Analysis of a Discontinuous Finite Element Method for the Coupled Stokes and Darcy Problems

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1 Journal of Scintific Computing, Volums and 3, Jun 005 ( 005) DOI: 0.007/s Analysis of a Discontinuous Finit Elmnt Mthod for th Coupld Stoks and Darcy Problms Béatric Rivièr Rcivd July 8, 003; accptd (in rvisd form) May 5, 004 Th coupld Stoks and Darcy flows problm is solvd by th locally consrvativ discontinuous Galrkin mthod. Optimal rror stimats for th fluid vlocity and prssur ar drivd. KEY WORDS: Surfac and subsurfac flow; rror stimats; intrfac conditions. INTRODUCTION Modling th intraction btwn surfac and subsurfac flow is a challnging nvironmntal problm. On such xampl is th simulation of transport of contaminants through rivrs into th aquifrs. Mathmatically, this complx problm can b modld by th coupld systm of Stoks and Darcy quations. Th aim of this papr is to formulat and analyz a discontinuous finit lmnt mthod for th coupld Stoks and Darcy problms. Th physical domain is dcomposd into two rgions: th rgion filld with an incomprssibl fluid modld by th Stoks quations and th porous mdium rgion modld by Darcy s law. Th intrfac conditions consist of th Bavrs Josph Saffman condition, th continuity of flux and th balanc of forcs. Th matching condition of mshs at th intrfac can b rlaxd. Th unknowns, namly th fluid vlocity and prssur in th fluid rgion, and th fluid prssur in th porous mdium, ar approximatd by totally discontinuous polynomials of diffrnt ordr. Th discontinuous Galrkin (DG) mthods considrd hr, ar basd on th 30 Thackray, Univrsity of Pittsburgh, Pittsburgh, PA 560, USA. rivir@math.pitt.du /05/ /0 005 Springr Scinc+Businss Mdia, Inc.

2 480 Rivièr Ω Γ Ω Fig.. Exampl of domain. Non-symmtric Intrior Pnalty Galrkin (NIPG) mthod [8, 9] and th Symmtric Intrior Pnalty Galrkin (SIPG) mthod []. DG mthods ar attractiv mthods bcaus thy ar lmnt-wis consrvativ, thy ar high-ordr mthods, and thy ar asily implmntabl on unstructurd mshs. A fw algorithms for th coupling of Stoks and Darcy can b found in th litratur. In a papr by Layton t al. [5], th xistnc and uniqunss of a wak solution to th coupld systm is provd and th proposd schm combins th continuous finit lmnt mthod for th Stoks problm with th mixd finit lmnt mthod for th Darcy quations. Th study of continuous finit lmnt mthods for both rgions can b found in th work of Discacciati t al. [8,9]. In a mor rcnt work [0], th DG mthod is usd in th Stoks rgion whil th mixd finit lmnt mthod is usd for Darcy rgion. Finally, th radr can rfr to [4,0,6, 7] for analysis of similar coupld modls. Th outlin of th papr is as follows. Sction contains th modl problm and notation. Th numrical schm is introducd in Sc. 3. Th a priori rror stimats ar provd in Sc. 4 and followd by som concluding rmarks.. MODEL PROBLEM AND NOTATION Lt Ω b a polygonal domain in R, subdividd into two subdomains Ω,Ω, with intrfac Γ = Ω Ω (s Fig. ). Dfin Γ i = Ω i Γ for i =,. Th physical quantitis ar th fluid vlocity u and prssur p. Dnot u i = u Ωi and p i = p Ωi. W assum that th flow satisfis th Stoks quations on Ω and th singl phas quations on Ω. (µd(u ) p I) = f in Ω, ()

