Uniformly stable discontinuous Galerkin discretization and robust iterative solution methods for the Brinkman problem

Transcription

1 Uniformly stabl discontinuous Galrin discrtization and robust itrativ solution mtods for t Brinman problm Q. Hong, J. Kraus RICAM-Rport

2 UNIFORMLY STABLE DISCONTINUOUS GALERKIN DISCRETIZATION AND ROBUST ITERATIVE SOLUTION METHODS FOR THE BRINKMAN PROBLEM QINGGUO HONG AND JOHANNES KRAUS Abstract. W considr robust itrativ mtods for discontinuous Galrin (DG) H(div, Ω)- conforming discrtizations of t Brinman quations. W dscrib a simpl Uzawa itration for t solution of tis problm, wic rquirs t solution of a narly incomprssibl linar lasticity typ quation wit mass trm on vry itration. W prov t uniform stability of t DG discrtization for bot problms. Tn, w analyz variabl V-cycl and W-cycl multigrid mtods wit nonnstd bilinar forms. W prov tat ts algoritms ar robust, and tir convrgnc rats ar indpndnt of t paramtrs in t Brinman problm and of t ms siz. T tortical analysis is confirmd by numrical rsults. 1. Introduction T Darcy-Stos-Brinman quations provid a unifid matmatical modl for on-pas flow of a visccous fluid in porous mdia (Darcy problm) coupld wit fr flow in a cavity (Stos problm). Dpnding on t coic of paramtrs, ts quations in t limiting cass rduc to itr t Darcy, or t Stos modl. Ty also allow for trating t Darcy-Stos intrfac problm wit on st of quations by introducing a jump discontinuity of t cofficints across t intrfac. T solution of t govrning quations of flows in porous mdia migt b sougt at a coars scal (.g., if only t global prssur drop for a givn flow rat is ndd), a coars scal nricd wit som dsirabl fin-scal dtails, or at t fin scal (if computationally affordabl and practically dsirabl). A subgrid (variational multiscal (VMS)) mtod for Brinman s problm allowing to comput a two-scal (nricd coars scal) solution as bn introducd in [1] togtr wit a subgrid-basd two-lvl domain dcomposition mtod for solving bot Darcy s and Brinman s problm in igly trognous porous mdia at t fin scal (somtims calld itrativ upscaling). T aim of t prsnt wor, owvr, is to propos a uniformly stabl discrtization at coars scal, and to dvis and analyz optimal and robust itrativ solution mtods for tis discrtization. Diffrnt stabl discrtizations of tis problm av bn considrd in litratur, wic ar itr basd on stabl Stos lmnts, or on H(div, Ω)-conforming lmnts, or on a stabilization of t wa formulation, s,.g., [2]. Nonconforming lmnts for t Brinman problm av bn studid for xampl in [3], s also t rfrncs trin. T y point in tis approac is to guarant t robustnss wit rspct to t limiting cas of Darcy flow. Using t framwor givn in [2], ncssary and sufficint conditions can b formulatd to nsur t robustnss of finit lmnt mtods for t Brinman problm using (Stos) stabl finit lmnt pairs. Dat: Today is Novmbr 29, Matmatics Subjct Classification. 65F10, 65N20, 65N30. Ky words and prass. finit lmnt mtod, discontinuous Galrin mtod, Brinman problm, multigrid mtod. T rsarc of Qingguo Hong and Joanns Kraus was supportd by t Austrian Scinc Fund Grant P

3 2 QINGGUO HONG AND JOHANNES KRAUS For mixd H(div, Ω)-conforming scms,.g., Raviart-Tomas (RT) or Brzzi-Douglas-Marini (BDM) lmnts for t flux and discontinuous picwis polynomials for t prssur, and using a postprocssing procdur for t prssur, optimal convrgnc rats wr provd bot in t Darcy and in t Stos rgim in [4] undr propr rgularity assumptions. Quit rcntly, a novl mixd formulation of t Brinman problm as bn proposd in [5] in wic t flow vorticity is introducd as an additional unnown. Tis formulation allows for a uniformly stabl and conforming discrtization by standard finit lmnts (Ndlc, Raviart- Tomas, discontinuous picwis polynomials). T obtaind stability stimats rsult in provably uniform scalabl bloc-diagonal prconditionrs in t constant cofficint cas, s [6]. T arising subproblms can tn b solvd using optimal mtods,.g., auxiliary spac algbraic multigrid (AMG) solvrs for H(curl; Ω) and H(div; Ω) problms. Tr ar multigrid mtods for Stos typ problms, wic can rougly b classifid into two catgoris: coupld and dcoupld mtods, cf. [7]. A wll-nown coupld approac is basd on solving small saddl point systms at vry grid point or on appropriat patcs, cf. [8]. On classical dcoupld approac is t Uzawa mtod [9]. Tr is also a wll dvlopd tory of multigrid mtods for H(div; Ω) problms, s,.g. [10], wic in particular guarants optimal solvrs for t Darcy problm. Tr is, owvr, to t bst of t autors nowldg, no rigorous analysis proving t optimality of multigrid mtods for uniformly stabl DG discrtizations of t 3D Brinman problm up to now. For rlatd rsults, for t 2D Brinman problm discrtizd by H 1 -conforming lmnts w rfr to [11]. In t prsnt papr a family of discontinuous Galrin H(div, Ω)-conforming discrtizations is analyzd. A main fatur of ts mtods is tat ty prsrv divrgnc-fr solutions of t Stos problm. First introducd in [12], mtods of tis family wr also dscribd in [13] and, mor rcntly, studid in t contxt of auxiliary spac prconditionrs for t Stos problm, s [14], and optimal multigrid mtods for Stos and lasticity typ quations, s [15]. T goals of tis wor ar to prov t uniform stability of tis family of DG discrtizations for t Brinman problm, and furtr, to dvis and analyz multigrid mtods tat ar basd on nonnstd bilinar forms. Using t local boundd cocain projctions constructd in [16], w prov t uniform stability of a class of DG discrtizations for t Brinman problm as wll as for a linar lasticity typ problm wit mass trm, wic ariss in vry stp of a simpl Uzawa itration wn usd to solv t Brinman problm. T proof of t robust (paramtr-indpndnt) convrgnc of t considrd multigrid mtods is basd on a spcial (subspac-) dcomposition of H(div, Ω). Similar dcompositions av alrady bn succssfully applid in multigrid analysis in [17] and [15]. T rmaindr of t papr is organizd as follows. In Sction 2 w stat t Brinman problm in strong and wa form. T discontinuous Galrin discrtizations undr considration ar discussd in Sction 3 along wit tir approximation and stability proprtis and an a priori rror stimat. In Sction 4 w discuss multigrid mtods tat ar basd on nonnstd bilinar forms. T main rsult is a proof of tir robust and optimal convrgnc in t prsnt contxt. W prsnt som numrical rsults in Sction 5 confirming our tortical findings. Finally, w draw som concluding rmars in Sction Problm formulation In tis sction, w formulat of t Brinman problm. Lt Ω R d (d = 2, 3) b a polygonal domain wit boundary Ω, f L 2 (Ω) d, and H 1 0 (Ω) = {u L2 (Ω) : u L 2 (Ω), u Ω = 0}. W nd t standard Sobolv spacs L 2 (Ω), H 1 (Ω), H 2 (Ω), and t corrsponding norms u = ( u 2 dx ) 1/2, u 1 = ( α u Ω α 1 Ω x α 2 dx ) 1/2, u 2 = ( α u α 2 Ω x α 2 dx ) 1/2.

