Discontinuous Galerkin and Mimetic Finite Difference Methods for Coupled Stokes-Darcy Flows on Polygonal and Polyhedral Grids

Transcription

1 Discontinuous Galrkin and Mimtic Finit Diffrnc Mthods for Coupld Stoks-Darcy Flows on Polygonal and Polyhdral Grids Konstantin Lipnikov Danail Vassilv Ivan Yotov Abstract W study locally mass consrvativ approximations of coupld Darcy and Stoks flows on polygonal and polyhdral mshs. Th discontinuous Galrkin DG) finit lmnt mthod is usd in th Stoks rgion and th mimtic finit diffrnc mthod is usd in th Darcy rgion. DG finit lmnt spacs ar dfind on polygonal and polyhdral grids by introducing lifting oprators mapping mimtic dgrs of frdom to functional spacs. Optimal convrgnc stimats for th numrical schm ar drivd. Rsults from computational xprimnts supporting th thory ar prsntd. 1 Introduction Coupld Stoks-Darcy flows occur in various physical procsss of significant importanc. Blood motion in th vssls, th intraction of ground and surfac watr, and nginring filtration problms ar just a fw xampls that involv such flows. Our modl consists of a fluid whos motion is govrnd by th Stoks quation and a porous mdium saturatd by th sam fluid, in which th Darcy s law is valid. Th two quations ar coupld through transmission conditions that must b satisfid on th intrfac btwn th fr fluid rgion and th porous mdium rgion. Ths conditions ar continuity of flux and normal strss, as wll as slip with friction condition for th Stoks vlocity known as th Bavrs-Josph-Saffman law [5, 48]. In this papr, w considr th surfac-subsurfac watr flow as an application of th modl. Thr ar numbr of stabl and convrgnt numrical mthods dvlopd for th coupld Stoks-Darcy flow systm, s.g., [34, 19, 39, 45, 4, 5, 54, 41]. Oftn it is of intrst to study contaminant transport in such flows, which ncssitats mploying numrical schms that consrv mass locally. In this papr w us th discontinuous Galrkin DG) and th mimtic finit diffrnc MFD) mthods to discrtiz th Stoks and Darcy quations, rspctivly. Both mthods ar locally mass consrvativ. W considr vry gnral polygonal or polyhdral grids, as thy allow for modling complicatd gomtris with rlativly fw dgrs of frdom. Th local mass consrvation proprty of th DG mthod stms from th fact that discontinuous functions ar usd to approximat th solution on a givn msh. Th original DG mthod was introducd in th arly svntis for solving th nutron transport quation [43, 33]. Sinc that tim, svral DG schms hav bn introducd, including th Bassy-Rbay mthod [4], th intrior pnalty Galrkin mthods [, 18, 44, 53], th Odn-Babuška-Baumann mthod [4], and th local discontinuous Galrkin LDG) mthod [15]. A unifying DG framwork for lliptic problms is studid in [3]. DG mthods hav bn usd to solv a wid rang of problms, including comprssibl [4] and incomprssibl [38, 7, 45] fluid flows, magnto-hydrodynamics 1 Applid Mathmatics and Plasma Physics Group, Thortical Division, Mail Stop B84, Los Alamos National Laboratory, Los Alamos, NM 87545, mail: lipnikov@lanl.gov; partially supportd by th DO Offic of Scinc Advancd Scintific Computing Rsarch ASCR) Program in Applid Mathmatics Dpartmnt of Mathmatics, 301 Thackray Hall, Univrsity of Pittsburgh, Pittsburgh, PA 1560, mail: dhv1@pitt.du; yotov@math.pitt.du; partially supportd by th DO grant D-FG0-04R5618 and th NSF grant DMS

2 [5], and contaminant transport [18]. In [51], th LDG mthod is mployd for transport coupld with Stoks-Darcy flows. Th MFD mthod is a rlativly nw discrtization tchniqu originating from th support-oprator algorithms [49, 31]. Th mthod has bn succssfully applid to problms of continuum mchanics [40], lctromagntics [30], linar diffusion [31, 36], and rcntly fluid dynamics [6, 7]. Th goal of th MFD discrtization is to incorporat in th numrical modl th mathmatical and physical principls consrvation laws, solution symmtris) of th undrlying systm. This is achivd by approximating th diffrntial oprators in th govrning quations by discrt oprators that satisfy discrt vrsions of th fundamntal idntitis of vctor and tnsor calculus. Th MFD mthod can handl polygonal in -D and polyhdral in 3-D mshs with curvd boundaris and possibly dgnrat clls, which ar wll-suitd to rprsnt th irrgular faturs of th porous mdium. For simplicial and quadrilatral mshs, an quivalnc btwn th MFD mthod and th lowst ordr Raviart-Thomas MF mthod has bn stablishd in [8] and [9], rspctivly. For polyhdral mshs, a rlationship btwn th MFD mthod and th multipoint flux approximation MPFA) has bn studid in [37]. A strong connction btwn th MFD family of mthods and a family of th gradint-typ finit volum mthods [] and th mixd-finit volum mthods [0] has bn stablishd in [1]. In this papr, w formulat th DG mthod on polygonal or polyhdral mshs by using on of th MFD tools, a lifting oprator from mimtic dgrs of frdom to a functional spac. In particular, constant flux valus on acdg or fac) of an lmnt ar xtndd into a picwis linar function insid th lmnt. This allows us to formulat a DG-MFD mthod for coupld Stoks-Darcy flows on polygonal or polyhdral mshs. Th mthod is htrognous in th sns that discrt mimtic dgrs of frdom in th Darcy domain ar coupld with picwis polynomial finit lmnt spacs in th Stoks rgion. Th mshs from th two rgions may b non-matching on th intrfac and th continuity of flux condition is imposd through a Lagrang multiplir spac. This spac is dfind on an intrfac msh that is th trac of th msh of th Darcy rgion and it is also usd to approximat th normal strss on th intrfac. A global inf-sup condition is stablishd that implis th wll-posdnss of th coupld schm. For this w construct an intrpolant in th spac of DG-MFD vlocitis with wakly continuous normal componnts. W also stablish optimal ordr convrgnc for th approximat vlocity and prssur filds. Numrical calculations in -D ar prsntd to support th thory. Th papr outlin is as follows. In Sction w formulat th coupld problm. Its discrtization is prsntd in Sction 3. In Sction 4 w construct som intrpolants that will b usd in th analysis of th mthod. Sction 5 dals with th wll-posdnss of th mthod. rror stimats ar drivd in Sction 6. In Sction 7, w discuss som implmntation dtails and provid rsults from computational tsts that vrify th thortical rror bounds. Th coupld Stoks-Darcy problm Th following modl dscribs flow of incomprssibl fluid from a domain Ω 1 across an intrfac Γ I, into a porous mdium domain Ω. W assum that both Ω 1 and Ω ar Lipschitz polyhdral domains in R d, d =, 3, sparatd by a simply connctd intrfac Γ I s Fig. 1). Lt Γ k = Ω k \ Γ I, k = 1,, and n k b th xtrior unit normal vctor to Ω k. W dnot th fluid vlocity in domain Ω k by u k, th fluid viscosity by µ, and th prssur by p k. Th strss tnsor is givn by T 1 = p 1 I + µ Du 1 ), Du 1 ) = 1 u 1 + u T 1 ).

