Discontinuous Galerkin approximation of flows in fractured porous media

Transcription

1 MOX-Rport No. 22/2016 Discontinuous Galrkin approximation of flows in fracturd porous mdia Antonitti, P.F.; Facciola', C.; Russo, A.;Vrani, M. MOX, Dipartimnto di Matmatica Politcnico di Milano, Via Bonardi Milano (Italy)

2 Discontinuous Galrkin approximation of flows in fracturd porous mdia Paola F. Antonitti, Chiara Facciolà, Alssandro Russo # and Marco Vrani May 15, 2016 MOX- Laboratory for Modling and Scintific Computing Dipartimnto di Matmatica Politcnico di Milano Piazza Lonardo da Vinci 32, Milano, Italy paola.antonitti@polimi.it, chiara.facciola@polimi.it, marco.vrani@polimi.it # Dipartimnto di Matmatica Applicazioni Univrsità dgli Studi di Milano-Bicocca Via Cozzi 55, Milano, Italy alssandro.russo@unimib.it Abstract W prsnt a numrical approximation of Darcy s flow through a fracturd porous mdium which mploys discontinuous Galrkin mthods. For simplicity, w considr th cas of a singl fractur rprsntd by a (d 1)-dimnsional intrfac btwn two d-dimnsional subdomains, d = 2, 3. W propos a discontinuous Galrkin finit lmnt approximation for th flow in th porous matrix which is coupld with a conforming finit lmnt schm for th flow in th fractur. Suitabl (physically consistnt) coupling conditions complt th modl. W thortically analys th rsulting formulation and prov its wll-posdnss. Morovr, w driv optimal a priori rror stimats in a suitabl (msh-dpndnt) nrgy norm and w prsnt two-dimnsional numrical xprimnts assssing thir validity. Introduction Modlling flows in fracturd porous mdia has rcivd incrasing attntion in th past dcads, bing fundamntal for addrssing many 1

3 2 nvironmntal and nrgy problms, such as watr rsourcs managmnt, oil migration tracmnt, isolation of radioactiv wast, ground watr contamination. In ths applications th flow is strongly influncd by th prsnc of fracturs, which can act as prfrntial paths (whn thir prmability is highr than that of th surrounding mdium), or as barrirs for th flow (whn thy ar filld with low prmabl matrial). A fractur is typically dfind as a rgion charactrizd by a small aprtur compard to its lngth and th siz of th domain and with a diffrnt porous structur than th surrounding mdium. Th task of ffctivly modlling th intraction btwn th systm of fracturs and th porous matrix is particularly challnging. In th following, lt us brifly commnt on a popular modlling choic to handl such a problm, s.g [27, 5, 20]. To rduc th complxity of th problm, a common modlling choic consists in trating fracturs as (d 1)-dimnsional intrfacs btwn d-dimnsional porous matrics, d = 2, 3. Th dvlopmnt of this kind of rducd modls, which can b justifid in cas of fracturs with vry small width, has bn addrssd for singl-phas flows in svral works, s.g. [2, 1, 27, 23]. In this papr w adopt th prspctiv of th singl fractur modl dscribd in [27]. A first vrsion of this modl has bn introducd in [2] and [1] undr th assumption of larg prmability in th fractur. In [27] th modl has bn furthr gnralisd to handl also fracturs with low prmability. On th othr hand, th flow in th porous mdium is assumd to b govrnd by Darcy s law and a suitabl rducd vrsion of this law is formulatd on th surfac modlling th fractur. Physically consistnt coupling conditions ar addd to account for th xchang of fluid btwn th fractur and th porous mdium. Th xtnsion of such a coupld modl to th cas of two-phas flow has bn addrssd in [24] and [26], whil a totally immrsd fractur has bn considrd in [3]. Various numrical mthods hav bn mployd in th litratur for th approximation of th rsulting coupld bulk-fractur flow. Roughly spaking, thy can b classifid dpnding on th condition (matching or non-matching) btwn th bulk and th fractur mshs. A traditional approach combining mixd finit lmnts and bulk mshs conforming to th fractur msh was adoptd for xampl in [2, 23, 27]. Th us of non-matching grids coupld with th Xtndd Finit Elmnt Mthod (XFEM) has bn proposd in [24, 20]. Mor rcntly, an approximation basd on th us of conforming polygonal mshs and Mimtic Finit Diffrncs (MFD) has bn xplord in [5]. W also mntion a promising framwork to trat flows in systms of fractur ntworks introducd in [12, 13, 14, 15, 11]. Th aim of this papr is to mploy discontinuous Garlrkin (DG) finit

