On 2D Ellptc Dscontnuous Galrkn Mthods S.J. Shrwn Dpartmnt of Aronautcs Impral Collg London, UK J. Pró Dpartmnt of Aronautcs Impral Collg London, UK R.L. Taylor R.M. Krby School of Computng Unvrsty of Utah, USA Dpartmnt of Cvl and Envronmntal Engnrng Unvrsty of Calforna at Brkly, USA O.C. Znkwcz Dpartmnt of Cvl Engnrng Unvrsty of Wals, Swansa, UK Abstract W dscuss th dscrtsaton usng dscontnuous Galrkn (DG) formulaton of an llptc Posson problm. Two commonly usd DG schms ar nvstgatd: th orgnal avrag flux proposd by Bass and Rbay [] and th local dscontnuous Galrkn (LDG) [2] schm. In ths papr w xpand on prvous xpostons [3, 4, 5] by adoptng a matrx basd notaton wth a vw to hghlghtng th stps rqurd n a numrcal mplmntaton of th DG mthod. Through consdraton of standard C -typ xpanson bass, as opposd to lmntally -mal: s.shrwn@mpral.ac.uk -mal: krby@cs.utah.du -mal: j.pro@mpral.ac.uk -mal: rlt@c.vultur.brkly.du. Vstng Profssor, CIMNE, UPC, Barclona, Span. Unsco Profssor, CIMNE, UPC, Barclona, Span.
orthogonal xpansons, wth th matrx formulaton w ar abl to apply statc condnsaton tchnqus to mprov ffcncy of th drct solvr whn hgh ordr xpansons ar adoptd. Th us of C -typ xpansons also prmts th drct nforcmnt of Drchlt boundary condtons through a lftng approach whr th LDG flux dos not rqur furthr stablsaton. In our constructon w also adopt a formulaton of th contnuous DG fluxs that prmts a mor gnral ntrprtaton of thr numrcal mplmntaton. In partcular t allows us to dtrmn th condtons undr whch condtons th LDG mthod provds a nar local stncl. Fnally a study of th condtonng and th sz of th null spac of th matrx systms rsultng from th DG dscrtsaton of th llptc problm s undrtakn. 2
Introducton Although th orgnal thrust of most dscontnuous Galrkn rsarch was th soluton of hyprbolc problms, th gnral prolfraton of th DG mthodology has also sprad to th study of parabolc and llptc problms. For xampl, works such as [], n whch th vscous comprssbl Navr-Stoks quatons wr solvd, rqurd that a dscontnuous Galrkn formulaton b xtndd byond th hyprbolc advcton trms to th vscous trms of th Navr-Stoks quatons. Concurrntly, othr dscontnuous Galrkn formulatons for parabolc and llptc problms wr proposd [2, 6]. In an ffort to classfy xstng DG mthods for llptc problms, Arnold t al. publshd, frst n [3] and thn mor fully n [7], an unfd analyss of dscontnuous Galrkn mthods for llptc problms. Thr hav subsquntly bn svral attmpts to provd prformanc nformaton concrnng th choc of contnuous fluxs usd n ths mthods, both by th dvloprs of dffrnt flux chocs (.g. [2, 6]) and by thos ntrstd n flux choc comparsons (.g. [8, 9,, 4, ]). For an ovrvw of many of th proprts of th dscontnuous Galrkn mthod, from th thortcal, prformanc and applcaton prspctvs w rfr th radr to th rvw artcl [2] and th rfrncs thrn. Followng th formulaton of Arnold t al. [7], th scond ordr llptc DG matrx systms can b rcast n trms of a largr frst ordr systm through th ntroducton of an auxlary varabl. Ultmatly th systm can b rcombnd to obtan th so-calld prmal form of problm. Snc ths approach fts vry naturally nto th way th DG formulaton s appld to frst ordr hyprbolc problms w wll follow ths constructon n our xposton. In dong so w stll nd to dfn a sngl valud flux at th lmntal ntrfacs whch n trms dtrmns th typ of DG formulaton. W wll concntrat on two of th mor commonly usd formulatons currntly bng adoptd from th st of numrcal fluxs ndpndnt of u h [7], namly: Bass-Rbay [] and Local Dscontnuous Galrkn (LDG) [2]. Spcfc dtals rgardng th thortcal foundatons for th mult-dmnsonal LDG hav bn prsntd n [3, 4, 4]. W also not that th LDG has bn succssfully appld n th soluton of non-trval llptc problms such as th Stoks systm [5] and th Osn quatons [6]. Buldng upon [5], th prmary motvaton bhnd ths papr s to llustrat how to ffcntly mplmnt th mult-dmnsonal DG schms and to compar th formulaton wth th standard contnuous Galrkn mplmntaton. W hav addrssd th followng ssus n our nvstgatons th numrcal mplmntaton of DG mthods for llptc problms: 3
Whlst th papr by Arnold t al. [7] s vry comprhnsv and rgorous, t s drctd towards th mathmatcal undrstandng rathr than th numrcal mplmntaton. For xampl thr dfnton of th contnuous lmntal fluxs for a componnt of th auxlary varabls s basd on an avrag of lmntal contrbutons of all componnts of th auxlary varabls coupld through th dg normal. In sctons 3.2.3 and 3.2.4 w ntroduc an quvalnt numrcal flux dfnton whr th flux for a componnt of th auxlary varabl s only dpndnt upon gomtrc nformaton and th sam componnt of th auxlary varabl and so s amnabl to matrx mplmntaton. Th DG formulaton prmts lmntally dscontnuous xpansons to b adoptd and so w can consdr usng lmntally orthogonal xpansons whch ar numrcally attractv snc lmntal mass matrcs ar dagonal. Howvr th us of polynomal xpansons wth boundary-ntror dcompostons, dsgnd to nforc C contnuty n contnuous Galrkn mthods, provd th followng numrcal bnfts. Unlk th lmntal orthogonal xpanson, th C -typ xpansons ar amnabl to th applcaton of statc condnsaton tchnqus whch lad to Schur complmnt matrx systms wth mprovd condtonng, partcular for hghr ordr polynomal approxmatons. Whn usng an orthogonal bass, th common practs s to us a pnalty mthod to mpos Drchlt boundary condtons. Whn applyng a C -typ bass wth a boundary-ntror dcomposton Drchlt condtons can b drctly mposd through global lftng (or homogonsaton) as s commonly appld n contnuous Galrkn mthods. Fnally w hav obsrvd that stablsaton s not rqurd n an LDG schm whn Drchlt boundary condtons ar drctly nforcd through a lftng typ opraton applcabl whn usng C - typ xpanson. Th papr s organsd as follows: Scton 2 provds a summary of th notaton adoptd throughout th papr. Scton 3 prsnts a full drvaton of th dscontnuous Galrkn formulaton appld to th llptc dffuson oprator wth a varabl dffusvty tnsor. Ths prsntaton starts from th contnuous formulaton to ntroduc topcs such as stablsaton, flux avragng for th Bass-Rbay and local dscontnuous Galrkn (LDG) formulatons and boundary condton nforcmnt. Aftr th ntroducton of th 4
contnuous problm w formulat n Scton 3.3 th dscrtsaton n trms of a matrx rprsntaton whch s mor amanabl to a numrcal mplmntaton of ths schms. In Scton 3.4 w dscuss dffrnt lmntal polynomal xpansons that can b appld n th DG formulatons. Ths allows us to consdr how th statc condnsaton tchnqu can b appld n a DG formulaton n Scton 3.5. In Scton 4 w analys th ncrasd null spac dmnson of th DG formulaton and assocatd condtonng. Fnally n Scton 5 w dscuss som numrcal solutons of smooth and non-smooth llptc problms. 2 Notaton Th followng notaton wll b adoptd n ths papr. Rgons x Ω Ω Ω D Ω N Ω Ω Ω n Cartsan coordnats, x = [x, x 2 ] T Global computatonal doman Boundary of computatonal doman Ω Boundary of computatonal doman Ω wth Drchlt boundary condtons Boundary of computatonal doman Ω wth Numann boundary condtons Elmntal rgon n Ω, Ω = N l = Ω Boundary of lmnt th boundary sgmnt of boundary Ω, whr Ω = Nb = Ω and Ω Ω j = whn j Outr normal to th boundary of lmnt, n = [n, n 2] T Varabls u (x) Prmary soluton varabl n lmnt q (x) Auxlary soluton varabl n lmnt,.. q = u x q Vctor of auxlary functons on an lmnt Ω,.. q = [q, q 2, q 3 ]T q Vctor of auxlary functons on an lmnt Ω whch ar contnuous ovr adjacnt lmnts D dffusvty tnsor D[, j] = D j,, j 3 Intgrs N l Numbr of lmntal rgons 5