3 Γ 1 Γ 1 Γ n Ω 1 fluid rgion) Ω n saturatd porous mdium) 1 Γ I Γ 1 Γ Γ Figur 1: A -D modl of th coupld Stoks-Darcy flow. Flow in th Stoks domain is govrnd by th consrvation of momntum and mass laws. Considring no slip boundary conditions for simplicity, w hav div T 1 = f 1, u 1 = 0 in Ω 1, u 1 = 0 on Γ 1..1) Flow in th Darcy domain is govrnd by Darcy s law and th consrvation of mass law: u = K p, u = f in Ω, u n = 0 on Γ,.) whr for simplicity w assum no-flow boundary conditions. In th abov, K is a uniformly positiv dfinit and boundd full tnsor rprsnting th rock prmability dividd by th fluid viscosity. Th abov problms ar coupld across Γ I through thr intrfac conditions rprsnting mass consrvation, balanc of normal strss, and th Bavrs-Josph-Saffman law [5, 48]: u 1 n 1 = u n,.3) T 1 n 1 ) n 1 = p,.4) u 1 τ j = G j Du 1 )n 1 ) τ j, j = 1,, d 1,.5) whr τ j, j = 1,, d 1, is an orthonormal systm of tangntial vctors on Γ I. Condition.5) modls slip with friction, whr G j = µkτ j ) τ j /α and α > 0 is an xprimntally dtrmind friction constant. xistnc of a uniqu wak solution to th coupld problm.1).5) is shown in [34]. 3 Coupling of two discrtization mthods In this sction w dscrib coupling of two discrtization mthods, th discontinuous Galrkin DG) mthod in th Stoks domain and th mimtic finit diffrnc MFD) mthod in th Darcy domain. 3.1 Admissibl mshs Lt Ω h k b a partition of Ω k, k = 1, into polygonal in -D and polyhdral in 3-D lmnts with diamtr h. Th mshs may b non-matching on th intrfac Γ I. Lt h k = max Ω h k h. Hraftr, w shall us 3

4 th trm fac, dnotd by, for both a fac in 3-D and an dg in -D. W will dnot dgs in 3-D by l. Lt x, x, and x l b th cntroids of lmnt, fac, and dg l, rspctivly. Lt b th volum of and b th ara of fac. Lt C dnot a gnric constant indpndnt of h and. W assum that partitions Ω h k ar shap-rgular in th following sns. Dfinition 3.1 Th polygonal polyhdral) partition Ω h k is shap-rgular if aclmnt has at most N facs, whr N is indpndnt of h 1 and h. aclmnt is star-shapd with rspct to a ball of radius ρ h cntrd at point x, whr ρ is indpndnt of h 1 and h. Morovr, ach fac of and acdg l of in 3-D is star-shapd with rspct to a ball of radius ρ h cntrd at th point x and x l, rspctivly. Thus, C h d h d, C h d 1 h d ) Not that mshs with non-convx lmnts may b shap-rgular in this sns. Lt h k b th st of intrior facs of Ωh k. For vry fac, w dfin a unit normal vctor n that will b fixd onc and for all. If blongs to Γ k, w choos th outward normal to Ω k. If blongs to Γ I, w choos th outward normal to Ω. Lt n b th outward unit normal vctor to, so that χ n n is ithr 1 or Discrtization in th Stoks domain Lt D b a domain in R d and W s,p D), s 0, p 1, b th usual Sobolv spac [1] with a norm s,p,d and a sminorm s,p,d. Th norm and sminorm in th Hilbrt spac H s D) W s, D) ar dnotd by s,d and s,d, rspctivly. Th uclidan norm of algbraic vctors is dnotd by, i.. without a subscript. W xtnd th formulation in [7, 45] on simplicial lmnts to gnral polyhdra. Lt X 1 and Q 1 b Sobolv spacs for th vlocity and prssur, rspctivly, in th Stoks domain: X 1 = {v 1 L Ω 1 )) d : v 1 W,3/ )) d Ω h 1, v 1 = 0 on Γ 1 }, Q 1 = {q 1 L Ω 1 ) : q 1 W 1,3/ ) Ω h 1}. Th functions in X 1 and Q 1 hav doubl valud tracs on th intrior lmnt facs. Th trac inquality and th Sobolv imbdding thorm imply th q 1 L ). For a function w, w dfin its avrag {w} and its jump [w] across an intrior fac 1 h as follows: {w} = 1 w w, [w] = w 1 w, whr 1 and ar two lmnts that shar fac and such that n is dirctd from 1 to. For Ω h 1, th avrag and th jump ar qual to th valu of w. W introduc th norms v 1 0,Ω 1 = v 1 0,, v 1 X 1 = v 1 0,Ω 1 + h 1 Γ 1 Ω h 1 [v 1 ] 0, + Γ I q 1 Q1 = q 1 0,Ω1, d 1 j=1 µ G j v 1 τ j 0,, 4

5 whr > 0 is a paramtr that is a constant on. Th DG mthod is basd on th bilinar forms a 1 : X 1 X 1 R and b 1 : X 1 Q 1 R dfind as follows: a 1 u 1, v 1 ) = µ Du 1 ) : Dv 1 ) dx + [u 1 ] [v 1 ] ds Ω h 1 1 h Γ 1 µ {Du 1 )}n [v 1 ] ds + µε {Dv 1 )}n [u 1 ] ds h 1 Γ 1 + d 1 Γ I b 1 v, q) = Ω h 1 µ G j=1 j h 1 Γ 1 u 1 τ j )v 1 τ j ) ds, u 1, v 1 X 1 q 1 div v 1 dx + h 1 Γ 1 {q 1 }[v 1 ] n ds, v 1 X 1, q 1 Q 1. Th jump trm involving is addd for stabilization. W assum that for all facs σ 0 > 0, 3.) whr σ 0 is chosn to b sufficintly larg according to Lmma 5.3 in ordr to guarant th corcivity of a, ). Th paramtr ε controls th symmtry of th bilinar form and taks valu 1, 0 or 1 for th symmtric intrior pnalty Galrkin SIPG) [, 53], th incomplt intrior pnalty Galrkin IIPG) [18], and th non-symmtric intrior pnalty Galrkin NIPG) [4, 44] mthods, rspctivly. Following closly th -D proof in Lmma.5 of [45], w obtain th following rsult. Lmma 3.1 Th solution u, p) = u 1, u ; p 1, p ) to.1).3) satisfis a 1 u 1, v 1 ) + b 1 v 1, p 1 ) + p v 1 n 1 ds = f 1 v 1 dx, v 1 X 1, 3.3) Γ I Ω 1 b 1 u 1, q 1 ) = 0, q 1 Q ) Th cas of simplicial lmnts has bn studid xtnsivly in th litratur. Lt P r dnot th spac of polynomials of dgr at most r. Th DG discrt spacs X1 h and Qh 1 for th vlocity and prssur, rspctivly, ar dfind as X h 1 = {v h 1 : v h 1 P r )) d Ω h 1}, Q h 1 = {q h 1 : q h 1 P r 1 ) Ω h 1}. W considr th cass r = 1,, 3 in -D and r = 1 in 3-D. To dvlop a lowst ordr r = 1) DG mthod for gnral polyhdra, w follow th mimtic approach and considr a lifting oprator from dgrs of frdom dfind on msh facs to a functional spac. For vry lmnt and vry fac of, w associat d dgrs of frdom a vctor in R d ) rprsnting th man vlocity on : V1, = 1 v 1 ds. Lt X h 1,MF D b th vctor spac with th abov dgrs of frdom. For a vctor V 1 X h 1,MF D, lt V 1, b its rstriction to lmnt. On ach, w dfin a lifting oprator R 1, acting on a vctor V 1, and rturning a function in H 1 )) d. W impos th following two proprtis on th lifting oprator: 5

6 L1) Th man valu of th liftd function on facs of is qual to th prscribd dgrs of frdom: 1 R 1, V 1, ) ds = V 1,. L) Th lifting oprator is xact for linar functions. Mor prcisly, if V1, L valus of a linar function v1 L, thn R 1, V1,) L = v1 L. is th vctor of fac man Using th lmntal lifting oprators R 1,, w dfin th following finit lmnt spacs: X h 1,LIF T = {v h : v h = R 1, V 1, ), Ω h 1, V 1, X h 1,MF D )}, Q h 1,LIF T = {q h : q h P 0 ), Ω h 1 }. Whn is a ttrahdron, th lifting oprator can b chosn to b th lowst ordr Crouzix-Raviart finit lmnt [17]. In this cas, th DG spacs X1 h Qh 1 coincid with Xh 1,LIF T Qh 1,LIF T. A constructiv mthod for building a lifting oprator for a polyhdron is prsntd in Sction 4. Th spacs X1,LIF h T Qh 1,LIF T ar nw DG spacs for Stoks on polygons or polyhdra. To kp th notation simpl, for th rst of th papr w will dnot th DG spacs for both simplicial and polyhdral lmnts by X1 h Qh 1, Rmark 3.1 Du to proprty L1), th DG spacs on polygons and polyhdra dfind abov hav continuous fluxs. This is dsirabl whn th computd Stoks flow fild is coupld with a transport quation. W ar now rady to formulat th DG mthod in Ω 1. Givn an approximation λ h of p on Γ I to b dfind latr), th DG solution on Ω 1, u h 1, ph 1 ) Xh 1 Qh 1, satisfis a 1 u h 1, vh 1 ) + b 1v1 h, ph 1 Γ ) + λh v1 h n 1 ds = f 1 v1 h dx, v1 h X1 h, 3.5) I Ω 1 b 1 u h 1, qh 1 ) = 0, qh 1 Qh ) 3.3 Discrtization in th Darcy domain Lt X and Q b th Sobolv spacs for th vlocity and prssur in Ω, rspctivly, dfind as follows: X = {v L s Ω )) d, s > : div v L Ω ), v n = 0 on Γ }, Q = L Ω ). W introduc th following L -norms: v X = v 0,Ω, q Q = q 0,Ω. It is asy to s that th solution to.1).5) satisfis K 1 u v dx p div v dx + p v n ds = 0, v X, 3.7) Ω Ω Γ I q div u dx = f q dx, q Q. 3.8) Ω Ω Not that th boundary intgral in 3.7) is wll dfind if p H 1 Ω ). 6