4 3 lmnts to discrtiz th coupld bulk-fractur problm stmming from th modlling of flows in fracturd porous mdia. Th inhritd flxibility of DG mthods in handling arbitrarily shapd, non-ncssarly matching, grids rprsnts th idal stting to handl such kind of problms that typically fatur a high-lvl of gomtrical complxity. Discontinuous Galrkin mthods wr first introducd in th arly 1970s (s for xampl [28, 22, 8, 31, 6]) as a tchniqu to numrically solv partial diffrntial quations. Thy hav bn succssfully dvlopd and applid to hyprbolic, lliptic and parabolic problms arising from a wid rang of applications: various xampls can b found, for xampl, in [7, 9, 18, 19, 16, 29, 25, 21]. Th primary motivations for th us of DG mthods ar th nhancd flxibility affordd by discontinuous lmnts [6] and th possibility of handling mshs mad of arbitrarily shapd lmnts and with hanging nods. Morovr, th local natur of th trial spac allows lmntwis variabl polynomial ordrs which nabls mor accurat approximation of solutions which vary in charactr from on part of th domain to anothr. W rfr to [7] for a unifid prsntation and analysis on DG mthods for lliptic problms. Mor spcifically, th choic of DG mthods for addrssing th problm of th flow in a fracturd porous mdium ariss quit spontanously in viw of th discontinuous natur of th solution at th matrix-fractur intrfac. Howvr, this is not th only motivation to mploy DG mthods in this spcific contxt. Indd, our diffrntial modl is basd on th primal form of th Darcy s quations for both th bulk and fractur flows, which ar coupld with suitabl conditions at th intrfac. Ths coupling conditions can b naturally formulatd using jump and avrag oprators, so that DG mthods turn out to b a vry natural and powrful tool for fficintly handling th coupling of th two problms, which is indd naturally mbddd, in wak form, in th variational formulation. In this papr w propos a discrtization which combins a DG approximation for th problm in th bulk with a conforming finit lmnt approximation in th fractur. Th us of conforming finit lmnts to discrtiz th quations in th fractur is mad just for th sak of simplicity, othr discrtization tchniqus can b mployd and our approach is as gnral to tak into account straightforwardly also such cass. W analys th rsulting mthod and prov a priori rror stimats, which w numrically tst in a two-dimnsional stting. Th papr is structurd as follows. In Sction 1 w introduc th govrning quations for th coupld problm. Th problm is thn writtn in a wak form in Sction 2, whr w also prov its wll-posdnss. In Sction 3 w

5 4 introduc th DG discrtization of th coupld problm and w show som tchnical rsults ndd to prov its wll-posdnss. In Sction 4 w driv a priori rror stimats in a suitabl (msh-dpndnt) norm, whil in Sction 5 w prsnt two-dimnsional numrical xprimnts assssing th validity of th thortical rror stimats. 1 Modl problm Throughout th papr w will mploy th following notation. For an opn, boundd domain D R d, d = 2, 3, w dnot by H s (D) th standard Sobolv spac of ordr s, for a ral numbr s 0. Th usual norm on H s (D) is dnotd by H s (D) and th usual sminorm by H s (D). Furthrmor, w will dnot by P k (D) th spac of polynomials of dgr lss than or qual to k 1 on D. Throughout th papr th symbol (and ) will signify that th inqualitis hold up to multiplicativ constants which ar indpndnt of th discrtization paramtr. In th following w prsnt th govrning quations for our modl, which is a variant of th modl drivd in [27]. Th flow of an incomprssibl fluid through a fracturd d-dimnsional porous mdium, d = 2, 3, can b dscribd by th following thr ingrdints: 1. th govrning quations for th flow in th porous mdium; 2. th govrning quations for th flow in th fracturs; 3. a st of physically consistnt conditions which coupl th problms in th bulk and fracturs along thir intrfacs. For simplicity, w will assum that thr is a uniqu fractur in th porous mdium and that th fractur cuts th domain xactly into two disjoint connctd subrgions (s Figur 1 for a two-dimnsional xampl), following th approach of [5] and [20]. Th xtnsion to multipl fracturs can b tratd analogously, whil th cas of an immrsd fractur is mor complx to b analysd [3] and will b th subjct of futur rsarch. Mor prcisly, lt Ω R d, d = 2, 3, b an opn, boundd, convx polygonal/polyhdral domain rprsnting th porous matrix. W suppos that th fractur is a (d 1)-dimnsional C manifold Γ R d 1, d = 2, 3, whos masur is uniformly boundd (i.., Γ = O(1)), and assum that Γ sparats Ω into two connctd subdomains, which ar disjoint, i.., Ω \ Γ = Ω 1 Ω 2 with Ω 1 Ω 2 =. For i = 1, 2, w dnot by γ i th part of boundary of Ω i shard

6 5 Γ γ 2 Ω 2 n Γ γ 1 Ω 1 Figur 1: Th subdomains Ω 1 and Ω 2 sparatd by th fractur Γ considrd as an intrfac. with th boundary of Ω, i.., γ i = Ω i Ω. W dnot by n i, i = 1, 2 th unit normal vctor to Γ pointing outwards from Ω i and, for a (rgular nough) scalar-valud function v and a (rgular nough) vctor-valud function τ, w dfin th standard jump and avrag oprators across Γ as {v} = 1 2 (v 1 + v 2 ) v = v 1 n 1 + v 2 n 2, {τ } = 1 2 (τ 1 + τ 2 ) τ = τ 1 n 1 + τ 2 n 2, (1) whr th subscript i = 1, 2 dnots th rstriction to th subdomain Ω i. Morovr w dnot by n Γ th normal unit vctor on Γ with a fixd orintation from Ω 1 to Ω 2, so that w hav n Γ = n 1 = n Govrning quations According to th abov discussion, w suppos that th flow in th bulk is govrnd by Darcy s law. Lt K = K(x) R d d b th bulk prmability tnsor, which satisfis th following rgularity assumptions: (i) K is a symmtric, positiv dfinit tnsor whos ntris ar boundd, picwis continuous ral-valud functions; (ii) K is uniformly boundd by blow and abov, i.., x T x x T Kx x T x x R d. (2)