Nb Nq N u Numbr of boundary sgmnts (or facs) n lmnt Numbr of lmntal dgrs of frdom n auxlary varabl q (x) Numbr of lmntal dgrs of frdom n prmary varabl u (x) Innr products (u, v) Ω Innr product of th scalar functons u(x) and v(x) ovr lmnt,.. u(x) v(x) dx Ω (a, b) Ω Innr product of two vctor functons a(x) and b(x) ovr lmnt,.. a(x) b(x) dx Ω u, v Ω Innr product along th boundary Ω of lmnt,.. u, v Ω = u(s) v(s) ds = Nb Ω = u, v Ω u, v Ω Innr product along th th dg (fac) of lmnt boundary Ω Dscrt matrcs and vctors φ j(x) j th xpanson bass n lmnt Ω ; j =,..., Nu (or Nq dpndng on varabl). û [j] j th xpanson coffcnts for th prmtv varabl n lmnt Ω such that u (x) = Nu j= û [j] φ j (x) ˆq [j] j th xpanson coffcnts for th th auxlary varabl n lmnt Ω such that q (x) = Nq j= ˆq [j] φ j (x) M Elmntal mass matrx.. M [, j] = (φ, φ j ) Ω D k Elmntal wak drvatv matrx of th xpanson bass wth rspct to th x k drcton,.. D k[, j] = (φ, φ j x k ) Ω ˆD k Elmntal wak matrx of th k th componnt of th gradnt of th xpanson ( bass multpld by dffusvty ) tnsor,.. ˆD k [, j] = φ, D k x φ j + D k2 x 2 φ j Ω D k Adjont oprator of ˆD ( k [, j].g. D [ ] [ k[, j] = φ, Dk x φ j + D2k x 2 φj] ) Ω E,f kl Elmntal matrx of th nnr product ovr Ω l (dg l of lmnt ) of bass φ n lmnt wth bass φ f j of lmnt f wghtd wth th k th componnt of n,.. E,f kl [, j] = φ, φf j n k Ω l F,f kl Elmntal matrx (for th k th flux ) of th nnr product ovr Ω l such that F,f kl [, j] = (n D k + n 2D 2k )φ, φ f j Ω l Elmntal notaton () Th lmntal ndx of th lmnt adjacnt to dg of lmnt 6
(, j) Th lmntal ndx of th lmnt adjacnt to dg j of lmnt (),.. th lmnt adjacnt to dg of lmnt. û () Th xpanson coffcnts assocatd wth lmnt () û (,j) Th xpanson coffcnts assocatd wth lmnt (, j) ζ Unqu dg vctor usd n th LDG flux assocatd wth dg of lmnt. 3 Formulaton In ths scton w ntroduc th dscontnuous Galrkn formulaton of th llptc stady dffuson problm wth a varabl dffusvty tnsor. In Scton 3. w dfn th strong and auxlary forms of th dffuson problm. In Scton 3.2 w construct th wak form of th auxlary problm for th global doman. As s typcal for a dscontnuous Galrkn formulaton w thn consdr th wak constructon at an lmntal lvl as dscussd n Scton 3.2.. In th lmntal formulaton w not that a contnuous flux at lmntal boundars s rqurd and ths typ flux s dfnd n Scton 3.2.3 for two dffrnt DG mthods: th classcal Bass-Rbay and th local dscontnuous Galrkn mthods. Fnally n Scton 3.2.4 w dscuss th quvalnc btwn th flux formulaton adoptd n th currnt work as compard to th flux dfnton dscussd n th wdly ctd work of Arnold t al. [7]. 3. Problm Dfnton W consdr th followng stady dffuson or Posson problm n a doman Ω wth boundary Ω, whch s dcomposd nto a rgon of Drchlt boundary condtons Ω D and a rgon of Numann boundary condtons Ω N, (D u(x)) = f(x) x Ω () u(x) = g D (x) x Ω D (2) [D u(x)] n = g N (x) x Ω N (3) whr Ω D ΩN = Ω and Ω D ΩN =. In th abov w also consdr th dffusvty tnsor D to b a symmtrc postv dfnt matrx whch may vary n spac,.. [ ] D D D = D(x) = 2 D 2 D 22 and D 2 = D 2. 7
3.. Auxlary Formulaton Equaton () can b wrttn n auxlary or mxd form as two frst ordr dffrntal quatons by ntroducng an auxlary flux varabl q, such that q = D u(x). (4) Substtutng dfnton (4) nto quaton () w obtan q = f(x) x Ω (5) q = D u(x) x Ω (6) u(x) = g D (x) x Ω D (7) q n = g N (x) x Ω N (8) 3.2 Wak Form of th Auxlary Formulaton Takng th nnr product of quatons (5) and (6) wth tst functons v and w rspctvly ovr th soluton doman Ω w obtan: v q dx = v f(x) dx (9) Ω Ω w q dx = w [D u(x)] dx Ω Ω = Dw u(x) dx () Ω whr n th last quaton w rcall that D = D T. W assum T (Ω) s a two or thr-dmnsonal tssllaton of Ω. Lt Ω T (Ω) b a non-ovrlappng lmnt wthn th tssllaton such that f 2 thn Ω Ω 2 =. Lt Ω dnot th boundary of th lmnt Ω and N l dnot th numbr of lmnts (or cardnalty) of T (Ω). For a two-dmnsonal problm w dfn th followng two spacs V h := {v L 2 (Ω) : v Ω P(Ω ) Ω T } Σ h := {τ [L 2 (Ω)] 2 : τ Ω Σ(Ω ) Ω T } whr P(Ω ) = T P (Ω ) s th lnar polynomal spac n a trangular rgon and P(Ω ) = Q P (Ω ) s th blnar polynomal spac for a quadrlatral rgon, dfnd as T P (Ω ) = {x p x q 2; p + q P; (x, x 2 ) Ω } Q P (Ω ) = {x p xq 2 ; p, q P; (x, x 2 ) Ω } 8
Smlarly Σ(Ω ) = [T P (Ω )] 2 or Σ(Ω ) = [Q P (Ω )] 2. For curvlnar rgons th xpansons ar only polynomals whn mappd to a straght sdd standard rgon [7]. Lt v V h and w Σ h dnot scalar and vctor tst functons, rspctvly, dfnd on an lmnt Ω. Th ntgral form of quatons (9) and () thn rduc to fndng u V h and q Σ h such that: v q dx = v f(x) dx v V h () Ω Ω w q dx = Dw u (x) dx w Σ h (2) Ω Ω W not that th abov systm s not solvabl as vry lmnt s now ndpndnt of ach othr. In th standard Galrkn approach th global xpanson s chosn to nforc suffcnt contnuty, whch s typcally C for scond ordr problms, and thn a global assmbly procdur [7] s ncssary to combn th lmntal contrbutons nto a global dscrpton. Howvr n th dscontnuous Galrkn formulaton, contnuty of flux of th prmtv and auxlary varabls s nforcd btwn th lmntal boundars. To llustrat th typ of flux contnuty adoptd n th DG mthod w contnu th problm formulaton at an lmntal lvl as gvn n quatons () and (2). 3.2. Elmntal Formulaton Th applcaton of th dvrgnc thorm to th ndvdual lmntal contrbutons gvn by quatons () and (2) lads to v q dx v (q n ) ds = v f dx (3) Ω Ω Ω w q dx = ( Dw ) u dx + ([Dw ] n ) u ds (4) Ω Ω Ω In quatons (3)-(4), w not that th valus of u and q ar rqurd on th boundary of ach lmnt. In th absnc of a drct nforcmnt of lmntal contnuty through th xpanson spac dfnton, th local approxmaton wll b dscontnuous at th boundary btwn two lmnts. W thrfor dnot a contnuous flux on th boundary as ũ for th flux of th varabl u and q for th flux of th varabl q. Th dscontnuous Galrkn formulaton on vry lmnt can now b xprssd as 9
(3) l=3 l= = () l=2 (2) () Fgur : Dfnton of lmnt numbrng (l) whch shar an dg l wth lmnt. ( v q ) dx v (n q ) ds = v f dx (5) Ω Ω Ω (w q ) dx = ( Dw ) u dx + ([Dw ] n ) ũ ds. (6) Ω Ω Ω Altrnatvly w can apply th dvrgnc thorm gong back th othr way to obtan an quvalnt analytc form v ( q ) dx + v (n [q q ]) ds = v f dx (7) Ω Ω Ω (w q ) dx = w D u dx + ([Dw ] n ) [ũ u ] ds. (8) Ω Ω Ω 3.2.2 Stablsaton As w shall xplor furthr n Scton 4., stablsaton s ncssary whn usng th Bass-Rbay boundary fluxs. It can, howvr, also b ntroducd whn othr fluxs ar appld. Typcally stablsaton can b ntroducd as a arbtrary pnalsaton of th jump btwn th soluton along lmntal boundars. To ncorporat ths nto th formulaton w ntroduc th notaton (l); l Nb to dnot th lmnts adjacnt lmnts to dg l. Ths notaton s hghlghtd n Fgur whr w show th adjacnt lmnt numbrs (), (2) and (3) assocatd wth dgs l =, 2 and 3 of trangular lmnt = ().