4 STABLE AND ROBUST SOLUTION METHODS FOR THE BRINKMAN PROBLEM 3 W sall considr uniformly stabl discontinuous Galrin mtods and robust itrativ solution mtods for t following Brinman problm: ɛ 2 div ε(u) + ρ 2 u + p = f in Ω, div u = g in Ω, u = 0 on Ω. Hr, wit t usual notation, u is t vlocity fild, p is t prssur, and ε(u) L 2 (Ω) sym d d is t symmtric (linarizd) strain rat tnsor dfind by ε(u) = u+ ut 2. W also assum tat t cofficints ɛ and ρ satisfy (2.1) 0 < ɛ 2 1 and 0 ρ 2 1. Assumption (2.1) on t cofficints is rasonabl sinc for ɛ 2 ρ 2, on can divid t quation by ɛ 2, and if ɛ 2 < ρ 2, on can divid t quation by ρ 2. In t cas ɛ = 1 and ρ = 0 on obtains t Stos problm. On can also considr t non-dimnsional momntum quation 1 t u 1 R div ε(u)+ p = f arising from tim discrtization of t Navir-Stos problm, wr R is t Rynolds numbr and t is t tim stp siz. Lt δ = min{ t, R}, tn on can multiply t quation wit δ and satisfy t assumption (2.1) for t rscald quation ɛ 2 div ε(u) + ρ 2 u + p = f wit p = δp. Finally, lt g satisfy t compatibility condition Ω gdx = 0. T variational formulation of t Brinman problm can b writtn as: Find (u, p) H0 1(Ω)d (Ω) suc tat L 2 0 (2.2) { a(u, v) + b(v, p) = (f, v), for all v H 1 0 (Ω) d, b(u, q) = (g, q), for all q L 2 0 (Ω), T bilinar forms a(, ) and b(, ) ar dfind by a(u, v) := ɛ 2 ε(u) : ε(v)dx + ρ 2 uvdx, Ω Ω (2.3) for all u, v H0 1 (Ω) d, b(u, q) := q div udx, for all u H0 1 (Ω) d, q L 2 0(Ω). Ω By Korn s inquality for functions in H 1 (Ω) d, t conditions for t xistnc and uniqunss of t solution (u, p) of (2.2) (2.3) ar wll nown and undrstood, s,.g. [18]. For convninc, in tis papr, w assum tat t domain Ω is suc tat t following rgularity stimat olds for f L 2 (Ω) d and g = 0 (s.g. [18, 19]) : (2.4) ɛ 2 u 2 + ɛρ u 1 + p 1 f. Lmma 2.1. Lt Ω b an opn boundd domain wit C 1,1 -boundary or a convx polydron in R d (d = 2, 3).Tn t solution of t problm (2.2) (2.3) as t rgularity proprty (2.4). In quation (2.4) and trougout t prsntation tat follows, t iddn constants in, and ar indpndnt of ɛ, ρ, λ and t ms siz. 3. Discontinuous Galrin discrtization In tis sction, w first giv som prliminaris and notation for a DG formulations. Nxt, w driv t DG discrtization of t Brinman problm and dscrib t Uzawa mtod for solving t saddl piont systm (2.2). Finally, w analyz t stability and approximation proprtis of tis discrtization.

5 4 QINGGUO HONG AND JOHANNES KRAUS 3.1. Prliminaris and notation. W dnot by T a sap-rgular triangulation of ms-siz of t domain Ω into triangls {K}. W furtr dnot by E I t st of all intrior dgs (or facs) of T and by E B t st of all boundary dgs (or facs); w st E = E I EB. For s 1, w dfin H s (T ) = {φ L 2 (Ω), suc tat φ K H s (K) for all K T }. Lt us rcall t dfinitions of t spacs to b usd rin: wit t norm H(div; Ω) := {v L 2 (Ω) : div v L 2 (Ω)}, v 2 H(div;Ω) := v 2 + div v 2. As commonly usd wit DG mtod, w dfin som trac oprators. Lt = K 1 K 2 b t common boundary (intrfac) of two subdomains K 1 and K 2 in T, and n 1 and n 2 b unit normal vctors to pointing to t xtrior of K 1 and K 2, rspctivly. For any dg (or fac) E I and a scalar q H 1 (T ), vctor v H 1 (T ) d and tnsor τ H 1 (T ) d d, w dfin t avrags and jumps {v} = 1 2 (v K 1 n 1 v K2 n 2 ), {τ } = 1 2 (τ K 1 n 1 τ K2 n 2 ), τ = 1 2 (τ K 1 + τ K2 ), [q] = q K1 q K2, [v] = v K1 v K2, [v ] = v K1 n 1 + v K2 n 2, wr v n = 1 2 (vnt + nv T ) is t symmtric part of t tnsor product of v and n. Wn E B tn t abov quantitis ar dfind as {v} = v n, {τ } = τ n, τ = τ, [q] = q, [v] = v, [v ] = v n. Sinc n 1 = n 2, {ε(u)} = ε(u) n 1 and [[v ] = [v] n 1, it follows tat (3.1) ε(u) : [v ] = trac([v ] T ε(u) ) = trac([v]{ε(u)} T ) = {ε(u)} [v], for all u, v H 1 (T ) d. If w dnot by n K t outward unit normal to K, it is asy to cc tat (3.2) v n K qds = {v}[q]ds, for all v H(div; Ω), for all q H 1 (T ). K T K E Also for τ H 1 (Ω) d d and for all v H 1 (T ) d, w av (3.3) (τ n K ) vds = {τ } [v]ds. K T K T finit lmnt spacs ar dnotd by E V = {v H(div; Ω) : v K V (K), K T ; v n = 0 on Ω}, S = {q L 2 (Ω) : q K Q(K), K T ; qdx = 0}. For t DG mtod, w dnot t RT pair RT l (K)/P l (K) or t BDM pair BDM l (K)/P l 1 (K) or t BDFM pair BDF M l (K)/P l 1 (K) as V (K)/Q(K). Not tat div V (K) = Q(K), wic implis tat t divrgnc-fr vlocity filds ar prsrvd (s [14]). W rcall t basic approximation proprtis of ts spacs: for all K T and for all v H s (K) d, tr xists v I V (K) suc tat (3.4) v v I 0,K + K v v I 1,K + 2 K v v I 2,K s K v s,k, 2 s l + 1. Ω

6 STABLE AND ROBUST SOLUTION METHODS FOR THE BRINKMAN PROBLEM DG formulations. W not tat according to t dfinition of V, t normal componnt of any v V is continuous on t intrnal dgs and vaniss on t boundary dgs. Trfor, by splitting a vctor v V into its normal and tangntial componnts v n and v t (3.5) v n := (v n)n, v t := v v n, w find tat for all E, (3.6) [v n ] wds = 0, for all w H 1 (T ) d, v V, and tus (3.7) [v] wds = [v t ] wds = 0, for all w H 1 (T ) d, v V. A dirct computation, similar to t on givn in (3.1), sows tat implying tat [u t ] n : [v t ] n = (([u t ] n)n) [v t ] = 1 2 ( ([ut ]n T + n[u t ] T ) n ) [v t ] = 1 2 ([u t] + ([u t ] n)n) [v t ] = 1 2 [u t] [v t ], (3.8) [u t ] : [v t ] = 1 2 [u t] [v t ]. Trfor, t discrtization of t Brinman problm (2.2) is givn by: Find (u, p ) V S suc tat { a (u (3.9), v ) + b (v, p ) = (f, v ), for all v V, b (u, q ) = (g, q ), for all q S, wr (3.10) (3.11) a (u, v) = ɛ 2( b (u, q) = K T K ε(u) : ε(v)dx {ε(u)} [v t ]ds E {ε(v)} [u t ]ds + Ω E uqdx, E ) η 1 [u t ] [v t ]ds + ρ 2 Ω uvdx, and η is a proprly cosn pnalty paramtr indpndnt of t ms siz suc tat a (, ) is positiv dfinit. Rmar 3.1. Noting t idntitis (3.1) and (3.8), w can rwrit a (, ) as a (u, v) = ɛ 2( ε(u) : ε(v)dx ε(u) : [v t ]ds (3.12) K T K E ε(v) : [u t ]ds + E E ) 2η 1 [u t ] : [v t ]ds + ρ 2 Ω uvdx, wic gnralizs t bilinar form in [12] sinc t normal componnt of u V is continuous. Wn compard to t bilinar form in [14], w can s tat t only diffrnc is tat in (3.12) t jumps of u t on t boundary dgs ar includd for t Diriclt boundary, wic is du to t fact tat t condition u n = 0 is includd in t dfinition of t spac V.

