DG Methods for Elliptic Equations

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1 DG Mthods for Elliptic Equations Part I: Introduction A Prsntation in Profssor C-W Shu s DG Sminar Andras löcknr <klocknr@dambrowndu> Tabl of contnts Tabl of contnts 1 Sourcs 1 1 Elliptic Equations 1 11 A littl bit of Thory Dipping into DG 1 Why considr DG for Elliptic Equations? What is a Pnalty Mthod? And why do w nd on? 3 3 Obtaining a Wak Formulation 3 4 Discrtizing th Wak Formulation 4 41 Function Spacs 4 4 A Global Viw 4 5 Jumps and Avrags 5 51 Intgration by Parts using Jumps and Avrags 5 6 Final Touchs to th Framwork 6 3 A Closr Look at th Intrior Pnalty Mthod 6 31 Obtaining a Bilinar Form 6 3 Basics for a Dtaild Analysis 7 33 Th Innr Workings of our First Estimat 8 34 Consistncy 9 35 Bounddnss 9 36 Corcivity Approximation 10 4 Closing Rmarks 11 Bibliography 11 Sourcs This prsntation is basd largly on [3], with som influnc from [6] [7] builds on [6], but is rarly rfrrd to Nothing in hr is original 1 Elliptic Equations Elliptic partial diffrntial quations ar quit diffrnt from th othr PDEs tratd in this sminar in on major aspct: Th solution is globally coupld On cannot hop to solv th problm just on a subdomain This mans spcially: Thr is no innat tim dpndncy, and hnc no tim stpping in a numrical solution

2 Sction Thr ar no charactristics Considr for a minut that DG as w hav sn it havily rlis on both ths faturs Poisson s quation is th prim xampl of an lliptic quation, and its Dirichlt problm is what w will b trating hr Lt R n b a boundd, opn, polygonal domain u = f on, u = g on In 1D, this boils down to th boundary valu problm for u = f Wlog, g 0 othrwis, continu g onto in an arbitrary fashion and solv for ũ 4 u g, which lads to ũ = f + g Put wakly, w want a u H 1 0 ) such that Bu, v) 4 u v = f v v H 1 0 ) 11 A littl bit of Thory Thorm 1 Lax-Milgram) 77) in [ 5]) Lt V b a Hilbrt spac For a bilinar form B: V V R and a linar functional l: V R, is uniquly solvabl if B is continuous: Bu, v) C 1 u v, B is corciv: Bv, v)c v, l is continuous: lv)c 3 v Bu, v)=lv) v V This thorm guarants solvability of both th continuous and th discrt flavor of th problm W will com up with a bilinar form, and w will hav to show that it satisfis th assumptions of Lax-Milgram Thorm Rgularity) [ 4], Thm 7) Lt B: H 0 1 ) H 0 1 ) R b a corciv bilinar form with sufficintly smooth cofficnt functions, and lt b convx Thn th variational problm Bu, v)=f,v) 0 with H 0 1 ) V H 1 ) has a solution u H ) and u c f 0 v V Not that for highr ordr rror stimats, you nd bttr rgularity, which may or may not b availabl, dpnding on your domain Thorm 3 Trac Thorm) [ 9], Thm 11 or [ 1]) Lt b a Lipschitz domain, k N, and l {0,, k 1} Thn thr xists a continuous map γ l : H k ) L ) with ) l γ l ϕ)= ϕ ϕ C k ) n Dipping into DG 1 Why considr DG for Elliptic Equations? For most rlvant cass, th abov rgularity thorm stats that our solution will b in H, which is fairly smooth Why, thn, do w vn considr discontinuous approximations to this solution? Th answr lis mostly in th difficulty of constructing smooth mthods Th usual finit lmnt mthod chooss an approximation spac V h H 0 1 = : V Mthods with this proprty ar calld conforming Non-conforming mans that V h V DG is clarly non-conforming