W can now ntroduc th stablsaton factor η v (u u (l) )ds whch Ω l pnalss th jump of th prmtv functon btwn lmntal rgons, whr Ω l s th boundary of dg l of lmnt. Ths typ of stablsaton was prvously adoptd n th LDG formulaton prsntd n [2], whr η was also consdrd as a functon of th dg on whch th jump was bng pnalsd. Th wak dscontnuous Galrkn lmntal formulaton (5-6) s modfd to ( v q ) dx v (n q ) ds + Ω Ω N b η v (u u (l) )ds = l= Ω l (w q ) dx = ( Dw ) u dx + Ω Ω v f dx (9) Ω ([Dw ] n ) ũ ds.(2) Ω whr f η = quaton (9) rducs to quaton (5). A smlar modfcaton can also b appld to quaton (7). Throughout ths work, w wll consdr only ths form of stablsaton. W not, that n [8] an altrnatv jump trm was prsntd usng th lftng oprator r dfnd as r (φ) τ dx = φ {τ} ds τ Σ h, φ [L ()] 2 Ω Ω l whr Ω s th doman ovr whch th tssllaton T h s dfnd, Ω l dnots an dg wthn that tssllaton (whch may b ownd by on lmnt or may b shard by two adjacnt lmnts) and th curly brackts ndcat a jump. Th pnalsaton of a varabl φ s gvn by th xprsson η r (φ) whch uss th aformntond oprator. Onc agan, a fr paramtr η allows ths stablzng factor to b tund for th problm undr consdraton. As w wll dmonstrat n th nxt scton, ths nw stablsng factor can b ncorporatd nto th flux dfnton. 3.2.3 Elmntal Boundary Flux Dfnton Smlar to th work of Arnold t al. [7], w dfn th local dscontnuous Galrkn (LDG) flux from whch w can automatcally obtan th Bass- Rbay choc. Howvr, n contrast wth th form adoptd n [7] whch coupls vry componnts of th auxlary varabls q, w construct an altrnatv approach whr ach componnt of q rmans uncoupld.
W can now dfn th boundary fluxs ũ and q as ũ Ω = α u u Ω + β u u() Ω (2) and q Ω = α q q Ω + β q q() (22) Ω t can b asly vrfd that ths fluxs ar contnuous across th dg shard by and () f α, β, β, α satsfy th followng condtons: α u and β u ar ral-valud scalars appld n lmnt along dg wth th constrant that α u + β u = and α q and β q ar dagonal matrcs (of th dmnson of q ) appld n lmnt along dg wth th constrant that α q + β q = I whr I s th dntty matrx. Undr th abov constrant, w dfn α, β, β, α by frst ntroducng a rfrnc vctor ζ along ach dg of lmnt whch s unqu along an dg n th sns that ζ = ζf j f dg n lmnt s adjacnt to dg j of lmnt f. W can adopt th followng form for th coffcnts α u = 2 ζ n Ω (23) β u = 2 ζ n() Ω (24) α q = α q I whr α q = 2 + ζ n Ω (25) β q = β q I whr β q = 2 + ζ n () Ω. (26) whr n quatons (25) and (26) w hav assumd that th avragng s th sam on ach componnt of th flux q. W obsrv that th sgn of th avragng n α and α (as wll as β and β ar rvrsd to ntroduc a flp-flop natur of th fluxs whr th bas of th contnuous flux for ũ s rvrsd to that for th contnuous flux of q. Fnally w also not that th stablsaton dscrbd n Scton 3.2.2 can b ncorporatd drctly nto th contnuous flux by th followng modfcaton to quatons (25) and (26) α q β q = 2 + ζ n Ω η u Ω n j Ω qj j =, 2, 3 (27) Ω = 2 + ζ n() Ω η u() Ω n () j Ω j =, 2, 3. (28) q () j Ω Th us of quatons (27) and (28) n quaton (5), or (7), s quvalnt to applyng quatons (25) and (26) n quaton (9). 2
W can furthr xprss ζ (brfly droppng th subscrpt and suprscrpt notaton) n trms of ts magntud ζ and an angl vctor θ = [cos θ sn θ] T whr θ = tan (ζ /ζ 2 ),.. ζ = ζ θ. Adoptng ths form whn ζ = and η =, w rgan th classc (unstablsd) Bass-Rbay schm. Altrnatvly f ζ and η = w obtan th famly of LDG schms. Sttng ζ = /2 and { n θ = Ω f < () n (29) Ω f > () rcovrs th flp-flop natur of th LDG flux (s Scton 3.3.) as xhbtd n th dscontnuous Galrkn LDG formulaton for on-dmnsonal problms [] and as dscussd for mult-dmnsonal problms n [3]. For a gvn dg thr ar stll two chocs of th vctor gvn by quaton (29) and ts ngatv. As w shall dmonstrat n Scton 4. dfnng ζ = /2 θ as pr quaton (29) can lad to a LDG schm whch has a null spac largr than on for th soluton of th Posson quaton n a prodc rgon. From th analyss n ths scton th ncras n th dmnson of th null spac appars to b rlatd to stuatons whr all th local dg vctors θ n a sngl lmnt pont outwards. To avod ths lmtaton w can dtrmn th drcton of ζ by projctng on to an arbtrary global vctor g,.. θ f g ζ ζ = 2 (3) θ f g ζ 2 < For a curvd dg a smlar phlosophy to th abov can b appld but th valu of α q and β q wll now vary along th dg as th valu of th normal vars. 3.2.4 Equvalnc of Fluxs Th contnuous auxlary fluxs q, gvn by quaton (22) togthr wth (25) and (26), ar not dntcal to th thos proposd n th papr of Arnold t al. [7] and usd by othr rsarchrs. Howvr, quvalnc to th flux form prsntd n th papr of Arnold t al. [7] s obsrvd whn both fluxs ar projctd nto th normal lmntal drcton as rqurd by quaton (5) and (7). Usng th prvously ntroducd notaton, th auxlary fluxs of th papr of Arnold t al. [7], dnotd hr by a subndx A, can b wrttn as q A Ω = 2 ( q Ω + ) ( q() Ω + ζ n Ω ) q Ω + n () Ω q () Ω (3) 3
Insrtng quatons (25) and (26) nto quaton (22) w can wrt th auxlary flux adoptd n th currnt work as q Ω = ( q Ω 2 + ) ( q() Ω + ζ ) n Ω q Ω + ( ζ ) n() Ω q () Ω (32) Rcallng that n Ω = n () Ω th nnr product of th normal n Ω wth quaton (3) as s n Ω q A Ω = ( ) 2 n Ω q Ω + q () () Ω + ( n Ω ζ)( n Ω ) q Ω ( ( n Ω ζ) n Ω ) q() Ω (33) Equvalntly th nnr product of th normal, n Ω, wth quaton (32) n Ω q A Ω = ( ) 2 n Ω q Ω + q () () Ω + ( ζ )( ) n Ω n Ω q Ω ( ζ n Ω ) ( n Ω q () Ω () ). (34) Clarly th frst trm corrsponds to th classcal Bass-Rbay avragng and ( s th sam n quatons (33) and (34). Th ndvdual contrbutons, ζ n Ω q ) ( ) Ω and ζ n Ω q Ω, to th avrag fluxs (3) and (32) ar not dntcal, but thr projcton nto th normal dg drcton of ths trms ar quvalnt as llustratd n Fgur 2. It s mportant to strss that th contnuous auxlary flux valuaton q Ω, usng quaton (32) s mor asly mplmntd than th corrspondng contnuous auxlary flux valuaton q A Ω, usng quaton (3) snc t only nvolvs th nnr products ( ζ n Ω ) and ( ζ n() Ω ) and thrfor lavs th vctor componnts dcoupld. In contrast, quaton (3) coupls th dffrnc componnts of vctor q Ω and q () Ω and so dos not prmt ach componnt of q Ω to b ndvdually valuatd. 3.2.5 Boundary Condton Enforcmnt Up to ths pont w hav only consdrd th lmntal formulaton and how to nforc flux contnuty btwn th soluton varabl and th auxlary fluxs. It s mportant to undrstand how boundary condtons may b nforcd, as t modfs som of th proprts of th fnal prmal matrx systm whch s solvd. 4