3 Analysis of a Discontinuous Finit Elmnt 48 u = 0 in Ω, () u = K p in Ω, (3) u = f in Ω. (4) Hr, f and f ar xtrnal forcs acting on th fluid, µ>0 is th constant fluid viscosity, D(u) is th strain tnsor dfind by D(u) = (/)( u + u T ) and K is th prmability tnsor. W assum that K is symmtric, positiv dfinit tnsor, boundd blow and abov uniformly, and that th forc satisfis th solvability condition Ω f =0. Th boundary conditions ar u = 0 on Γ, (5) K p n = 0 on Γ, (6) whr n is th outward normal to th boundary Ω. As th prssur is uniqu up to an additiv constant, w assum that p = 0. (7) Ω Lt n (rspctivly τ ) b th unit normal (rspctivly tangntial) vctor to Γ outward of Ω. Th conditions at th intrfac Γ ar u n = u n, (8) p µ(d(u )n ) n = p, (9) u τ = G(D(u )n ) τ. (0) Intrfac conditions (8) and (9) rprsnt th mass consrvation and th balanc of forcs, rspctivly, across th intrfac. Th Bavr Josph Saffman law (0) is th most accptd condition [3,4,] and includs a friction constant G>0 that can b dtrmind xprimntally. Th xistnc of a uniqu wak solution of () (0) was shown in [5]. W assum hr that th solution (u,p) is rgular nough, and is a strong solution of () (0). W now dfin th functional spacs. For i =,, lt Eh i b a nondgnrat quasi-uniform subdivision of Ω i, lt Γ i h b th st of intrior dgs and lt h i dnot th maximum diamtr of lmnts in Eh i. W assum that th mshs at th intrfac ar non-matching in th following sns: any dg = E Γ whr E blongs to Eh, blongs to on lmnt E Eh and on only. For any non-ngativ intgr k and numbr r, th classical Sobolv spac [] on a domain O is dnotd by W k,r (O) ={v L r (O) : D m v L r (O), m k}, whr

4 48 Rivièr D m v ar th partial drivativs of v of ordr m. Th associatd Sobolv norm (rspctivly, smi-norm) is dnotd by k,r,o (rspctivly, k,r,o ), or by k,o (rspctivly, k,o ) if r =. W us th usual notation H k (O) for W k, (O) and L 0 (O) for th spac of squarintgrabl functions with zro avrag. Th L innr-product will b dnotd by (, ). Throughout th papr, C will dnot a gnric positiv constant whos valu may vary but will b indpndnt of th msh sizs h and h. Our schm rquirs that th trac of p and th trac of th normal drivativs of u and p ar wll dfind, and ar squar-intgrabl. Thrfor, w dfin th following functional spacs: X ={v (L (Ω )) : E E h, v E (W,4/3 (E)) }, () M ={q L (Ω ) : E E h, q E W,4/3 (E)}, () M ={q L (Ω ) : E E h, q E W, (E)}. (3) W associat to ths spacs th following norms X, M and sminorm M : v X = v 0,Ω + Γ h Γ q M = q 0,Ω, q M = q 0,Ω + σ, [v ] 0, + G Γ h σ, [q ] 0,. Γ v τ 0,, Hr, dnots th masur of ach dg, th paramtrs σ, and σ, ar positiv constants that may vary on ach dg, and that will b dfind in Sc. 3. Finally, th norm is th usual brokn norm: w m,ω i = E Eh i w m,e for i =, and for w any scalar or vctor function. Givn a fixd normal vctor n on ach dg = E E, pointing from E to E, th avrag {w} and jump [w] of function w ar uniquly dfind {w}= (w E ) + (w E ), [w] = (w E ) (w E ), {w}=w E, [w] = w E. Th aim of this papr is to formulat an algorithm that uss totally discontinuous approximating spacs. Lt k, k b two positiv intgrs and lt th finit-dimnsional subspacs X h X, M h M and M h M, with

5 Analysis of a Discontinuous Finit Elmnt 483 th inducd norms, b dfind as follows: X h ={v X : E Eh, v (P k (E)) }, Mh ={q M : E Eh, q P k (E)}, Mh ={q M : E Eh, q P k (E)}. W assum that th discrt spacs satisfy th optimal approximation proprtis. In particular, thr xists an oprator R h L(H (Ω ) ; X h ) such that for any E E h, v H (Ω ), q P k (E), q (R h (v) v) = 0, (4) E v H (Ω ), Γ h, q P k (), q [R h (v)] = 0, v H 0 (Ω ), Ω, q P k (), v H (Ω ), Γ h Γ Γ, (5) q R h (v) = 0, (6) (R h (v) v) = 0, (7) v H0 (Ω ), v R h (v) X C (v R h (v)) 0,Ω, (8) v W s,t (Ω ), s [,k + ], m= 0,, v R h (v) m,t,e Ch s m E v s,t, E, (9) whr E is a suitabl macro-lmnt containing E. Not that proprty (8) is an asy consqunc of (7) (s Rf. []). Thr xists also an oprator r h L(L (Ω); Mh M h ) such that for s = k,k and for any z L (Ω) H s+ (Ω) q P s (E), q(r h (z) z) = 0, E Eh E h, (0) E z r h (z) m,e Ch s+ m E z s+,e, E Eh E h, m= 0,. () Not that th xistnc of R h for k =, or 3 follows from th nonconforming lmnts of Crouzix t al. [7,6] and Fortin t al. [], and th oprator r h is th wll-known L projction oprator. W rcall a rsult provd in Rf. [] that gnralizs a Sobolv imbdding. For any ral numbr s [, ), thr is a constant C indpndnt of h such that v X h, v L s (Ω ) C v X. ()