7 6 Givn a function f L 2 (Ω) rprsnting a sourc trm and g H 1/2 ( Ω), th motion of a incomprssibl fluid in ach domain Ω i, i = 1, 2, with prssur p i is dscribd by: (K i p i ) = f i in Ω i, i = 1, 2, (3) p i = g i on γ i, i = 1, 2. (4) Hr w hav dnotd by K i and f i, th rstrictions of K and f to Ω i, i = 1, 2, rspctivly, and by g i th rstriction of g to γ i, i = 1, 2 (for simplicity, w hav imposd Dirichlt boundary conditions on both γ 1 and γ 2 ). Th scond ingrdint for th modl is rprsntd by th govrning quations for th fractur flow. In our modl th fractur is tratd as a (d 1)-dimnsional manifold immrsd in a d-dimnsional objct. If w assum that th fracturs ar filld by a porous mdium with diffrnt porosity and prmability than th surroundings, Darcy s law can b usd also for modlling th flow along th fracturs [10]. Th rducd modl is thn obtaind through a procss of avraging across th fractur: in th bginning th fractur is assumd to b a d-dimnsional subdomain of Ω, that sparats it into two disjoint subdomains. Thn Darcy s quations ar writtn on th fractur in th normal and tangntial componnts and th tangntial componnt is intgratd along th thicknss l Γ = l Γ (x) > 0 of th fractur domain, which is typically som ordrs of magnitud smallr than th siz of th domain. W rfr to [27] for a rigourous drivation of th rducd mathmatical modl. Not that in [27] this avraging procss is carrid out for th flow quations writtn in mixd form. Hr, w rwrit th modl in primal form. Th fractur flow is thn charactrizd by th fractur prmability tnsor K Γ, which is assumd to satisfy th sam rgularity assumptions as thos satisfid by th bulk prmability K and to hav a block-diagonal structur of th form [ ] K n K Γ = Γ 0 0 KΓ τ, (5) whn writtn in its normal and tangntial componnts. Hr K τ Γ R(d 1) (d 1) is a positiv dfinit tnsor (it rducs to a positiv numbr for d = 2) that rprsnts th tangntial componnt of th prmability of th fractur. Lt us assum that f Γ L 2 (Γ). Stting Γ = Γ Ω, and dnoting by

8 7 p Γ th fractur prssur, th govrning quations for th fractur flow rad τ (K τ Γl Γ τ p Γ ) = f + K p in Γ, (6) p Γ = g Γ on Γ, (7) whr g Γ H 1/2 ( Γ) and τ and τ dnot th tangntial gradint and divrgnc oprators, rspctivly. Equation (6) rprsnts Darcy s law in th dirction tangntial to th fractur, whr a sourc trm K p is introducd to tak into account th contribution of th subdomain flows to th fractur flow [27]. For th sak of simplicity, w impos Dirichlt boundary conditions at th boundary Γ of th fractur Γ. Finally, following [27], w provid th intrfac conditions to coupl problms (3)-(4) and (6)-(7). Lt ξ b a positiv ral numbr, ξ 1 2, that will b chosn latr on. Th coupling conditions ar givn by whr 2{K p} n Γ = β Γ (p 1 p 2 ) on Γ, (8) K p = α Γ ({p} p Γ ) on Γ, (9) β Γ = 1 2η Γ, α Γ = 2 η Γ (2ξ 1) (10) and η Γ = l Γ KΓ n, KΓ n bing th normal componnt of th fractur prmability tnsor, s (5). Not that th coupling conditions ar formulatd mploying jump and avrag oprators. This turns out to b convnint for mploying DG mthods in th discrtization. In conclusion, th coupld modl problm rads: (K i p i ) = f i in Ω i, i = 1, 2, p i = g i on γ i, i = 1, 2, τ (KΓl τ Γ τ p Γ ) = f + K p in Γ, p Γ = g Γ on Γ, 2{K p} n Γ = β Γ (p 1 p 2 ) on Γ, K p = α Γ ({p} p Γ ) on Γ. (11) Not that th introduction of th paramtr ξ yilds a family of modls, s [27] for mor dtails.