3 Dipping into DG 3 Considr th following difficultis: h-adaptivity: Hanging nods p-adaptivity: Non-matchd polynomial dgrs on lmnt intrfacs C 1 lmnts xist, but ar prtty awkward to construct g th Argyris lmnt with 1 dgrs of frdom, cf 310) in [5]) Th asy simplical or quadrilatral lmnts ar only C 0 Crtain discrtizations of th biharmonic problm u = f rquir H, and hnc C 1 approximations DG mthods allviat this by only rquiring us to b abl to comput a boundary intgral, nothing mor w ar not confind by continuity or diffrntiability rquirmnts at lmnt intrfacs Lik in th hyprbolic cas, numrical fluxs will hlp us nforc th rgularity rquirmnts that w ar choosing to not build into th approximation spac For th most part but not in all mthods), this will happn by mans of a pnalty mthod Ths advantags com at a pric, howvr A calculation using DG typically has twic th numbr of dgrs of frdom of a conforming on, for no dirct gain in th accuracy and hnc rror) stimats It dpnds on th individual application whthr this pric is worth paying What is a Pnalty Mthod? And why do w nd on? Th whol point of our mthod is that w will not forc th intr-lmnt jump u h,1 u h, to b zro W will us a softr approach instad: Our bilinar form will contain a trm that looks lik Bu, v)= + 1 α u h,1 u h, )v, whr rprsnts an lmnt intrfac, its n 1-dimnsional masur, and α 0 is th ordr of th pnalty trm A similar mthod nforcs our zro boundary condition What dos this trm do? For α = 0, a condition lik u h,1 u h, )v =0 v nsurs u h,1 u h, = 0 But, w did not add this as a sparat condition W just addd th condition to our xisting quation, which might yild a diffrnt solution altogthr Th saving grac is th division by α If w lt h 4 max h 0 whr h is th diamtr of th lmnt ), automatically α 0, and thus α, so allowing u h,1 u h, to b nonzro bcoms mor and mor xpnsiv as th msh is rfind Curiously though, in practic, α 1 is sufficint [6], Thm ), which mans 1 α u h,1 u h, )v = O1) Highr powrs of α would crtainly work, but lad to an incrasingly largr condition numbr of th stiffnss matrix 3 Obtaining a Wak Formulation Sinc intgration by parts yilds no asy way to dal with functions whos drivativs hav jumps and rmmbr, w ar daling with scond drivativs hr), w rphras th Poisson quation as a systm of first-ordr quations u = u σ 4 u, σ = f W can imagin ths quations to spcify th divrgnc of a flux σ, and solving for a potntial that gnrats this gradint, somwhat lik a consrvation law Lt b a compact st Considring th two narly idntical qualitis u) τ + σ v)+ u τ Gauß = v σ Gauß = uτ) =? start hr? start hr σv) = uτ n, vσ n,