6 484 Rivièr W now finish this sction with som trac and invrs inqualitis ndd for th analysis. Lt E b a msh lmnt with diamtr h E. Thn, thr xists a constant C indpndnt of h E such that φ H (E), E, φ 0, C(h E φ 0,E + h E φ,e ), (3) φ H (E), E, φ n 0, C(h E φ,e + h E φ,e ), (4) φ P k (E), E, φ n 0, Ch / E φ,e. (5) 3. SCHEME W introduc th following bilinar forms a : X X R, b : X M R and a : M M R: a (u, v ) = D(u ) : D(v ) + σ, [u ] [v ] E Eh E Γ Γ h {D(u )n } [v ] + ɛ {D(v )n } [u ] + G Γ h Γ Γ b (v,p ) = E E h a (p,q ) = E Eh Γ Γ h Γ (u τ )(v τ ) + u v, (6) Γ h E E p v + Γ h Γ K p q + Γ h {p }[v ] n, (7) σ, [p ][q ] {K p n }[q ] + ɛ {K q n }[p ]. (8) Γ h By introducing th paramtrs ɛ,ɛ that tak th valu ±, w allow for non-symmtric or symmtric bilinar forms a and a. Throughout th papr, w assum that: Hypothsis A: w assum that th paramtr σ, is boundd blow by a sufficintly larg positiv valu. For th paramtr σ,,ifɛ =, thn on

7 Analysis of a Discontinuous Finit Elmnt 485 can choos σ, =, but if ɛ =, σ, must b boundd blow by a sufficintly larg positiv valu. Th constraint on th paramtrs σ, and σ, is standard for th SIPG mthod, but hr ncssary for th NIPG (for σ, ) bcaus of th gnralizd Korn s inquality (s Lmma 3.). W nxt dfin a bilinar form Λ : M X R acting on th intrfac Γ. Λ(q, v ) = q v n, (q, v ) M X. (9) Γ With ths forms, w propos th following variational problm of () (0): Find (u,p,p ) X M M, solution of µa (u, v ) + b(v,p ) + Λ(p, v ) = (f, v ), v Xh, (30) b(u,q ) = 0, q Mh, (3) a (p,q ) Λ(q, u ) = (f,q ), q Mh, (3) p + Ω p = 0. Ω (33) Lmma 3.. If (u,p,p ) is th solution of th coupld Stokssingl phas flow problm () (0), thn (u,p,p ) is th solution of (30) (33). Proof. Multiply th Stoks quation () by v, intgrat by parts ovr on lmnt E Eh, and sum ovr all lmnts in E h E E h E ( p I + µd(u )) : v [( p I + µd(u ))n v ] Γ h ( p I + µd(u ))n v ( p I + µd(u ))n v Γ = f v. Ω Γ

8 486 Rivièr Noting that D(u ) : v = D(u ) : D(v ) and I : v = v, w can rwrit th quation E E h E (µd(u ) : D(v ) p v ) Γ h Γ {( p I + µd(u ))n } [v ] [( p I + µd(u ))n ] {v } Γ h Γ ( p I + µd(u ))n v = (f, v ). By rgularity of th tru solution and with th boundary condition (5), w hav E E h E (µd(u ) : D(v) p v ) + µ Γ Γ h Γ Γ h Γ {D(u )n } [v ] + µɛ {p }[v ] n Γ h Γ {D(v )n } [u ] ( p I + µd(u ))n v = (f, v ). (34) By dcomposing v and µd(u )n into thir normal and tangntial componnts, th intrfac intgral is rducd to = Γ Γ ( p I + µd(u ))n v (p µ(d(u )n ) n )(v n ) Γ µ(d(u )n ) τ (v τ ).

