Adrian Lew, Patrizio Neff, Deborah Sulsky, and Michael Ortiz

Transcription

1 AMRX Applid Mathmatics Rsarch Xprss 004, No. 3 Optimal V stimats for a Discontinuous Galrkin Mthod for Linar lasticity Adrian Lw, Patrizio Nff, Dborah Sulsky, and Michal Ortiz 1 Introduction Discontinuous Galrkin DG finit-lmnt mthods for scond- and fourth-ordr lliptic problms wr introducd about thr dcads ago. Ths mthods stm from th hybrid mthods dvlopd by Pian and his coworkr [5]. At th tim of thir introduction, DG mthods wr gnrally calld intrior pnalty mthods, and wr considrd by akr [4], Douglas Jr. [14], and Douglas Jr. and Dupont [15] for fourth-ordr problms, whr C 1 continuity was imposd on C 0 lmnts. For scond-ordr quations, Nitsch [1] appars to hav introducd th idas of imposing Dirichlt boundary conditions wakly and of adding stabilization trms to obtain optimal convrgnc rats. Th sam ida of pnalizing jumps along intrlmnt facs ld to th intrior pnalty mthods of Prcll and Whlr [4] and Whlr [30]. Mthods for a scond-ordr, nonlinar, parabolic quation appard in [1]. According to [3], intrst in DG mthods for solving lliptic problms wand bcaus thy wr nvr provn to b mor advantagous than traditional conforming lmnts. Th difficulty in idntifying optimal pnalty paramtrs and fficint solvrs may also hav contributd to th lack of intrst [3]. Rcntly, howvr, intrst has bn rkindld by dvlopmnts in DG mthods for convction-diffusion problms; s, for xampl, Cockburn and Shu [1, 13], Odn, abuška, and aumann [], Castillo, Cockburn, Rcivd Fbruary 004. Communicatd by Thomas Yizhao Hou.

2 74 Adrian Lw t al. Prugia, and Schötzau [9], and Houston, Schwab, and Süli [18], whr th scalar Poisson quation is analyzd. assi and Rbay [5] applid a similar tchniqu for th solution of th Navir-Stoks quations. rzzi, Manzini, Marini, Pitra, and Russo [7, 8] analyzd th mthod of assi and Rbay for stability and accuracy, as it applis to th scalar Poisson quation. Arnold, rzzi, Cockburn, and Marini [, 3] providd a common framwork for all of ths mthods and showd th intrconnctions by casting thm into th form of th local discontinuous Galrkin LDG mthod of Cockburn and Shu. W ar intrstd in a DG mthod for studying th mchanical bhavior of solids. In this papr, w analyz th linar lasticity problm, with an y toward a formulation for nonlinar lastic-plastic problms and cohsiv lmnts [3]. Thr ar svral bnfits of such an approach, including th potntial for fficint hp-adaptivity, for xampl, using adaptiv msh rfinmnt on mshs with hanging nods, and th prospct of rigorously handling problms with discontinuous displacmnts as aris in th study of fractur. Rivièr and Whlr [6] formulat and analyz a mthod for linar lasticity basd on a gnralization of th nonsymmtric intrior pnalty Galrkin NIPG mthod prsntd in [] for th diffusion quation. Th rsulting bilinar form is nonsymmtric. As an altrnativ, w follow th analysis of rzzi, Manzini, Marini, Pitra, and Russo [7, 8] quit closly in our gnralization from th scalar Poisson quation to th linar lasticity problm. In this cas, th bilinar form is symmtric. rror stimats for DG mthods ar usually obtaind in trms of msh-dpndnt norms. It is, a priori, not clar how to compar norms corrsponding to mshs of diffrnt siz. In this papr, w show that th traditional rror stimats xprssd in mshdpndnt norms can b usd to driv rror stimats in th msh-indpndnt D and V norms, liminating th ambiguity. Sction bgins with a statmnt of th problm and its formulation using th DG approach. A nw drivation of th quations is basd on a discrt variational principl for lasticity which naturally xtnds to finit dformations. Th variational approach lads to a formulation analogous to th on utilizd in [5]. Stabilization trms of th form considrd in [7, 8] ar addd to obtain a wll-posd discrt problm. In Sction 3, w show optimal convrgnc rats in a msh-dpndnt norm similar to th on usd by rzzi t al. This msh-dpndnt stimat is immdiatly strngthnd to a msh-indpndnt D stimat in Sction 3.. Th classical analysis of th quations of linar lasticity nds a global vrsion of Korn s first inquality to insur corcivnss of th bilinar form. In contrast to th standard approach, in Sction 3.3, w prov a gnralization of Korn s scond inquality on th lmnt lvl, which allows us to obtain an improvd msh-dpndnt stimat. Finally, in Sction 3.5, w show uniform convrgnc in th V norm, an optimal

3 Discontinuous Galrkin Mthod 75 msh-indpndnt stimat. Sinc th discrt solutions ar allowd to hav jumps in displacmnt but th classical solution is smooth, gradints can at most convrg in masur, and indd thy do. Formulation Th linar lasticity problm is dscribd by th following st of quations for a body R d, whr d =, 3: C s u = f in, u = ū on D, C s u n = T on N..1 Th body is assumd to b a boundd, polyhdral domain. Th function u : R d is th displacmnt, and C is th fourth-ordr lasticity tnsor with major and minor symmtris. In ordr to avoid tchnical difficultis that do not provid any additional insight, w tak C to b constant. W also assum that C is uniformly positiv dfinit, that is, c >0: γ C γ cγ γ. for all γ in th spac of d d symmtric tnsors, which implis that C is invrtibl on this spac. Th notation s u dnots th symmtric gradint of th displacmnt, s u = 1/ u+ u T. Th boundary of th domain,, is dcomposd into two disjoint sts, D and N. Th body is actd upon by body forcs, f : R d, and surfac tractions, T : N R d. Th displacmnt, ū : D R d, is prscribd on th part of th boundary indicatd by D..1 Strss-displacmnt formulation Th two-fild, strss-displacmnt formulation of th linar lasticity problm is σ C s u = 0 in, σ = f in, u = ū on D,.3 σ n = T on N.