(n.q) ζ (n.q)(n. ζ) ( ζ.n)(n.q) a) ζ n (n.q) q b) ζ n ( ζ.n) q ( ζ.n) q Fgur 2: Graphcal quvalnc of th projctd two-dmnsonal LDG componnt of th auxlary fluxs such that th qualty (n q)(n ζ ) = (ζ n)(n q) s numrcally prsrvd n our mplmntaton. (a) form adoptd n Arnold t al.[7] (b) form adoptd n ths work. Thr ar two fundamntal mchansms to nforc Drchlt boundary condtons. Th frst s strong nforcmnt or lftng and th scond s wak nforcmnt. Strong nforcmnt of th boundary condtons s accomplshd through a global lftng of th boundary condtons. W can dmonstrat th global lftng procss by th followng xampl: Assum w wsh to solv th matrx systm Ax = f whr x( Ω) = g( Ω) (35) Snc ths s a lnar problm w can dcompos x nto a known soluton x D and an unknown homognous soluton x H such that x = x D + x U x D ( Ω) = g( Ω), x H ( Ω) =. W can thn nsrt ths dcomposton nto quaton (35) and snc x D s a known soluton (typcally rprsntd n th dscrt approxmaton spac) w can put ths contrbuton on th rght hand sd to obtan th lftd or homognous problm A HH x H = f H (A HH + A HD )x D whr x D ( Ω) = (36) whr th matrx suprscrpts H and D dnot th dgrs of frdom assocatd wth th homognous (zro on Drchlt boundars) and Drchlt 5
(non-zro on Drchlt boundars) dgrs of frdom, rspctvly. In solvng th homognous problm (36) w thrfor only consdr dscrt xpansons whch ar dfnd to b zro on a Drchlt boundary. Ths rducs th numbr of global dgrs of frdom whn compard to th wak nforcmnt of Drchlt boundary condtons. As n standard contnuous Galrkn fnt lmnts, th valu of th approxmaton on th boundary s xact up to th lftng oprator projcton rror. Wak nforcmnt s accomplshd through th ntroducton of th boundary flux trm ũ n quaton (6) for thos lmnts adjacnt to a Drchlt boundary. Ths can b mplmntd thr by drct substtuton or through th us of ghost lmnts surroundng th boundary lmnts n whch th soluton on th ghost lmnts s qual to th boundary condton. Th potntal mplmntaton advantag of ghost lmnts s that th boundary flux trm ũ s computd as t would b for any othr lmnt/lmnt ntrfac. A fw thngs to not concrnng th wak nforcmnt ar that: () Unlk th strong nforcmnt, wak nforcmnt dos not rmov boundary dgrs of frdom from th rsultant matrx problm whch has to b solvd. (2) Wak nforcmnt (as th nam suggsts) dos not rqur that th boundary condton b mt xactly, but consstntly approxmatd. That s, th boundary condton valu s rachd n th lmt of ncrasng spatal rsoluton and th ordr of convrgnc s th sam as th convrgnc of th ntror schm. (3) On must tak car to guarant that th boundary condton valu s appld n an LDG formulaton,.. that th flp-flop s dsgnd so that th spcfd boundary valus ar ntroducd nto th matrx problm. Arnold t al. [7], as wll as most DG practtonrs, typcally apply a wak nforcmnt of Drchlt boundary condtons. Numann or natural boundary condtons ar handld n an analogous mannr to th wak nforcmnt n gnral. Th Numann boundary valu s substtutd drctly nto lmnts adjacnt to Numann boundars as th boundary flux trm q n quaton (5). 3.3 Dscrt Matrx Rprsntaton To gt a bttr apprcaton of th mplmntaton of th dffrnt DG approachs w now consdr th matrx rprsntaton of quatons (5)-(6) 6
and (7)-(8) whch s mor amnabl to a numrcal mplmntaton of th mthod. W start by approxmatng u (x) and q (x) = [q, q 2 ] T by a fnt xpanson n trms of th bass φ j(x) of th form N u u (x) = φ j (x)û [j] j= q k N q (x) = j= φ j (x) ˆq k [j] 3.3. Matrx Form of th Auxlary quatons Followng a standard Galrkn formulaton w st th scalar tst functons v to b rprsntd by φ (x) whr =,...,N u, and lt our vctor tst functon w b rprsntd by k φ whr = [, ] T and w = [, ] T. Insrtng th fnt xpanson of th tral functons nto quaton (5) wth th flux form (25)-(26), th quaton for vry tst functon φ bcoms N q j= N b l= [( ) ( ) ] φ φ, φ j ˆq x [j] +, φ j ˆq 2 Ω x [j] 2 Ω N b l= β q l α q l N q j= Nq ] φ [n, q [j] + n2 q2 [j] j= Ω l [ ] φ, n q (l) [j] + n 2 q (l) 2 [j] Ω l = (φ, f) Ω. (37) for =, 2. Hr w rcall that (l) dnots th adjacnt lmnts whch shar a common dg wth lmnt as llustratd n Fgur. Introducng th matrcs ( ) D k[, j] = φ, φ j x k E,f kl [, j] = φ, φf j n k and dfnng f [] = (φ, f) Ω w can wrt quaton (37) n matrx form as [ (D ) T (D 2) T][ ˆq ˆq 2 ] N b l= N b l= Ω Ω l α q l [E, l E, 2l ] [ ˆq ˆq 2 β q l 7 [ E,(l) l ] ] [ E,(l) ˆq (l) 2l ˆq (l) 2 ] = f (38)
β u = β u = β q = β q = q β = u = β g a) b) Fgur 3: Schmatc llustratng th rol of α q = β q and β q n quaton (38) whn (a) Bass-Rbay and (b) normalsd drcton LDG fluxs ar mployd. A complt xplanaton of th dagram s provdd n th txt. In th matrx systm (38) th matrx D dnots th lmntal wak drvatv commonly usd n standard Galrkn mplmntatons. On th othr hand, th matrx E,f kl s a typ of mass matrx valuatd on an lmnt dg and projctd n th normal componnt drcton n k. W also not that whn f ths lmntal mass matrx nvolvs th nnr product of two, potntally compltly dffrnt dg xpansons, φ ( Ω l ) and φf ( Ω l ) (not Ω l = Ω f m whn dg l of lmnt s adjacnt to dg m of lmnt f). Th two DG formulatons w ar consdrng dffr n thr support n th computatonal doman. Fgur 3 shows th rol of α q and β q n quaton (38) whn Bass-Rbay and a normalsd drcton LDG fluxs ar mployd. In Fgur 3(a) w prsnt th contrbuton to th lmnt dnotd by a crcl whn usng Bass-Rbay fluxs. In ths cas th boundary flux valuaton uss nformaton from both sds of an dg and so th shadd lmnts ar nvolvd. In Fgur 3(b) w llustrat th nflunc on th lmnt dnotd by a crcl of adoptng th normalsd drcton LDG fluxs. In ths cas th orntaton of th normals (nward or outward facng) s basd upon projcton aganst a globally dfnd orntaton vctor g. Th sngl lmnt dagram on th rght of ths fgur provds th LDG valus of β u and β q on ach dg whn quatons (29) and (3) ar mployd. Not that α u and α q ar mmdatly dducbl du to th convx combnaton rqurmnts. Onc agan th shadd lmnts surroundng th lmnt undr consdraton dnot th lmnts from whch non-zro contrbutons ar obtand n quaton (38) (du to th partcular α q and β q valus). W not that not all adjacnt lmnts ar nvolvd n th LDG valuaton as dctatd by ts flp-flop natur. 8