4 4 Sction and plugging in th rwrittn systm in th appropriat spots, w gt σ τ = u τ + uτ n, σ v = v f + vσ n, whr w sk solutions u V, σ V n, for som V H 1 ), that satisfy ths quations for all v V, τ V n At this point, it is appropriat to not that th Trac Thorm allows us to safly talk about th boundary valus usd in ths xprssions, sinc thy ar dfind at last in an L -function sns 4 Discrtizing th Wak Formulation For th rst of this prsntation, w spcializ to n = Whn discrtizing th wak formulation abov, w run into on problm: W nd th valus of u and σ on in th boundary trms Th problm is that ths functions ar potntially doubl-valud thr Lik with hyprbolic problms, this problm is rsolvd by picking numrical fluxs û and σˆ: σ h τ h = u h τ h + û h τ h n, σ h v h = v h f + v h σˆh n Obsrv that only th normal componnt of σˆh is vr usd in our mthod 41 Function Spacs Lt s worry for a minut about th spacs to which ths functions blong To that nd, lt T h b a triangulation of with h 4 max Th h, whr h 4 diam) for T h V h 4 {v L ): v P) T h }, Σ h 4 V h, whr P) is a suitabl finit-dimnsional approximation spac on th lmnt, such as th polynomials P p of up to dgr p A ky point hr is that th local spac P) is allowd to vary dpnding on W assum u h, v h V h and σ h, τ h Σ h Notic that this is th point whr w dpart from th saf grounds of conforming mthods, sinc for v V h, v 1 P 1 ) and v P ) do not nd to agr on 1, and thus v H 1 ) For th purposs of our rror analysis, w also nd continuous spacs that admit discontinuitis of th kind that occur in V h Th appropriat spac for this is H l T h ) 4 H l ) T h Naturally, V h H l T h ) for any l To finish off our dalings with function spacs, w also dfin trac spacs, that is spacs for th function valus on th boundaris If w lt 4 T h and 0 4 \, thn Thorm 3 allows us to dfin tracs of a function v H l T h ), which ar guarantd to b in a trac spac, which w dfin analogously to H l T h ), lik this T) 4 L ) T h v T) may b doubl-valud on th innr boundary 0 and is singl-valud on \ 0 4 A Global Viw If w add ovr all lmnts in th formula abov, w gt σ h τ h = u h h τ h + û h, τ h, n, σ h h v h = v h f + v h, σˆh, n

5 Dipping into DG 5 On small pic of magic was smuggld in hr, namly th introduction of h and h, which w ndd sinc th jumpy functions v h and τ h do not actually hav H 1 -drivativs on lmnt boundaris Thus, w dfin h to us th H 1 -drivativs on th lmnt intrior, and lav it undfind lswhr 5 Jumps and Avrags W will now dfin two quantitis, th jump and th avrag W will writ down our fluxs in trms of ths Stp 1: Dfinition on th intrior boundary Lt 1, T h b two lmnts sharing an dg 1, and lt n b th outward normal of 1 Thn for a scalar quantity v T), w dfin v 1 + v {v} 4, v 4 nv 1 + n) v, whr v i i =1, ) is th part of v associatd with i Not how w clvrly skirt having to fix a sign or dg orintation) convntion for th jump by noting that it dos not mattr which lmnt w call 1 and which This is th major rason to dfin ϕ as a vctor quantity, vn though it strictly only rprsnts a scalar valu For a vctor quantity τ T), w dfin {τ } 4 τ 4 τ 1 + τ, τ 1 n+τ n) Similar commnts as abov apply hr Stp : On th outr boundary, w only dfin v 4 vn, {τ } 4 τ 51 Intgration by Parts using Jumps and Avrags Th sums at th nd of ach of th abov two trms look vry much alik Lt us dvlop a formula to dal with that kind of sum, using th abov notation for th jump and th avrag Lt v T) and τ T), and lt E h dnot all dgs of lmnts in T h, whil E h,0 dnots all intrior dgs, i such dgs along wich v and τ can b doubl-valud v τ n = [v 1 τ 1 n v τ n] + [v τ n],0 = =,0 [ v 1 n v n) τ 1 + τ v {τ }+ {v}τ 0 \E h,0 + v 1 +v τ 1 τ ) n ] + \E h,0 In this calculation, w hav assumd n =n 1 for brvity W can milk this formula vn furthr, by applying Gauß s Thorm: v {τ } + {v}τ = v τ n = v τ ) = 0 = h v τ + v h τ [v n τ ] h vτ) Naturally, hr w nd to hav v H 1 T h ) and τ H 1 T h ) So w hav gaind an asy intgration-byparts formula involving only th jump and avrag trms