7 W us th mimtic finit diffrnc mthod [13, 14] to dfin discrt forms of 3.7) 3.8). Th first stp in th MFD mthod is th dfinition of dgrs of frdom. For ach fac in Ω h, w prscrib on dgr of frdom V rprsnting th avrag flux across. Lt Xh b th vctor spac with ths dgrs of frdom. Th dimnsion of X h is qual to th numbr of facs in Ωh. For any v X, w dfin its intrpolant v I Xh by v) I = 1 v n ds. 3.9) Lmma.1 in [37] guarants th xistnc of this intgral for vry v X. For any V X h, lt V, dnot th vctor of dgrs of frdom associatd only with an lmnt. W dnot its componnt associatd with fac by V,. To approximat th prssur, on aclmnt Ω h, w introduc on dgr of frdom P, rprsnting th avrag prssur on. Lt Q h b th vctor spac with ths dgrs of frdom. Th dimnsion of Q h is qual to th numbr of lmnts in Ωh. For any p Q, w dfin its intrpolant p I Qh by p I ) = 1 p dx. 3.10) W also nd to dfin a discrt mimtic spac for th approximation of th prssur on th intrfac Γ I. This spac will also srv th rol of a Lagrang multiplir spac for imposing th continuity of normal flux across Γ I. For ach fac Γ h I = Ωh Γ I w introduc on dgr of frdom λ rprsnting th avrag prssur on. Lt Λ h I b th vctor spac with ths dgrs of frdom. Not also that Λh I = Xh Γ I and its dimnsion is qual to th numbr of facs of Γ I. Th scond stp in th MFD mthod is to quip th discrt spacs Q h, Xh, and Λh I with innr products. Th innr product in th spac Q h is rlativly simpl: [P, Q] Q h = P Q, P, Q Q h. 3.11) Ω h This innr product can b viwd as a mid-point quadratur rul for L -product of two scalar functions. Th innr product in X h can b dfind formally as [U, V] X h = U T M V, U, V X h, 3.1) whr M is a symmtric positiv dfinit matrix. It can b viwd as a quadratur rul for th K 1 - wightd L -product of two vctor functions. Th mass matrix M is assmbld from lmnt matrics M, : U T M V = U T M, V. Ω h Th symmtric and positiv dfinit matrix M, inducs th local innr product [U, V ] X h, = UT M, V. 3.13) Th construction of matrix M, for a gnral lmnt is at th hart of th mimtic mthod [14]. Th innr product in Λ h I is dfind as λ, µ Λ h = λ µ, λ, µ Λ h I I. 3.14) Γ h I 7

8 Sinc V ΓI Λ h I for vry V Xh, 3.14) can also b usd to dfin V, µ Λ h: I V, µ Λ h = V µ, V X I h, µ Λh I. Γ h I Th third stp in th mimtic mthod is discrtization of th gradint and divrgnc oprators. Th dgrs of frdom hav bn slctd to provid a simpl approximation of th divrgnc oprator. Th Gauss divrgnc thorm naturally lads to th following formula: DIV V) = 1 χ V. 3.15) W hav a usful commutativ proprty of th intrpolants: DIV v I ) = 1 v n ds = 1 div v dx = div v) I, v X. 3.16) Th discrt gradint oprator must b a discrtization of th continuous oprator K. To provid a compatibl discrtization, th mimtic mthod drivs this discrt oprator from a discrt Gauss-Grn formula: [U, GRAD P, λ)] X h = [DIV U, P] Q h U, λ Λ h I U X h, P Q h, λ Λ h I. This quation mimics th continuous Gauss-Grn formula u K 1 K p) dx = p div u dx p u n dx, u X, p H 1 Ω ). Ω Ω Γ I Non-homognous vlocity boundary conditions would rquir additional trms that rprsnt non-zro boundary trms in th continuous Gauss-Grn formula [9]. Th construction of an admissibl matrix M, is basd on th consistncy condition s [14] for dtails). Lt K b th man valu of K on lmnt. Thn, w rquir [V, K p l ) I ] X h, = [DIV V, pl ) I ] Q h, χ V p l ds, p l P 1 ). 3.17) Th introducd innr products dfin th following norms: P Q h = [P, P ] Q h and V X h = [V, V ] X h. Lmma 3. [14]). Undr assumptions of Dfinition 3.1, thr xists th local innr product 3.13) such that 1 C V [V, V ] X h, C V, 3.18) whr th constant C dpnds only on shap rgularity of th auxiliary partition of. In th following, for consistncy btwn th DG and th mimtic notations, w will dnot a vctor V X h by vh, a vctor Q Q h by qh, and a vctor λ Λh I by λh. Givn an approximation λ h Λ h I of p on Γ I, th mimtic approximation of 3.7) 3.8) rads: Find u h, ph ) Xh Qh such that a u h, vh ) + b v h, ph ) + vh, λh Λ h I = 0, v h Xh, 3.19) b u h, qh ) = [f I, qh ] Q h, qh Qh, 3.0) whr a u h, v h ) = [u h, v h ] X h and b v h, q h ) = [DIV v h, q h ] Q h. 8

9 3.4 Discrt formulation of th coupld problm In th prvious two subsctions w prsntd partially coupld discrtizations for th Stoks and th Darcy rgions, 3.5) 3.6) and 3.19) 3.0), rspctivly. Th approximations λ h and λ h of p on intrfac Γ I ar appard in 3.5) and 3.19), rspctivly. W impos th continuity of normal strss condition.4) by taking λ h to b th picwis constant function on Γ h I satisfying λ h = λ h ), Γ h I. W impos th continuity of th flux.3) in a wak sns, using Λ h I as th Lagrang multiplir spac. Th wak continuity is mbddd in th dfinition of th global vlocity spac. Mor prcisly, lt X h = X1 h Xh, Qh = Q h 1 Qh, and { } V h = v h X h : v1 h n 1 µ h ds + v, h µ h Λh = 0, µ h Λ h I I. 3.1) Γ I W also dfin th composit bilinar forms au h, v h ) = a 1 u h 1, vh 1 ) + a u h, vh ), uh, v h X h, bv h, q h ) = b 1 v h 1, qh 1 ) + b v h, qh ), vh X h, q h Q h. Th wak formulation of th coupld problm is: find th pair u h, p h ) V h Q h such that au h, v h ) + bv h, p h ) = f 1 v1 h dx, v h V h, 3.) Ω 1 bu h, q h ) = [f I, qh ] Q h, qh Q h. 3.3) Rmark 3. W usd a lifting oprator from dgrs of frdom to a functional spac to dfin th DG spacs for th Stoks domain. A similar lifting oprator can b usd to dfin th MFD mthod in th Darcy domain as a finit lmnt mthod. 4 Trac inqualitis and intrpolation rsults Throughout this articl, w us a fw wll known inqualitis. Th Young inquality rads: ab ɛ a + 1 ɛ b, a, b 0, ɛ > ) A numbr of trac inqualitis utilizd in [45] on triangular mshs can b xtndd to polyhdral mshs using th auxiliary partition of an lmnt into shap-rgular simplics. In particular, for any fac of lmnt, w hav φ 0, C h 1 φ 0, + h φ 1,), φ H 1 ), 4.) and its immdiat consqunc φ n 0, C h 1 φ 1, + h φ,), φ H ). 4.3) For polynomial functions, w hav th trac inquality φ n 0, Ch 1/ φ 1,, φ P r ). 4.4) For φ H s )), 0 s 1, with div φ L ) w us Lmma 3.1 from [37] that givs φ n s 1, C h 1 φ 0, + h s 1 φ s, + h div φ 0,). 4.5) Th proof of th following lmma givs a constructiv way for building a lifting oprator. 9