4 76 Adrian Lw t al. Th first quation is th constitutiv quation that rlats th strss tnsor σ to th strain ε u = s u. Th scond quation xprsss forc quilibrium, and th final two quations giv th prscribd boundary conditions. Th problm dscribd by quation.3 has solutions u, σ with componnts in H m+1 and H m, rspctivly, for m 1, dpnding on th smoothnss of th data and th domain. Nominally, f L d. Th quations.3 ar th ulr-lagrang quations that rsult from taking fr variations of th Hllingr-Rissnr nrgy, I : H m+1 d H m d d R, whr 1 I[u, σ] = σ C 1 σ σ s u + f u + n σ u ū + T u. D N.4 Th discrt quations in th nxt sction ar drivd using a discrtization of this variational principl.. Th discrt schm A subdivision, T h of, is a finit numbr of sts, such that = Th. A subdivision, T h, is calld admissibl in th sns of [10, pag 38] if ach is closd and has nonmpty intrior, th intriors of th sts of T h ar pairwis disjoint, and th boundary,, of ach is Lipschitz continuous. W assum th family of admissibl subdivisions T h, with h 0, is quasi-uniform [6, pag 106] so that max { diam : T h } = h;.5 ρ >0: min { diam : T h } ρh, h >0,.6 whr is th largst ball containd in. Thrfor, it follows that thr xist positiv constants c and C such that ch d Ch d.7 for vry lmnt T h and vry h>0, whr is th masur of. In addition, w rquir all finit lmnts within th family of subdivisions to b affin quivalnt [6, pag 80] to a finit numbr of polyhdral rfrnc finit lmnts, ach with a finit numbr of facs. Hnc th rfrnc lmnts possss Lipschitz boundaris, th masur of ach fac of an arbitrary lmnt in T h is finit, and thr xists an uppr bound on th Lipschitz constant of th boundary for all lmnts in th family T h, indpndnt of h.

5 Discontinuous Galrkin Mthod 77 Morovr, with.5, w infr that thr xists a constant C>0such that h C,.8 for all h>0, and for any fac of any lmnt T h. vn though DG mthods can potntially b usd on mshs with hanging nods, w considr, for simplicity, only conforming mshs, so that a fac of an lmnt is ithr also a fac of anothr lmnt or part of. Wnot, howvr, that most of th thortical dvlopmnt dos not rly on this assumption. Considr a givn subdivision T h of. achlmnt T h has an orintabl boundary,, with unit, outward normal dnotd by n. Dfin th st of intrnal facs I h = { \ : T h },.9 th st of Dirichlt facs D h = { D : T h },.10 and th st of Numann facs N h = { N : T h }..11 Th st of all facs is dnotd by h = I h D h N h. Corrsponding to this st of facs, dfin th combind intrnal and xtrnal boundary to b Γ = h..1 Lt Ṽ = Π H 1 d b th spac of functions on whos rstriction to ach lmnt blongs to th Sobolv spac H 1 d. Thrfor, th tracs of functions in Ṽ blong to TΓ = Π Th L d. Functions in TΓ ar multivalud on Γ \ and singlvalud on. Th spac L Γ d can b idntifid with th subspac of TΓ consisting of functions for which th possibl multipl valus agr on all intrnal facs. Similarly, lt W = Π Th H 1 d d b th spac of functions on whos rstriction to ach lmnt blongs to th Sobolv spac H 1 d d. A tnsor τ W has d componnts. Th d tracs, th componnts of τ, ar dfind, and ach blongs to L. In particular, th linar combination of tracs τ n is in TΓ. 1 1 Th spac of strsss W could b takn to b largr; howvr, this is unncssary sinc w considr xact solutions u, σ in H d H 1 d d.

6 78 Adrian Lw t al. Nxt, w introduc two finit-lmnt spacs of scalar functions ovr an lmnt, Vh and W h, with V h W h. Ths lmntal spacs contain th polynomials and hav minimal smoothnss ovr th lmnt, P k Vh,W h H1, k 1, whr P k dnots th spac of polynomials of dgr at most k on. Th finit-lmnt spacs for th displacmnts, V h, and displacmnt gradints, W h, ar constructd so that ach componnt is in Vh or W h on th lmnt, V h = Π Th Vh d and W h = Π Th Wh d d. Consquntly, w hav V h Ṽ. W also assum that gradints of th displacmnt ar in th spac of displacmnt gradints, [Vh d ] Wh d d. Furthrmor, w rquir th lmntal finit-lmnt spacs to coincid ovr common facs. Mor prcisly, lt I h b th fac common to two lmnts, + and, thn {φ : φ Vh + } = {φ : φ Vh } and {φ : φ Wh + } = {φ : φ Wh }. This rquirmnt insurs that th trac of a is also th trac of a function in V function in Vh + W+ h with Wh s th spac of symmtric tnsors in W h. h W h, on. Lastly, w dnot W assum that th discrt spacs, V h and W h, ar finit dimnsional. Obsrv that th functions in both discrt spacs can b discontinuous across lmnt boundaris. Th conditions spcifid hr ar satisfid by many standard finit-lmnt spacs, such as thos constructd from Lagrang simplics of various dgrs and thos constructd with bilinar quadrilatrals or trilinar bricks. Rmark.1. Most of th proofs in this articl immdiatly gnraliz to th cas of isoparamtric lmnts, though som adjustmnt of th assumptions on th finitlmnt spacs might b rquird. In particular, th spcial tratmnt of Korn s inquality also applis to isoparamtric lmnts. W wish to formulat a discrtizd vrsion of.4 subordinat to th subdivision. To this nd, w dfin th avrag oprator, { } : TΓ L Γ d, and th jump oprator, [[ ]] : TΓ L Γ d. ach fac, I h, is shard by two lmnts, + and ; lt v ± = v ± for v Ṽ. Dfin th avrag, for I h, by {v} = 1 v + v +.13 and th jump by [[v]] = v v For D h, put {v} = v, [[v]] = v;.15

7 Discontinuous Galrkin Mthod 79 and for N h, assign {v} = v, [[v]] = In th squl, w choos an orintation, n, for ach fac I h, as th unit normal pointing toward +.For, n is th unit outward normal to. Forσ W, lt σ ± = σ ±.On I h, th avrag of th vctor σ n mans {σ n} = 1 σ + + σ n,.17 with n givn uniquly on th fac. Th dfinition of {σ n} on boundary facs, D h N h, is clar. Now, spcializ.4 to ach individual lmnt as follows: 1 I = σ 1 C 1 σ σ s u + f u + \ n σ u u xt + n σ u ū + T u, D N.18 whr u xt is th trac of u on th lmnts adjacnt to \.Th1/ factor in th scond trm accounts for th fact that for a givn fac, two adjacnt lmnts contribut to th potntial nrgy. A global discrt functional, I h : V h Wh s R, is dfind simply by summing ovr all lmntal contributions: I h = I..19 Th corrsponding ulr-lagrang quations that rsult from taking fr variations of I h ar δσ C 1 σ δσ s u + {n δσ} [[u]] n δσ ū = 0, Γ D σ s δu + f δu + {n σ} [[δu]] + T δu = 0. Γ N.0