7 6 QINGGUO HONG AND JOHANNES KRAUS Noting tat div V = S w can rwrit t abov systm in t following quivalnt form: Find (u, p ) V S suc tat { a (u (3.13), v ) + λb (u, div v ) + b (v, p ) = (f, v ) + λ(g, div v ), for all v V, b (u, q ) = (g, q ), for all q S. By t dfinition of b (, ), namly (3.11), t Uzawa mtod for (3.13) rads: Givn (u l, pl ), t nw itrat (u l+1, pl+1 ) is obtaind by solving t following systm: { a (u l+1 (3.14), v ) + λ(div u l+1, div v ) = (f, v ) b (v, p l ) + λ(g, div v ), v V, p l+1 = p l λ div ul+1 λg, wr λ > 1 is a damping paramtr. Convrgnc of tis mtod as bn discussd in svral wors, s,.g., [9, 20, 21, 22] indicating tat for larg λ, t itrats convrg rapidly to t solution of problm (3.9). As a consqunc, t major computational cost lis in solving t discrt problm arising from t following linar lasticity typ quation wit mass trm: Find u V suc tat (3.15) wr A (, ) is dfind by (3.16) and a (u, v ) is givn in (3.10). A (u, v ) = (F, v ), for all v V, A (u, v ) = a (u, v ) + λ(div u, div v ), Rmar 3.2. In fact, (3.15) is t DG discrtization of t following problm: Find u H 1 0 (Ω) suc tat (3.17) a(u, v) + λ(div u, div v) = (F, v), for all v H 1 0 (Ω). Noting tat div V = S, it is immdiatly sn tat problm (3.15) (3.16) as t following quivalnt formulation: Find (u, p ) V S suc tat { a (u (3.18), v ) + b (v, p ) = (F, v ), for all v V, b (u, q ) (λ 1 p, q ) = 0, for all q S, wr a (u, v ) and b (u, q ) ar dfind by (3.10) and (3.11), rspctivly. It is clar tat (3.15) is t DG discrtization of t linar lasticity quation wit a mass trm wic rads: Find (u, p) H0 1(Ω)d L 2 0 (Ω) suc tat { a(u, v) + b(v, p) = (F, v), for all v H 1 (3.19) 0 (Ω) d, b(u, q) (λ 1 p, q) = 0, for all q L 2 (Ω), wr a(u, v) and b(u, q) ar dfind in (2.3) Approximation and stability proprtis. In tis subsction, w analyz t approximation and stability proprtis of t discrt problms (3.9) and (3.15) (3.16). For any u H 1 (T ) d, w dfin t ms dpndnt norms: u 2 = ε(u) 2 0,K + 1 [u t ] 2 0,, K T E u 2 1, = u 2 0,K + 1 [u t ] 2 0,. K T E Nxt, for u H 2 (T ) d, w dfin t DG -norm, t ɛ,ρ -norm and t ɛ, -norm as follows (3.20) u 2 DG = u 2 1, + K T 2 K u 2 2,K,

8 STABLE AND ROBUST SOLUTION METHODS FOR THE BRINKMAN PROBLEM 7 (3.21) u 2 ɛ,ρ = ɛ 2 u 2 DG + ρ 2 u 2, (3.22) u 2 ɛ, = u 2 ɛ,ρ + div u 2. From t discrt vrsion of t Korn s inquality (s [23, Equation (1.12)]) w av t following norm quivalnc rsult. Lmma 3.1. T norms DG,, and 1, ar quivalnt in V, namly (3.23) u DG u u 1,, for all u V. Proof. By t invrs inquality, w clarly av tat u DG u 1,. W sow now tat u u 1,. From t dfinitions w immdiatly gt u u 1,. To prov t inquality in t otr dirction, w us [23, Equation (1.22)], namly, u 2 + ( u 2 0,K ε(u) 2 0,K + sup u mds) 2 m RM(Ω) K T K T Ω (3.24) + ) 1 π [u] 2 0,. E I m L 2 ( Ω)=1 ( Hr RM(Ω) dnots t spac of rigid body motions, { RM(Ω) = a + Ax } a R d, A R d d, A = A T, and t oprator π is t L 2 ()-ortogonal projction oprator onto (P 1 ()) d, t spac of vctorvalud linar polynomials on. For t scond trm on t rigt and sid of (3.24) w av sup ( m RM(Ω) m L 2 ( Ω)=1 Ω u mds) 2 E I Ω u 2 ds = E B u 2 0, E B 1 [u] 2 0,. Sinc π is an ortogonal projction, for t tird trm on t rigt and sid of (3.24) w obtain 1 π [u] 2 0, 1 [u] 2 0,. E I Finally, combining t two inqualitis abov complts t proof. Morovr, bot bilinar forms, a (, ) and b (, ), introducd abov ar continuous and w av a (u, v) u ɛ, v ɛ,, for all u, v H 2 (T ) d, b (u, q) u ɛ, q, for all u V, q L 2 0(Ω). For our coic of t finit lmnt spacs V and S, noting tat div u u 1, and t proof of Lmma 3.1, wic implis u u 1,, w av t following inf-sup condition for b (, ) (s,.g., [14, 24]). Lmma 3.2. Tr xist positiv constants β 0, β 1 indpndnt of ɛ, ρ and t ms siz, suc tat (div u, q ) (3.25) inf sup q S u V u 1, q β (div u, q ) 0 and inf sup q S u V u ɛ, q β 1. Now for any givn g L 2 0 (Ω), w dfin (3.26) Z (g) = {u V : b (u, q ) = (g, q ), q S }. Noting tat div V = S, it follows tat a (, ) is corciv on Z (0), namly, w av t following Lmma wos proof follows t lins of similar argumnts in [14, 25].

9 8 QINGGUO HONG AND JOHANNES KRAUS Lmma 3.3. For sufficintly larg η, indpndnt of t ms siz, w av (3.27) a (u, u ) u 2 ɛ,ρ, for all u V, and nc (3.28) a (u, u ) u 2 ɛ,, for all u Z (0). By t quivalnc of t norms sown in (3.23) and also by t standard tory for solvability of mixd problms [26], w obtain t following torm tat is similar to a rsult in [27]. Torm 3.1. T discrt problm (3.9) as a uniqu solution (u, p ) V S tat satisfis (3.29) u u ɛ,ρ inf u v ɛ,ρ, p p inf p q + v Z (g) q S wit (u, p) bing t solution of (2.2). inf u v ɛ,ρ v Z (g) T bilinar forms a (, ) and A (, ) ar corciv and also dfin norms on V, i.., u 2 a = a (u, u), u 2 A = A (u, u). W now introduc t canonical intrpolation oprator Π div : H 1 (Ω) d V and dnot t L 2 -projction on S by Q. T following Lmma summarizs som of t proprtis of Π div and Q ndd latr. Lmma 3.4. For all w H 1 (K) d w av wr r 1 = sup χ H 1 (χ,r) χ 1. div Π div = Q div ; Π div w 1,K w 1,K ; w Π div w 2 0, K K w 2 1,K; div(w Π div w) 1 K div w, Proof. T proof of t commutativity of Π div and div and t first two inqualitis ar wll nown and w rfr t radr to [28] for t dtails. T last inquality follows from t approximation proprtis of t L 2 -ortogonal projction, tat is, div w div Π div w ((I Q ) div w, χ) (div w, (I Q )χ) 1 = sup = sup χ H 1 χ 1 χ H 1 χ 1 div w (I Q )χ) sup K div w. χ H 1 χ 1 Rmar 3.3. For all w H 0 (div; Ω) H 1 (T ) d, by noting tat H 0 (div; Ω) H 1 (T ) d is a subst of L 3 (Ω) H(div; Ω), w conclud tat Π div is wll-dfind for w, s,.g., [28] and furtr by Lmma 3.4, w av t stimat (3.30) Π div w 1, w 1,. T abov canonical intrpolation is not boundd in H(div) norm. For tat rason, w will nd to us t following local boundd cocain projction oprators. For any K T, w dnot by (K) t st of all subsimplxs of K, and by m (K) all subsimplxs of dimnsion m. W furtr dnot by (T ) t st of all subsimplxs of all dimnsions of t triangulation T, and corrspondingly by m (T ) t st of all subsimplics of dimnsion m. Now for ac f m (T ), w lt Ω f b t associatd macrolmnt consisting of t union of t lmnts of T containing f, i.., (3.31) Ω f = {K K T, f (K)}.

10 STABLE AND ROBUST SOLUTION METHODS FOR THE BRINKMAN PROBLEM 9 In addition to macrolmnts Ω f it will also b convnint to introduc t notion of an xtndd macrolmnt Ω f dfind for f (T ) by (3.32) Ω f = i 0 (f) W lt T f, dnot t rstriction of T to Ω f. Dfin D m,k Ω by (3.33) D m,k = {D m 1,K K T f,, f m (K)}, D 0,K = Ω K. and finally lt (3.34) D K = D d,k (d = 2, 3). Lmma 3.5. Tr xist projction oprators π div : H 0 (div; Ω) V and π : L 2 (Ω) S satisfying div π div = π div. T construction of π div and π in diffrntial form can b found in [16]. Now w sow t local bounddnss of ts oprators. Lt D K b dfind in (3.34). Tn t following stimats old, s [16]. Lmma 3.6. ([16]) For any u H 0 (div, Ω) and p L 2 0 (Ω), w av Ω i. (3.35) π div u 0,K u 0,DK + K div u 0,DK, K T, π p 0,K p 0,DK, K T. Using Lmma 3.6 and Lmma 3.4, w can prov t following bounddnss of π div in DG norm. Lmma 3.7. Lt u H 0 (div, Ω) H 1 (T ) d and dnot u 1,DK = u 1,K. Tn w av K D K (3.36) Furtrmor, (3.37) π div u 1,K u 1,DK, K T, u π div u 0, K 1 2 K u 1,DK, K T. π div u 1, u 1,. Proof. W prov t first inquality first. Noting tat π div is a projction, by t triangl inquality, it follows tat (3.38) π div u 1,K π div (u Πdiv u) 1,K + Π div u 1,K. Now by a standard invrs inquality and t H 1 -stability of Π div givn in Lmma 3.4, w av (3.39) π div u 1,K 1 K πdiv (u Πdiv u) 0,K + u 1,K. Nxt w us Lmma 3.6 and t rror stimat (3.4) for Π div in ordr to obtain (3.40) π div u 1,K 1 K ( u Πdiv u 0,D K + K div u Q div u 0,DK ) + u 1,K u 1,DK + div u 0,DK + u 1,K u 1,DK + u 1,DK + u 1,DK u 1,DK.