10 Lmma 4.1 For vry lmnt Ω 1 h, thr xists a lifting oprator R 1, satisfying L1) and L) such that R 1, V 1, ) m, Ch d m V 1,, V 1,, 4.6) whr m = 0, 1. Morovr, th liftd function satisfis th trac inquality 4.4) for vry fac of. Proof. W considr an auxiliary partition of lmnt into simplxs. For vry fac of, w connct its cntroid x with its vrtics. This splits boundary into pics t k that ar triangls in 3-D or sgmnts in -D. Th auxiliary simplicial partition is obtaind by conncting th cntroid x with th points x and th vrtics of. Du to th msh assumptions in Dfinition 3.1, this is a shap rgular partition. W construct a continuous picwis linar lifting function R 1, V 1, ). Proprty L1) givs th following systm of linar quations for th valus of R 1, V 1, ) at th nods of th auxiliary partition on : 1 d, t k R 1, V 1, ))a i k d ) = V 1,, t k i=1 whr a i k ar th vrtics of t k. Sinc th unknowns associatd with vrtics x ar not connctd to ach othr and thir numbr is qual to th numbr of quations, th matrix of this systm has a full rank. Thrfor, thr xists a family of solutions, whr th unknowns corrsponding to th lmnt cntroid x and th vrtics of ar fr paramtrs. W populat th fr paramtrs by th valus of a vctor linar function Lx) that minimizs th quadratic functional Lx ) V1,. This dfins a continuous picwis linar function R 1, V 1, )x). Proprty L1) is satisfid by construction. Proprty L) also holds, sinc, if V1, = 1 v1 L ds for a linar vctor v1 L, thn Lx) = vl 1 is th minimizr of quadratic functional. Th lattr follows from th fact that for all facs v1 L x ) = 1 v1 L ds = V 1,. Th shap rgularity of implis that th fr paramtrs ar boundd by C V 1,. Th shap rgularity of t k and implis that / t k C. Thus, th valus of th liftd function at points x ar boundd by th sam norm. W hav R 1, V 1, ) 0, C h d max x R 1,V 1, )x) C h d V 1,. Th stimat for th gradint of th liftd function follows from th invrs inquality and th shap rgularity of th auxiliary partition. Finally, th shap rgularity of th auxiliary partition implis that th trac inquality 4.4) holds for vry t k and hnc for vry fac. This provs th assrtion of th lmma. Lmma 4. Lt v 1 H 1 Ω 1 )) d. Thr xists an intrpolant π 1 h : H1 Ω 1 )) d X h 1 such that b 1 πh 1 v 1) v 1, q h ) = 0, q h Q h 1, 4.7) [π1 h v 1 ] w ds = 0, w P r 1 )) d, 4.8) 10

11 for vry fac h 1 Γ 1, and π h 1 v 1 ) 1,Ω1 C v 1 1,Ω1. 4.9) Th intrpolant has optimal approximation proprtis for v 1 H s Ω 1 )) d, 1 s r + 1: π h 1 v 1 ) v 1 m, Ch s m v 1 s, δ), m = 0, 1, 4.10) whr ithr δ) is th union of with all its closst nighbors in th cas of simplics or δ) = in th cas of th liftd DG spacs on polygons and polyhdra. Furthrmor, th following stimats hold for v 1 H s Ω 1 )) d, 1 s r + 1: πh 1v 1) v 1 X1 Ch s 1 1 v 1 s,ω1, 4.11) πh 1v 1) X1 C v 1 1,Ω1. 4.1) Proof. On triangls for r = 1,, 3 and ttrahdra for r = 1 th xistnc of such an intrpolant is shown in [17, 3, 16, 44, 45]. It rmains to considr th cas of polygonal and polyhdral mshs with r = 1. Lt v 1 H 1 Ω 1 )) d and lt V 1 b th corrsponding vctor of dgrs of frdom. W introduc th intrpolant π1 h such that π1 hv 1) = R 1 V 1 ). Thn, for vry q h Q h 1, lifting proprty L1) givs b 1 π1 h v 1 ) v 1, q h ) = q R 1, V 1, ) v 1 ) n ds = ) Ω h 1 Du to lifting proprty L1), w immdiatly gt condition 4.8) with w P 0 )) d. To show 4.9), lt v1 c b th L -orthogonal projction of v 1 onto th spac of picwis constant functions on Ω h 1. Thn, w hav v c 1 0, v 1 v c 1 0, + v 1 0, Ch v 1 1, + v 1 0, C v 1 1,. For vry lmnt, th triangl inquality and lifting proprtis L) and 4.6) giv π1 hv 1) 0, πh 1 v 1 v1 c) 0, + vc 1 0, ) 1 C v 1 v c 1 ds + v 1 Applying th trac inquality 4.) to ach componnt of v 1 and using th standard approximation proprty of th L -projction, w bound ach of th dg intgrals: v 1 v1 ds) c v 1 v1 c ds ) 4.14) C h 1 v 1 v1 c 0, + h v 1, C h v 1,. Combining th last two inqualitis and using th shap rgularity of 3.1), w gt ) π1 h v 1 ) h 0, C v 1, + v 1, C v 1,. To bound th H 1 -sminorm of π h 1 v 1), w us 4.6) to obtain π h 1 v 1 ) v c 1 1, Ch d V 1, V c 1, Ch d 11 1, ) 1 v 1 v c 1 ds),.

12 whr V1, c is th vctor of dgrs of frdom for th constant function vc 1. Combining th abov inquality with 4.14), and using th shap rgularity of 3.1), w conclud that π1 hv 1) 1, C v 1 1,, which complts th proof of 4.9). Sinc L) implis that π1 h is xact for all linar functions on, an application of th Brambl-Hilbrt lmma [11] givs 4.10). It rmains to show 4.11) and 4.1). Not that L1) implis that for all facs of π1 h v 1 v 1 ) ds = 0, v 1 H 1 )) d. Thrfor w can mploy Lmma 3.9 of [44] to conclud that π 1 h v 1) v 1 X1 C π 1 h v 1) v 1 ) 0,Ω1, which, combind with 4.10), implis 4.11). Th continuity bound 4.1) follows from th triangl inquality, 4.11), and th bound v 1 X1 C v 1 1,Ω1. This provs th assrtion of th lmma. 5 Stability and wll-posdnss of th discrt problm In this sction w prov a discrt inf-sup condition and show that th discrt problm 3.) 3.3) has a uniqu solution. Lt X = X 1 X and Q = Q 1 Q. W introduc th composit norms q h Q h = q h 1 0,Ω 1 + q h Q h, qh = q h 1, qh ) Qh, v h X h = v h 1 X 1 + v h div, vh = v h 1, vh ) Xh, whr v h div = vh X h + DIV v h, v Q h h X h. Lmma 5.1 Lt v H 1 Ω)) d and v i = v Ωi, i = 1,. Thn, thr xists an oprator π h : X H 1 Ω)) d V h, π h v) = π h 1 v 1), π h v )), such that bπ h v) v, q h ) = 0, q h Q h, 5.1) and π h 1 v 1 ) X1 C v 1 1,Ω1, π h v ) X h C v 1,Ω. 5.) Proof. Lt π h 1 b th oprator dfind in Lmma 4.. Th proprty 4.7) givs 5.1) for any qh = q h 1, 0). Du to 4.1), w gt automatically th first inquality in 5.). To construct π h v ), w solv th following boundary valu problm: ϕ = 0 in Ω, ϕ n = 0 on Γ, ϕ n = v π h 1 v 1)) n 1 on Γ I, and dfin π h v ) = v I + ϕ)i. By lliptic rgularity [35], 5.3) ϕ H θ Ω ) C v π1 h v 1 )) n 1 H θ 1/ Γ I ), 0 θ 1/. 5.4) For all q h Qh, using dfinition of πh and th commutativ proprty 3.16), w gt b π h v) v I, q h ) = b ϕ) I, q h ) = [DIV ϕ) I, q h ] Q h = [ ϕ) I, q h ] Q h = 0. 1