8 80 Adrian Lw t al. Thus, w obtain th gnral problm which is to find u h V h and σ h Wh s such that γh C 1 σ h γ h s u h + { n γh } [ uh ] = = D n γ h ū γ h W s h; σ h s v h Γ { n σh } [ ] f v h + T v h v h V h. N Γ.1. quations.1 and. constitut th flux form of th discrt problm. Nxt, w dfin th lifting oprator Rū : L Γ d Wh s by Rūv γ = {n γ} v + n γ ū γ Wh. s.3 Γ D This oprator will now b usd to driv th primal form [3] of th discrtization, whr a singl quation is obtaind by liminating σ h btwn.1 and.. In trms of.3, quation.1 is th sam as γh C 1 [ ] σ h γ h s u h Rū uh γh = 0 γh Wh. s.4 Sinc w rquir th lmntal finit-lmnt spacs to satisfy [Vh d ] Wh d d, this quation allows us to idntify [ ] σ h = σ h uh = C s u h + C Rū uh in W s h..5 This constitutiv quation for th discrt strss can b viwd as a strss-strain rlation whr th strain involvs th usual dpndnc on th displacmnt gradint, plus a linar contribution that ariss from jumps in displacmnt. Nxt, tak γ h = C s v h in quation.1 to gt s v h σ h s v h C s u h + { n C s v h } [ uh ] = D Γ n C s v h ū..6

9 Finally, substitut quation. toobtain = Discontinuous Galrkin Mthod 81 { } [ ] { } [ ] s v h C s u h n C s v h uh + n σh Γ f v h + T v h n C.7 s v h ū. N D If u h,σ h V h Wh s solvs.1 and., thn u h solvs.7, with σ h = σ h u h givn by.5. quation.7 is calld th primal formulation. Rcall th dfinition of Rū in.3 and introduc th notation R = R 0.Using.3 and.5, th primal form.7 can also b writtn as = s v h + R [ ] C s u h + R [ uh ] f v h + T v h n C s v h + R [.8 ] ū. N D W rmark that our physically basd drivation of this quation, obtaind by discrtizing th variational principl, producs an analogous discrtization to that usd by assi and Rbay in [5, 8]. rzzi, Manzini, Marini, Pitra, and Russo [7, 8] propos a stabilizing trm for th scalar cas which naturally xtnds to linar lasticity. Th stabilization is givn in trms of r,ū : L Γ d Wh s. Dfin r,ū for I h, r,ū v γ = {n γ} v γ Wh, s.9 whil for D h, r,ū v γ = {n γ} v + n γ ū γ Wh, s.30 and for N h, r,ū = 0. As bfor, st r = r,0.notthatr,ū v vanishs outsid th union of lmnts containing, and that for any lmnt T h, Rūv = r,ū v.31 on. Th stabilizing trm is β r,ū[[u h ]] C r [[v h ]], with β>0th stabilization paramtr. Th rsulting primal form with th stabilizing trm is s v h +R [ ] C s u h +R [ ] [ [ ] uh +β r uh ] C r = h f v h + N T v h D n C s v h + R [ ] + βr [ ] ū..3

10 8 Adrian Lw t al. Th form.3, which drivs dirctly from th variational principl, is stabl for any β > 0.InSction 3, w analyz in dtail a modification proposd by rzzi, Manzini, Marini, Pitra, and Russo [7, 8] that omits th quadratic trm in R, making th mthod stabl for β>n, whr N is th maximum numbr of facs in an lmnt of th subdivision. Th advantag of dropping this quadratic trm is that th sparsity of thstiffnss matrix is incrasd. Th analysis of th proposd mthod rlis on lliptic rgularity, so w rstrict it to Dirichlt boundary conditions on th ntir boundary,. Thus, N =, N h =, and, without loss of gnrality, ū = 0 on. Accordingly, th complt discrt problm statmnt, with ths modifications, is to find u h V h such that a h uh,v h = f v h v h V h,.33 whr th bilinar form a h is givn by a h uh,v h = s v h C s u h + s v h C R [ ] [ ] uh + R C s u h + β h [ ] [ ] r uh C r..34 Rmark.. Th bilinar form.34 can b writtn altrnativly using.3 as a h uh,v h = s v h C s u h Γ + β { n C s v h } [ uh ] + { n C s u h } [ ] h [ ] [ ] r uh C r..35 Ths two forms ar quivalnt for u h, v h V h..3 Notation In Sction 3, a convrgnc proof will b givn for d = and 3, simultanously. In th proofs, th lttr C indicats a gnric constant whos valu can chang in ach occurrnc. W also mploy th standard notation p,ω to dnot th usual norm on H p Ω,

11 Discontinuous Galrkin Mthod 83 and p,ω to dnot th H p Ω sminorm, whras dnots th uclidan norm for vctors or tnsors. Whn othr standard norms ar usd, thy will b indicatd xplicitly with a subscript; for xampl, L1 Ω indicats th L 1 Ω-norm..4 Summary of th thortical rsults Th convrgnc proof utilizs two rlvant msh-dpndnt norms on V = H 1 0 d +V h givn by v s = v = s v + 0, v 0, + h h r [[v]] 0,, v V,.36 r [[v]] 0,, v V..37 Proposition 3.4 stablishs that s is a norm on V.Alsonotthat v s v, v V,.38 which shows that is also a norm on V. Although on might xpct th scond trm in th dfinitions of norms.36 and.37 to act as an L -lik contribution, w can only assrt that ths ar sminorms on Ṽ. In th cas of th scalar Poisson quation [, 3, 7, 8, 9, 18, ], thr is no nd to distinguish btwn th norms.36 and.37. Following th idas in [7, 8], it is straightforward to obtain bounddnss and corcivity of th bilinar form a h with rspct to th msh-dpndnt norm, s Proposition 3.5, which lads to convrgnc of th discrt solutions in th s -norm and in L Thorms 3.14 and Th convrgnc in th s -norm is sufficint for a mshindpndnt D stimat Thorm 3.19; howvr, th s norm dos not provid control ovr th antisymmtric part of th displacmnt gradint. If th displacmnts ar in H 1 0, th quivalnc of th two norms, s and, rlis on Korn s first inquality; for nonconforming lmnts, Korn s inquality may not b valid [17]. ordr to obtain convrgnc in th norm,, Thorm 3.3, w prov a gnralizd vrsion of Korn s scond inquality for th subdivision, Corollary 3.. Th proof of this inquality rlis on obsrvations about how Korn s inquality for an lmnt bhavs undr distortion Thorm 3.0 and scaling Thorm 3.1. Finally, Thorm 3.6 shows that th msh-dpndnt norm,, stimats th V norm, and as a consqunc, Corollary 3.7, w obtain convrgnc in V, an optimal msh-indpndnt rsult.