9 Analysis of a Discontinuous Finit Elmnt 487 With th intrfac conditions (9) and (0), th intgral bcoms ( p I + µd(u ))n v = Γ Γ p (v n ) + µ G Γ (u τ )(v τ ). (35) Th continuity of u, th boundary condition (5), and Eqs. (34) and (35) yild: µa (u, v ) + b(v,p ) + Λ(p, v ) = (f, v ). Equation (3) is a consqunc of (), (5) and th fact that [u] n = 0on ach dg. Now for th singl phas flow part, w rpat th procss with (4): multiply by a tst function q, and intgrat by parts and sum ovr all lmnts in Eh. Dfinition (3) of u, th rgularity of p and th boundary condition (6) giv K p q {K p n }[q ] E E h E Γ h Γ +ɛ {K q n } [p ] + Γ h Γ (u n )q = (f,q ). Γ h σ [p ][q ] Using th intrfac condition (8) in th quation abov, givs (3). Finally, Eq. (33) is just condition (7). Th discrt schm is: Find (U,P,P ) X h M h M h such that µa (U, v ) + b(v,p ) + Λ(P, v ) = (f, v ), v X h, (36) b(u,q ) = 0, q Mh, (37) a (P,q ) Λ(q, U ) = (f,q ), q Mh, (38) P + Ω P = 0. Ω (39) Bfor addrssing th xistnc and uniqunss of a solution to th numrical schm, w rcall th fact that th bilinar forms a and a ar corciv with rspct to X and M, rspctivly.

10 488 Rivièr Lmma 3.. Undr Hypothsis A, thr xist two positiv constants C and C such that C v X a (v, v ), v X h, (40) C q M a (q,q ), q Mh. (4) Proof. Inquality (40) is asily drivd from th Korn s inquality (.9) in [5] for picwis H functions, obtaind by Brnnr. Th proof of (4) is straightforward for th nonsymmtric bilinar forms, and is standard for th symmtric bilinar forms (Whlr []). Lmma 3.3. Th discrt schm has a uniqu solution. Proof. Sinc (36) (39) is a squar systm of linar quations in finit dimnsion, it suffics to prov that (f,f ) = (0, 0) implis (U,P,P )=(0, 0, 0). W choos v =U in (36), q =P in (37) and q = P in (38). Adding th rsulting quations givs µa (U, U ) + a (P,P ) = 0. From Lmma 3., this implis U X = P M = 0, which mans that U = 0 and P is a global constant ovr Ω. Equation (36), with (39) bcoms b(v,p ) ( ) Ω v n = 0, v X h. Ω P Γ W now writ P = P + P, whr abov bcoms Not that b(v, P ) + b(v, P ) Ω P Ω So that, w hav b(v, P ) P = (/ Ω ) Ω P. Th quation Γ b(v, P ) = P ( + Ω ) Ω P Γ Γ v n = 0, v X h. v n. v n = 0, v X h. (4)

11 Analysis of a Discontinuous Finit Elmnt 489 Sinc P blongs to L 0 (Ω ) and th spacs H0 (Ω ),L 0 (Ω ) satisfy th xact inf-sup condition (s for xampl [3]), thr xists ṽ H0 (Ω ) such that ṽ = P. Choos v = R h (ṽ) in (4), thn w hav from proprtis (4) (6) P 0,Ω = 0. This implis that P = 0, and from (4), w hav that mans that P = 0. P = 0, which also Rmark. Th approximation U of th Darcy vlocity u is a discontinuous picwis polynomial vctor of dgr k and is dirctly obtaind from th discontinuous approximation P, by th formula U = K P. 4. ERROR ESTIMATES In this sction, w driv first optimal rror stimats in th nrgy norm for th Stoks vlocity and in th L norm for th Darcy prssur. A scond stimat givs an optimal convrgnc rat for th L norm of th prssur in th Stoks rgion. Thorm 4.. Lt (u,p) b th solution of th coupld problm () (4), such that u Ω H k+ (Ω ), p Ω H k (Ω ) and p Ω H k+ (Ω ). Th discrt solution (U,P) of (36) (39) satisfis th rror stimat: u U X + p P M Ch k + Ch k whr C is a constant indpndnt of h and h. ( u k +,Ω + p k,ω ) ( ) + h / h / p k +,Ω, Proof. Lt ũ = R h (u ), p = r h (p ) and p = r h (p ) b intrpolants of u,p and p, rspctivly. Dfin χ = U ũ and ξ = P p.from (30) (3) and (36) (38), th rror quations ar µa (χ, v ) + b(v,p p ) + Λ(ξ, v ) = µa (u ũ, v ) + b(v,p p ) + Λ(p p, v ), v Xh, (43) b(χ,q ) = b(u ũ,q ), q Mh, (44) a (ξ, q ) Λ(q, χ) = a (p p,q ) Λ(q, u ũ ), q Mh. (45)