13 To prov th scond inquality in 5.), w start with th triangl inquality π h v) X h v I X h + ϕ) I X h 5.5) and bound vry trm. From th stability stimat 3.18), th trac inquality 4.), and th shap rgularity stimats 3.1), w obtain v I = [v X, I v] I h X h C v I ) Ω h C Ω C Ω h C v 1,Ω. h 1 v 0, + h v ) 1, v 0, + h v 1,) 5.6) Using th sam argumnts plus inquality 4.5) with s = 1/, w gt ϕ) I X h C Ω h C C Ω ) 1 ϕ n ds ϕ 0,Ω + h ϕ 1,Ω h 1 ϕ 0, + ϕ 1, ) ). 5.7) To bound th first and th scond trm on th right hand sid in 5.7) w apply 5.4) with θ = 0 and θ = 1/, rspctivly: ) ϕ) I C v X h 1 π1 h v 1 )) n 1 + h v 1 1 π h,γi 1 v 1 )) n 1 0,ΓI 5.8) C v 1 π1 h v 1 )) n 1 0,Γ I. Using th trac inquality 4.) for vry Γ h I and th approximation rsult 4.10), w hav that ) v 1 π1 h v 1 )) n 1 L ) C h 1/ v 1 π1 h v 1 ) 0, + h 1/ v 1 π1 h v 1 ) 1, Thus, ϕ) I X h Ch s 1/ v 1 s,δ), 1 s r ) Ch s 1/ 1 v 1 s,ω1, 1 s r ) Combining 5.5) witstimats 5.6) and 5.10), w conclud that π hv) X h C v 1,Ω. It rmains to show that π h v) V h. Lt µ h Λ h I. From dfinition of th innr product 3.14), dfinition of th intrpolant 3.9), th boundary conditions in 5.3), and th rgularity assumption v H 1 Ω)) d, it 13

14 follows that π h v, µ h Λ h = v I I, µ h Λ h + ϕ) I, µ h I Λ h I = µ h ) v n ds + µ h ) ϕ n ds Γ h I Γ h I = v n µ h ds + Γ I v 1 n 1 µ h ds Γ I π1 h v 1 ) n 1 µ h ds Γ I = π1 h v 1 ) n 1 µ h ds. Γ I Thrfor π h v) V h. This provs th assrtion of th lmma. Lmma 5. Thr xists a positiv constant β such that inf q h Q h sup b 1 v1 h, qh 1 ) + b v h, qh ) v h V h v h X h q h Q h β. 5.11) Proof. For a givn q h Q h, lt us dfin w L Ω) by w = w 1, w ), whr w 1 = q h 1 and w = q h ), Ω h. Not that w I = qh and w 0,Ω = q h Q h. W can construct construct v H1 Ω)) d [6] for which div v = w and v 1,Ω C w 0,Ω. 5.1) Lt π h v) = π h 1 v 1), π h v )) b th intrpolant constructd in Lmma 5.1. Using 5.1) and th commutativ proprty 3.16), w gt b 1 π1 h v 1 ), q1 h ) + b π h v ), q h ) = b 1 v 1, q1 h ) + b v, I q h ) = div v 1 ) q1 h dx [DIV v, I q h ] Q h Ω 1 = q1 h 0,Ω 1 + q h = q h Q h Q. h 5.13) Th dfinition of π h and 3.16) imply that DIV π h v)) = DIV v I + ϕ) I ) = div v ) I + ϕ) I = q h. Using stimat 5.) from Lmma 5.1), w bound π h v): π h v) X = π1 h v 1 ) X 1 + π h v ) + DIV π h X h v )) Q h ) C v 1,Ω + q h Q h ) C q1 h 0,Ω + q h C q h Q Q. h 5.14) Combining 5.13) and 5.14) yilds b 1 π h 1 v 1 ), q h 1 ) + b π h v ), q h ) C π h v) X q h Q, 5.15) which provs th assrtion of th lmma. To prov that th mthod is wll-posd w nd th corcivity proprty stablishd in th nxt lmma. 14

15 Lmma 5.3 Assuming 3.), thr xists a positiv constant α c dpndnt on σ 0 but indpndnt of h 1 such that a 1 v h 1, v h 1) α c v h 1 X h 1, v h 1 X h ) Proof. Lt v1 h Xh 1. From th dfinition of a 1, ) w hav a 1 v1, h v1) h = µ Dv1) h : Dv1) h dx + µ1 ε) Ω h 1 1 h Γ 1 h 1 Γ 1 {Dv1) h n } [v1] h ds + d 1 Γ I µ [v h 1] [v h 1] ds G j=1 j v h 1 τ j )v h 1 τ j ) ds. Sinc v1 h is continuous and picwis linar on a shap rgular auxiliary partition of, th following Korn s holds [10]: v1 h 0,Ω 1 K 0 Dv 1) h 0,Ω [v h h 1] 0,, v1 h X1 h. 5.17) Thus, a 1 v h 1, v h 1) µ K 0 v h 1 1,Ω 1 + µ1 ε) 1 h Γ 1 h f Γ 1 h 1 Γ 1 µ [v h 1] 0, {Dv1) h n } [v1] h ds + d 1 µ v1 h τ j G 0,. j Γ I Clarly, th corcivity proprty holds whn ε = 1 and σ 0 = µ/α for som 0 < α < 1. To addrss th cas whn ε = 1 or 0, w us th trac inquality 4.4) and th Young s inquality 4.1) to stimat th third trm. Lt b th lmnt with fac. Thn, {Dv1) h n } [v1] h ds C 1 v1 h [v1] h 0, C v C 1 h 0, + C C 3 [v 3 h 1] h 0,. Thn, a 1 v1, h v1) h µ C ) 1 ε) v K 0 C 1 h 1,Ω µ + C C 3 1 ε))) h 1 Γ 1 [v h 1 ] 0, + Γ I j=1 d 1 j=1 µ G j v h 1 τ j 0,. Stting C 3 = K 0 C nsurs that th first trm is positiv for both ε = 0 and ε = 1. Thn to control th scond trm it is sufficint to choos σ 0 = µ1 + C C 3 )/α = µ1 + K 0 C )/α for som 0 < α < 1. Thorm 5.1 Th problm 3.) 3.3) has a uniqu solution. Proof. It is sufficint to show that solution of th homognous problm 3.) 3.3) is zro. By choosing v h = u h and q h = p h w gt a 1 u h 1, u h 1) + a u h, u h ) = 0, which combind with 5.16) and 3.18) implis that u h = 0. Th rmaindr of 3.) togthr with th inf-sup condition 5.11) imply that p h = 0. 15

16 6 rror analysis Lt th pair u, p) b th solution to.1).3) and lt u i = u Ωi, i = 1,. W dfin functions ũ V h and p Q h as follows: ũ = ũ 1, ũ ) = π h 1 u 1 ), π h u )), p = p 1, p ), whr π h is th oprator introducd in Lmma 5.1), p = p I Qh is th intrpolant of p introducd in 3.10) and p 1 is th L -projction of p 1 : p 1 p 1 ) q 1 dx = 0, q 1 P r 1 ), Ω h ) For any p 1 H s Ω 1 ) w hav th approximation rsult: p 1 p 1 m, Ch s m p 1 s,, m = 0, 1, 1 s r. 6.) W also nd th following approximation rsult [11]: for any φ H s ), 1 s, thr xists a linar function φ 1 such that φ φ 1 m, Ch s m φ s,, m = 0, ) Applying 4.) to φ φ 1 and using 6.3), w obtain th stimat for fac : Similarly, 4.) and 4.10) imply that φ φ 1 0, C h s 1 φ s,. 6.4) u 1 ũ 1 0, C h s 1 u 1 s,δ), 1 s r ) Lt K b a picwis constant tnsor qual to K on lmnt. Rcall that K is th man of valu of K on. W assum that K W 1, )) d d, Ω h, and that max Ω h K 1,, is uniformly boundd indpndnt of h, whr K 1,, = max 1 i,j d K i,j W 1, ). From Taylor s thorm it follows that max K ijx) K,ij Ch K ij W x 1, ). 6.6) 6.1 rror quation Subtracting th variational quations 3.3) 3.4) from th discrt quation 3.) 3.3), w obtain a 1 u h 1 u 1, v1) h + b 1 v1, h p h 1 p 1 ) p v1 h n 1 ds Γ h I + a u h, v h ) + b v h, p h ) = 0, v h V h, 6.7) b 1 u h 1 u 1, q1 h ) + b u h, q h ) = [f I, q h ] Q h, q h Q h. If w tak q1 h = 0 in th scond quation, w rcovr th wak form of th mass balanc quation for th Darcy rgion 3.0). Using this, plus adding and subtracting ũ 1, p 1, and u I in th appropriat trms of 6.7), 16

