Γ (T ), and rmark that all th bubbls of P k (T ) can b writtn as b T P k 3 (T ). Sinc P k 3 (T ) Γ (T ), all th bubbls of P k (T ) blong to b T Γ (T ). Hnc a st of dgrs of frdom for θ Θ(T ) consists of th valus of ach componnt of θ at th vrtics of T, th momnts of dgr at most k 2 of ach componnt of θ on ach dg of T, and th intgrals T θ η dx for ach η in a basis for Γ (T ). Hnc, w can st Θ h = { θ H 0(Ω) θ T Θ(T ), T T h }, and us th abov dgrs of frdom to construct a projction oprator C(T ) Θ(T ), and so an oprator θ θ I from C(Ω) H 0(Ω) Θ h. It is thn clar that for this oprator (36) holds. Bcaus of th continuity of th lmnts of Θ h and th nar continuity of th lmnts of W h, all th trms involving dg intgrals in (2) (3) vanish, and th mthod may b simply writtn (C ε(θ h ), ε(η)) + (γ h, h v P h η)) = (g, v), (η, v) Θ h W h, (57) ( h w h P h θ h, τ ) t 2 (γ h, τ ) = 0, τ Γ h. (58) Rmark 5. In th lowst ordr cas, k =, Γ h = Γ h is just th spac of picwis constants and P h th L 2 -projction into this spac, Θ h is th usual spac of conforming picwis linars augmntd by bubbls, and W h th usual spac on nonconforming picwis linars, so this mthod is xactly that of Arnold and Falk [5]. Rmark 5.2 Th choic of Θ h was mad in ordr to obtain (36) asily, rathr than to simplify th implmntation of th mthod. From th lattr point of viw, th altrnativ choic basd on Θ(T ) := P k (T ) + b T P k (which coincids with our choic only for k = ) sms natural, but w shall not considr this possibility hr. 7
Equation (58) may b writtn in strong form as t 2 γ h = h w h P h θ h, (59) which maks it asy to s that th mthod admits a uniqu solution. Morovr, (59) and (44) togthr giv t 2 γ δ = h w δ P h θ δ (60) (with th notation givn by (46)). W now turn to th rror analysis, assuming that k 3 (sinc th cas k = is handld in [5]). Th analysis procds along th sam lins as in th prvious sction. Following (48) and (49) w obtain α θ δ 2 Θ T + T 2 + (γ h γ, P h θ δ ), with T and T 2 still givn by (5) and (52). W can now us (60), add and subtract γ I, us (57) and (0) (both with η = 0), and th dfinition of j givn by (23) to obtain: (γ h γ, P h θ δ ) = t 2 (γ h γ, γ δ ) + (γ h γ, h w δ ) = t 2 γ δ 2 0 t 2 (γ I γ, γ δ ) + (γ h γ, h w δ ) =: t 2 γ δ 2 0 + T 3 + (γ h γ, h w δ ) = t 2 γ δ 2 0 + T 3 j(γ, w δ ), whr th first trm in th right-hand sid is ngativ, and will go to th lft in th final stimat, and T 3 is givn by (53). It rmains to bound th last trm. Lt γ M dnot th BDM intrpolant of γ dgr k (s,.g., [3] or [4]). Thus γ M L 0 k (T h ) satisfis: i) its normal componnt is continuous across intrlmnt boundaris, ii) div γ M = P k 2 div γ = P k 2 g whr P k 2 dnots th L 2 -projction onto L 0 k 2(T h ), and iii) γ γ M is orthogonal to L 0 0. Using th dfinition (23) of j(, ), thn (56), thn (2) and Grn formula in ach T, thn (60) and ii), thn iii) and ii), thn Cauchy-Schwarz, th arithmtic-gomtric man inquality and finally standard intrpola- 8
tion stimats, w gt j(γ, w δ ) = j(γ γ M, w δ ) = (γ γ M, h w δ ) + (div γ div γ M, w δ ) = (γ γ M, t 2 γ δ + P h θ δ ) + (g P k 2 g, w δ ) = t 2 (γ γ M, γ δ ) + (γ γ M, (I P 0 )P h θ δ ) + (g P k 2 g, (I P 0 )w δ ) 2ε t2 γ γ M 2 0 + ε 2 t2 γ δ 2 0 + γ γ M 0 (I P 0 )P h θ δ 0 + g P k 2 g 0 (I P 0 )w δ 0 2ε t2 h 2k γ 2 k + ε 2 t2 γ δ 2 0 + C h k γ k h θ δ + C h k g k h w δ W. Th rmaindr of th rror analysis follows th lins of th prvious sction, arriving finally to th rror bound θ θ h Θ + w w h W + t γ γ h Γ C h k ( θ k+,ω + w k+,ω + t γ k + γ k + g k ). Rfrncs [] Agmon, S. (965). Lcturs on lliptic boundary valu problms, Van Nostrand Mathmatical Studis, Princton, NJ. [2] Arnold, D. N. (982). An intrior pnalty finit lmnt mthod with discontinuous lmnts, SIAM J. Numr. Anal. 9, 742 760. [3] Arnold, D.N., Brzzi, F., Cockburn, B., and Marini, L.D. (2000). Discontinuous Galrkin mthods for lliptic problms, in: Discontinuous Galrkin mthods (Nwport, RI, 999), Lct. Nots Comput. Sci. Eng., Springr, Brlin, 89 0. [4] Arnold, D.N., Brzzi, F., Cockburn, B., and Marini, L.D. (2002). Unifid Analysis of Discontinuous Galrkin Mthods for Elliptic Problms, SIAM J. Numr. Anal. 39, 749 779. [5] Arnold, D.N. and Falk, R.S. (989). A uniformly accurat finit lmnt mthod for th Rissnr Mindlin plat, SIAM J. Numr. Anal. 26, 276 290. 9
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