12 84 Adrian Lw t al. 3 Thortical rsults 3.1 Convrgnc in th msh-dpndnt symmtric norm In this sction, w obtain th convrgnc of th discrtizd solutions in th mshdpndnt norm s. Our analysis follows th outlin in [7, 8] for th two-dimnsional Poisson quation, but th dtails of most proofs diffr. Th first thr lmmas charactriz proprtis of th jumps. In th subsqunt proposition, our analysis starts by stablishing that s is in fact a norm on V. Lmma 3.1 xtnsion of tracs. Lt b a fac of an lmnt T h. For any φ in th trac spac, T = {φ L d d : φ = γ,γ Wh d d }, thr xists P φ Wh d d such that P φ = φ. Morovr, for all φ T, C >0: P φ 0, Ch 1/ φ 0, 3.1 for all h>0and for all T h. Proof. First xamin a rfrnc lmnt. Lt ê Ê b a fac of on of th rfrnc lmnts, Ê, and lt φ Tê. Thr xists C>0such that sup inf γh <C. 3. 0,Ê φ T ê, φ 0,ê =1 γ h WÊh d d,γ h =φ Sinc γ h WÊh d d is a linar combination of basis functions on Ê, γ h 0, Ê is a quadratic form in a finit-dimnsional spac. Thrfor, thr is a minimizr, Pêφ, of γ h subjct to th linar constraint γ h = φ Tê, which dpnds continuously on φ. Thus, 0,Ê Pêφ is boundd on th compact st φ 0,ê = 1, and 3. follows. Nxt, not that Pêλφ = λpêφ for λ R, which implis Pêφ 0,Ê C φ 0,ê 3.3 for all φ Tê. Sinc th numbr of rfrnc lmnts is finit, as is th numbr of facs pr lmnt, w can choos C in 3.3 indpndnt of th rfrnc lmnt and th fac. Now, lt b any lmnt in th family of subdivisions T h, and lt b any on of its facs. Lt F b th affin transformation such that = FÊ for on of th rfrnc lmnts Ê, and lt ê b th corrsponding fac in th rfrnc lmnt, = Fê. Givn

13 Discontinuous Galrkin Mthod 85 φ T, th dfinition of affin quivalnc implis φ = φ F Tê. Dfin P φ = Pê φ F 1 Wh d d, and not P φ = φ.thn, us 3.3 and F h/ ρ s,.g.,[10, pag 10], whr ρ is th diamtr of th largst ball containd in Ê, to obtain P φ = dt F Pê φ Ê C dt F φ ê C dt F φ dt F 1 F C F φ C h ρ φ. 3.4 Th lmma follows. Lmma 3. trac inquality for r. Thr xists a constant C>0, indpndnt of th fac h and of h, such that r v 0, Ch 1/ r v 0, 3.5 for all v L d. Proof. Th inquality 3.5 is actually a statmnt about tnsors γ Wh d d, whr γ = r v. Th proof follows a scaling argumnt. Lt ê Ê b a fac of on of th rfrnc lmnts, Ê.Thn, thr xists a constant C>0such that γ 0,ê C γ 0, Ê 3.6 for all γ WÊh d d. Inquality 3.6 is a dirct consqunc of th continuity of th trac in WÊh H1 Ê s,.g., [6, pag 37] and th fact that in a finit-dimnsional spac, all norms ar quivalnt. Sinc thr ar a finit numbr of rfrnc lmnts, ach with a finit numbr of facs, th constant C can b chosn indpndnt of th rfrnc lmnt and of its fac. Now, considr γ Wh d d, whr is an lmnt affin quivalnt to Ê. Thn, thr xists an affin mapping F such that = FÊ, and γ WÊh d d such that γ = γ F 1. Not that γ 0, = γ γ = dt F γ γ = dt F γ 0,Ê, Ê γ 0, = γ γ = F 1 n dt F γ γ F dt F γ, 0,ê ê

14 86 Adrian Lw t al. whr n is th unit outward normal to ê. Thrfor,3.6 and 3.7 combin to yild γ 0, C F 1 1/ γ 0, Cĥ1/ ρ 1/ h 1/ γ 0,. 3.8 Th last part of th bound uss th fact that F 1 ĥ/diam ĥ/ρh s,.g.,[10, pag 10]. Lmma 3.3 jump bound. Thr xist two positiv constants C 1 and C, indpndnt of th fac h and of h, such that [ ] 0, C 1 h 1/ [ ] r 0, v h V h ; 3.9 [ ] r 0, C h 1/ [ ] 0, v h V h Proof. Lt b a fac of lmnt.givn[[v h ]] L d, lt γ h L d d b such that γ h n = [[v h]]. Not that it is possibl to choos γ h so that γ h C [[v h]]. Forth tnsor γ h dfind only on, construct an xtnsion to th lmnt, γ h = P γ h, as in Lmma 3.1. Tak γ h Wh s to b γ h = P γ h on, γ h = 0 lswhr, and v = [[v h ]] in quation.9 to gt 1 [ ] = 1 [ ] [ ] 0, [ ] r P γ h r [ ] 0, P γ h 0, 3.11 Ch 1/ r [ ] 0, [ ] 0,. In th nontrivial cas in which [[v h ]] 0, 0, inquality 3.9 follows from 3.11 by dividing through by [[v h ]] 0,. To prov 3.10, tak γ = r [[v h ]] and v = [[v h ]] in quation.9 to gt r [ ] 0, = { n r [ ]} [ ] [[v]] 0, { r [ ]} 0, 3.1 C h 1/ [ ] 0, r [ ] 0,. W hav usd th linarity of r and 3.5 in th last stp. Th rsult,3.10, follows.