17 w obtain a 1 u h 1 ũ 1, v h 1) + b 1 v h 1, p h 1 p 1 ) + a u h u I, v h ) + b v h, p h ) = a 1 u 1 ũ 1, v1) h + b 1 v1, h p 1 p 1 ) + p v1 h n 1 ds a u I, v), h v h V h, Γ h I 6.8) 6. Vlocity stimat b 1 u h 1 ũ 1, q h 1 ) = b 1 u 1 ũ 1, q h 1 ), q h Q h. Thorm 6.1 Lt u, p) b th solution to.1).5) and u h, p h ) b th solution to 3.) 3.3). Furthrmor, lt u 1 H r+1 Ω 1 )) d, p 1 H r Ω 1 ), u H 1 Ω )) d, and p H Ω ). Thn, th following rror bound holds u h 1 u 1 X1 + u h u I X h C ε 1 + ε ), 6.9) whr ε 1 = h r 1 u 1 r+1,ω1 + p 1 r,ω1 ) ) ε = h p 1,Ω + p,ω + u 1,Ω ) + h 1/ h h 1/ 1 + h 1/ 1 p 1,Ω. Proof. W choos th tst functions in 6.8) to b v h = u h ũ and q h = p h p. Th dfinition of π h 1 u 1) implis that th right-hand sid of th scond quation in 6.8) is zro: b 1 u h 1 ũ 1, p h 1 p 1 ) = 0. Using th commutativ proprty 3.16) and 5.3) w conclud that DIV u h ũ ) = DIV u h u I ϕ) I ) = DIV u h div u ) I ϕ) I = f I f I 0 = 0. Plugging th last two rsults in th first quation of 6.8), w liminat th trms in th lft-hand sid that contain th bilinar forms b 1 and b. Using th dfinition of ũ, w brak th third trm in th lft-hand sid into thr pics: a 1 u h 1 ũ 1, u h 1 ũ 1 ) + a u h u I, u h u I ) = a 1 u 1 ũ 1, u h 1 ũ 1 ) + b 1 u h 1 ũ 1, p 1 p 1 ) + p u h 1 ũ 1 ) n 1 ds a u I, u h u I ) + a u I, ϕ) I ) Γ h I + a u h u I, ϕ) I ) T 1 + T + T 3 + T 4 + T 5 + T ) 17

18 To bound T 1, w follow th analysis of a similar trm in [45]. W xpand it as follows: a 1 u 1 ũ 1, u h 1 ũ 1 ) =µ Du 1 ũ 1 ) : Du h 1 ũ 1 ) dx Ω h 1 µ + µε + h 1 Γh 1 h 1 Γh 1 + h 1 Γh 1 d 1 µ {Du 1 ũ 1 )}n [u h 1 ũ 1 ] ds {Du h 1 ũ 1 )}n [u 1 ũ 1 ] ds [u 1 ũ 1 ] [u h 1 ũ 1 ] ds G Γ h j=1 j I T 11 + T 1 + T 13 + T 14 + T 15. u 1 ũ 1 ) τ j u h 1 ũ 1 ) τ j ds 6.11) To stimat T 11, w apply th Cauchy-Schwarz inquality, th Young inquality 4.1), and th approximation proprty 4.11): T 11 µ u 1 ũ 1 ) 0, u h 1 ũ 1 ) 0, Ω 1 h C u 1 ũ 1 ) 0,Ω uh 1 ũ 1 ) 0,Ω 1 6.1) C h r 1 u 1 r+1,ω uh 1 ũ 1 ) 0,Ω 1. To bound T 1, w introduc th Lagrang intrpolant L h u 1 ) of dgr r satisfying u 1 L h u 1 ) m, C h s m u 1 s,, s r + 1, m = 0, 1,. 6.13) Lt δ) b th union of lmnts having th fac. W split split T 1 in two pics T1 a and T 1 b by adding and subtracting L h u 1 ) insid th avrag factor { }. Using th Cauchy-Schwarz inquality, th Young inquality 4.1), th trac inquality 4.4), and 6.13), w obtain T1 a = {DL h u 1 ) ũ 1 )}n [u h 1 ũ 1 ] ds C h 1 Γh 1 h 1 Γh 1 h 1 Γh 1 h 1/ σ 1/ Ch r 1 u 1 r+1,ω σ 1/ {DL h u 1 ) ũ 1 )}n 0, L h u 1 ) ũ 1 1,δ) h 1 Γh 1 h 1/ h 1 Γh 1 [u h 1 ũ 1 ] 0,. [u h 1 ũ 1 ] 0, [u h 1 ũ 1 ] 0, 6.14) 18

19 Th othr trm is stimatd similarly using th trac inquality 4.3): T1 b = {Du 1 L h u 1 ))}n [u h 1 ũ 1 ] ds C h 1 Γh 1 h 1 Γh h h 1 Γh 1 h 1 C h r 1 u 1 r+1,ω u 1 L h u 1 ) 1,δ) + u 1 L h u 1 ),δ) [u h 1 ũ 1 ] 0, h 1 Γh 1 [u h 1 ũ 1 ] 0,. ) 6.15) W conclud that T 1 C h r 1 u 1 r+1,ω h 1 Γh 1 [u h 1 ũ 1 ] 0,. 6.16) For simplicial mshs, th third trm in 6.11) is zro, T 13 = 0, du to th continuity of u 1 and th proprty 4.8). For polygonal and polyhdral mshs, w us th Cauchy-Schwarz inquality, th Young inquality 4.1), th trac inquality 4.4), and approximation rsult 6.5) to obtain T 13 µ Du h 1 ũ 1 ) n 0, [u 1 ũ 1 ] 0, µ h 1 Γh 1 h 1 Γh 1 h C Duh 1 ũ 1 ) n 0, + C ) [u 1 ũ 1 ] 0, 1 8 uh 1 ũ 1 ) 0,Ω 1 + Ch u 1,Ω 1. Th fourth trm is stimatd applying th Cauchy-Schwarz inquality, th approximation proprty 4.10), and th trac inquality 4.): T 14 C h 1 Γh 1 u 1 ũ 1 0, C h r 1 u 1 r+1,ω Using th sam argumnts, w bound th fifth trm: h 1 Γh 1 h 1 Γh 1 [u h 1 ũ 1 ] 0,. [u h 1 ũ 1 ] 0, 6.17) T 15 d 1 Γ h j=1 I µ G j u 1 ũ 1 0, u h 1 ũ 1 ) τ 0, C h r 1 u 1 r+1,ω 1 + d 1 Γ h j=1 I µ G j u h 1 ũ 1 ) τ 0,. 6.18) 19

20 To handl th trm T, w us th proprty 6.1) of th L -projction p 1 : b 1 u h 1 ũ 1, p 1 p 1 ) = p 1 p 1 ) div u h 1 ũ 1 ) dx Ω h 1 + {p 1 p 1 }[u h 1 ũ 1 ] n ds 1 h Γh 1 = {p 1 p 1 }[u h 1 ũ 1 ] n ds. 6.19) h 1 Γh 1 Thus, using th trac inquality 4.) and th proprty 6.) of th L projction p 1, w gt T C h r 1 p 1 r,ω [u h 1 ũ 1 ] ds. 6.0) 8 h 1 Γh 1 For th rmaining trms in th rror quation 6.10) w us argumnts dvlopd for th analysis of mimtic discrtizations of lliptic quations [1, 37]. W us th picwis constant tnsor K dfind at th bginning of this sction. Lt p 1 b a discontinuous picwis linar function dfind on Ωh such that 6.3) holds on vry lmnt Ω h. Thn, adding and subtracting K p1, w obtain T 4 = a u + K p 1 ) I, u I u h ) a K p 1 ) I, u I u h ) T 41 + T ) Applying th Cauchy-Schwarz inquality, th stability assumption 3.18), and th trac inquality 4.), w gt T 41 u + K p 1 ) I X h u h u I X h C 1 u + K p 1 ) n ds ) 1/ u h u I X h Ω h C Ω h [ u + K p 1 0, + h u ] ) 1/ 1, u h u I X h. 6.) Using th triangl inquality and thn stimats 6.6) and 6.3), w obtain u + K p 1 0, K p p 1 ) 0, + K K) p 1 0, C h p, + h p 1 0, ) Ch p, + p 0, + p p 1 ) 0, ) Ch p, + p 1, ). Combining th two last inqualitis and applying th Young inquality 4.1), w gt T 41 C h ) 1 p 1,Ω + p,ω + u 1,Ω + 8 uh u I. 6.3) X h 0