12 490 Rivièr Choosing v = χ in (43), q = P p in (44), q = ξ in (45) and summing th rsulting quations givs µa (χ, χ) + a (ξ, ξ) = µa (u ũ, χ) + a (p p,ξ)+ b(χ,p p ) b(u ũ,p p ) + Λ(p p, χ) Λ(ξ, u ũ ). (46) W now bound ach of th trms on th right-hand sid of (46). W first rwrit a (u ũ, χ) µa (u ũ, χ) = µ E Eh E µ +µε Γ h Γ +µ Γ h Γ Γ h Γ D(u ũ ) : D(χ) {D(u ũ )}n [χ] σ, {D(χ)}n [u ũ ] [u ũ ] [χ] + µ ((u ũ ) τ )(χ τ ) G Γ +µ (u ũ ) χ = T + +T 6. Γ Using th Cauchy Schwarz inquality, and th approximation rsult (9), w hav T Cµ E E h (u ũ ) 0,E χ 0,E µ 8 χ 0,Ω + C (u ũ ) 0,Ω µ 8 χ 0,Ω + Ch k u k +,Ω. (47) Lt L h (u) dnot th standard Lagrang intrpolant of dgr k dfind in Ω and lt us insrt it in th scond intgral trm. For a sgmnt of

13 Analysis of a Discontinuous Finit Elmnt 49 Γ h Γ,whav {D(u ũ )}n [χ] = {D(u L h (u ))}n [χ] + {D(L h (u ) ũ )}n [χ]. Expanding th first intgral, w obtain from th trac inquality (4) and from th fact that th Lagrang intrpolant satisfis th optimal approximation rsult (9) µ {D(u L h (u ))}n [χ] Γ h Γ Γ h Γ Γ h Γ µ 6 Γ h Γ σ /, / [χ] 0, / σ /, {D(u L h (u ))}n 0. σ, [χ] 0, + Chk u k +,Ω. Similarly, if w dnot by E th lmnts sharing th dg, and w us th trac inquality (5), a triangl inquality and th approximation rsults, thn w hav µ {D(L h (u ) u )}n [χ] µ σ, 6 [χ] 0, Thrfor, T µ 8 Γ h Γ Γ h Γ + µ 6 Γ h Γ Γ h Γ h u L h (u ) σ,e, σ, [χ] 0, +Ch k u k +,Ω. σ, [χ] 0, + Chk u k +,Ω. (48) Th third trm vanishs bcaus of th proprtis (5), (6) satisfid by ũ. T 3 = 0. (49)

14 49 Rivièr Using th Cauchy Schwarz inquality, th jump trm T 4 is boundd by virtu of (3) and (9) T 4 µ 8 µ 8 Γ h Γ Γ h Γ σ, [χ] 0, + C Γ h Γ σ, [u ũ ] 0, σ, [χ] 0, + Chk u k+,ω. (50) Th fifth trm is boundd using th trac inquality (3) and th bound (9) T 5 µ u ũ 0, χ τ 0, G Γ µ χ τ 0, G + C (h u ũ 0,E + h u ũ,e ) Γ Γ µ χ τ 0, G + Chk u k +,Ω. (5) Γ Finally th last trm is boundd using th proprty (9) and (). T 6 Ch k +/ u k +,Ω h / χ 0,Ω Ch k u k +,Ω χ X µ 8 χ X + Chk u k +,Ω. (5) Combining (47) (5), w hav a (u ũ, χ) 3µ 4 χ 0,Ω + 3µ 4 + µ G Γ h Γ σ, [χ] 0, Γ χ τ 0, + Chk u k +,Ω. (53)