11 10 QINGGUO HONG AND JOHANNES KRAUS Now w prov t scond inquality. By t trac torm, inquality (3.40), and using t fact tat π div is a projction and also Lmma 3.4 and Lmma 3.6, w gt (3.41) u π div u 2 0, K K u π div u 2 1,K + 1 K u πdiv u 2 0,K K u 2 1,K + K u 2 1,D K + 1 K (I πdiv )(u Πdiv u) 0,K K u 2 1,D K + 1 K u Πdiv u 2 0,K + 1 K πdiv (u Πdiv K u 2 1,D K + K u 2 1,K + 1 K ( u Πdiv u 2 0,D K + 2 K div u Q div u 2 0,D K ) K u 2 1,D K. Finally w prov t tird inquality. From t inquality (3.40), it is obvious tat (3.42) π div u 2 1,K u 2 1,K. K T K T Now st = K 1 K 2. Tn from (3.41), w av (3.43) [π div u] 2 0, [π div u u] 2 0, + [u] 2 0, π div u u 2 0, K 1 + π div u u 2 0, K 2 + [u] 2 0, K1 u 2 1,D K1 + K2 u 2 1,D K2 + [u] 2 0,. u) 2 0,K In viw of t dfinition of t norm 1,, combining t two inqualitis (3.42) and (3.43) complts t proof of (3.37). T following rsult sows tat t approximation proposd by t DG discrtization (3.15) is uniform. Torm 3.2. Lt u b t solution of (3.19) and u b t solution of (3.15). Tn w av t following stimat ( u u 2 ɛ,ρ + λ div(u u ) 2 inf u v 2 ɛ,ρ + λ div(u v) 2). v V Proof. If (u, p) is t solution of t continuous problm (3.19) and (u, p ) is t solution of t discrt problm (3.18) w av tat p = λ div u, and, sinc div V = S w also av tat p = λ div u. T lft and sid of t first quation in (3.18) tn is givn by t bilinar form (3.16), and, sinc tis discrt problm is consistnt, w av A (u u, v) = 0, for all v V. Considr now t intrpolation π divu V of u and st q = λ div π div u. Rcall tat p = λ div u, and p = λ div u and nc (by Lmma 3.5) q = λπ div u = π p. Tn wit = (u π divu) from t corcivity of a (, ) w gt (3.44) wic mans tat (3.45) 2 ɛ,ρ + λ 1 p q 2 = ɛ 2 2 DG + ρ λ div 2 A (, ) = A (u π div u π div u ɛ,ρ ɛ,ρ + λ 1 p q p q u, ) = a (u π div u, ) + λ(div(u π div u), div ) ( u π div u 2 ɛ,ρ + λ 1 p q 2 ) 1/2 ( 2 ɛ,ρ + λ 1 p q 2 ) 1/2, u u 2 ɛ,ρ + λ 1 p p 2 u π div u 2 ɛ,ρ + λ 1 p q 2.

12 STABLE AND ROBUST SOLUTION METHODS FOR THE BRINKMAN PROBLEM 11 Hnc for any v V, w av (3.46) u u 2 ɛ,ρ + λ div(u u ) 2 u v π div (u v) 2 ɛ,ρ + λ div(u v) π div(u v) 2 u v 2 ɛ,ρ + λ div(u v) 2 + π div (u v) 2 ɛ,ρ + λ π div(u v) 2. By t dfinition of t norm ɛ,ρ and Lmma 3.7, w gt (3.47) π div (u v) 2 ɛ,ρ + λ π div(u v) 2 ɛ 2 π div (u v) 2 1, + ρ2 π div (u v) 2 + λ π div(u v) 2 ɛ 2 u v 2 1, + ρ2 π div (u v) 2 + λ π div(u v) 2. Now, sinc 1, ρ 1, λ 1 w s tat (3.48) π div (u v) 2 ɛ,ρ + λ π div(u v) 2 ɛ 2 u v 2 1, + ρ2 ( u v div(u v) 2 ) + λ div(u v) 2 ɛ 2 u v 2 1, + ρ2 u v 2 + ρ 2 2 div(u v) 2 + λ div(u v) 2 ɛ 2 u v 2 1, + ρ2 u v 2 + λ div(u v) 2 u v 2 ɛ,ρ + λ div(u v) 2, wr w av usd Lmma 3.6. Taing t infimum ovr v tn yilds t dsird rsult. Rmar 3.4. Lt us st a s (u, v) = ε(u) : ε(v)dx {ε(u)} [v t ]ds {ε(v)} [u t ]ds+ K T K E E E and B λ ((u, p ), (v, q )) = a s (u, v ) (div u, q ) (div v, p ) λ 1 (p, q ). η 1 [u t ] [v t ]ds, Tn for any givn (u, p ), coosing (v, q ) = (u, p ), by t corcivity of a (, ) wit ɛ = 1, ρ = 0, it is straigtforward to sow tat t inf-sup condition for B λ (, ) olds, namly, for any (u, p ) V S w av (3.49) sup (v,q ) V S B λ ((u, p ), (v, q )) v 1, + λ 1/2 q u 1, + λ 1/2 p. For t Stos quation, w av from [26, Torm 8.2.1] and [29, 30] tat B ((u, p ), (v, q )) (3.50) sup u 1, + p. (v,q ) V S v 1, + q 3.4. An a priori stimat for t discrt problm. T nxt lmma is an a priori stimat of t L 2 -norm of t solution of a discrt problm wic is latr usd to prov t so calld smooting proprty an ssntial part in t multigrid convrgnc analysis. W stat and prov tis stimat r (bfor t multigrid analysis), sinc it could b of indpndnt intrst. W considr t finit lmnt spacs introducd arlir: V H(div; Ω) and S L 2 0 (Ω). Lt w 1 V and w 2 V b givn and lt ũ V, p S solv t discrt problm (3.51) a s (ũ, v) (div v, p) = as (w 1, v), for all v V, (div ũ, q) = (div w 2, q), for all q S.

13 12 QINGGUO HONG AND JOHANNES KRAUS W not tat t inf-sup condition (3.50) implis tat (3.52) a s (ũ, v) (div v, p) (div ũ, q) ũ 1, + p sup (v,q) V S v 1, + q a s = sup (w 1, v) (div w 2, q) w 1 1, + div w 2. (v,q) V S v 1, + q Lmma 3.8. For t solution of (3.51) w av t stimat: (3.53) ũ w 1 + div w 2 1. Proof. W considr t following dual problm: Find φ (H0 1(Ω))d and θ L 2 0 (Ω) suc tat (3.54) a s (v, φ) (div v, θ) = (ũ, v), for all v (H 1 0 (Ω))d, (div φ, q) = 0, for all q L 2 0 (Ω), wr a s (v, φ) = Ω ε(v) : ε(φ)dx. Lt Π div b t intrpolation oprator introducd arlir in Sction 3.3. Rcall tat div φ = 0 and nc (div Π div φ, p) = 0. From quations (3.51) w tn av (3.55) 0 = a s (w 1, Π div φ) as = a s (ũ, Πdiv (w 1, φ) a s (w 1, φ Π div φ) + (div Πdivφ, p) φ) as (ũ, Πdiv Obsrving tat a s (φ, v) = a s (φ, v) for all v V, from (3.54) and (3.55) w obtain (3.56) ũ 2 = a s (φ, ũ) (div ũ, θ) + as (w 1, φ) a s (w 1, φ Π div φ) as (ũ, Πdiv φ). Combining t first and t last trm, using t triangl inquality and t continuity of a s (, ) tn sows tat ũ 2 (div ũ, θ) + a s (w 1, φ) + a s (w 1, φ Π div φ) + as (ũ, φ Πdiv φ) (div ũ, θ) + a s (w 1, φ) + ( w 1 1, + ũ 1, ) φ Π div φ 1,. As w av tat div ũ = div w 2 for t first trm on t rigt sid w gt (div ũ, θ) = (div w 2, θ) θ 1 sup χ H 1 (div w 2, χ) χ 1 = div w 2 1 θ 1. For t scond trm, by t rgularity stimat (2.4) wit ɛ = 1, ρ = 0, w av tat φ (H 2 (Ω)) d, and, tus, φ is continuous and [φ] = 0. Now, intgrating by parts and combining t intrfac trms from nigboring lmnts sows tat a s (φ, w 1) = ε(φ) : ε(w 1 )dx {ε(φ)} [(w 1 ) t ]ds K T K E {ε(w 1 )} [φ t ]ds + = E K T = T T K E φ). η 1 [φ t ] [(w 1 ) t ]ds, ε(φ) : ε(w 1 )dx {ε(φ)} [(w 1 ) t ]ds T E div ε(φ) w 1 φ 2 w 1. Finally, t dsird rsult follows from t intrpolation stimats in Lmma 4.3, t rgularity stimat φ 2 + θ 1 ũ, inquality (3.52), and t invrs inqualitis w 1 1, 1 w 1 and div w 2 1 div w 2 1.