21 Th consistncy condition 3.17) and continuity of p allow us to rwrit T 4 as follows: T 4 = χ u h u I ) p 1, ds Ω = Ω χ u h u I ) p 1, p ) ds + χ u h u I ) p ds Γ h I 6.4) T a 4 + T b 4. W stimat T4 a using 6.4) and th stability proprty 3.18): T4 a 1/ u h u I ) p 1, p 0, Ω C Ω h h u h u I ) ) 1/ p, 6.5) C h p,ω u h u I X h C h p,ω uh u I. X h Th trm T4 b will b combind with othr trms latr. Now w procd with th fifth trm in th rror quation. Adding and subtracting K p 1, w gt T 5 = a u + K p 1 ) I, ϕ) I ) a K p 1 ) I, ϕ) I ) T 51 + T ) Th trm T 51 is similar to T 41 ; thrfor, w us th sam approach to bound it: ) T 51 C h p 1,Ω + p,ω + u 1,Ω ϕ) I X h. Using stimat 5.10), w conclud that T 51 C h h 1 p 1,Ω + p,ω + u 1,Ω ) u 1 3/,Ω1. 6.7) For th trm T 5, w apply stimat 6.4) and th consistncy condition 3.17): T 5 = χ ϕ) I ) p 1, ds = Ω Ω χ ϕ) I ) p p 1,) ds χ ϕ) I ) p ds Γ h I 6.8) T a 5 + T b 5. To stimat T a 5, w rpat argumnts usd for trms T a 4 and T 51. W obtain T a 5 C h p,ω ϕ) I X h C h h 1 p,ω u 1 3/,Ω1. 6.9) Th trm T5 b will b combind with othr trms latr. Th sixth trm in th rror quation is boundd using th Cauchy-Schwarz inquality and stimat 5.10): T 6 u h u I X h ϕ) I X h 1 8 uh u I X h 1 + C h 1 u 1 3/,Ω )

22 Finally, th third trm in th rror quation 6.10) is combind with T4 b and T 5 b. Lt p Λh I such that p ) is th L -projction of p on P 0 ) and lt p b th picwis constant function on Γh I satisfying p = p ), Γ h I. Bcaus u h ũ h V h, Γ I p u h 1 ũ 1 ) n 1 ds + p, u h ũ Λh I = 0. Using th abov quation, th dfinition of oprator π h and th proprty of th L projction, w obtain T 3 + T4 b + T5 b = p u h 1 ũ 1 ) n 1 ds + χ u h u I ϕ) I ) p ds Γ h I = Γ h I Γ h I p u h 1 ũ 1 ) n 1 ds + u h ũ ) p ds = p p )u h 1 ũ 1 ) n 1 ds + u h ũ ) p p ) ) ds = p p )u h 1 ũ 1 ) n 1 ds. Γ h I For ach fac Γ h I w dfin c to b th L -projction of u h ũ on P 0 ). Lt us assum that = n i=1 1,i, whr Ωh, and 1,i Ωh 1 for i = 1,..., n. Using th orthogonality and approximation proprtis of th L -projction, and th trac inquality 4.), w obtain T 3 + T4 b + T 5 b = p p )u h 1 ũ 1 c ) n 1 ds whr C Γ h I C Γ h I Γ h I h 1/ p 1, h 1/ p 1, n i=1 n i=1 h 1/ 1 u h 1 ũ 1 c 0, 1,i + h 1/ 1 u h 1 ũ 1 1, 1,i ) h h 1/ 1 + h 1/ 1 ) u h 1 ũ 1 1, 1,i ) Ch h h 1/ 1 + h 1/ 1 p + 1 1,Ω 8 uh 1 ũ 1 ) 0,Ω 1, h = maxh 1, h ). 6.31) If h = h 1, thn th trms h h 1/ 1 and h 1/ 1 can b combind. Othrwis, w hav th xtra trm h h 1/ 1. Collcting th stimats of all trms in th right hand sid of rror quation 6.10), using corcivity Lmma 5.3, thn th triangl inquality u h 1 u 1 X1 u h 1 ũ 1 X1 + ũ 1 u 1 X1, and finally th intrpolant proprty 4.11), w prov th assrtion of th thorm.

23 6.3 Prssur stimats Thorm 6. Undr th assumptions of Thorm 6.1, th following rror bound holds: whr ε 1 = h r 1 p 1 r,ω1 + u 1 r+1,ω1 ), ε = h p 1,Ω + p,ω + u 1,Ω ) + h 1/ p h p Q h Cε 1 + ε ) 6.3) Proof. Taking q h = p h 1 p 1, p h p ) in th inf-sup condition 5.11), w gt From 6.8), w gt ) h h 1/ 1 + h 1/ 1 p 1,Ω. p h p Q 1 β sup b 1 v1 h, ph 1 p 1) + b v h, ph p ) v h V h v h. 6.33) X h b 1 v1, h p h 1 p 1 ) + b v, h p h p ) = a 1 u 1 u h 1, v1) h + b 1 v1, h p 1 p 1 ) + p v1 h n 1 ds a u h, v) h b v, h p ) Γ h I J 1 + J + J 3 + J 4 + J 5. By adding and subtracting trms, and using th consistncy condition 3.17), w obtain J 4 + J 5 = a u + K p 1 ) I, v h ) + a K p 1 ) I, v h ) + [DIV v, h p p 1 ) I ] Q h + [DIV v h, p 1 ) I ] Q h a u h u I, v) h = a u + K p 1 ) I, v) h + χ v) p 1 ds 6.34) + [DIV v h, p p 1 ) I ] Q h a u h u I, v h ) = J 6 + J 7 + J 8 + J 9. Thus, w nd to stimat svn trms. W xpand J 1 as follows: J 1 = a 1 u 1 u h 1, v1) h =µ Du 1 u h 1) : Dv1) h dx Ω h 1 µ + µε + h 1 Γh 1 h 1 Γh 1 + h 1 Γh 1 d 1 µ G Γ h j=1 j I {Du 1 u h 1)}n [v1] h ds {Dv1)}n [u 1 u h 1] ds [u 1 u h 1] [v1] h ds u 1 u h 1) τ v1 h τ ds = J 11 + J 1 + J 13 + J 14 + J ) 3

24 From Cauchy-Schwarz inquality, w immdiatly gt bounds for thr trms: J 11 + J 14 + J 15 C u 1 u h 1 X1 v h 1 X ) W bound J 1 by taking similar approach as th on usd for T 1, J 1 C h 1 Γh 1 C C h 1 Γh 1 h ) 1/ ) 1/ u 1 u h σ 1) 0, [v h 1] h 0, h ) ) 1/ v u 1 ũ 1 ) 0, + ũ 1 u h σ 1) h 0, 1 X1 h r 1 u 1 r+1,ω 1 + ũ 1 u h 1 X 1 ) 1/ v h 1 X ) To bound th trm J 13, w us th trac inquality 4.4), and shap rgularity of lmnt having fac : J 13 C {Dv1)}n 0, [u 1 u h 1] 0, C h 1 Γh 1 h 1 Γh 1 h 1/ h ) 1/ v h 1 0, σ ) 1/ [u 1 u h 1] 0, 6.38) C v h 1 X h 1 u 1 u h 1 X1. W procd with J by applying th trac inquality 4.) and th proprty 6.) of th L projction: J = b 1 v1, h p 1 p 1 ) = {p 1 p 1 } [v1] h n ds h 1 Γh 1 h Ch r 1 p r,ω1 v h 1 X1. h 1 Γh 1 ) 1/ {p 1 p 1 } 0, σ ) 1/ v h 1 0, By combining J 3 with J 7 and rpating th stps w followd to bound T 3 and T 4, w gt ) ) J 3 + J 7 C h p,ω v h X h + h 1/ h h 1/ 1 + h 1/ 1 p 1,Ω v1 h 0,Ω1. Sinc J 6 is similar to T 51, w can writ: 6.39) J 6 C h p 1,Ω + p,ω + u 1,Ω ) v h X h. 6.40) Th trm J 8 is stimatd by using Cauchy-Schwartz inquality and th approximation proprtis 6.3): J 8 C h v h div p,ω. 6.41) Nxt, for th trm J 9, using Cauchy-Schwarz inquality and th vlocity stimats, w find that J 9 C h 1 u 1,Ω1 + p 1 1,Ω1 ) + h p 1,Ω + p,ω + u 1,Ω ) ) ) + h 1/ h h 1/ 1 + h 1/ 1 p 1,Ω1 v h X h. Combining all th bounds and dividing by v h X yilds th assrtion of th thorm. 6.4) 4