9 8 2 Wak formulation and its wll-posdnss In this sction w prsnt a wak formulation of our modl problm (11) and prov its wll-posdnss. For th sak of simplicity w will assum that homognous Dirichlt boundary conditions ar imposd for both th bulk and fractur problms. Th xtnsion to th gnral cas is straightforward. W introduc th following spacs V b = {p = (p 1, p 2 ) V b 1 V b 2 }, V Γ = H 1 0 (Γ) H s (Γ), (12) whr w dfin, for i = 1, 2 and s 1, Vi b H0,γ 1 i (Ω i ) = {q H 1 (Ω i ) s.t. q γi = 0}. = H s (Ω i ) H 1 0,γ i (Ω i ), with Nxt w introduc th bilinar forms A b : V b V b R, A Γ : V Γ V Γ R and I : (V b V Γ ) (V b V Γ ) R, dfind as follows 2 A b (p, q) = K p i q i, i=1 Ω i A Γ (p Γ, q Γ ) = KΓl τ Γ p Γ q Γ, Γ I((p, p Γ ), (q, q Γ )) = β Γ p q + α Γ ({p} p Γ )({q} q Γ ). Γ Clarly, th bilinar forms A b (, ) and A Γ (, ) tak into account th problms in th bulk and in th fractur, rspctivly, whil I(, ) taks into account th intrfac conditions (8)-(9). W also introduc th linar functionals L b : V b R and L Γ : V Γ R dfind as 2 L b (q) = fq i, i=1 Ω i L Γ (q Γ ) = fq Γ, which rprsnt th sourc trms in th bulk and fractur, rspctivly. With th abov notation, th wak formulation of our modl problm rads as follows: Find (p, p Γ ) V b V Γ such that, for all (q, q Γ ) V b V Γ Γ Γ A ((p, p Γ ), (q, q Γ )) = L(q, q Γ ), (13) whr A : (V b V Γ ) (V b V Γ ) R is dfind as th sum of th bilinar forms just introducd: A ((p, p Γ ), (q, q Γ )) = A b (p, q) + A Γ (p Γ, q Γ ) + I((p, p Γ ), (q, q Γ )), (14)

10 9 and th linar oprator L : V b V Γ R is dfind as L(q, q Γ ) = L b (q) + L Γ (q Γ ). (15) Nxt, w show that formulation (13) is wll-posd. introduc th following norm on V b V Γ : To this aim w (q, q Γ ) 2 E = 2 i=1 K 1/2 i q i 2 L 2 (Ω i ) + (Kτ Γl Γ ) 1/2 q Γ 2 L 2 (Γ) + β Γ q 2 L 2 (Γ) + α Γ {q} q Γ 2 L 2 (Γ). (16) 2 This is clarly a norm if α Γ 0. Sinc α Γ = η Γ (2ξ 1), from now on, w will assum that ξ > 1/2. W rmark that th sam condition on th paramtr ξ has bn found also in [27] and [5]. Thorm 2.1. Lt ξ > 1/2. Thn, problm (13) is wll-posd. Proof. W show that A(, ) is continuous and corciv on V b V Γ quippd with th norm (16), as wll as L( ) is continuous on V b V Γ with rspct to th sam norm. Thn, xistnc and uniqunss of th solution, as wll as linar dpndnc on th data, follow dirctly from Lax-Milgram s lmma. Corcivity is straightforward, as w clarly hav that A((q, q Γ ), (q, q Γ )) = (q, q Γ ) 2 E for any (q, q Γ ) V b V Γ. On th othr hand, continuity is a dirct consqunc of Cauchy-Schwarz inquality, whil continuity of L( ) on V b V Γ is guarantd by th rgularity of th forcing trm f. 3 Numrical discrtization In this sction w prsnt a numrical discrtization of our problm which combins a Discontinuous Galrkin approximation for th problm in th bulk with a conforming finit lmnt approximation in th fractur (s Rmark 2 blow). DG mthods rsult to b vry convnint for handling th discontinuity of th bulk prssur across th fractur, as wll as th coupling of th bulk-fractur problms, which has bn formulatd using jump and avrag oprators. As a rsult, w can mploy th standard tools offrd by DG mthods to prov th wll-posdnss of our discrt mthod (Proposition 3.3). W start with th introduction of som usful notation. W considr a family of shap-rgular, not-ncssarily matching, simplicial mshs T h which ar alignd with th fractur Γ, so that any triangl E T h cannot b cut by Γ.