15 Discontinuous Galrkin Mthod 87 Proposition 3.4 symmtric norm. Lt v h V = H 1 0 d + V h.thn s : V R as dfind in.36 is a norm on V. Proof. It is immdiat that λv s = λ v s for all λ R, and that th triangl inquality holds sinc r is linar. W show that v s = 0 implis v = 0 in V. Noticthat v s = 0 if and only if s v 0, = 0 for all T h and r [[v]] 0, = 0 for all h.ltv = v 1 + v V, with v 1 H 1 0 d and v V h.ylmma 3.3, w hav that [[v ]] 0, Ch 1/ r [[v ]] 0,. Thrfor, [[v ]] 0, = 0. Sinc also [[v 1 ]] 0, = 0, w hav [[v]] 0, = 0. So v H 1 0 d by [7, Thorm 1.3]. Korn s first inquality for homognous boundary data applid to v H 1 0 d thn shows that v = 0. Nxt, w show that th bilinar form.34 is continuous and corciv with rspct to th norm, s. Th proofs follow [7, 8] almost xactly. Proposition 3.5 continuity and corcivity of th bilinar form. Lt N b a bound on th numbr of facs in an lmnt. Thn, thr xists a constant M>0, indpndnt of h, such that i a h u h,v h M u h s v h s for all u h,v h V. Morovr, for β>n, thr xists a constant µ>0, indpndnt of h, such that ii a h u h,u h µ u h s for all u h V. Proof. W first prov th following inquality, a consqunc of quation.31: R [ ] 0, N [ ] r , W hav R [ ] 0, = r [ ] N 1 [ ] r r [ ] r [ ] r [ ] + [ ] r 0, 0, r [ ] 0,. 3.14

16 88 Adrian Lw t al. Nxt, th continuity of th bilinar form.34 follows from stimating ach trm. s u h C s v h C s u h 0, s v h 0,, s u h C R [ ] C s u h 0, R [ ] 0, C s u h 0, [ r [ uh ] C r [ ] C N r [ ] 0, ] 1/, r [ uh ] 0, r [ ] 0, Adding ach trm ovr all lmnts and using th Cauchy-Schwartz inquality yilds i. Th constant M dpnds on C, N, and β, but is indpndnt of h. Now w show corcivity,ii. To simplify th notation, dfin γ 0,,C = γ C γ γ Wh. s 3.16 Du to 3.13, w gt a h uh,u h = s u h 0,,C + s u h C R [ uh ] + β r [ uh ] 1 ε s u h 1 R [ ] uh 0,,C ε + β [ ] r 0,,C uh 1 ε s u h + β N [ ] r uh, 0,,C ε 0,,C 0,,C 0,,C 3.17 whr w usd th standard inquality, ab εa + b /ε, for all ε > 0.Anyβ > N guarants that β N /ε >0whnvr N /β<ε<1. Sinc ach trm is positiv, w can invok. to dduc ii with µ = cβ N /ε >0. Rmark 3.6. As suggstd in [7, 8], following th sam stps as in th prvious proof stablishs th continuity and corcivity of th bilinar form givn by th lft-hand sid of quation.3, but for any β>0.

17 Discontinuous Galrkin Mthod 89 Th following lmma is a prliminary to proving convrgnc of th discrt displacmnt, first in s, and subsquntly in L. Lmma 3.7. Lt u H 1 d with C u L d, and lt v h V h, thn n C u v h = h [ ] {n C u} Proof. Th assumd rgularity of u implis that n C u is continuous across intrlmnt boundaris.g.,[7, Thorm 1.3]; that is, 0 = n C u C u + on any fac in I h. Thrfor = n C u v h I h n C u + v + h + n C u v h + = 1 n C u + + n C u v + h I h + 1 n C u + + n C u v h + = h {n C u} [ ]. D h D h n C u v h n C u v h 3.19 Th nxt componnt of th convrgnc proof is a bound on th approximation rror u u I s whn u I is a suitabl intrpolant of th xact solution u. Arnold, rzzi, Cockburn, and Marini [3] not that discontinuous intrpolants can b mployd if thy satisfy a local approximation proprty summarizd in th nxt thorm. Thorm 3.8 local intrpolation-rror stimat. For v H k+1 d, lt v I b th P k - intrpolant of v on T h. Thr xists C>0, indpndnt of T h and thrfor of h, such that v vi q, Ch k+1 q v k+1,, k+ 1 q 0, 3.0 providd P k V h Hq. Proof. Th proof is givn by Ciarlt [10, Thorm 3.1.5].

18 90 Adrian Lw t al. Thorm 3.9 intrpolation-rror stimat. Lt u H m d for som m such that m k + 1, and lt u I V h b th P k -intrpolant of u ovr ach lmnt in T h. Thn th following intrpolation inquality holds: u ui s Ch m 1 u m,, 3.1 whr C>0is a constant dpnding only on d, m, and th uppr bound on th Lipschitz constant of th boundary for vry lmnt T h, but not on h or th function u. Proof. From th prvious thorm, w hav u ui q, Ch m q u m,, m q. 3. In addition, th trac inquality [16, pag 133] togthr with a scaling argumnt givs u 0, C h 1 u 0, + h u 1, u H 1, 3.3 whr th constant C dpnds only on th Lipschitz constant of th boundary of th lmnt, and can b chosn to b th sam for all lmnts in th family of subdivisions T h undr considration. Following [3], th intrpolation inquality 3.1 is stablishd using th inquality 3.3, th bound 3., and th invrs inquality Starting from th dfinition of s, th thorm is obtaind as follows: u ui s = C s u ui 0, + h u ui 0, + u ui 1, + h m u m, Ch m u m,. h h Ch 1 r [ u ui ] 0, r [ u ui ] 0, [ ] u ui 0, 3.4 Again, th constant C is positiv and dpnds only on d, m, and th uppr bound on th Lipschitz constant of th boundary for vry lmnt T h, but not on h or th function u.

19 Discontinuous Galrkin Mthod 91 Th last ingrdint for th convrgnc proof is an analysis of th consistncy rror in th bilinar form.34 for functions in V h. Rmark Th bilinar form.35, which coincids with.34 on V h, is consistnt but continuity dos not hold with rspct to th norm s for functions in V h.howvr, it can b shown that.34 is continuous with rspct to a diffrnt norm on a diffrnt function spac in which som additional rgularity is rqustd [3]. Thorm 3.11 bound on consistncy rror. Lt u b th xact solution to.1, with u H m d for som m such that m k + 1.Thn a h u, f v h Ch m 1 u m, s 3.5 for all v h V h. In addition, if v h is continuous, thn a h u, = f v h. 3.6 Proof. Rcall that [[u]] = 0, and us th dfinition of th bilinar form a h,,.34, intgration by parts, and application of Lmma 3.7 to obtain a h u, f v h = s u C s v h + T h = u C v h + s u C R [ ] T h = = h = h C u + f + [ ] { n C s u } + s u C R [ ] f v h f v h [ ] { n C s u ui } + n C u v h + s u C R [ ] s u C R [ ] s u ui C R [ ]. 3.7 Th last lin coms from th dfinition,.3, of Rv with γ h = C u I and v = [[v h ]]. W hav also usd th symmtry of C to intrchang gradints and symmtric gradints. Not that if v h is continuous, thn [[v h ]] = 0, and th bilinar form is consistnt.