14 STABLE AND ROBUST SOLUTION METHODS FOR THE BRINKMAN PROBLEM Multigrid mtod In tis sction, w dsign a multigrid algoritm to solv t discrt systm (3.15) (3.16) arising from t DG discrtization of t quation ɛ 2 div ε(u) + ρ 2 u + λ u = F. W will sow tat t algoritm is robust wit rspct to t paramtrs ɛ, ρ, λ wn ɛ 2 or λ min{ɛ 1, 1 }. Hnc, by combining it wit t Uzawa mtod and coosing λ to b larg noug, w can also solv t discrt systm (3.9) arising from t DG discrtization of t Brinman problm vry fficintly Prliminaris. Lt {T } J =0 b a family of partitions of Ω and dnot t finst partition by T = T J. T dgs (facs) of T ar dnotd by E. W assum tat all t partitions T, = 0, 1,..., J, ar quasi-uniform wit caractristic ms siz and = γ 1, γ (0, 1), wr 0 = O(1). Not tat t pnalty trm in t bilinar form a (, ) dpnds on t ms siz of t partition. Tus, for vry partition T w nd to spcify t spac V at lvl. A natural coic is t spac M dfind as follows: M = {v H(div; Ω) : v K V (K), K T ; v n = 0 on Ω}. Morovr, w dnot t prssur spac S at lvl by { S = q L 2 (Ω) : q K Q(K), K T ; Ω } qdx = 0. Tus, corrsponding to t st of rfind triangulations {T } J =0, w also av a squnc of nstd H(div, Ω)-conforming finit lmnt spacs M 0 M 1 M 2 M J H(div, Ω). Wit vry spac w associat a bilinar form a (, ) wic corrsponds to a (, ) at lvl, i.., a (u, v) = ɛ 2( ε(u) : ε(v)dx {ε(u)} [v t ]ds K T K {ε(v)} [u t ]ds + E E E ) η 1 [u t] [v t ]ds + ρ 2 (u, v). Adding t divrgnc trm tn givs t bilinar form corrsponding to A (, ) at lvl i.., A (u, v) = a (u, v) + λ(div u, div v), for all u, v M. Our goal is to analyz t variabl V-cycl and W-cycl multigrid algoritms for t solution of t following problm: Givn F M J, find v M J satisfying (4.1) A J (v, φ) = (F, φ), for all φ M J. To dfin t algoritm, lt us first clarify t notation. For = 0,, J, dfin t oprator A : M M by (A w, φ) = A (w, φ), for all φ M. T norms on M inducd by A (, ) and a (, ) ar dnotd by 2 A, and 2 a rspctivly, i.., u 2 A = A (u, u), u 2 a = a (u, u), for all u M. W also nd t L 2 -ortogonal projctions on M, and S, wic will b dnotd by Q : L 2 (Ω) M and Q : L 2 (Ω) S and t canonical intrpolation Π : [H0 1(Ω)]2 M. According to t notation of t prvious sction, Π and Q ar just sortands for Π div and Q, i..

15 14 QINGGUO HONG AND JOHANNES KRAUS Π = Π div, Q = Q, and w rcall tat Q div = div Π. Furtr, w introduc t oprator P 1 : M M 1 dfind by (4.2) A 1 (P 1 w, φ) = A (w, φ), for all φ M 1. Noting tat t bilinar forms A ar nonnstd, P 1 is not a projction. Finally, w dnot t norm 1, at lvl as 1,. To dfin t smooting procss, w rquir linar oprators R : M M for = 1,, J. Ts oprators may b symmtric or nonsymmtric wit rspct to t innr product (, ). If R is nonsymmtric, tn w dfin R t to b its adjoint and st R (l) = R if l is odd, R t if l is vn Multigrid algoritm. T multigrid oprator B : M M is dfind by induction and is givn as follows, s,.g., [31]. Multigrid algoritm St B 0 = A 1 0. Assum tat B 1 as alrady bn dfind, lt B g for g M b dfind as follows: (1) St x 0 = 0 and q 0 = 0. (2) Dfin x l for l = 1,, m() by (4.3) x l = x l 1 + R (l+m()) (g A x l 1 ). (3) St y m() = x m() + q p, wr q i for i = 1,, p is dfind by (4.4) q i = q i 1 + B 1 [Q 1 (g A x m() ) A 1 q i 1 ]. (4) Dfin y l for l = m() + 1,, 2m() by y l = y l 1 + R (l+m()) (g A y l 1 ). (5) St B g = y 2m(). In tis algoritm, m() is a positiv intgr wic may vary from lvl to lvl and dtrmins t numbr of smooting itrations at tat lvl and p is anotr positiv intgr. W sall study t cass p = 1 and p = 2, wic corrspond to t symmtric V - and W -cycls of t multigrid algoritm, rspctivly Multigrid convrgnc. St K = I R A, tn K = I Rt A is t adjoint of K wit rspct to A (, ). Furtr, st K (m) (K = K ) m/2 if l is odd, (K K ) (m 1)/2 K if l is vn, (m) and dnot by ( K ) (m) t adjoint of K wit rspct to A (, ). For convrgnc stimats, w sall ma t following a priori assumptions: (A0) T spctrum of K K is in t intrval [0, 1). In ordr to analyz t approximation proprty and t smooting proprty of t multigrid algoritm, for any u M, w nd to dfin a norm at lvl wic is similar to t on in [17] and givn by (4.5) u 2,0 := ɛ 2 u 2 + ρ 2 2 u 2 + λ 2 div u 2 + λ 2 max{ 2, ɛ2 } Q 1 div u 2 if ɛ 2, ɛ 2 u 2 + ρ 2 2 u 2 + λ 2 div u 2 + λ 2 ɛ 2 2 Q 1 div u 2 if ɛ 2.

16 STABLE AND ROBUST SOLUTION METHODS FOR THE BRINKMAN PROBLEM 15 T scond assumption is an approximation proprty in,0 -norm (also nown as approximation and rgularity assumption, cf [31]), i.., (A1) (I P 1 )u,0 u A, for all u M. T tird assumption is a rquirmnt on t smootr (smooting proprty) and rads, (m) (A2) ( K ) u A m 1/4 1 u,0, for all u M. T nxt Lmma is an analogu of a rsult givn in Brambl, Pascia, Xu [31, Lmma 4.1]. W includ its sort proof for t sa of slf-containdnss. (m) Lmma 4.1. Assum tat (A0), (A1) and (A2) old and lt ũ = K u. Tn w av t stimat A ((I P 1 )ũ, ũ) m 1/4 u 2 A, for all u M. Proof. By t Caucy-Scwarz inquality and assumption (A2), w av A ((I P 1 )ũ, ũ) = A ((I P 1 ) = A (( (m) (m) K u, K u) (m) K ) (I P 1 ) K (m) u, u) (m) ( K ) (I P 1 )ũ A u A m 1/4 1 (I P 1)ũ,0 u A. Nxt, by assumptions (A1) and (A0) (applid in tat ordr) w av A ((I P 1 )ũ, ũ) m 1/4 1 (I P 1)ũ,0 u A m 1/4 ũ A u A m 1/4 u 2 A. T stimat in Lmma 4.1 provids t prrquisit to apply t gnral tory dvlopd in [31]. Indd, according to [31], assumptions (A0), (A1) and (A2) and Lmma 4.1 ar sufficint to sow spctral quivalnc for t variabl V-cycl multigrid prconditionr (Torm 4.1) and a uniform convrgnc rsult for t W-cycl multigrid mtod (Torm 4.2). T first rsult is just a rstatmnt of [31, Torm 6] undr full rgularity. Torm 4.1 (Torm 6 in [31]). Assum tat (A0), (A1) and (A2) old and dfin B j in Algoritm 4.2 wit p = 1. Furtr assum tat t numbr of smooting stps m() satisfis β 3 m() m( 1) β 4 m() wit β 3 1 and β 4 > 1 indpndnt of. Tn t following spctral quivalnc olds (4.6) η 0 A (u, u) A (B A u, u) η 1 A (u, u) for all u M. wit constants η 0 and η 1 suc tat η 0 m()α M + m() α and η 1 M + m()α m() α, wr M is indpndnt of ɛ, ρ, λ and, and α dnots t rgularity indx. T convrgnc analysis of t W -cycl is also bn conductd in [31]. Torm 4.2 (Torm 4 in [31]). Assum tat (A0), (A1) and (A2) old and tat t numbr of smooting stps m() = m is constant for all. Tn, for sufficintly larg m, B dfind via t W-cycl algoritm (p=2) satisfis M A ((I B A )u, u) M + m α u 2 A for all u M wit M indpndnt of ɛ, ρ, λ and, wr α dnots t rgularity indx.