6 6 Sction 3 6 Final Touchs to th Framwork Using th formula from Sction 51 with th quations abov yilds σ h τ h = u h h τ h + û h {τ h }+ {û h }τ h, 0 σ h v h = v h f + v h {σˆh}+ {v h }σˆh 0 W will now hav to worry about how w can choos our fluxs It turns out that our choic will fall into on of two catgoris, dpnding on whthr th vctor flux σˆh dpnds on σ h If this is not th cas, thn w can asily liminat σ h : w pick τ h = h v h, us th quality of th lft hand sids of th abov quations Thr is a small catch, howvr W can only do this if : V h V h, and that map is also onto On way to not hav this condition is to hav polynomial spacs with bubbl functions Th onto condition says that w ar still using all possibl tst functions on th first quation Th polynomial spacs P k satisfy ths conditions If w can us this shortcut, thn our mthod is calld a primal mthod If not, th mthod is calld a flux mthod cf [8]) For flux mthods, it sms that w truly hav a systm of quations But it turns out that by invsting a littl bit mor work, w can sidstp this rquirmnt, as Dan will show latr 3 A Closr Look at th Intrior Pnalty Mthod For th rst of this prsntation, lt us focus on on spcific mthod, namly th Intrior Pnalty Mthod W obtain it from th abov dduction if w choos û 4 {u h }, σˆh 4 { h u h } η h u h, whr h is th lngth of th dg at which σˆh is valuatd, and η is som larg positiv constant Notic that th last trm in σˆh is th pnalty trm mntiond in Sction 31 Obtaining a Bilinar Form W quat both right hand sids from abov, substituting in our fluxs in th procss: { u h h τ h + {u h } {τ h } + {{u h }}τ h = v h f + v h { h u h } η } u h + 0 h {v h } { h u h } η u h 0 h Now us { } =0, =0, {{ }} = { } and { } = : u h h τ h + {u h }τ h = v h f + 0 v h { h u h } η ) u h 1) h Nxt, us th intgration-by-parts formula on th first trm, to avoid gnrating h h v h, which would b problmatic: h u h τ h u h {τ h } {u h }τ h + {u h }τ h = v h f + v h { h u h } η ) u h 0 0 h Finally, w substitut τ h = h u h as indictatd arlir: h u h h v h [ u h { h v h } +v h { h u h } v h η h u h ] = v h f

7 A Closr Look at th Intrior Pnalty Mthod 7 W call th lft hand sid th bilinar form B h u h, v h ) 3 Basics for a Dtaild Analysis W will now procd with th analysis of this mthod, with th goal of proving a first rror bound It turns out that a good function spac for this analysis is V h) 4 V h +H ) H 0 1 ) H T h ) A convnint norm with which to carry out this analysis is th following: v 4 + h v, + h T h v 1, 1 v 0, which is quivalnt to th usd in [3] by Formula 45) in that sam papr It is obvious that, bing composd of sminorms, is also a sminorm But it is also a norm on V h) In ordr to s this, rmmbr that th main ingrdint in proving that 1 is a norm on H 0 1 ) is th Poincaré Inquality v 0 C v 0 Th Poincaré Inquality also ntails that th convntional FEM bilinar form B u, v) 4 u v is corciv Such an stimat is not radily availabl to us, sinc V h) H 0 1 ), so w will hav to prov on for ourslvs Lmma 4 [ 3], Lmma 1) Lt T h b a msh on whos intrior angls and adjacnt-dg ratios ar boundd blow Thn thr xists a constant C dpnding only on and ths lowr bounds such that ϕ 0 C h ϕ 0 + ) for ϕ H 1 T h ) h 1 v 0, Proof Dfin ψ H ) H 1 0 ) by ψ = ϕ Thn by Thorm thr xists a constant C 1 dpnding only on such that ψ C 1 ϕ 0 Using our intgration-by-parts formula and th Cauchy- Schwarz Inquality, w obtain ϕ 0 = ϕ, ψ) = h ϕ, ψ) = h ϕ, ψ) h ϕ 0 + Now, w mploy th trac inquality h n ψ 0, C h 1 ϕ n {h n ψ} 1 ϕ n 0, ϕ { ψ} + ) 1/ h 1 ψ 1,,0 ψ 0 + ) + h ψ,, {ϕ} ψ? =0 h n ψ 0, for E h and an adjacnt triangl T h If th inquality did not contain th h trms, it would b implid by Thorm 3, abov This particular trac inquality can b found as Formula 5) in [] Mor spcifically, w obtain h n ψ 0, C ψ 1, ) +h ψ, C ψ 1, ) + ψ, C ψ 0, + ψ 1, ) 1/ ) + ψ, = C ψ,,