11 10 Not that, sinc Ω 1 and Ω 2 ar disjoint, ach lmnt E blongs xactly to on of th two subdomains. Clarly ach msh T h inducs a subdivision of th fractur Γ into dgs (dg mans fac for d = 3), that w will dnot by Γ h. Morovr, th st of all dgs of th dcomposition T h is dnotd by E h and w hav E h = E I h EB h Γ h, whr Eh B is th st of boundary dgs and EI h is th st of intrior dgs not blonging to th fractur. Not that, sinc th prsnc of hanging nods is prmittd, an intrior dg is dfind as th (non-mpty) intrsction of th boundaris of two nighbouring lmnts of T h. For ach lmnt E T h, w dnot by h E its diamtr and w st h = max E Th h E. W mak th assumption that, for all Eh I, if = E+ E, it holds h E + h and h E h, whr h is th diamtr of. Finally, givn an lmnt E T h, for any dg E w dfin n as th unit normal vctor on that points outsid E. W can thn dfin th standard jump and avrag oprators across an dg E h for (rgular nough) scalar and vctor-valud functions similarly to (1). Givn a partition T h of th domain, w dnot by H s (T h ), s 0, th standard brokn Sobolv spac. For s = 0 w writ L 2 (Ω) in plac of H 0 (T h ). With th aim of building a DG-conforming finit lmnt approximation, w choos to st th discrt problm in th finit-dimnsional spacs V b h = {q h L 2 (Ω) : q h E P k (E) E T h }, V Γ h = {qγ h C0 (Γ) : q Γ h P k () Γ h }. W rmark that th polynomial dgr in th bulk and fractur discrt spacs just dfind can b chosn indpndntly. Hr, for th sak of simplicity, w dnot both by k. Nxt, w introduc th bilinar forms A DG b : Vh b V h b R and I DG : (V b h V Γ h ) (V h b A DG b (p h, q h ) = E T E h V h Γ ) R, dfind as follows K p h q h {K p h } q h E h \Γ h E h \Γ h {K q h } p h + E h \Γ h σ p h q h, I DG ((p h, p Γ h ), (q h, qh Γ )) = β Γ p h q h + α Γ ({p h } p Γ h )({q h} qh Γ ). Γ h Γ h

12 11 Th cofficint σ is dfind by σ = σ0 for any β 0 EI h EB h. Th pnalty paramtr σ 0 > 0 and th positiv numbr β 0 will b dfind latr on. Finally w dfin th linar functional L DG b : Vh b R as L DG b (q h ) = fq h. E T E h Rmark 1. Sinc w ar imposing homognous boundary conditions L DG b has th sam structur of th linar functional L b prviously dfind. In gnral, for g 0, L DG b contains som additional trms: L DG b (q h ) = E T h E fq h + Eh B ( K q h n + σ q h )g. Th DG discrtization of problm (13) rads as follows: Find (p h, p Γ h ) V h b V h Γ such that A h ( (ph, p Γ h ), (q h, q Γ h )) = L h (q h, q Γ h ) (q h, q Γ h ) V b h V Γ h, (17) whr A h : (Vh b V h Γ) (V h b V h Γ ) R is dfind as ( A h (ph, p Γ h ), (q h, qh Γ )) = A DG b (p h, q h ) + A Γ (p Γ h, qγ h ) + IDG ((p h, p Γ h ), (q h, qh Γ )), (18) and L h : Vh b V h Γ R is dfind as L h (q h, qh Γ ) = LDG b (q h ) + L Γ (qh Γ ). (19) Not that th discrt bilinar form A h has th sam structur as th bilinar form A b,γ prviously dfind, bing th sum of thr diffrnt componnts, ach rprsnting a spcific part of th problm. Bfor proving that formulation (17) is wll-posd, w stat (and prov) som auxiliary rsults, s Lmma 3.1 and 3.2 blow. W primarily introduc th following norm on V b h V Γ h (q h, q Γ h ) 2 E h = q h 2 DG + q Γ h 2 Γ + (q h, q Γ h ) 2 I, (20) whr q h 2 DG = K 1/2 h q h 2 L 2 (Ω) + E h \Γ h σ 1/2 q h 2 L 2 (), qh Γ 2 Γ = (KΓl τ Γ ) 1/2 h qh Γ 2 L 2 (Γ), (q h, qh Γ ) 2 I = β Γ q h 2 L 2 () + α Γ {q h } qh Γ 2 L 2 (). Γ h Γ h

13 12 It is asy to show that DG is a norm if σ 0 > 0 for all and that I is a norm if α Γ 0 (that is ξ > 1/2). Nxt w rcall two wll-known trac inqualitis (s for xampl [6]) for a polynomial q P k (E) ovr an dg E, whr E is a shap-rgular lmnt: q L 2 () h 1/2 E q L 2 (E), (21) q n L 2 () h 1/2 E q L 2 (E). (22) Ths inqualitis ar instrumntal to prov th subsqunt rsults. Lmma 3.1. Th bilinar form A DG b (, ) is corciv and continuous on Vh b V h b with rspct to th norm DG. Proof. Th proof is rathr standard in th analysis of Discontinuous Galrkin mthods (s for xampl [29]). W rcall that in ordr to prov corcivity, w us th trac inqualitis (21)-(22) and th bounddnss of th prmability tnsor K, s (2), to show that, for any q h V b h, E T h E K q h q h 2 E h \Γ h E T h {K q h } q h + E K q h q h + E h \Γ h σ E h \Γ h σ q h q h q h q h. Not that th inquality holds if th stabilization cofficints σ 0 ar chosn larg nough and β 0 (d 1) 1. Lmma 3.2. Th bilinar form A Γ (, ) is corciv and continuous on Vh Γ V h Γ with rspct to th norm Γ Proof. Sinc A Γ (q Γ h, qγ h ) = qγ h 2 Γ q Γ h V Γ h, A Γ (, ) is clarly corciv. Continuity follows dirctly from Cauchy-Schwarz inquality. Employing Lmma 3.1 and Lmma 3.2, w can asily prov th wllposdnss of th discrt problm (17). Proposition 3.3. Problm (17) is wll-posd.