If w now consdr quaton (6) for th k-th componnt of th auxlary flux, q k, w obsrv that, on nsrton of th fnt tral bass xpanson, w hav for vry tst functon φ N q N (φ, φ j ) u Ω ˆq k [j] = j= + + N b l= N b l= α u l β u l N u j= N u j= j= [( [D k φ x ] + ) ] [D 2k φ x ],φ j û [j] 2 Ω [ (n ] D k + n 2D 2k )φ, φ j û [j] (39) Ω l [ ] (n D k + n 2 D 2k)φ, φ(l) j û (l) [j] Ω l and ntroducng th matrcs M [, j] = ( φ, ) φ j Ω ( D k[, j] = φ [ ], Dk φ [ j + D2k φ j]) x x 2 Ω F,f kl [, j] = (n D k + n 2 D 2k)φ, φf j w can wrt quaton (39) as M ˆq k = ( D k )T û + N b l= α u l F, kl û + N b l= Ω l β u l F,(l) kl û (l). (4) In quaton (4), M s th lmnt mass matrx usd n standard Galrkn formulatons. Matrx D k dnots th nnr product of th dvrgnc of dffusvty tnsor by th vctor xpanson bass for th k-th componnt of th auxlary flux. W not that, f th dffusvty tnsor has th smplfd form D = νi whr ν s a constant, thn D k = νd k. Fnally th matrx F,f kl s anothr dg matrx dnotng a typ of lmntal mass matrx wghtd wth th k-th componnt of th dffusvty tnsor and dg normals. Onc agan f D = νi thn F,f kl = νe,f kl. W not that th matrx opraton (M ) F, kl rprsnts th dscrt lmntal lftng opraton (s scton 3.2.5) whch lfts or xtnds th nformaton from dg l of th soluton u nto th ntror of th lmnt through th acton of th nvrs mass matrx. Smlarly to th xampl of Fgur 3 t s ntrstng to not th lmntal couplng of quaton (4). Thrfor n Fgur 4 w prsnt a schmatc llustratng th rol of α u and β u n quaton (4) whn both Bass-Rbay 9
β u = β u = β q = g β q = βu = β q = a) b) Fgur 4: Schmatc dmonstratng th rol of α u = β u and β u n quaton (4) whn LDG basd upon quaton (29) s mployd. A complt xplanaton of th dagram s provdd n th txt. and normalsd drcton LDG fluxs ar mployd. As ndcatd by th shadd trangls n Fgur 4(a), th rgon of nflunc of th Bass-Rbay fluxs on an lmnt of ntrst (dnotd by th crcl) s dntcal to th stncl shown n Fgur 3. For th normalsd drcton LDG flux w rcall th drcton of ζ s ndcatd by th dg arrows n Fgur 4(b) and s a consqunc of applyng quaton (3) aganst a globally dfnd orntaton vctor g. Th LDG valus of α u and β u n ths xampl ar such that only on shadd lmnt contrbuts to th lmnt of ntrst dnotd by th crcl n quaton (4). Ths s th only lmnt whch was not usd n th LDG flux of Fgur 3(b). Fnally w can also dfn th lmntal matrx rprsntng th nnr product wth th k-th componnt of th gradnt of th xpanson bass multpld by th dffusvty tnsor,.. ( ˆD k [, j] = φ, D φ j φ ) j k + D k2 x x 2 In th absnc of any ntgraton rrors w not that th adjont rlatonshp btwn ˆD k and D k can b xprssd as: N b ˆD k = ( D k )T û + F, kl û. (4) Thrfor nsrtng quaton (4) nto quaton (4) w obtan N b l= M ˆq k = ˆD kû + (α u l )F, kl û + l= 2 N b l= Ω β u l F,(l) kl û (l) (42)
whch s th dscrt matrx rprsntaton of quaton (8). It s usful to not, at ths pont n our drvaton, that th mass matrx M n th auxlary flux quatons gvn by quaton (42) s dcoupld at an lmntal lvl. Hnc an lmntal nvrson of th mass matrx allows us to wrt an xplct quaton for th auxlary flux varabl. Ths wll b usd n th nxt scton for th drvaton of a matrx form of th prmal quaton. W also rcall that th multplcaton by th nvrs mass matrx acts as a local lftng oprator lftng th nflunc of th boundary flux trms across th ntr lmntal xpanson. 3.3.2 Matrx Form of th Prmal Equaton Buldng upon th matrx rprsntatons of th Scton 3.3. w can obtan th matrx form of th prmal quaton (). To procd w nsrt quaton (42) nto quaton (38) to obtan th two-dmnsonal prmal form that corrsponds to th contnuous lmntal auxlary quatons (5) and (8). Ths matrx systm can b wrttn as K û + N b m= N b β u m K2mû(m) β q l K 3 l û(l) l= N b l= Nb (l) m= K 4 lmû(l,m) = f (43) whr w hav usd th dfntons dscrbd n th followng. K dnots th contrbuton of th lmntal rgon and s gvn by N b N K = (D )T α q l E, (M ) ˆD + b (α u l )F, + (D 2 )T l= N b l= l α q l E, (M ) 2l b l= N ˆD 2 + l= l (α u l )F 2l. Th contrbuton 2 = (D ) T (M ) ˆD corrspond to th lmntal contrbuton whch typcally arss n a standard Galrkn formulaton and s rcovrd whn E,f kl = F,f kl =. K 2 m dnots th contrbutons from th lmnts mmdatly adjacnt to th dgs of lmnt and s gvn by K 2 m = N b m= β u m (D 2) T (D ) T N b l= N b l= α q l E, (M ) F,(m) α q l E, (M ) F,(m). 2l 2 l 2m m +
Ths contrbuton arss from th contrbuton of th thrd trm n quaton (42) bng nsrtd nto th frst two trms of quaton (38). K 3 l dnots th contrbutons from th lmnts mmdatly adjacnt to th dgs of lmnt and s gvn by K 3 l = N b l= E,(l) 2l β q l E,(l) l (M (l) ) (M (l) ) ˆD(l) 2 + ˆD(l) + b N (l) m= (α u (l) m b N (l) m= (α u (l) m )F(l),(l) 2m )F(l),(l) m. + Ths matrx arss du to th frst two trms of quaton (42) bng nsrtd nto th thrd trm trms of quaton (38). K 4 lm dnots th contrbutons to th prmal form of lmnts adjacnt to th dgs of th lmnts adjacnt to lmnt and arss whn th thrd trm of quaton (42) s nsrtd nto th thrd trm of quaton (38), t s dfnd as N K 4 lm = b l= β q l E,(l) 2l b N (l) m= β u (l) m [ E,(l) l ]] (M (l) ) F (l),(l,m). 2m (M (l) ) F (l),(l,m) m + In ths matrx dfnton w hav xtndd th us of th prvously dfnd suprscrpt notaton () to th form (, j). Ths xtnson (, j) s to b undrstood as th lmnt ndx of th nghbourng lmnt adjacnt to dg j of () whch w rcall s th lmnt adjacnt to dg of lmnt. Thrfor ths nvolvs th lmnts n two halos surroundng lmnt. Fnally w not that f [] = (φ, f) Ω. In a contnuous Galrkn formulaton th global contnuty btwn lmntal rgons nforcs that th nformaton from lmnt s coupld to th lmnts adjacnt to ts mmdat dgs, although th C contnuous vrtx mods typcally coupl furthr nformaton from all nghbourng lmnts. W thrfor not that th matrcs K 2 m, K3 l and K 4 lm rprsnt th non-local contrbutons to th prmal form. Havng constructd th matrx form (43) of th prmal quaton w can now obsrv that to gnrat a local dscontnuous Galrkn mthod whch has a local nflunc on adjacnt lmnts w rqur that thr β q or β u m (m =,..., N b must b dntcally zro to mak K4 lm zro (or quvalntly thr α q or α u m (m =,..., Nb ) hav a valu of ). Rcallng 22
