MORTAR ADAPTIVITY IN MIXED METHODS FOR FLOW IN POROUS MEDIA

Transcription

1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volum 2, Numbr 3, Pags c 25 Institut for Scintific Computing and Information MORTAR ADAPTIVITY IN MIXED METHODS FOR FLOW IN POROUS MEDIA MA LGORZATA PESZYŃSKA Abstract. W dfin an rror indicator for mixd mortar formulation of flow in porous mdia. Th mixd mortar domain dcomposition mthod for singlphas flow problms was dfind by Arbogast t al; it rlis on coupling of subdomain problms using mortar Lagrang multiplirs dfind as continuous picwis linars on th subdomain intrfac. Th accuracy and fficincy of th rsulting intrfac formulation dpnds on th numbr of mortar dgrs of frdom which w propos to adapt using rror indicators involving jump of th flux across th intrfac. Rigorous a-postriori analysis and proof of rliability of th stimator ar stablishd for singl-phas 2D flow problms with diagonal cofficints for RT [] spacs on rctangular grids. Computational xprimnts dmonstrat th application of th stimator. Nxt, th algorithm and indicator ar xtndd to th two-phas flow cas which is illustratd with numrical xampls. W focus on adapting th mortar grid whil kping subdomain grids fixd. Full mortar adaptivity is discussd lswhr. Ky Words. Singl-phas flow in porous mdia, multi-phas flow, mixd finit lmnts, a-postriori rror stimation, mortars, domain dcomposition, adaptivity 1. Introduction This papr is dvotd to grid adaptivity for a family of htrognous domain dcomposition mthods basd on th mixd mortar finit lmnt mthod. Th mthod was introducd in [7] and it provids a rigorous optimally convrgnt discrtization tchniqu for th lliptic quation (1) (K p) = f, x Ω, with Ω R d, d = 2, 3. Hr K dnots th diffusion cofficint, f dnots th sourc/sink trms, and p is th unknown prssur. In th mortar domain dcomposition mthod, th rgion Ω is dcomposd into individual non-ovrlapping subdomains Ω i, i = 1,... n which ar sparatd by th union of intrfacs Γ on which mortar grids and unknowns ar introducd. Th subdomains ar griddd indpndntly; subdomain problms which ar th local countrparts of (1) can b solvd ssntially indpndntly from on anothr but ar coupld by mortars; s [14, 12] for mortar formulation whn subdomain problms ar solvd with standard Galrkin (conforming) mthods. In th mixd mortar mthod th subdomain problms ar solvd using mixd finit lmnt mthods thrby providing a locally consrvativ approximation to Rcivd by th ditors April 15, 24 and, in rvisd form, Octobr 22, Mathmatics Subjct Classification. 65N3,65M6,65N5,65N55,76S5,76M1. This rsarch was partly supportd by th DOE grant DE-FG3-99ER25371, and th NSF grants: SBR , ITR EIA , NPACI

2 242 M. PESZYŃSKA both prssur and vlocity unknowns u := K p. Th mthod rlis on introducing mortar Lagrang multiplirs on th intrfac Γ which provid Dirichlt boundary conditions for th subdomain problms. Additionally, th subdomain problms ar coupld by th rquirmnt that th global vlocitis b wakly continuous across Γ which rlaxs th global continuity (of normal componnts across any smooth surfac) of xact vlocitis. This wak-continuity condition avrags th jumps of vlocitis and is dfind rlativ to th discrt spac of Lagrang multiplirs on th intrfac which ar dfind on a mortar grid charactrizd by th paramtr h m, or by th numbr of mortar dgrs of frdom n m O( 1 h m ). Lt us b givn a collction of rctangular partitions of Ω i with associatd grid paramtr h = max i h i. In principl, h m can b slctd indpndntly of h, as long as crtain lowr and uppr bound conditions hold. Ths guarant, rspctivly, th uniqu solvability of th mixd mortar formulation, and th optimal approximation proprtis of wakly continuous vlocitis which in turn ar ncssary for th optimal rat of convrgnc of th mthod, th sam as for discrtization without mortars. For this optimal convrgnc rat which, for lowst ordr Raviart- Thomas spacs RT [], is O(h) in both th prssur and vlocity unknowns [46, 17], h m should dpnd linarly on h; th approximation rror incrass, in gnral, with th proportionality constant. Th numbr of mortar unknowns n m on Γ dtrmins th complxity of th intrfac problm. Rcall that, in a classical domain dcomposition stting, th algorithm for approximation of (1) can b writtn in trms of th intrfac unknowns and as such solvd by an itrativ algorithm which rquirs, in ach itration, solution of subdomain problms which ar rsponding to th currnt guss of Dirichlt data. In th mixd mortar algorithms th numbr of itrations on th intrfac in gnral grows with n m, unlss optimal prconditionrs can b applid. Th mixd mortar mthod has bn th cornrston of svral major rsrvoir simulation projcts. Rcall that (1) can b usd as a modl for singl-phas flow in a rsrvoir Ω. Its natural xtnsion is to th multi-phas flow; th algorithm has bn intgratd within th IPARS (Intgratd Paralll Accurat Rsrvoir Simulator) framwork [51, 44, 49, 41, 35]. Th attractivnss of th mortar approach lis in that it maks th subdomain problms indpndnt from on anothr. It is only th intrfac Lagrang multiplirs (Dirichlt data) and th rsulting fluxs (th Numann data) which provid communication btwn subdomains. As such, th subdomain problms can b considrd as black-boxs thrby allowing for local adaptivity of tim-stpping [43], grids and solvrs, and mainly, th physical modls [37]. Th lattr coupling is a form of multiphysics and is an instanc of htrognous domain dcomposition; s [5]. Th difficultis in practical application of th ovrall procdur li in finding optimal prconditionrs for th gnral multi-phas solvr on th intrfac; s [57]. Howvr, in spit of th larg complxity of th intrfac solvr, th mixd mortar approach has bn xtrmly succssful whn applid to a larg class of ral rsrvoir problms. It is important to not that in all th succssful cass w found a rlativly small n m sufficint for a good lvl of accuracy and at th sam tim mandatory for an accptabl dgr of computational complxity. In [45] th mixd mortar mthod gav ris to th mortar upscaling mthod. Hr th subdomain grids ar kpt fixd but n m varis, thrby providing a variabl dgr of local consrvation of mass or of wak-continuity of th fluxs. It is in th rsarch rportd in [43, 37, 45] that th nd to dfin th right mortar grid and to control th rror du to only wak continuity of fluxs bcam