8 8 Sction 3 whr w hav usd h diam), which subsquntly got swallowd up in th constant Thus, ϕ 0 h ϕ 0 + ) 1/ 1 ϕ n 0, C ψ ϕ 0 C h h ϕ 0 + h h ϕ 0 + h ) 1/ 1 ϕ n 0, C ϕ 0 ) 1/ 1 ϕ n 0, Thus, w know that for v V h) v 0 C v, so that if v 0 in L -sns, thn v 0, making a norm 33 Th Innr Workings of our First Estimat In ordr to prov our first rror stimat, w nd a fw ingrdints: Consistncy: This coms in in th form of Galrkin orthogonality, which mans B h u h u, v)=0 v V h), with u h th numrical and u th xact solution, as dfind latr Bounddnss: B h v, w) C v w for all v, w V h) Corcivity/Stability: C v B h v, v) for all v V h) Th main part of this is showing it for v h V h Approximation: W assum a projction oprator P:H p+1 V h, v Pv Ch p v p+1 v H p+1, whr p is som numbr that is a proprty of our approximation spac To show how vrything works togthr, w will prov th stimat now and go through all th componnts latr Bounddnss and corcivity togthr lt us apply Lax-Milgram on V h, yilding a numrical solution u h V h givn by B h u h, v h )= f v h v h V h ) Lt u H b th xact solution of th original Dirichlt problm W ar going for a fairly standard H 1 - typ finit lmnt stimat, using th following chain of inqualitis Pu u h stab consis = bound approx C B h Pu u h, Pu u h ) C B h Pu u, Pu u h ) C Pu u Pu u h C h p u p+1 Pu u h Onc w gt this far, w us th triangl inquality and finish off: u u h u Pu + Pu u h C h p u p+1 +C h p u p+1 p=1) C h f 0

9 A Closr Look at th Intrior Pnalty Mthod 9 Obviously, w will hav to go through th ingrdints of this stimat on by on and vrify that thy do indd hold 34 Consistncy Just lik w applid Lax-Milgram to th problm on th approximation spac, w may ask ourslvs what happns if w do th sam on V h), i sk a ũ V h) such that B h ũ, v)= f v v V h) 3) Do ũ and u match? Considr our bilinar form applid to u, kping in mind that u is smooth, and us th intgration-by-parts formula: [ B h u, v) = h u h v u { v}+v { h u } v η ] u h = h u h v v { h u } IBP = h h u v + v { h u }+ {v} h u v { h u } 0 = f v For th last stp, rcall that u = f Mor gnrally, this gos back to a proprty of th fluxs that is calld consistncy Dan will say mor about that Now, do ũ and u match? Wll, u and ũ both satisfy Equation 3) By Lax-Milgram, th solution to 3) is uniqu, so th answr is ys, thy do match Finally, subtracting Equation 3) from Equation ) givs us Galrkin orthogonality which w usd abov B h u h u, v)=0 v V h), 35 Bounddnss Showing bounddnss amounts to showing ach trm in th bilinar form abov can b boundd by th -norm Lt v, w V h) For th first trm, w us th Cauchy-Schwarz inquality and obtain h v h w hv h w v w For th scond trm, w rus th trac inquality from abov togthr with Cauchy-Schwarz to obtain for any w H ) and v L ) for an dg adjacnt to T h : n w v C which mans that for v, w V h) v { h w} = [ C w 1, { n w}v n w 1, C w v ) + h 1/h w 1/, v, 0, Th sam argumnt gos through for th third trm Lastly, [ ] η v 1/ h w C v w 1/ h ) ] 1/[ ] + h 1/ w, h 1 v 0,