15 Analysis of a Discontinuous Finit Elmnt 493 Lt us now xpand a (p p,ξ). a (p p,ξ) = K (p p ) ξ + E Eh E Γ h {K (p p ) n }[ξ] Γ h +ɛ {K ξ n }[p p ]. Γ h σ, [p p ][ξ] Clarly, ths trms ar boundd in a similar fashion as th trms T,...,T 4, using in particular th approximation rsult (). a (p p,ξ) 8 ξ 0,Ω Γ h σ, [ξ] 0, + Chk p k +,Ω. (54) Using proprty (0) of th oprator r h, th third trm on th right-hand sid of (46) rducs to b(χ,p p ) = {p p }[χ] n, Γ h Γ which is boundd using th Cauchy Schwarz inquality, trac inquality (3) and th stimat () by b(χ,p p ) µ 6 Γ h Γ σ, [χ] 0, + Chk p k,ω. (55) Th trm b(u ũ,p p ) vanishs bcaus of th proprtis (4) (6) of ũ. It suffics thn to bound th intrfac trms Λ(p p, χ) and Λ(ξ, u ũ ) in (46). Using th trac inquality (3), th approximation rsult () and th bound (), w obtain Λ(p p, χ) Γ p p 0, χ 0, C χ 0,Ω h k +/ h / p k +,Ω µ 6 χ X + Ch k + h p k +,Ω. (56)

16 494 Rivièr W now associat for ach dg = Γ E, for som E Eh, th constant c = (/ ) ξ. From th proprty (7) and (9), w can writ Λ(ξ, u ũ ) = (ξ c )(u ũ ) n Γ Γ ξ c 0, u ũ 0, Ch k +/ h / ξ 0,Ω u k +,Ω 8 ξ M + Ch k u k +,Ω. (57) Collcting th bounds (53) (57) and using Lmma 3. yilds χ X + ξ M Ch k ( u k +,Ω + p k,ω ) +Ch k ( + h h ) p k +,Ω. Th final rsult is obtaind with a triangl inquality and approximation rsults. Rmark. Clarly, Thorm 4. givs an stimat of th rror in th L norm of th Darcy vlocity. Th convrgnc rat is optimal. Thorm 4.. Undr th assumptions and notation of Thorm 4., w hav p P 0,Ω Ch k ( u k +,Ω + p k,ω ) +Ch k ( + h / h / ) p k +,Ω. whr C is a constant indpndnt of h and h. Proof. Subtracting (30) from (36) givs µa (U u, v ) + b(v,p p ) + Λ(P p, v ) = 0, v X h. As in Thorm 4., lt p = r h (p ), p = r h (p ), ζ = P p and ξ = P p. Th rror quation bcoms b(v,ζ) = µa (U u, v ) + Λ(ξ, v ) Λ(p p, v ) b(v,p p ), v X h.

17 Analysis of a Discontinuous Finit Elmnt 495 If w dcompos ζ = ζ + ζ, whr ζ = (/ Ω ) Ω ζ, and ξ = ξ + ξ, whr ξ = (/ Ω ) Ω ξ, thn (39) and (33) with proprty (0) yild Ω ζ + Ω ξ = 0. Th rror quation is thn rwrittn b(v, ζ)+ ( + Ω Ω ) ζ v n = µa (U u, v ) Γ +Λ( ξ,v ) Λ(p p, v ) b(v,p p ), v X h. (58) Sinc ζ blongs to L 0 (Ω ), thr xists ṽ H 0 (Ω ) such that ṽ = ζ and ṽ,ω C ζ 0,Ω. Choos v = R h (ṽ) in (58). From proprtis (4) (6), (58) is rducd to ζ 0,Ω = µa (U u, v ) + Λ( ξ,v ) Λ(p p, v ) b(v,p p ). (59) W xpand and bound ach trm on th right-hand sid of (59). a(u u, v ) = µ D(U u ) : D(v ) E Eh E + σ, [U u ] [v ] Γ Γ h µ {D(U u )n } [v ] +µɛ + µ G Γ h Γ Γ h Γ Γ = Q + +Q 6. {D(v )n } [U u ] (U u ) τ (v τ ) + (U u ) v Th bounds for Q,Q,Q 4,Q 5 ar asily obtaind Γ Q + Q + Q 4 + Q 5 C U u X v X.