14 13 Proof. In ordr to apply Lax-Milgram s lmma, w show that A h (, ) and L h ( ) ar continuous and corciv on Vh b V h Γ quippd with th norm (20). In ordr to prov corcivity w considr for (q h, qh Γ) V h b V h Γ, th trm I DG ((q h, qh Γ), (q h, qh Γ )). Clarly w hav I DG ((q h, q Γ h ), (q h, q Γ h )) = (q h, q Γ h ) 2 I. Morovr from Lmma 3.1 and Lmma 3.2 w know that A DG b (q h, q h ) q h 2 DG and A Γ(qh Γ, qγ h ) = qγ h 2 Γ, rspctivly. Thrfor ( A h (qh, qh Γ ), (q h, qh Γ )) (q h, qh Γ ) 2 E h (q h, qh Γ ) V h b V h Γ. (23) Nxt w prov continuity. Lt (q h, qh Γ), (w h, wh Γ) V h b V h Γ. Thn, from Lmma 3.1, w know that A DG b (q h, w h ) q h DG w h DG (q h, q Γ h ) E h (w h, w Γ h ) E h. Similarly w hav from Lmma 3.2 that A Γ (q Γ h, wγ h ) qγ h Γ w Γ h Γ (q h, q Γ h ) E h (w h, w Γ h ) E h. Finally, from Cauchy-Schwarz inquality, w gt I DG ((q h, qh Γ ), (w h, wh Γ )) β Γ q h 2 L 2 () w h 2 L 2 () Γ h + α Γ In conclusion w hav provd that Γ h {q h } q Γ h 2 L 2 () {w h} w Γ h 2 L 2 () (q h, q Γ h ) E h (w h, w Γ h ) E h. A h ( (qh, q Γ h ), (w h, w Γ h )) (q h, q Γ h ) E h (w h, w Γ h ) E h. Th continuity of L h ( ) on Vh b V h Γ can b asily provd using Cauchy- Schwarz inquality, thanks to th rgularity assumptions on th forcing trm f. Rmark 2. Th choic of mploying a conforming finit lmnt approximation for th flow in th fractur has bn mad in ordr to kp th analysis of th numrical mthod as clar as possibl. W rmark that DG mthods could b mployd for th fractur problm, too. Howvr, th analysis in th lattr cas would rquir unncssary tchnical dtails without giving any dpr undrstanding of th problm.

15 14 4 Error stimats In ordr to prov rror stimats, w assum that th xact solution (p, p Γ ) blongs to th spac V b V Γ dfind in (12), and w assum that in th dfinition of Vi b, i = 1, 2, w hav s 2. W will show that th discrt solution (p h, p Γ h ) to problm (17) convrgs to th xact solution (p, p Γ ), driving an a priori stimat for th rror in th norm (20). To this aim w introduc th spac V b (h) Vh Γ = (V h b + V b ) Vh Γ ndowd with th norm (p, p Γ h ) 2 = p 2 DG + p Γ h 2 Γ + (p, p Γ h ) 2 I, (24) whr DG : V b (h) R is dfind as q DG = q 2 DG + h 2 E q 2 H 2 (E) E Th 1/2. (25) Using classical intrpolation stimats (s for xampl [7]), it can b shown that, for any p H k+1 (T h ), k 1, thr xists p I b V h b such that p p I b DG E T h h k E p H k+1 (E). (26) Analogously, for any p Γ H k+1 (Γ h ), k 1, thr xists p I Γ V h Γ such that p Γ p I Γ Γ h k p Γ H k+1 (). Γ h (27) Th nxt continuity rsult will b crucial for th proof of our a priori rror stimat. Lmma 4.1. Th bilinar form A DG b (, ) is continuous on V b (h) V b (h) quippd with th norm DG. Proof. Lt q, w V b (h), thn w hav A DG b (v, w) = K q w E T E h {K w} q + E h \Γ h = T 1 + T 2 + T 3 + T 4. E h \Γ h {K q} w E h \Γ h σ q w