20 9 Adrian Lw t al. To complt th proof, w bound th consistncy rror as follows. Using th trac inquality,3.3, for u u I H, and 3.9, w hav h [ ] { n C u ui } C C C Ch m 1 u m, s. h 1/ r [ ] 0, u u I 0, h 1/ [ ] r 0, h 1 u ui + h 1, u ui 1/, [ ] r 0, u u 1, I + h u u, I 3.8 Morovr, s u ui C R C C s u ui 0, R v h 0, u u I 1, Ch m 1 u m, s. r 0, 3.9 Corollary 3.1. Lt u b th xact solution to.1, with u H m d for som m such that m k + 1, and lt u h b th solution of.33.thn a h u uh,v h Ch m 1 u m, v h s 3.30 for all v h V h. In addition, if v h is continuous, thn a h u uh,v h = Proof. Not that a u u h,v h = ah u, ah uh,v h = a h u, f v h 3.3 for all v h V h. Th conclusion follows from th prvious thorm.

21 Discontinuous Galrkin Mthod 93 Rmark quation 3.31 xprsss a Galrkin orthogonality which follows from consistncy. At this point w hav gathrd all th ncssary ingrdints to prov convrgnc of th discrt solutions in s and 0,, which is th contnt of th nxt two thorms. Thorm 3.14 convrgnc in th msh-dpndnt norm s. Lt u b th xact solution to.1, with u H m d for som m such that m k + 1, and lt u h b th solution of.33, thn th following stimat holds: u u h s Ch m 1 u m,, 3.33 whr C is a positiv constant indpndnt of h. Proof. From Proposition 3.5, w hav µ ui u h a s h ui u h,u I u h = a h ui u, u I u h + ah u uh,u I u h M ui u h s ui u s + ah u uh,u I u h Th last trm in th abov quation is du to th gnral lack of Galrkin orthogonality, but it is appropriatly boundd using Corollary 3.1. Thrfor, µ ui u h s M ui u h s ui u s + Ch m 1 u m, ui u h s Th conclusion 3.33 follows using 3.1 abov. Thorm 3.15 convrgnc in L. Lt u b th xact solution of.1, with u H m d for som m such that m k + 1, and lt u h b th solution of.33, thn th following stimat holds: u uh 0, Ch m u m, Proof. Th proof follows a standard duality argumnt. Considr th adjoint problm. Find w H d such that C s w = u u h in, w = 0 on. 3.37

22 94 Adrian Lw t al. Sinc u u h L d, th following standard lliptic rgularity stimat holds s,.g.,[19]: w, C u u h 0, 3.38 for som constant C>0.Forw H, lt w I V h b th continuous, picwis linar intrpolant of w ovr ach lmnt. Apply 3.5 to w, with v h = u u h and m =, or a h w, u uh u uh Ch w, u uh s 3.39 u u h 0, a h w, u uh + Ch w, u u h s Sinc w I is continuous, Corollary 3.1 shows a h w I,u u h = 0. This fact and continuity of th bilinar form, Proposition 3.5, allow us to conclud that u uh 0, ah w wi,u u h + Ch w, u uh s M w wi s u uh s + Ch w, u uh s 3.41 Ch w, u uh s, whr w hav usd Thorm 3.9 for th intrpolation rror stimat w w I s. Th conclusion of th proof follows from 3.38 and Thorm Corollary 3.16 convrgnc of th strss in L. Lt σ b th xact solution with componnts in H m 1 for som m such that m k + 1, and lt σ h b givn by.5, thn th following stimat holds: σ σh 0, Ch m 1 u m,. 3.4 Proof. For th xact solution, th displacmnt is continuous, [[u]] = 0, which implis R[[u]] = 0. So w can writ σ = C s u + R[[u]]. Thrfor, σ σ h = C s u uh + C R [ u uh ]. 3.43

23 Discontinuous Galrkin Mthod 95 It follows that σ σh 0, = = C σ σh 0, C s u uh + C R [ u uh ] 0, s u uh + N 0, C u uh s Chm u m,. r [ u uh ] 0, 3.44 Not that this corollary givs L convrgnc of th strss, vn though no such rsult holds for th strain. This discrpancy is possibl bcaus th discrt strss is givn by.5, and is not, in gnral, proportional to th strain. Rmark Again, as suggstd in [7, 8], it can also b provd that th sam rror stimats hold for th problm dirctly drivd from th variational principl, quation Th natural suboptimal but msh-indpndnt D-stimat Possibl discontinuitis in th displacmnt across lmnt boundaris naturally lad to sking rror stimats in D, th spac of boundd dformations. This spac is dfind as th st of functions u L 1 whos symmtric part of th distributional drivativ Du, Du = 1/Du + Du T, is a matrix-valud boundd Radon masur. For a function u D, lt Du dnot th total symmtric variation masur of Du. A gnral Poincaré-typ stimat for D-functions holds in th following form. Thorm 3.18 Poincaré inquality for D. Lt R d b a boundd domain with Lipschitz boundary. Thn thr xists C>0such that for all u D, u = 0, u L1 C Du, 3.45 whr u dnots th gnralizd trac. Proof. Th proof is givn by Tmam [9, Rmark II..5, pag 189].

24 96 Adrian Lw t al. Thorm 3.19 natural D stimat. Thr xists C>0such that for all u V, u D C u s, 3.46 withc indpndnt of h. Proof. Rcall th dfinition of th D norm u D = u L 1 + Du, 3.47 whr { Du = sup u Ψ T + Ψ : Ψ C 1 0, R d d }, Ψ L Th proof continus mutatis mutandis as in Thorm 3.6. Using th stimat 3.46 for th diffrnc u u h togthr with Thorm 3.9 shows that convrgnc of th mthod is immdiatly strngthnd from th s -norm to a msh-indpndnt stimat in th spac D. It is clar that any optimal stimat in th symmtric norm, drivd undr lss smoothnss assumptions on th undrlying continuous problm [6], translats into a corrsponding optimal msh-indpndnt D stimat. It is worth rmarking that th drivation of th D stimat dos not mak us of Thorm 3.15 that additionally stablishs convrgnc of th discrt solutions in L. Th occurrnc of th spac D is, strictly spaking, an artifact of th linarizd tratmnt, whr only th symmtrizd infinitsimal strains ε u appar. Sinc this D stimat dos not control th antisymmtric part of th displacmnt gradint, w ar intrstd in obtaining convrgnc in th spac V. Howvr, sinc V is strictly smallr than D, thr is no obvious way to procd dirctly from th D stimat to a V stimat. Instad, w will first strngthn Thorm 3.14 to th -norm. Not that for a givn msh siz h>0, givn th finit dimnsionality of V h and th fact that both and s ar norms in V h, w hav, for u h V h, u h D u h V C u h ch u h s, 3.49