10 10 Sction 3 is immdiat, complting th bounddnss proof 36 Corcivity For notational convninc, lt 1 v 4 h v 0, and v 4 Th v h 1, T h for v V h) and rmmbr that w provd v { h v}c w v in th prcding sction Also not that for functions v h V h which w assumd finit-dimnsional), w may us an invrs quality for xampl, Thm 4511) in [5]) to stimat so that h v h, C v h 1,, v h C [ v h Th + v h ] W bgin by showing corcivity for v h V h : B h v h, v h ) = ε<! 1, η C ) C C ε! 1 [ h v h h v h v h { h v h } v h η v h h + η v h + v h { h v h } = v h 1, T h v h Th + η v h C v h v h ) v h Th + η v h C ε v h + v h ε v h Th + 1 v h 1 v h Cε v h + v h η C )) ε ) C Cε This corcivity stimat on V h xtnds naturally to V h) sinc any lmnt v V h) can b dcomposd into v = v h + ṽ, with v h V h and ṽ H H 1 0 It thus only rmains to show that corcivity holds for ṽ Rmbring that ṽ is smooth and alrady satisfis a Poincaré inquality, w gt [ B h ṽ,ṽ) = h ṽ h ṽ ṽ { h ṽ } + ṽ { h ṽ } ṽ η ] ṽ h = h ṽ h ṽ C ṽ 0 C ṽ ] 37 Approximation To show approximation, w would vry much lik to rus th approximation thory for continuous lmnts Thus w assum th projction oprator P: H p+1 V h projcts its argumnt onto a continuous function Thn, for v H p+1 with 4 v Pv v Pv = T h [ 1, + h, ] C 1, T h Ch p v p+1 v H p+1,

11 Bibliography 11 whr w hav usd th dfinition of, an invrs inquality lik abov, and a standard approximation rsult lik Thm 64 in [4] 4 Closing Rmarks For thos intrstd in furthr study, I would rcommnd th survy papr [3], to whos notation I hav trid to stay as clos as possibl Thr rmain many aras which wr not vn touchd upon by this brif ovrviw, such as: Spcial proprtis of fluxs Consistncy, Consrvativity) and consquntly L rror stimats, Othr mthods spcially non-primal mthods ths will rquir slight additions to th proof of bounddnss), Numann boundaris, Mor gnral lliptic oprators, Convrgnc proofs, sp suprconvrgnc Som of this will b covrd in Dan s prsntation Bibliography [1] Robrt Adams Sobolv spacs Acadmic Prss, 1975 [] Douglas N Arnold An Intrior Pnalty Finit Elmnt with Discontinuous Elmnts SIAM J Numr Anal, 194):74 760, August 198 [3] Douglas N Arnold, Franco Brzzi, Brnardo Cockburn, and L Donatlla Marini Unifid Analysis of Discontinuous Galrkin Mthods for Elliptic Problms SIAM J Numr Anal, 395): , 00 [4] Ditrich Brass Finit Elmnts Thory, fast solvrs and applications in solid mchanics Cambridg Univrsity Prss, nd dition, 001 [5] Susann C Brnnr and L Ridgway Scott Th mathmatical thory of finit lmnt mthods Springr, 1994 [6] Paul Castillo, Brnardo Cockburn, Ilaria Prugia, and Dominik Schötzau An A Priori Error Analysis of th Local Discontinuous Galrkin Mthod for Elliptic Problms SIAM J Numr Anal, 385): , 000 [7] Brnardo Cockburn, Guido anschat, Ilaria Prugia, and Dominik Schötzau Suprconvrgnc of th Discontinuous Galrkin Mthod for Elliptic Problms on Cartsian Grids SIAM J Numr Anal, 391):64 85, 001 [8] L Donatlla Marini Discontinuous FEM for Elliptic Problms Prsntation at Cambridg, 8 May 003 [9] Rüdigr Vrfürth Numrisch Bhandlung von Diffrntialglichungn II Finit Elmnt) Lctur nots Grman)

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