16 15 W bound ach of th trms sparatly. Using Cauchy-Schwarz inquality, w hav T 1 E T h K 1/2 q L 2 (E) K 1/2 w L 2 (E) q DG w DG, T 4 E h \Γ h σ q L 2 () w L 2 () q DG w DG q DG w DG. In ordr to bound T 2 and T 3 w will mak us of th following wll-known trac inquality (s [6]), for a function v H s (E), s 2, and E: v n 2 L 2 () ( 1 v 2 H 1 (E) + v 2 H 2 (E) ). In fact combining th abov rsult with (2) it follows that, for vry q L 2 (), K w n q w n L 2 () q L 2 () and this lattr implis that T 3 = {K w} q E h \Γ h [ ] 1/2 w 2 H 1 (E) + h2 E w 2 1/2 H 2 (E) h E q L 2 (), w 2 H 1 (E) + h2 E w 2 H 2 (E) E Th 1/2 w DG q DG w DG q DG. E h \Γ h σ q 2 Similarly T 2 q DG w DG w DG q DG. Summing all th contributions concluds th proof. 1/2 W now hav all th ingrdints to prov th following rror stimat: Thorm 4.2. Lt (p, p Γ ) b th solution of problm (13) and lt (p h, p Γ h ) V h b V h Γ b its approximation obtaind with th mthod (17). Thn, if (p, p Γ ) H k+1 (T h ) H k+1 (Γ h ), k 1, it holds (p, p Γ ) (p h, p Γ h ) E h h k ( p H k+1 (T h ) + p Γ H k+1 (Γ)). (28)

17 16 Proof. First w obsrv that th bilinar form A h (, ) is continuous on (V b (h) Vh Γ) (V b (h) Vh Γ ) with rspct to th norm (24). Rcalling that A h ( (q, q Γ h ), (w, w Γ h )) = A DG b (q, w) + A Γ (q Γ h, wγ h ) + IDG ((q, q Γ h ), (w, wγ h )), w considr ach of th trms sparatly. Lmma 4.1 implis A DG b (q, w) q DG w DG (q, q Γ h ) (w, wγ h ). Using Lmma 3.2 w also know that for all (q, q Γ h ), (w, wγ h ) V b (h) V Γ h A Γ (q Γ h, wγ h ) (q, qγ h ) (w, wγ h ). Finally w considr th trm I((q, qh Γ), (w, wγ h )). W hav, using Cauchy- Schwarz inquality, that I((q, q Γ h ), (w, wγ h )) (q, qγ h ) I (w, w Γ h ) I (q, q Γ h ) (w, wγ h ). Combining all th prvious bounds w obtain that A h ( (q, q Γ h ), (w, w Γ h )) (q, q Γ h ) (w, wγ h ). Nxt, lt p I b and pi Γ b th two intrpolant functions of stimats (26) and (27), rspctivly. Thus, w hav (p p I b, p Γ p I Γ) 2 I β Γ p p I b 2 L 2 () Γ h + α Γ {p p I b } 2 L 2 () + α Γ p Γ p I Γ 2 L 2 (). Γ h Γ h Using wll known intrpolation rror stimats, w hav β Γ p p I b 2 L 2 () β Γ p p I b 2 L 2 ( E) Γ h Similarly, α Γ Γ h {p p I b } 2 L 2 () E Γ E Γ p p I b 2 L 2 ( E) Morovr, from (27) w know that α Γ Γ h p Γ p I Γ 2 L 2 () Γ h h 2k E Γ E Γ p Γ 2 H k+1 (). h 2(k+1/2) E p 2 H k+1 (E). h 2(k+1/2) E p 2 H k+1 (E).

18 17 If w assum that for any E T h and for any E it holds h h E, combining all prvious bounds with (26) and (27), w obtain that ( ) (p, p Γ ) (p I b, pi Γ) h k p H k+1 (T h ) + p Γ H k+1 (Γ). In conclusion, sinc w hav alrady provd that A h (, ) is corciv on th discrt spac Vh b V h Γ (s (23)), and sinc Galrkin orthogonality trivially holds, th thsis follows (s [7]). 5 Numrical rsults In this sction w prsnt som two-dimnsional numrical xprimnts to confirm th validity of th a priori rror stimats that w hav drivd for our mthod. Th numrical rsults hav bn obtaind in Matlab R. Throughout this sction w st th fractur thicknss (apparing in th coupling conditions (8)-(9)) qual to l Γ = = η Γ and KΓ τ = 1. Morovr in our tsts w tak th dgr k of polynomials in th bulk and fractur finit dimnsional spacs qual to 1. W hav also implmntd a vrsion of our mthod which is abl to dal with mshs mad of gnral polygonal lmnts. Th analysis of th aformntiond mthod, which is basd on th framwork dvlopd in [17, 4], will b th objct of a futur publication. For th gnration of polygonal mshs conforming to th fracturs w hav usd a modification of th Matlab R cod calld PolyMshr implmntd by G.C. Paulino and collaborators [30]. In ach of our xampls w also rport th rsults of th numrical xprimnts prformd on polygonal mshs. 5.1 Exampl 1 In this first tst cas w tak Ω = (0, 1) 2, and choos as xact solutions in th bulk and in th fractur Γ = {(x, y) Ω : x + y = 1} p = { x+y in Ω 1, x+y + 4η Γ 2 in Ω 2, p Γ = + 2η Γ 2. It is asy to prov that p and p Γ satisfy th coupling conditions (8)-(9) with ξ = 1 and K = I. Not that w nd to choos f = 0 on Γ sinc th solution is constant and K p = 0.