quatons (24) and (26) w dduc that a ncssary condton for th LDG formulaton to mantan a local structur s that ζ n Ω = ± 2. Th most obvous choc for ζ along dg Ω was prvously gvn n quaton (29) and s ζ = ± 2 n Ω. (44) Clarly an arbtrary componnt of th tangnt can b addd to ζ. It would appar that ths rlatvly spcfc choc of ζ to achv a local schm has not bn wdly dscussd. For xampl Arnold t al. [7] suggst any vctor ζ s sutabl. Howvr Cockburn t al. [3] hav prvously suggstd a vctor smlar to th on dfnd abov. As w shall dmonstrat n Scton 4. th choc of sgn of ζ should b normalsd usng a projcton to a global drcton smlar to quaton (3) to avod gnratng undsrabl ncrass n th dmnson of th null spacs of th oprator. W not that th dfnton of th LDG vctor usng quatons (29) and (3) dos not absolutly guarant a local schm (whr only lmnts adjacnt to ar usd). Ths arss du to th rol of β u n th nnr summaton of th lst two lns of quaton (43) whr a β u can ars on any non-local dg and b coupld through β q to lmnt. To llustrat ths pont w buld upon th xampls of Fgurs 3 and 4. In Fgur 5 w schmatcally prsnt th stncl or rgon of nflunc wth rspct to quaton (43) for th classc Bass-Rbay (lft) and th LDG, basd upon quatons (29) and (3) (rght), schms. As wth prvous llustraton xampls, th trangl at th top of th dagram s to rmnd th radr of th ffct of th normalsd drcton LDG. Snc th classc Bass-Rbay mploys a factor of /2 for all α and β valus, svral obsrvatons and dductons can b mad. Frst, th classc Bass-Rbay stncl s qut larg: a total footprnt of lmnts. Scondly, th LDG stncl basd upon quatons (29) and (3) has a far mor compact stncl than th Bass-Rbay stncl. Ths s du to th flp-flop natur of th convx combnaton. Thrdly, th LDG stncl s not guarantd to us nformaton only from nghbourng lmnts n contrast to th formulaton by for nstanc Baumann-Odn [9]. Th choc as to whthr to st β u or β q to zro along a soluton boundary s mportant whn consdrng Numann boundary condtons. In ths cas w rqur that β q = othrws th Numann boundary flux wll not b ncorporatd nto th wak problm. Th analogous ssu for mplmntaton of Drchlt boundary condtons dpnds on whthr ths condton s nforcd through thr a pnalty or a lftng approach. 23
a) b) g β u = β q = β u = β q = β u = β q = Fgur 5: Schmatc dmonstratng th stncl (rgon of nflunc) wth rspct to quaton (43) classc Bass-Rbay (lft) and th normalsd drcton LDG basd upon quatons (29) and (3) (rght). Th trangl at th top of th dagram s to rmnd th radr of th consquncs of th LDG choc. Th classc Bass-Rbay schm mploys on-half for all α and β valus. A complt xplanaton of th dagram s provdd n th txt. 3.4 Polynomal xpanson bass In th followng numrcal mplmntaton w hav appld a spctral/hp lmnt typ dscrtsaton whch dscrbd n dtal n [7]. In ths scton w dscrb th orthogonal and C contnuous quadrlatral and trangular xpansons wthn th standard rgons whch w hav adoptd. For a standard quadrlatral rgon x, x 2 a P-th ordr orthogonal polynomal xpanson can b dfnd as th tnsor product of Lgndr polynomals L p (x) such that φ (pq) (x, x 2 ) = L p (x )L q (x 2 ) p, q P whr th par (pq) rprsnts th unqu ndxng of th D ndcs p, q to th conscutv lst. Analogously th most commonly usd hrarchcal C polynomal xpanson [7] s basd on th tnsor product of th ntgral of Lgndr polynomals (or quvalntly gnralsd Jacob polynomals P, p whr (x)) such that φ (pq) (x, x 2 ) = ψ p (x )ψ q (x 2 ) ψ p (x) = x p, q P p = 2 +x P, 2 2 p (x) < p < P p = P 2 x +x 24
Intror mods Fgur 6: Trangular xpanson mods for a P = 4 ordr xpanson usng an orthogonal xpanson (lft) and a C contnuous xpanson (rght). Th mods n th C xpanson can b dntfd as thr ntror (bng zro on all boundars) or boundary mods. Wthn a trangular doman w can us an orthogonal xpanson dscrbd by, amongst othrs, Prorol [2], Koornwndr [2] and Dubnr [22]. Ths xpanson s llustratd n Fgur 6 (lft) and s xplctly dfnd for P-th ordr xpansons n x, x 2 ; x + x 2 as φ (pq) (x, x 2 ) = Pp, (x ) ( ) x 2 p P 2p+, 2 q (x 2 ) p, q; p + q P. A C xtnson of ths xpanson was proposd by Dubnr [22] and s also dtald n [7]. Th C xpanson for an xpanson of ordr P = 4 s shown n Fgur 6 (rght). As w wll dmonstrat n Scton 3.5, thr ar dffrnt numrcal consdratons assocatd wth th choc of orthogonal or C contnuous xpansons. Th orthogonal xpanson, by dfnton, has a dagonal mass matrx n lmntal rgons wth lnar mappngs to th standard rgon. Ths proprty can b numrcally usful whn valuatng th auxlary varabl q. Altrnatvly, th us of a C xpanson allows th boundary condtons to b ffcntly nforcd through a global lftng typ opraton whr a known functon wth xact boundary condtons s lftd out of th problm as dscussd n Scton 3.2.5. Furthr, th natural dcomposton of th C bass nto ntror mods that ar zro on th lmntal boundars and boundary mods allows us to us a statc condnsaton tchnqu whr th structur of th global matrx systm arsng from th DG formulaton can b usd to ffctvly to solv th systm as dscussd n th nxt scton. 25
3.5 Statc condnsaton Whn th global matrx systm has a smpl block structur, such as a srs of dcoupld sub-matrx systms, th statc condnsaton tchnqu s an algbrac manpulaton whch utlss ths structur to smplfy th soluton of th systm. Statc condnsaton or sub-structurng s a tchnqu commonly usd n contnuous Galrkn mthods, partcularly for th p-typ xpansons whr th constructon of many ntror or bubbl xpansons naturally lnd thmslvs to ths typ of dcomposton. If w consdr a symmtrc matrx problm such as [ A B B T C ] [ u u 2 ] = [ f th problm can b rstatd as ntally solvng for u by consdrng th sub-matrx problm f 2 ], Su = [ A BC B T] u = f BC f 2 whr S s rfrrd to as th Schur complmnt. Th vctor u 2 can thn b solvd va th sub-matrx problm Cu 2 = f 2 B T u. W obsrv that solvng th statcally condnsd problm s not mor ffcnt than consdrng th full problm unlss C s asy to valuat. In th cas of a C contnuous p typ lmnt xpanson, th ntror or bubbl mods hav a block dagonal structur n th global matrx and so ths matrx has a numrcally ffcnt nvrs whn compard to th full nvrs of th global matrx of qual rank. Ths pont s hghlghtd n Fgur 7 whr w schmatcally llustrat th structur of an llptc contnuous Galrkn matrx. Th matrx can b consdrd as bng constructd from block dagonal lmntal componnts whch hav bn ordrd nto sub-matrcs contanng just th boundary and ntror componnts. In constructng th global matrx systm, a drct assmbly procdur nvolvng th matrx A [7] can b appld whch nforcs th contnuty btwn th lmntal rgons on th boundary dgrs of frdom. Howvr snc ntror or bubbl mods ar, by dfnton, ar zro on lmntal boundars thy can b consdrd ndvdually as global dgrs of frdom and so th globally assmbld matrx mantans th block dagonal structur of th ntror-ntror sub-block. Ths matrx can thrfor b nvrtd at th lmntal lvl thrby dramatcally rducng th sz of th global matrx problm for hgh-ordr polynomal xpansons. It s also possbl to apply a smlar phlosophy to a clustr of 26