25 Discontinuous Galrkin Mthod 97 whr th stimat u h V C u h is obtaind in Thorm 3.6. Howvr, ch may not b boundd from blow away from zro for all h>0. Th failur to obtain a mshindpndnt stimat btwn u h s and u h is a manifstation of th possibl lack of a discrt Korn s first inquality for nonconforming mshs [17]. In ordr to obtain convrgnc in th -norm, followd by a V-stimat, and thn convrgnc in V, w first stablish a gnralizd vrsion of Korn s scond inquality at th lmnt lvl. 3.3 Korn s scond inquality for th subdivision In this sction, w invstigat an analog to Korn s scond inquality at th lmnt lvl, indpndnt of th lmnt shap and siz. Th drivation of this inquality rlis havily on how Korn s scond inquality scals undr uniform contractions. W st SLd, R = {X R d d dt X = 1}. Thorm 3.0 Korn s scond inquality undr distortion. Assum that Ω R d is a boundd rfrnc domain with Lipschitz boundary Ω and lt M = {X SLd, R : X K}, for som K>0.ForF M dfin Ω ξ = FΩ. Thn thr xists C>0such that for all F M, u H 1 Ω ξ, ξ u T + ξ u 0,Ω ξ + u 0,Ω ξ C u 1,Ω ξ Proof. W first translat th statmnt to th fixd rfrnc domain Ω.Thaffin transformation ξ = Fx togthr with th dfinition uξ = ufx = ũx and dt F = 1 lads to ξ u T + ξ u + u = Ω ξ Ω F T ũ T + ũf 1 + ũ W procd by contradiction. Assum, without loss of gnrality, that thr xists a squnc {ũ n } H 1 Ω with ũ n 1,Ω = 1 and a squnc F n M such that F T n ũ T n + ũ n F 1 n + ũn 1 ũn 0,Ω 0,Ω n = 1 1,Ω n. 3.5 Sinc F n is boundd, w may xtract a subsqunc which convrgs strongly to F M by olzano-wirstrass. It is radily sn by continuity and th bounddnss of ũ n that F T ũ T n + ũ n F 1 0,Ω + ũn ,Ω

26 98 Adrian Lw t al. Thus ũ n is a minimizing squnc. For fixd F, th quadratic xprssion is uniformly positiv gnralizd Korn s scond inquality, s [0]such that F T ũ T n + ũ n F 1 0,Ω + ũ n 0,Ω C F ũ n 1,Ω for som C>0, contradicting ũ n 1,Ω = Thorm 3.1 Korn s scond inquality undr scaling. Lt Ω R d b a boundd domain with Lipschitz boundary Ω and, without loss of gnrality, Ω = 1. Considr th scald domain Ω h = {hx : x Ω}, h>0. Thn thr xists CΩ >0such that for all u H 1 Ω h, u T + u 1 + 0,Ω h Ωh /d u 0,Ω h CΩ u 1 0,Ω h + Ωh /d u 0,Ω h, 3.55 whr th constant CΩ is indpndnt of h > 0 and coincids with th constant in Korn s scond inquality for Ω. Proof. Lt ũ H 1 Ω. From Korn s scond inquality s,.g.,[0], w gt ũ T + ũ 0,Ω + ũ 0,Ω CΩ ũ 0,Ω + ũ 0,Ω xprssing vry trm with rspct to th down-scald Ω h, whr ũx = uhx, and noticing that Ω h = h d, w gt 1 h d u T + u + 1 0,Ω h h d u 0,Ω h 1 CΩ h d u 0,Ω h + 1 h d u 0,Ω h, 3.57 from which w dduc th rquird rsult. Not that CΩ is just th constant in Korn s scond inquality. Corollary 3. uniformity in T h. Lt Ê b th rfrnc lmnt for an lmnt T h as dfind in Sction. Without loss of gnrality, tak Ê = 1. Thn thr xists C>0such that for all T h, u H 1, u T + u 0, + 1 /d u 0, C u 0, + 1 /d u 0,. 3.58

27 Discontinuous Galrkin Mthod 99 Proof. Lt F b an affin transformation such that = FÊ. Dcompos F = F V F into its isochoric and volumtric part, whr F V = dt F 1/d I, I is th scond-ordr idntity tnsor, and F = F/dt F 1/d.Notthat = dt F.Using[11, Thorm 3.1.3, pag 10] and th quasi-uniformity of th subdivision, w hav that F = F dt F 1/d h ρ 1, /d C ρ whr ρ is th diamtr of th largst ball containd in Ê and C is indpndnt of. Thrfor, by Thorm 3.0, w can stat Korn s scond inquality for ach domain FÊ in th subdivision with th sam constant C>0. Th corollary thn follows from Thorm Convrgnc in W can now obtain convrgnc of th squnc of discrt solutions in th mshdpndnt norm using our gnralizd Korn s scond inquality for th subdivision. V b a s- Thorm 3.3 convrgnc in th msh-dpndnt norm. Lt v h qunc such that v h s Ch m 1 and v h 0, Ch m for h 0.Thn Ch m for som C>0indpndnt of h. Proof. Us Corollary 3. and sum ovr th lmnts to obtain th stimat v T h + v h + 1 C 0, + 1 /d 0, 0, /d 0, 3.61 which, in light of quation.7, can b waknd to v T h + v h + 1 0, h C + 1 0, 0, h, 3.6 0,

28 100 Adrian Lw t al. whr C is indpndnt of h>0. Without loss of gnrality, assum 0<C 1. Adding th spcific jump contribution ovr th facs of ach lmnt shows that or v T h + v h 0, + 1 h v h 0, + C v h s + 1 h h v h 0, + 1 h v h 0, + v h C v 0, h 1 + whr, again, C>0is indpndnt of h>0. Thus r [ ] 0, h h 3.63 [ ] r 0, v h, , + 1 s h C 1 0, + h C 0, Using th convrgnc of v h and quation 3.65, w obtain C h m + 1 h hm = Ch m, 3.66 which complts th thorm. Rmark 3.4. As is vidnt from th statmnt of Thorm 3.3, th convrgnc in can only b shown for squncs convrging in both s and L with spcific rats in h, which w hav stablishd for v h = u u h undr appropriat hypothss. In gnral, for solutions of th continuous problm with lss rgularity, on might not hav such knowldg. 3.5 Convrgnc in V W prov that th msh-dpndnt norm stimats th V norm on V = V h +H 1 0 d and as a rsult, w obtain convrgnc in V. Rcall that V is th spac of functions u L 1 such that th distributional drivativ Du is a matrix-valud boundd Radon masur. For a function u V, Du dnots th total variation masur of Du. A gnral Poincaré-typ stimat for V-functions holds in th following form.