17 16 QINGGUO HONG AND JOHANNES KRAUS W rmar r tat modifying assumption (A1) on can prov t rsults abov for t cas of lss tan full lliptic rgularity. For dtails w rfr to Brambl, Pascia and Xu [31]. As w av sn, t stimats in Torms ar valid if assumptions (A0), (A1) and (A2) ar vrifid. In t nxt subsctions w sow tat ts assumptions old in our framwor Approximation proprty. In tis subsction, w vrify (A1). On of t difficultis in t analysis is tat t bilinar forms A (, ), = 1,, J ar not nstd. W now prov a simpl rlation btwn A (, ) and A 1 (, ). Lmma 4.2. If = γ 1, γ (0, 1), tn (4.7) u 2 A 1 u 2 A u 2 A 1, for all u M 1. Proof. Lt u M 1. Obsrv tat [u t ] = 0 for dgs (facs) E wic ar intrior to t lmnts in T 1, bcaus u is a continuous, in fact a polynomial, function in ac lmnt from T 1. Hnc, E 1 ηγ [u t] 2 ds = E and w av A (u, u) = A 1 (u, u) + ɛ 2( E = A 1 (u, u) + ɛ 2 (γ 1 1) η 1 [u t] 2 ds, for all u M 1 η 1 [u t] 2 ds E 1 T stimats in (4.7) tn asily follow from t idntity abov. E 1 η 1 1 [u t] 2 ds. Rmar 4.1. From Lmma 4.2, for any givn u M, w also av namly, ) η 1 1 [u t] 2 ds P 1 u 2 A 1 P 1 u 2 A = A (u, P 1 u) u A P 1 u A u A P 1 u A 1, (4.8) P 1 u A 1 u A. W now introduc t dual problm for t primal problm (3.17) : Find w H 1 0 (Ω)d suc tat (4.9) ɛ 2 (ε(v) : ε(w)) + ρ 2 (v, w) + λ(div v, div w) = (G, v), for all v H 1 0 (Ω) d undr t following rgularity assumption on w: (4.10) ɛ 2 w 2 + ɛρ w 1 + λ div w 1 G. Rmar 4.2. T rgularity proprty (4.10) olds if Ω satisfis t conditions of Lmma 2.1 (s [32, 33]). From t dfinitions of t bilinar forms A 1 (, ) and A (, ) w av t following simpl idntity for t solution w of (4.9): (4.11) A (v, w) = A 1 (v, w), for all v M 1. Tis follows immdiatly, sinc bot A 1 (, ) and A (, ) ar consistnt. M 1 M w av A (v, w) = (G, v) = A 1 (v, w), wic provs (4.11). T nxt lmma provids stimats on t intrpolation rror. Indd, for any v

18 STABLE AND ROBUST SOLUTION METHODS FOR THE BRINKMAN PROBLEM 17 Lmma 4.3. Lt w H l+1 (Ω) d, l = 0, 1, and Π 1 w b t intrpolant of w in M 1, tn (4.12) w Π 1 w 2 A 1 2l 1 (ɛ2 w 2 l+1 + ρ2 w 2 l + λ div w 2 l ), w Π 1 w 2 A 2l 1 (ɛ2 w 2 l+1 + ρ2 w 2 l + λ div w 2 l ). Proof. By t continuity of a (, ), t trac torm and t intrpolation rror stimat (3.4), w av w Π 1 w 2 a 1 ɛ 2 w Π 1 w 2 DG + ρ 2 w Π 1 w 2 2l 1 (ɛ2 w 2 l+1 + ρ2 w 2 l ). Noting div Π 1 w = Q 1 div w, by t standard approximation rror stimat of t projction Q 1, w av div(w Π 1 w) 2 = div w Q 1 div w 2 2l 1 div w 2 l. Combining t abov two inqualitis and noting t dfinition of t norm A 1, w gt t first inquality in (4.12). T proof of t scond inquality in (4.12) is carrid out in a similar fasion. For t vrification of t approximation proprty w will nd t two following two-lvl stimats. Torm 4.3. For all u M w av (4.13) ɛ (I P 1 )u u A, ρ (I P 1 )u u A. Proof. W first stimat ɛ (I P 1 )u using a standard duality argumnt. Lt w H 1 0 (Ω)d b t solution of t dual problm (4.9) wit G = u P 1 u. Sinc, A (, ) is a consistnt bilinar form, w av A (w, v) = (u P 1 u, v), for all v M. Now lt v = u P 1 u and Π 1 w b t intrpolant of w in M 1. Using t fact tat A (, ) is symmtric for all = 1,, J, idntity (4.11), and t dfinition of t oprator P 1, w av (4.14) u P 1 u 2 = A (w, u P 1 u) = A (u, w) A (w, P 1 u) = A (u, w) A 1 (w, P 1 u) = A (u, w) A 1 (P 1 u, w Π 1 w) A 1 (P 1 u, Π 1 w) = A (u, w) A 1 (P 1 u, w Π 1 w) A (u, Π 1 w) = A (u, w Π 1 w) A 1 (P 1 u, w Π 1 w). Applying t Caucy-Scwarz inquality to t rigt and sid of t idntity abov and using t approximation stimats (4.12), inquality (4.8), and t rgularity stimat (4.10), lads to u P 1 u 2 u A w Π 1 w A + P 1 u A 1 w Π 1 w A 1 1 ( u A + P 1 u A 1 )(ɛ 2 w ρ 2 w λ div w 2 1) 1/2 1 u A (ɛ 2 w ρ 2 w λ div w 2 1) 1/2 1 u A (ɛ w 2 + ρ w 1 + λ div w 1 ) ɛ 1 1 u A (ɛ 2 w 2 + ɛρ w 1 + ɛ λ div w 1 ) ɛ 1 1 u A u P 1 u, wic complts t proof of t first inquality in (4.13). By inquality (4.8), t scond inquality in (4.13) is obtaind from ρ 2 u P 1 u 2 ρ 2 u 2 + ρ 2 P 1 u 2 u 2 A + P 1 u 2 A 1 u 2 A + u 2 A, wic complts t proof.

19 18 QINGGUO HONG AND JOHANNES KRAUS T nxt two Lmmas vrify t approximation proprty (A1). Lmma 4.4. For all u M w av t stimat (4.15) λɛ Q 1 div(u P 1 u) max{ɛ 2, min{, ɛ}} u A. Proof. Lt q = Q 1 div(u P 1 u), tn tr is a w H 1 0 (Ω)d suc tat [18, 34] (4.16) div w = q and w 1 q. By t proprtis of t L 2 -projctions on S and S 1 and t fact tat S 1 S w av Q 1 Q = Q 1 and Q 2 1 = Q 1. Hnc, (4.17) q 2 = (q, div w) = (Q 1 div(u P 1 u), Q 1 div w) = (Q 1 Q div(u P 1 u), Q 1 div w) = (Q div(u P 1 u), Q 2 1 div w) = (div(u P 1u), Q 1 div w). Noting tat Q 1 div w = div Π 1 w, w av (4.18) q 2 = (div(u P 1 u), div Π 1 w). Now from t dfinition of P 1 in (4.2), w av (4.19) λ(div u, div Π 1 w) λ(div(p 1 u), div Π 1 w) = ɛ 2 a s (u, Π 1w) + ɛ 2 a s 1 (P 1u, Π 1 w) ρ 2 (u, Π 1 w) + ρ 2 (P 1 u, Π 1 w). Combining (4.18) and (4.19), w gt q 2 = λ 1 ( ɛ 2 a s (u, Π 1w) + ɛ 2 a s 1 (P 1u, Π 1 w) ) (4.20) wr λ 1 ρ 2 (u P 1 u, Π 1 w w) λ 1 ρ 2 (u P 1 u, w) = I 1 + I 2 + I 3, I 1 = λ 1 ( ɛ 2 a s (u, Π 1w) + ɛ 2 a s 1 (P 1u, Π 1 w) ) ; I 2 = λ 1 ρ 2 (u P 1 u, Π 1 w w); I 3 = λ 1 ρ 2 (u P 1 u, w). First, by t bounddnss of bot a s (, ) and as 1 (, ), t t H1 -stability of Π 1 givn in (3.30), and (4.8), w can stimat I 1 as follows I 1 λ 1 ɛ(ɛ u 1, Π 1 w 1, + ɛ P 1 u 1, 1 Π 1 w 1, 1 ) λ 1 ɛ( u A w 1, + P 1 u A 1 w 1, ) λ 1 ɛ u A w 1,. Noting tat w H 1 0 (Ω)d implis w 1, = w 1, and by (4.16), w obtain (4.21) I 1 λ 1 ɛ u A w 1, = λ 1 ɛ u A w 1 λ 1 ɛ u A q. Nxt, using t Caucy-Scwarz inquality, t approximation stimats of Π 1 givn in (4.12), ρ 1, (4.8) and (4.16), w obtain t following stimat for I 2 : (4.22) I 2 = λ 1 ρ 2 (u P 1 u, Π 1 w w) λ 1 ρ 2 u P 1 u Π 1 w w λ 1 ρ(ρ u + ρ P 1 u ) 1 w 1 λ 1 ρ 1 ( u A + P 1 u A 1 ) w 1 λ 1 1 u A w 1 λ 1 1 u A q.