18 496 Rivièr Th rmaining trms ar boundd by introducing th intrpolant ũ = R h (u ). Q 3 = µ {D(U ũ )n } [v ] Γ h Γ µ Γ h Γ {D(ũ u )n } [v ]. Th first part vanishs bcaus of (5) and (6) and th scond part is boundd lik T in Thorm 4.. Thrfor, w hav a (U u, v ) v X ( U u X + Ch k u k +,Ω ). Similarly, from (7) and (9), w hav Q 6 = (U ũ ) v + (ũ u ) v Γ Γ C( U ũ 0,Ω + h k u k +,Ω ) v X. Lt us dnot by E th lmnt in Eh that shars th dg of th intrfac Γ. Dfin also th constant c = (/ ) ξ. From proprty (6) of th oprator R h,whav ξv n = ( ξ c )v n Γ Γ By () and (3), w hav (p p )v n Γ C / ξ 0,E v 0, Γ Ch / ξ 0,Ω v 0, Γ / p p 0, v 0, Γ Ch k +/ p k +,Ω v 0, Γ. /.

19 Analysis of a Discontinuous Finit Elmnt 497 Now, sinc v satisfis v = 0 for any dg in Γ,whav / v 0, Ch / v 0,Ω Ch / v X. Γ Thrfor, th rsulting bound is Λ( ξ,v ) Λ(p p,v ) Ch / h / v X ( P p M +h k p k +,Ω ). Finally, th bound for b(v,p p ) is similar to th on obtaind in (55) b(v,p p ) C v X h k p k,ω. From proprtis (9) and (8), w hav v X R h (ṽ) ṽ X + ṽ X C ṽ,ω C ζ 0,Ω. Combining all th bounds abov givs ζ 0,Ω C U u X + C P p M +Ch k ( u k +,Ω + p k,ω ) + Ch k p k +,Ω. (60) To finish th proof of th thorm, it rmains to bound ζ 0,Ω. This is accomplishd by choosing a particular tst function v in (58). Lt ρ b a function in C ( Ω), with compact support in Ω such that ρ n =. Γ Dnot v = Ω ζ ρ and choos v = R h ( v). W first show that v X boundd by ζ 0,Ω. By proprty (8) and (9), w hav is v X R h ( v) v X + v X C v 0,Ω + v X. But, v X = v 0,Ω + Ω ζ (ρ τ ) C Ω ζ 0,Ω G. Γ Thus, v X C ζ 0,Ω. (6)

20 498 Rivièr With that tst function v, th rror quation (58) bcoms ( + Ω ) ζ 0,Ω Ω = µa (U u, v ) +Λ( ξ,v ) Λ(p p, v ) b(v,p p ) + b(v, ζ). (6) Excpt for th last trm, all th trms on th right-hand sid of (6) ar boundd xactly as in (59). W now rwrit th last trm. Clarly, b(v, ζ)= b( ζ,r h ( v) v) + b( v, ζ). b( v, ζ)= Ω ζ E E h E ζ ρ C Ω / ζ 0,Ω ζ 0,Ω. And it is asy to chck from (4) to (6) that b(r h ( v) v, ζ)= 0. Finally, w obtain from (6) and th bounds abov ζ 0,Ω C U u X + C P p M + C ζ 0,Ω +Ch k ( u k +,Ω + p k,ω ) + Ch k p k +,Ω. (63) Th bounds (59), (63), () and Thorm 4. giv th final rsult. Rmark. Th rsults statd in Thorms 4. and 4. hold tru in th cas, whr k and th paramtr σ, is qual to zro, for all dgs in Γ h Γ. In this cas, th proof diffrs in th choic of th intrpolant p, which now has to satisfy spcial flux proprtis (s [9] for furthr dtails). 5. CONCLUSION In this papr, a discontinuous Galrkin mthod is formulatd for th coupld Stoks and Darcy quations. Both symmtric and non-symmtric cass ar considrd. Optimal convrgnc rats ar obtaind for th fluid vlocity and prssur.

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