29 Discontinuous Galrkin Mthod 101 Thorm 3.5 Poincaré inquality for V. Thr xists C > 0 such that for all u VR d, u L d/d 1 R d C Du R d Proof. vans and Garipy [16, Thorm 1, pag 189]. Thorm 3.6 natural V stimat. Thr xists C>0such that for all u V, u V C u, 3.68 withc indpndnt of h. Proof. Rcall th dfinition of th V norm u V = u L1 + Du, 3.69 whr { Du = sup u Ψ : Ψ C 1 0, R d d }, Ψ L First obsrv that u Ψ = u Ψ = = = Ψ u n Ψ u n Ψ [[u]] h T h Ψ u Ψ u Ψ u ach trm in th two sums may b stimatd individually by sup Ψ L 1 sup Ψ L 1 [ [ n Ψ [[u]] ] ] Ψ u [[u]] [[u]] [[u]] [[u]] L1, u u u u L 1, 3.7

30 10 Adrian Lw t al. which yilds th prliminary stimat Du h [[u]] + L1 u L Applying Höldr s inquality to ach trm in th sum givs Du [[u]] 0, + h 1/ 1/ u 0, Taking th squar of both sids and using Young s inquality lads to Du [ h 1/ ] [ ] [[u]] 0, + 1/ u 0, Now w us th Cauchy-Schwartz inquality for th sums in th brackts to show Du 1/ 1/ h h [[u]] 0, 1/ + 1/ 1/ 1/ u 0, [[u]] + u 0, 0, h h 3.76 which, by Lmma 3.3, implis Du C h [ Ch h h h h r [[u]] 0, ] r [[u]] 0, + + u 0, u 0,, 3.77

31 Discontinuous Galrkin Mthod 103 withc indpndnt of h. From.8, Du C C [ h [ h C u. h ] r [[u]] 0, r [[u]] 0, + + u 0, u 0, ] 3.78 y hypothsis, u V h + H 1 0 d ; this implis u V sinc u L and Du is boundd by u. W may xtnd u toafunctionũ on all of R d by stting u to zro outsid of. From [16, Thorm 1, pag 183], w hav th quivalnc Dũ R d = Du Thus, by applying th Poincaré inquality for V, Thorm 3.5, w obtain u L d/d 1 = ũ L d/d 1 R d C Dũ R d = C Du C u 3.80 withc >0indpndnt of h. This stimat is ncssary sinc th msh-dpndnt norm dos not contain a contribution of th form u L. Corollary 3.7 optimal msh-indpndnt stimat. Lt v h V b a squnc such that v h s Ch m 1 and v h 0, Ch m for h 0.Thn V Ch m Proof. Apply Thorm 3.3 togthr with Thorm Final rmarks Optimal convrgnc of a stabilizd DG mthod for linar lasticity with Dirichlt boundary conditions has bn stablishd in th msh-indpndnt V norm. Unlik intrior pnalty mthods, th stabilization trm contains a constant factor β>n that is asy to dtrmin for a givn discrtization. Th finit-lmnt spacs composd of picwis polynomial functions ovr th lmnts ar also asy to implmnt. In futur work, w will xplor th numrical proprtis of th mthod and its xtnsions to finit lasticity, lastoplasticity, and fractur.

32 104 Adrian Lw t al. Acknowldgmnts Patrizio Nff and Dborah Sulsky acknowldg th kind hospitality of th Graduat Aronautical Laboratoris during thir visits. W thank Donatlla Marini, Ilaria Prugia, and Dominik Schötzau for commnts on an arlir draft of this papr. Rfrncs [1] D. N. Arnold, An intrior pnalty finit lmnt mthod with discontinuous lmnts, SIAM J. Numr. Anal , no. 4, [] D. N. Arnold, F. rzzi,. Cockburn, and D. Marini, Discontinuous Galrkin mthods for lliptic problms, Discontinuous Galrkin Mthods: Thory, Computation and Applications Nwport, RI, Cockburn, G.. Karniadakis, and C.-W. Shu, ds., Lct. Nots Comput. Sci. ng., vol. 11, Springr, rlin, 000, pp [3], Unifid analysis of discontinuous Galrkin mthods for lliptic problms, SIAM J. Numr. Anal , no. 5, [4] G. A. akr, Finit lmnt mthods for lliptic quations using nonconforming lmnts, Math. Comp , no. 137, [5] F. assi and S. Rbay, A high-ordr accurat discontinuous finit lmnt mthod for th numrical solution of th comprssibl Navir-Stoks quations, J. Comput. Phys , no., [6] S. C. rnnr and L. R. Scott, Th Mathmatical Thory of Finit lmnt Mthods, Txts in Applid Mathmatics, vol. 15, Springr-Vrlag, Nw York, [7] F. rzzi, G. Manzini, D. Marini, P. Pitra, and A. Russo, Discontinuous Galrkin approximations for lliptic problms, Numr. Mthods Partial Diffrntial quations , no. 4, [8] F. rzzi, G. Manzini, D. Marini, P. Pitra, and A. Russo, Discontinuous finit lmnts for diffusion problms, Atti Convgno in Onor di F. rioschi Milano, 1997, Istituto Lombardo, Accadmia di Scinz Lttr, Milano, 001, pp [9] P. Castillo,. Cockburn, I. Prugia, and D. Schötzau, An a priori rror analysis of th local discontinuous Galrkin mthod for lliptic problms, SIAM J. Numr. Anal , no. 5, [10] P. G. Ciarlt, Th Finit lmnt Mthod for lliptic Problms, North-Holland Publishing, Amstrdam, [11], Mathmatical lasticity: Thr-Dimnsional lasticity, Studis in Mathmatics and Its Applications, vol. 0, North-Holland Publishing, Amstrdam, [1]. Cockburn and C.-W. Shu, Th local discontinuous Galrkin mthod for tim-dpndnt convction-diffusion systms, SIAM J. Numr. Anal , no. 6, [13], Rung-Kutta discontinuous Galrkin mthods for convction-dominatd problms, J. Sci. Comput , no. 3,

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34 106 Adrian Lw t al. Adrian Lw: Dpartmnt of Mchanical nginring, Stanford Univrsity, Stanford, CA 94305, USA -mail addrss: lwa@stanford.du Patrizio Nff: Fachbrich Mathmatik, Tchnisch Univrsität Darmstadt, 6489 Darmstadt, Grmany -mail addrss: nff@mathmatik.tu-darmstadt.d Dborah Sulsky: Dpartmnt of Mathmatics and Statistics, Univrsity of Nw Mxico, Albuqurqu, NM 87131, USA -mail addrss: sulsky@math.nm.du Michal Ortiz: Th Graduat Aronautical Laboratoris, California Institut of Tchnology, Pasadna, CA 9005, USA -mail addrss: ortiz@aro.caltch.du