20 STABLE AND ROBUST SOLUTION METHODS FOR THE BRINKMAN PROBLEM 19 Finally, w av two stimats for I 3. First, by Caucy-Scwarz inquality, t L 2 -rror stimat for u P 1 u givn in (4.13), ρ 1 and (4.16), w gt (4.23) I 3 = λ 1 ρ 2 (u P 1 u, w) λ 1 ρ 2 u P 1 u w λ 1 ρ 2 ɛ 1 u A w λ 1 ɛ 1 u A w λ 1 ɛ 1 u A q. Scond, by Caucy-Scwarz inquality, ρ 1, (4.8) and (4.16), w obtain (4.24) I 3 = λ 1 ρ 2 (u P 1 u, w) λ 1 ρ 2 u P 1 u w λ 1 ρ(ρ u + ρ P 1 u ) w λ 1 ρ( u A + P 1 u A 1 ) w λ 1 u A w λ 1 u A q. Finally, in viw of (4.20), by combining t stimats for I 1, I 2 and I 3, namly (4.21), (4.22), (4.23) and (4.24), w gt q 2 = I 1 + I 2 + I 3 λ 1 ɛ u A q + λ 1 1 u A q + λ 1 min{ɛ 1, 1} u A q wic implis λ 1 max{ɛ,, min{ɛ 1, 1}} u A q, Q 1 div(u P 1 u) λ 1 max{ɛ,, min{ɛ 1, 1}} u A. Lmma 4.5. For all u M w av t stimat (4.25) λ div(u P 1 u) 2 u 2 A. Proof. Using (4.8), t inquality can b obtaind as follows λ div(u P 1 u) 2 λ div u 2 + λ div(p 1 u) 2 u 2 A + P 1 u 2 A 1 u 2 A. Combining t L 2 -stimat (4.13), and t stimats givn in Lmma 4.4 and Lmma 4.5, w obtain t following torm, wic vrifis (A1). Torm 4.4. T following approximation stimat olds for λ 1 and for all u M. (I P 1 )u,0 u A Smooting proprty. In tis subsction, w vrify t smooting proprty (A2). W considr only t 3D cas bcaus t 2D cas is similar and simplr. By V and E w dnot t sts of vrtics and dgs of t partition T, rspctivly. For ν V E w dfin T ν = {K T : ν K}, Ων = K T ν K, Ω ν = intrior( Ω ν ). Tus Ω ν is t subdomain of Ω formd by t patc of lmnts mting at ν, and T ν is t rstriction of t ms partition T to Ω ν. W now considr t dcomposition of t spacs M into sums of spacs supportd in small patcs of lmnts. Dfin Tn M ν = {r M : supp r Ω ν }, ν V E. M = i V M i = E M. For ac of ts dcompositions tr is a corrsponding stimat on t sum of t squars of t L 2 -norms of t componnts. For xampl, w can dcompos an arbitrary lmnt u M as u = i V u i wit u i M i suc tat t stimat (4.26) i V u i 2 u 2

21 20 QINGGUO HONG AND JOHANNES KRAUS olds wit a constant tat dpnds only on t sap rgularity of t ms. Sinc t rnl basis functions of t divrgnc oprator ar capturd by t abov subspacs M i, w must us a bloc dampd Jacobi smootr or a bloc Gauss-Sidl smootr wr t blocs corrspond to on of t abov L 2 -dcompositions in ordr to prsrv t structur of t rnl. For xampl, w can us a vrtx bloc dampd Jacobi smootr, a vrtx bloc Gauss- Sidl smootr, an dg bloc dampd Jacobi smootr, or an dg bloc Gauss-Sidl smootr. Rmar 4.3. W sould point out tat t bloc Gauss-Sidl smootr satisfis t assumption (A0). But for t bloc dampd Jacobi smootr, w nd to coos t damping paramtr suc tat t basic assumption (A0) is satisfid. A dampd Ricardson smootr I τa would nd a damping paramtr τ proportional to λ 1. Tus t componnts of t rror in t rnl of A would b smootd out vry slowly as for larg λ. W sould also point out tat in t 2-dimnsional cas, w can only us vrtx bloc smootrs. In t rst of tis subsction, w considr t vrtx bloc dampd Jacobi smootr sinc t otrs ar similar, and dfin t oprator P,i : M M i for i V by A (P,i u, v i ) = A (u, v i ) for all u M, v i M i. W us xact local solvs and nc t bloc dampd Jacobi smootr R = τ i V P,i A 1 :=, wr τ is t damping paramtr suc tat (A0) is satisfid. In tis cas, K = K and τd 1 K (m) = K m. By t assumption (A0), t stimat (4.27) K m u 2 A = (D 1 A K 2m u, u) D m 1 u 2 D olds, wic is wll nown in multigrid tory (s.g. Hacbusc [35]). By additiv Scwarz tcniqus [36, 37] t inducd norm u D = (D u, u) 1/2 can b writtn as (4.28) u 2 D = inf u= u i u i 2 A. i V On t otr and, coosing τ sufficintly small it is obvious tat K m u A u A (t assumption (A0) olds). Tn an intrpolation btwn tis stimat and t stimat (4.27) givs K m u A m 1/4 u [D,A ], wr u [D,A ] is t intrpolation norm btwn D and A wit paramtr 1/2. Tus, on way to vrify assumption (A2), is to sow tat (4.29) u [D,A ] 1 u,0, and t rst of tis sction is dvotd to tis. W now dfin a dcomposition of u M wic is stabl in t norm,0 and tn sow t stimats for t componnts of tis dcomposition. W considr tr solutions of problm (3.51) dfind as follows: (4.30) (4.31) (4.32) (u 1, p 1 ) is t solution of (3.51) wit w 1 = u, w 2 = 0. (u 2, p 2 ) is t solution of (3.51) wit w 1 = 0, w 2 = u Π 1 u. (u 3, p 3 ) is dfind as t solution of (3.51) wit w 1 = 0, w 2 = Π 1 u. It is straigtforward to cc tat u u 1 u 2 u 3 and p 1 + p 2 + p 3 satisfy t quation (3.51) wit w 1 = 0 and w 2 = 0 and trfor u 1 + u 2 + u 3 = u. Wit ts sttings in and, w av t following stability rsult. Lmma 4.6. For t dcomposition givn in (4.30) (4.32) w av (4.33) (4.34) u 1,0 + u 2,0 + u 3,0 u,0, u 2 λ 1 2 u,0.