19 18 K1/2 (p ph ) L2 (Ω) p ph L2 (Ω) 10 1 Γ q Γ qh L2 (Γ) h h Figur 2: Exampl 1: Computd rrors as a function of th invrs of th msh siz (loglog scal). In Figur 2 w plot th computd rrors K1/2 (p ph ) L2 (Ω) and q Γ qhγ L2 (Γ) as a function of th msh siz (loglog scal). In both cass th numrical rsults ar in agrmnt with th thortical stimats. In Figur 2 w also rport th bhaviour of th rror p ph L2 (Ω). On ordr of convrgnc is clarly gaind. (a) msh 1 (b) msh 2 (c) msh 3 Figur 3: Exampl 1: Thr rfinmnts of th polygonal msh grid conforming to th fractur. Finally, in Figur 4 w rport th computd rrors for th mthod implmntd on polygonal msh grids. Th sam convrgnc rats as in th cas of triangular mshs ar achivd. In Figur 3 w also rport thr rfinmnts of a polygonal msh conforming to th fractur.

20 K 1/2 (p p h ) L 2 (Ω) p p h L 2 (Ω) q Γ qh Γ L 2 (Γ) h h Figur 4: Exampl 1: Computd rrors as a function of th invrs of th msh siz (loglog scal) on polygonal grids. 5.2 Exampl 2 Hr w considr again Ω = (0, 1) 2 and Γ = {(x, y) Ω : x+y = 1}. W tak as xact solutions in th bulk and in th fractur th following functions p = { x+y in Ω 1, x+y 2 + ( η Γ 2 ) in Ω 2, p Γ = (1 + 2η Γ ). W choos ξ = 1 and tak again K = I. In this cas w st th sourc trm as 2 x+y in Ω 1, f = 2 in Γ, x+y in Ω 2. Notic that on th fractur th sourc trm satisfis f Γ = τ (K τ Γl Γ τ p Γ ) + K p, and, sinc p Γ is constant, it must b f Γ = K p. Figur 5 shows th computd rrors K 1/2 (p p h ) L 2 (Ω) for th bulk problm and th computd rrors q Γ q Γ h L 2 (Γ) in th fractur. Again th validity of th rror stimats is confirmd. Morovr from Figur 5 on can clarly s that also in this tst cas on ordr of convrgnc is gaind if w comput th rror p p h L 2 (Ω) for th bulk problm.

21 20 1/ K (p ph ) L 2 (Ω) p p h L 2 (Ω) 10 2 q Γ q Γ h L 2 (Γ) h h Figur 5: Exampl 2: Computd rrors as a function of th invrs of th msh siz (loglog scal). 1/2 K (p ph ) L 2 (Ω) 10 1 p p h L 2 (Ω) q Γ qh Γ L 2 (Γ) h h Figur 6: Exampl 2: Computd rrors as a function of th invrs of th msh siz (loglog scal) on polygonal grids. In Figur 6 w plot th computd rrors on polygonal mshs, which ar in agrmnt with th sam convrgnc rats as thos xpctd for simplicial mshs.

22 Exampl 3 In this last xampl w considr th circular fractur Γ = {(x, y) Ω : x2 + y 2 = R}, with R = 0.7 includd in th domain Ω = (0, 1)2. W choos th xact solutions in th bulk and in th fractur as follows ( 2 2 x +y in Ω1, 7 ηγ 2 pγ = 1 + p = x2r+y2, 1 3 4R + R ηγ + 2 in Ω2, 2R2 so that thy satisfy coupling conditions sourc trm is chosn as 4 R2 f = R1 2 R2 (a) msh 1 (8)-(9) with ξ = (b) msh and K = I. Th in Ω1, in Γ, in Ω2. (c) msh 3 Figur 7: Exampl 3: Thr rfinmnts of th msh grid. Figur 7 rports thr rfinmnts of th triangular msh grid. On can s that th fractur is approximatd by a polygonal lin. Analogously, Figur 8 shows thr rfinmnts of th polygonal msh grid mployd in our xprimnts. In Figur 9 w hav plottd th computd rrors on triangular mshs for th problms in th bulk and in th fractur, which clarly validat th thortical stimats. Figur 10 shows th computd rrors for th mthod implmntd on polygonal msh grids.

23 22 (a) msh 1 (b) msh 2 (c) msh 3 Figur 8: Exampl 3: Thr rfinmnts of th polygonal msh grid. K1/2 (p ph ) L2 (Ω) p ph L2 (Ω) 10 1 Γ L2 (Γ) q Γ qh h h Figur 9: Exampl 3: Computd rrors as a function of th invrs of th msh siz (loglog scal). 6 Conclusions In this work w proposd a numrical approximation of Darcy s flow in a fracturd porous mdium basd on DG mthods. In particular w combind a DG approximation for th flow in th porous matrix with a conforming finit lmnt schm for th fractur flow. Th mthod was formulatd in th cas of a non-immrsd singl fractur, taking as a rfrnc th modl proposd in [27], rwrittn in primal form. W analysd th proprtis of our discrt mthod and provd its wll-posdnss and convrgnc. In particular w obtaind optimal a priori rror stimats for both th problm in th bulk and in th fractur. Finally, w carrid out two dimnsional numrical

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