' & ' ' & ' A T!!!! + + * * + + - -,, - - "# "# # # $ % $ % ) ) ( ( ) ) I / /.. / / A Fgur 7: Schmatc constructon of contnuous matrx systm. In th contnuous Galrkn systm, th matrx can b ntrprtd as block dagonal systms whch ar globally assmbld through pr- and post multplyng by a rstrcton matrx A. lmnts whr th ntror dgrs of frdom ar dfnd to b mods whch ar zro on th boundary of th lmntal clustr. If w ar to consdr th drct nvrson of th dscontnuous problm thn applcaton of th statc condnsaton tchnqu may also b dsrabl. Howvr n th dscontnuous Galrkn formulaton thr s drct nforcmnt of th contnuty of th lmntal xpansons across lmnts. W thrfor mght consdr adoptng an xpanson whch has an dagonal lmntal mass matrx such as th tnsor product of Lgndr polynomals. W not that th global matrx n th DG schm s th sam sz as th sum of th lmntal dgrs of frdom and so w no longr hav a global assmbly procdur dnotd by A. Altrnatvly w ntroduc th lmntal boundary flux whch coupl adjacnt lmntal contrbutons lads to off-dagonal componnts n th matrx structur. At ths pont t s not vdnt whthr w can stll apply th statc condnsaton tchnqu to th dscontnuous Galrkn formulaton. Indd, t s not untl w adopt a C contnuous xpanson, typcally usd n th contnuous Galrkn formulaton, that w rcovr an approprat structur to apply ths tchnqu. To apprcat why t s possbl to us statc condnsaton w obsrv that th dfnton of E,f kl and F,f kl ar purly dpndnt upon th support of th xpanson along th lmntal boundars. Snc by dfnton all ntror mods ar zro along th lmntal boundars, E,f kl and F,f kl ar ncssarly zro for all ntror mods. 27
4 Matrx Analyss In Scton 4. w nvstgat th null spac of th prmal matrx quaton (43) and th condtonng of ths matrx n Scton 4.2. 4. Null Spac of th Laplacan oprator 4.. Bass-Rbay Flux It s wll known that th Bass-Rbay choc of boundary flux α u = β u = α q = β q = wth no stablsaton lads to a Laplac oprator wth spurous 2 mods du to an nrchd null spac [7]. In ths scton w solv th Posson problm usng a unform msh of trangular and quadrlatral lmnts n a prodc doman. Th null spac s valuatd usng doubl prcson wth th gnral matrx gnvalu routn n th LAPACK lbrary appld to quaton (43). An gnvalu was dfnd to b n th null spac f th magntud of th gnvalu was lss than 3. Th gnvctor assocatd wth th zro gnvalu provds a st of xpanson coffcnts and so th corrspondng gnfuncton was dtrmnd by valuatng th xpanson at a srs of quadratur ponts n th soluton doman. Fgur 8 shows rprsntatv null spac functons whch ars at dffrnt polynomal ordrs whn consdrng a prodc rgon [ x, x 2 ] subdvdd nto ght qually shapd trangls. In ths fgur w plot th functon and ts drvatvs wth rspct to x and x 2 whch ar usd to valuat th auxlary fluxs. Tabl shows th sz of th numrcal valuatd null spac for ths msh as a functon of polynomal ordrs and w not that for th trangular xpanson th dmnson of th null spac ncrass wth polynomal ordr. W rcall from quatons (2) and (22) that th prmtv and auxlary fluxs for th Bass-Rbay fluxs ar gvn by th avrag of th valus of th functon or th rlvant drvatv thr sd of an lmntal ntrfac. From an nspcton of Fgur 8 w obsrv that th avrag fluxs wll b zro at any pont on th lmntal boundary and so ths non-zro mods n th null spac hav no global couplng from ũ btwn lmnts. A smlar proprty appars to hold for th drvatvs of th null spac mods whch dcoupls th contrbuton of th contnuous flux of th auxlary varabl q. Fgur 9 shows som rprsntatv null spac functons for a quadrlatral dscrtsaton of th doman x, x 2 nto four quadrlatral lmnts. Ths null spac also contans th constant mod whch s not shown. In contrast to Fgur 8 w obsrv that, whlst th valu of th prmtv functons hav th proprty that th valus ar qual and oppost along th lmntal 28
u(x, x 2 ) u x u x 2 x 2.5.5 x x 2.5.5 x x 2.5.5 x x 2.5.5 x x 2.5.5 x x 2.5.5 x x 2.5.5 x Fgur 8: Rprsntatv null spac gnmods and thr drvatvs (arbtrarly scald) for P = (top), P = 2 (mddl), and P = 3 (bottom) polynomal ordr trangular xpansons. boundars (up to th constant mod), th drvatvs of th functon n th null spac no longr hav ths proprty. Thrfor th avrag auxlary flux, q along th lmntal boundars s not always dntcally zro. Howvr th ntgral Ω q n ds s zro snc th valu of th auxlary avrag fluxs s qual along boundars whr th lmntal normal s qual and oppost. Fnally w not from Tabl that th dmnson of th null spac for quadrlatral dscrtsatons consdrd dos not ncras wth polynomal ordr. 4..2 LDG Flux To complmnt our nvstgaton of th null spac of th Bass-Rbay flux w also consdr th null spac of th LDG flux wth no stablsaton. In Scton 3.3 w argud that th only choc of th dg vctor ζ that lads to a local dscontnuous Galrkn formulaton, whch has smlar couplng as th standard Galrkn mthod, s to dfn ζ as n quaton (44). Howvr n th followng tst w obsrv that th drcton of th unqu vctor along 29