22 STABLE AND ROBUST SOLUTION METHODS FOR THE BRINKMAN PROBLEM 21 Proof. Computing,0 for all t componnts sows tat (4.35) (4.36) u 1,0 ɛ u 1 + u 1, u 2,0 ɛ u 2 + u 2 + λ 1 2 div(u Π 1 u). T rst of t proof is immdiat from t dfinitions of t componnts (4.30) (4.31), t dfinition of t,0 norm, Lmma 3.8 and Lmma 3.4. Finally, using t triangl inquality, it is asily sn tat u 3,0 u, Smooting proprty via intrpolation. Dfin t H(curl; Ω)-conforming finit lmnt spac at lvl (s,.g., [10]) W = {w H(curl, Ω) : w K W (K), K T, w n Ω = 0}. Tn t tr spacs M, S and W ar rlatd by t xact squnc ([10]) Furtrmor, w dfin Tn curl div 0 W M S 0. W ν = {r W : supp r Ω ν }, ν V E. W = i V W i = E W. Not tat for any v M, w av tat v A v D and v D v D and tis implis tat (4.37) v [D,A ] v D. T nxt two lmmas bound only t D -norm, wic is sufficint in viw of (4.37). Lmma 4.7. Lt u 1 b dfind as in (4.30). Tn (4.38) u 1 D 1 u 1,0. Proof. Sinc div u 1 = 0, w av u 1 = curl w (s [10]), wr w W. Noting tat w = i V w i, wr wi W i and curl wi M i, by idntity (4.28) and inquality (4.26), w av u 1 2 D = inf u 1 = u i 1 2 u i A curl w i 2 A = curl w i 2 a 1 i V i V i V = u i 1 2 a 2 (ɛ 2 u i ρ2 u i 1 2 ) i V i V 2 (ɛ2 u ρ2 u 1 2 ) = 2 u 1 2,0. Lmma 4.8. Lt u 2 b dfind as in (4.31). Tn (4.39) u 2 D 1 u,0. Proof. By t idntity (4.28) and Lmma 4.6, w av u 2 2 D = inf u 2 = u i 2 2 u i A λ ui λ u u 2,0. 2 i V 2 i V

23 22 QINGGUO HONG AND JOHANNES KRAUS Corollary 4.1. From t inquality (4.37) and t Lmmas 4.7 and 4.8, w immdiatly av (4.40) u 1 [D,A ] 1 u 1,0, u 2 [D,A ] 1 u,0. Lmma 4.9. Lt u 3 b dfind as in (4.32), ɛ 2 or λ min{ɛ 1, 1 }. Tn (4.41) u 3 [D,A ] 1 u,0. Proof. By t inf-sup condition (3.52) w av u 3 1, + p 3 Q 1 div u. Furtrmor, from t dfinition of u 3, w av div u 3 = Q 1 div u. Ts togtr wit ɛ 1, ρ 1 and u 3 u 3 1, giv wic implis u 3 2 A (ɛ 2 u 3 2 1, + ρ2 u λ div u 3 2 ) Q 1 div u 2 + λ Q 1 div u 2 λ Q 1 div u 2 u 3 2 A λ 1 ɛ 2 2 u 2,0 if ɛ 2 and u 3 2 A λ 1 2 u 2,0 if ɛ 2. On t otr and, by t idntity (4.28) and invrs inquality, w av u 3 2 D = inf u 3 = u i 3 2 u i A 2 λ ui λ u i V i V If ɛ 2, by noting tat u 3,0 u,0, w av u 3 2 D 2 If ɛ 2, by Lmma 3.8, it follows tat u 3 2 D 2 λ u λɛ 2 u 3 2,0 λ u λ Q 1 div u λɛ 2 u 2,0. A standard intrpolation argumnt, s,.g., [38], complts t proof. W clos tis subsction by t following torm wic vrifis (A2). λ Q 1 div u 2 2 λ 1 min{ɛ 2, 2 } u 2,0. Torm 4.5. If ɛ 2 or λ min{ɛ 1, 1 }, t following stimat olds for all u M (4.42) ( K (m) ) u A m 1/4 1 u,0. Proof. By Lmma 4.6, inqualitis (4.40) and (4.41), w obtain t smooting proprty (4.42). 5. Numrical rsults To tst t prformanc of t multigrid algoritms tat w av proposd w prsnt tr sts of numrical tsts solving quation (3.17). For simplicity, w ta as computational domain Ω = [0, 1] [0, 1] and discrtiz quation (2.2) and (3.17) by H(div, Ω)-conforming BDM 1 finit lmnts (BDM 1 (K)/P 0 (K) pair for Brinman quation) on a uniform ms using t DG mtod dscribd in Sction 3. Our tsts ar aimd at vrifying t tortical rsults on t convrgnc of t multigrid algritms for t linar systm (3.15). W av tabulatd t rsults obtaind wit t multigrid mtod for mss wit ms sizs J = 2 J wr J = 2,..., 6. In addition, w av varid t paramtr λ. For t V (1, 1)- and W (1, 1)-cycls of t multigrid (MG) algoritm w av usd a vrtx bloc Gauss- Sidl smootr. In ordr to approximat t rror rduction factor of t MG itration, i.. t numbr ϱ = E J AJ := I B J A J AJ, w av st i = E J i 1 wit a random initial guss 0 and computd t ratio ϱ i := (A J i, i )/(A J i 1, i 1 ) for larg noug i. In all tabls J dnots t lvl of t finst discrtization and N dnots t numbr of dgrs of frdom for t displacmnt componnt (for BDM 1 lmnts, N is twic t numbr of dgs).

24 STABLE AND ROBUST SOLUTION METHODS FOR THE BRINKMAN PROBLEM 23 λ = 5 10 l J N l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 l = Tabl 5.1. Convrgnc rat E J AJ of V (1, 1)-cycl MG mtod (confirming Torm 4.1) in cas of ɛ 2 = 1, ρ 2 = 0. T data in Tabl 5.1 vrifis t convrgnc rsult in Torm 4.1 and t data in Tabls vrifis t rsult sown in Torm 4.2. W want to mpasiz tat altoug Torm 4.2 rquirs tat t numbr of smooting stps is sufficintly larg, t rsults sown in Tabls indicat tat on smooting stp is sufficint for a uniform convrgnc of t W -cycl MG mtod. λ = 5 10 l J N l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 l = Tabl 5.2. Convrgnc rat E J AJ of W (1, 1)-cycl MG mtod (confirming Torm 4.2) in cas of ɛ 2 = 1, ρ 2 = 0. λ = 5 10 l J N l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 l = Tabl 5.3. Convrgnc rat E J AJ of W (1, 1)-cycl MG mtod (confirming Torm 4.2) in cas of ɛ 2 = 1, ρ 2 = 10 6.

25 24 QINGGUO HONG AND JOHANNES KRAUS λ = 5 10 l J N l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 l = Tabl 5.4. Convrgnc rat E J AJ of W (1, 1)-cycl MG mtod (confirming Torm 4.2) in cas of ɛ 2 = 1, ρ 2 = λ = 5 10 l J N l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 l = Tabl 5.5. Convrgnc rat E J AJ of W (1, 1)-cycl MG mtod (confirming Torm 4.2) in cas of ɛ 2 = 1, ρ 2 = 1. λ = 5 10 l J N l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 l = Tabl 5.6. Convrgnc rat E J AJ of W (1, 1)-cycl MG mtod (confirming Torm 4.2) in cas of ɛ 2 = 10 3, ρ 2 = 1. As prdictd by t tory, t rsults prsntd in Tabls sow uniform convrgnc indpndnt of ɛ, ρ, λ and. Finally, to tst t augmntd Uzawa itration w av st t rigt and sid of quation (2.2) to f = (1 6x + 6x2 )(y 3y 2 + 2y 3 ) + (x 2 2x 3 + x 4 )( 3 + 6y), g = 0, (1 6y + 6y 2 )(x 3x 2 + 2x 3 ) (y 2 2y 3 + y 4 )( 3 + 6x)

26 STABLE AND ROBUST SOLUTION METHODS FOR THE BRINKMAN PROBLEM 25 λ = 5 10 l J N l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 l = Tabl 5.7. Convrgnc rat E J AJ of W (1, 1)-cycl MG mtod (confirming Torm 4.2) in cas of ɛ 2 = 10 6, ρ 2 = 1. and usd t corrsponding xact solution of t sub-problm for t displacmnt u (s (3.14)). T itration as bn initializd wit u 0 = 0 and p0 = 0 and trminatd aftr a rduction of t rror of t vlocity in nrgy norm by a factor of In Tabl 5.8 w vrify t convrgnc rsults of t augmntd Uzawa itration wic is givn in (3.14) (s also [20]). λ = 5 10 l J N l = 0 l = 1 l = 2 l = 3 l = 4 l = Tabl 5.8. Itration count for t augmntd Uzawa mtod. 6. Conclusions W av provd t stability of a family of discontinuous Galrin H(div; Ω)-conforming discrtizations of t Brinman quations and t corrsponding linar lasticity typ quation wit mass trm wic appars in t Uzawa itration for t Brinman problm. Furtr, w av prsntd a robust itrativ solution mtod for t arising discrt problms. Variabl V-cycl and W-cycl multigrid algoritms av bn analyzd in t prsnt situation of nonnstd bilinar forms. T convrgnc rat of t itrativ mtods av bn provd to b indpndnt of t paramtrs in t Brinman problm, i.., ɛ and ρ and of t ms siz, wic sows tat t itrativ algoritm is robust and optimal. T prsntd numrical rsults support t tortical findings. Rfrncs [1] O. Iliv, R. Lazarov, and J. Willms. Variational multiscal finit lmnt mtod for flows in igly porous mdia. Multiscal Modl. Simul., 9(4): , [2] Xiaoping Xi, Jincao Xu, and Guangri Xu. Uniformly-stabl finit lmnt mtods for Darcy-Stos-Brinman modls. J. Comput. Mat., 26(3): , 2008.