25 7 Numrical xprimnts 7.1 Implmntation dtails Th global vlocity spac V h, whicmbds th intrfac continuity constraint, is not convnint for a computr program. Instad, th continuity constraints on th vlocity ar imposd wakly and additional variabls, th Lagrang multiplirs ar addd to th systm. fficint solution of Darcy s law uss th hybridization procdur that is th standard in numrical mthod for mixd discrtizations. W rlax flux continuity condition on all msh facs in th Darcy rgion. Two flux dgrs of frdom U, 1 and U, ar prscribd to vry intrior fac and th xplicit continuity condition U, 1 + U, = 0 is addd to th systm. Th nw systm is algbraically quivalnt to th original systm; howvr, it has a spcial structur that allows to liminat fficintly th primary prssur and vlocity unknowns in th Darcy rgion. ach continuity constraint rsults in on Lagrang multiplir. W collct th Lagrang multiplirs in a singl vctor L = λ 1,..., λ J ), whr J is th numbr of th msdgs in Ω h. Lt us dfin th block-diagonal matrix M = diag{m,1,..., M,N } and th vlocity continuity matrix C = diag{ 1,..., J }. Lt A 1 and B 1 b th matrics associatd with th bilinar forms a 1, ) and b 1, ), rspctivly. Th matrix associatd with th intrfac trm is dnotd by C 1. Th matrix quations ar A 1 B C 1 B T M B C 0 0 B T 0 0 C T 1 0 C T 0 0 U 1 P 1 U P L = F F 0, 7.1) whr F is a vctor of siz N consisting of th cll avrags of th sourc trm. Th first pair of quations is th matrix form of discrt Stoks problm. Th scond pair of quations rprsnts lmntal quations for th Darcy rgion. Th last block quation rprsnts continuity of Darcy vlocitis and no-slip boundary conditions. Th matrix of systm 7.1) is symmtric. Th hybridization procdur rsults in th block-diagonal matrix B with as many blocks as th numbr of lmnts in Ω h. Thus, th unknowns U and P may b asily liminatd. Changing th ordr of rmaining unknowns, w gt th following saddl point problm: whr A 1 C 1 B 1 C T 1 A 0 B T U 1 L P 1 = F 1 G 0 A = C T M 1 M 1 B B T M 1 B ) 1 B T M 1 ) C, 7.) is a symmtric positiv dfinit SPD) matrix. This matrix is a spcial approximation of th lliptic oprator in th Darcy rgion. Not, that only M 1 is usd in th abov formula which suggsts its dirct calculation as dscribd in [14]. 5

26 Block-diagonal prconditionrs for saddl point problms ar discussd in [3, 47]. A propr candidat for a prconditionr in our cas could b A H = 0 A 0, 7.3) 0 0 S whr S is a suitabl diagonal matrix. Th analysis ndd to guarant that H rsults in msh indpndnt convrgnc of Krylov spac basd itrativ mthods is byond th scop of this articl. Th invrsion of A 1 and A can b prformd by using on V-cycl of th algbraic multigrid [50]. 7. Thr tst problms W prsnt thr computr xprimnts, th first two of which confirm th convrgnc of th mthod. Th third tst dmonstrats th ability of th mthod to b applid to surfac-subsurfac flow problms with ralistic gomtris. In th first two tsts th computational domain is Ω = Ω 1 Ω, whr Ω 1 = [0, 1] [ 1, 1] and Ω = [0, 1] [0, 1 ]. In th Stoks quation th strss tnsor is takn to b Tu 1, p 1 ) = p 1 I + µ u 1. It is asy to show that th thortical analysis from th prvious sctions still applis with this choic of Tu 1, p 1 ). ach convrgnc tst uss a manufacturd solution that satisfis th coupld systm.1)-.3) with Dirichlt boundary conditions on Ω. W considr th scalar prmability fild K = KI. To tst th convrgnc of th mthod, w solv th problm on a squnc of grids with dcrasing maximum lmnt siz, using both structurd and unstructurd grids. W us triangls with picwis linar vlocitis in th Stoks rgion and polygons rctangls if structurd) in th Darcy rgion. Th grids ar chosn to match on th intrfac. Th unstructurd grids ar not nstd - thy ar gnratd indpndntly on ach lvl. Th structurd grids ar obtaind by first partitioning Ω into rctangls and thn dividing ach rctangl in Ω 1 along its diagonal into two triangls. In Tst 1, th normal vlocity is continuous, but th tangntial vlocity is discontinuous, across th intrfac: x)1.5 y)y ξ) u 1 =, ξ + 1.5) 1.5ξy sinωx) whr u = y3 3 + y [ ω cosωx)y χy + 0.5) + sinωx) p 1 = sinωx) + χ K µ = 0.1, K = 1, α 0 = 0.5, G = ], + µ0.5 ξ) + cosπy), p = χ y + 0.5) K In Tst th vlocity fild is chosn to b smooth across th intrfac: [ sin x u 1 = u = G + ] ω)y/g cos x G +, ω)y/g sinωx)y K, µk, ξ = 1 G 30ξ 17, χ =, ω = 6. α G) 48 p 1 = G K µ G ) cos x G + ω)1/g) + y 0.5, p = G K cos x G + ω)y/g, 6

27 0.8 Rf. vctor 0.8 Rf. vctor 0.1 Y P Y P_rr X X Figur : Tst 1: computd solution lft) and rror right) on a msh with h 1 = 0.066, h = Tabl 1: Numrical rrors and convrgnc rats for Tst 1 on unstructurd grids. Stoks rgion: lmnts h 1 u 1 u h 1 1,Ω 1 rat p 1 p h 1 0,Ω 1 rat Darcy rgion: lmnts h u I uh X h rat p I ph Q h rat whr ω = 1.05 and µ, K, α 0, G ar th sam as in th Tst 1. Th computd solution along with th associatd numrical rror on th third lvl of unstructurd grids for th two tsts ar plottd in Figur and Figur 3, rspctivly. Th convrgnc rats basd on th unstructurd grids ar rportd in Tabl 1 and Tabl 3, rspctivly. Th convrgnc rats basd on th structurd grids ar rportd in Tabl and Tabl 4, rspctivly. Ths xprimntal rsults vrify th thortically prdictd convrgnc rat of ordr on. Th slight discrpancy in th convrgnc rat for th prssur in th Stoks rgion whn th coupld problm is solvd on unstructurd grids may b attributd to diffrnt shap rgularity constants of th unstructurd triangular mshs. Tabl 1 and Tabl 3 show suprconvrgnc of th prssur in Ω. Tabl and Tabl 4 show suprconvrgnc of both th vlocity and th prssur in Ω whn a rctangular msh is usd in th porous mdium. It is wll known that th MFD and th MF mthods for th Darcy quation alon ar suprconvrgnt on rctangular grids [9, 46]. Invstigation of th similar bhavior for th coupld Stoks-Darcy problm is a possibl topic of futur work. 7

28 Tabl : Numrical rrors and convrgnc rats for Tst 1 on structurd grids. Stoks rgion: lmnts h 1 u 1 u h 1 1,Ω 1 rat p 1 p h 1 0,Ω 1 rat Darcy rgion: lmnts h u I uh X h rat p I ph Q h rat Rf. vctor 0.8 Rf. vctor 0.1 Y P Y P_rr X X Figur 3: Tst : computd solution lft) and th associatd rror right) on a msh with h 1 = 0.066, h = Tabl 3: Numrical rrors and convrgnc rats for Tst on unstructurd grids. Stoks rgion: lmnts h 1 u 1 u h 1 1,Ω 1 rat p p h 1 0,Ω 1 rat Darcy rgion: lmnts h u I uh X h rat p I ph Q h rat

29 Tabl 4: Numrical rrors and convrgnc rats for Tst on structurd grids. Stoks rgion: lmnts h 1 u 1 u h 1 1,Ω 1 rat p 1 p h 1 0,Ω 1 rat Darcy rgion: lmnts h u I uh X h rat p I ph Q h rat Y X Figur 4: Tst 3: Computational domain and msh h 1 = , h = ). In Tst 3, w prsnt a mor ralistic modl of coupld surfac and subsurfac flows. Th flow domain is dcomposd into two subdomains as shown on Figur 4. Th top half rprsnts a lak or a slow flowing rivr th Stoks rgion) and th bottom half rprsnts an aquifr th Darcy rgion). Th surfac fluid flows from lft to right, with a parabolic inflow condition on th lft boundary, no flow on th top, and zro strss on th right outflow) boundary. No flow condition is imposd on th lft and right boundaris of th aquifr. Th prssur is spcifid on th bottom to simulat gravity. Th prmability of th porous mdia is htrognous and is shown in Figur 5 right). Th computd prssur and vlocity ar shown in Figur 5 lft). As xpctd, th prssur and th tangntial vlocity ar discontinuous across th intrfac, whil th normal vlocity is continuous. Aftr th surfac fluid ntrs th aquifr, it dos not mov as fast in th tangntial dirction, but prcolats toward th bottom. 9