Poly Ordr. P 3 5 7 9 3 5 Dm. of tr. null spac 3 3 3 5 5 5 7 Dm. of Quad. null spac 4 4 4 4 4 4 4 4 Poly Ordr. P 2 4 6 8 2 4 6 Dm. of tr. null spac 2 2 4 4 4 6 6 6 Dm. of quad. null spac 4 4 4 4 4 4 4 4 Tabl : Numrcal valuaton of th dmnson of th null spac ( λ 3 ) usng Bass-Rbay fluxs for dffrnt polynomal ordr xpansons P n th doman shown n fgurs 8 and 9 usng smlar shapd trangular and quadrlatral lmnts. u(x, x 2 ) u x u x 2 x 2.5.5 x x 2.5.5 x x 2.5.5 x x 2.5.5 x x 2.5.5 x x 2.5.5 x Fgur 9: Rprsntatv null spac gnmods (arbtrarly scald) for P = (top) and P = 3 (bottom) polynomal ordr Quadrlatral xpansons. a gvn dg s mportant snc a choc wr all vctors ar thr ntrnal or xtrnal to a local lmnt lads to an undsrabl ncras n th dmnson of th null spac. To llustrat ths pont w consdr th computatonal domans usd n th null spac studs th prvous scton. W thn prscrb th vctor ζ usng only quaton (29). Ths mans that th lmnt wth th lowst global dntty has ζ vctors whch ar all algnd wth th nwards normal drcton of ths lmnt and so β u =. Smlarly th lmnt wth largst global dntty has ζ vctors whch ar algnd wth th outwards normal to th lmnt and so β q =. As wth th Bass-Rbay fluxs and shown n Tabl 2, ths dfnton of ζ lads to a null spac whch ncrass n dmnson wth polynomal ordr for th trangular msh and gvs a fxd null spac of dmnson 4 for th quadrlatral msh consdrd. A rprsntatv mod 3
Poly Ordr. P 3 5 7 9 3 5 Dm. of tr. null spac 3 5 7 9 3 5 7 Dm. of Quad. null spac 4 4 4 4 4 4 4 4 Poly Ordr. P 2 4 6 8 2 4 6 Dm. of tr. null spac 4 6 8 2 4 6 8 Dm. of quad. null spac 4 4 4 4 4 4 4 4 Tabl 2: Numrcal valuaton of th dmnson of th null spac (λ 3 ) usng non-normalsd LDG fluxs for dffrnt polynomal ordr xpansons P n th doman shown n Fgurs 8 and 9 usng smlar shapd trangular and quadrlatral lmnts. of th null spac and th lmntal drvatvs whn P = 2 ar also shown n Fgur. W not that th null spac s non-zro n th lmnt wth hghst global numbr. u(x, x 2 ) u x u x 2 x 2.5.5 x x 2.5.5 x x 2.5.5 x x2.5.5 x x2.5 Fgur : Rprsntatv null spac gnmods (arbtrarly scald) for polynomal ordr P = 2 n a trangular lmnts (top) n a quadrlatral lmnts (bottom)..5 x x2.5.5 x 4.2 Condton Numbr Scalng To complt th numrcal nvstgaton of th contnuous and dscontnuous Galrkn formulatons w consdr th condtonng of both th matrcs of th systm and thr Schur complmnt. In all th followng computatons 3
(a) (b) Fgur : L 2 condton numbr scalng for th stablsaton factor η for polynomal ordrs of P = 2, 4 and 6 for th (a) Bass-Rbay flux and (b) th normalsd drcton LDG flux. Th doman conssts of four qual r quadrlatral lmnts n x, x 2. w hav agan consdrd th Laplacan oprator n th rgon x, x 2 wth prodc boundary condtons. W numrcally dtrmnd th L 2 condton numbr as th rato of th maxmum to mnmum gnvalus. W hav xcludd th zro gnvalu corrspondng to th constant soluton. As w hav obsrvd n scton 4., f stablsaton s not appld to th DG mthod wth Bass-Rbay fluxs, spurous mods xsts du to th prsnc of a non-physcal null spac of th dscrt systm. To supprss ths spurous mods w can apply stablsaton as dscuss n scton 3.2.2. W thrfor start by consdrng th rol of th stablsaton factor η n quatons (27) and (28) on th condton numbr of th systm. In Fgur w show plots of th L 2 condton numbr of th full systm as a functon of th stablsaton factor η for both th Bass-Rbay flux and th LDG flux whr th drcton normalsaton of quaton (3) has bn appld. In ths tst w dvdd th computatonal doman nto four quspacd quadrlatral lmnts. Smlar trnds wr obsrvd for a trangular dscrtsaton whr ach quadrlatral rgon was subdvdd nto two trangular lmnts. Castllo [4] thortcally and numrcally analysd a rang of DG mthod for dffrnt fluxs as a functon of th stablsaton factor normalsd by th msh spac h. Although ths work dd not consdr Bass-Rbay flux, t was notd that th condton numbr should asymptotcally vary lnarly wth condton numbr for largr stablsaton factors n th rlatvly smlar ntror pnalty (IP) approach. 32
(a) 2 2 (b)4 4 (c) 4 4 (d) 6 6 () AR = (f) AR = 2 (g)ar = 4 (h) AR = 8 Fgur 2: Computatonal domans n x, x 2 usd for condton numbr tsts. AR dnots th maxmum aspct rato of th lmnts. Ths proprty s obsrvd for both fluxs consdrd n Fgur. For th IP mthod h also found that th condton numbr was nvrsly proportonal to th stablsaton factor for small valus of η. Ths proprty would also appar to b prsnt n th Bass-Rbay flux. Crtanly w would xpct an ncras n th condton numbr as th stablsaton factor tnds to zro and mor spurous mods of th Bass-Rbay fluxs ar ntroducd nto th systm. Howvr, n contrast to th fndngs of Castllo [4], w obsrv that for th normalsd drcton LDG mthod th condton numbr s constant as th stablsaton factor tnds to zro. If th drcton s not normalsd w obsrvd n scton 4..2 that spurous mods can ntr th systm and thrfor w could xpct an ncras of th condton numbr for small stablsaton factors. Ths obsrvaton appars to b consstnt wth obsrvatons mad n [3, 4] whr stablsaton s rqurd n th wak nforcmnt of boundary condtons. As w ar xamnng a prodc doman and usng th drcton normalzd LDG as prsntd, w hav lmnatd th sourc of th condtonng problm, and hnc s that th condton numbr dos not grow as th pnalsaton s takn to zro. W hav obsrvd a smlar bhavour whn usng Drchlt boundary condtons drctly nforcd through a global lftng of a known functon satsfyng th boundary condtons. 33
(a) (b) Fgur 3: L 2 condton numbr scalng as a functon of msh spacng h for th full systm usng (a) Bass-Rbay flux wth η = and (b) th normalsd drcton LDG flux. Th computatonal doman adoptd ar shown n Fgur 2(a)-(d). In our nxt st of tsts w consdr th scalng of th L 2 condton numbr as a functon of th charactrstc h of th lmntal rgons, th aspct rato of lmntal rgons and th polynomal ordr appld wthn vry lmnt. W start by consdrng a srs of hrarchcal mshs as shown n Fgur 2. Th computatonal doman s now sub-dvdd nto 4, 6, 64 and 256 qual squar lmnts as shown n Fgurs 2(a)-(d). To analys th ffct of th aspct rato of dffrnt mshs w hav also usd a srs of mshs of quadrlatral lmnts whch ar rfnd nto th bottom lft hand cornr as shown n Fgur 2()-(h). Th smallst lmnts of ths mshs corrspond to th smallst lmnts of th unformly dscrtsd cass. Th mshs shown n Fgur 2()-(h) hav lmnts wth maxmum aspct ratos of, 2, 4 and 8, rspctvly. Fgurs 3 (a) and (b) show th L 2 condton numbr of both th Bass- Rbay flux and th normalsd drcton LDG flux for polynomal ordrs n th rang P 5. In ths plots w hav takn th sz of th lmnts along ach dg n Fgur 2 (a)-(d) as a masur of th lmnt spacng. Ovr th h-rang consdrd w obsrv that th Bass-Rbay flux scals as O(h 2 ) for lnar polynomal ordrs but somwhat slowr at hghr polynomal ordrs. Somwhat mor surprsngly w also obsrv that on ths mshs th LDG flux at a hghr than lnar polynomal ordr dos not vary wth h. For th form of xpanson bass prsntd n ths work, a slowr than 34