3 MORTAR ADAPTIVITY FOR FLOW 243 vry important. It was clar that th quantitativ critria to dtrmin th mortar grid, that is, th rror indicators, must somhow involv a masur of th dfct introducd by th wak continuity condition, and that ths masurs must involv th jump of th computd fluxs across Γ. This intuition was radily confirmd by numrical xprimnts. At that tim, svral nw rsults applicabl to a-postriori rror indicators for mixd mthods in R 2 [56, 18] and for mortar formulations for conforming mthods [54, 53, 55] bcam availabl. Ths hav providd th basis for our rigorous analysis. In this papr w combin th idas coming from [56, 18, 54, 53, 55, 43, 37, 45] to rigorously justify th construction of rror indicators for mixd mortar solution of singl-phas problms; w also xtnd ths idas to two-phas flow. W us th algorithmic and implmntation basis rfrncd abov to provid th computational rsults which dmonstrat th strngth of our approach. Th plan of th papr is as follows. In Sction 2 w rcall th original statmnt of th mixd mortar formulation for (1) and introduc a viw of mortar grid paramtrization which dparts from on in [7]. In Sction 3 w construct an rror indicator for RT [] mixd finit lmnt spacs ovr a quadrangulatd subdomain with non-homognous boundary conditions and diagonal K, and prov an uppr bound for th rror; ths rsults ar a dtaild but natural xtnsion of [56, 18] whr triangular grids, Raviart-Thomas RT spacs on triangls, and homognous boundary conditions wr usd. Sinc th mixd mortar algorithm coms from application of boundary conditions to local subdomains, th construction in Sction 3 provids th building blocks for th main thortical rsult of this papr dvlopd in Sction 4; it is th dfinition of a rliabl rsidual a-postriori rror stimator η for (1) in th mixd mortar form. In Sction 5 w illustrat th applicability of this rsult to singl-phas flow computations. Nxt, Sction 6 introducs brifly th mixd mortar algorithm for two-phas flow and appropriat xtnsions of η, and Sction 7 prsnts th computational rsults on mortar adaptivity for two-phas flow. In all computational rsults w ar only concrnd with adaptivity of mortars. In gnral, both subdomain and mortar grids should b adaptd as is don in [25, 3] for Galrkin mthods, but this will not b pursud hr. 2. Mixd mortar formulation for singl phas flow In this sction w rcall th statmnt and th main rsults concrning th mixd mortar formulation from [7]. W start by introducing notation, following [7, 56, 18, 54], and procd in Sction 2.2 to dfin th problm as in [7]. In Sction 2.3 w discuss th paramtrization of mortar grid Notation. Considr a st ω R 2 with ω x = (x 1, x 2 ). Dnot = ( x 1, x 2 ) and Curl := ( x 2, x 1 ) so that for ψ : ω R w hav ψ := ( ψ x 1, ψ x 2 ) = (ψ 1, ψ 2 ) and Curlψ = ( ψ x 2, ψ x 1 ) = (ψ 2, ψ 1 ). For a vctor valud function q : ω R 2, q = (q 1, q 2 ) w writ divq := q = q1 x 1 + q2 Also, w us th symbol curlq := q1 x 2 = q 1,1 + q 2,2. q2 x 1 = q 1,2 q 2,1. W also introduc [ ] x 2 1 T :=. thn w notic Curlφ = T φ, and curlq = Tq. 1 On th boundary ω of ω w dnot by n ω = (n 1, n 2 ) th unit outward normal to th boundary ω and by t ω := Tn = (n 2, n 1 ) th unit tangnt vctor whos orintation is consistnt with th clockwis orintation of ω. W drop th subscript whn it dos not lad to a confusion. Clarly Curlφ n = φ t.

4 244 M. PESZYŃSKA Whn intgrating ovr ω, w abbrviat dx 1 dx 2 by dx and us th symbol ds in boundary intgrals. Also, w us th notation < φ, ψ > ω := φ(s)ψ(s)ds for ω boundary intgrals. Now w rcall th Lbsgu and Sobolv function spacs [2]. As usual, L 2 (ω) dnots th spac of squar intgrabl functions (distributions) ovr ω; it is a Hilbrt spac with th product (φ, ψ) ω := φ(x)ψ(x)dx and with th norm ω φ,ω := (φ, φ) ω. (W shall drop th subscript ω whn this dos not lad to confusion.) Nxt, H 1 (ω) dnots th spac of functions which, along with thir (wak) drivativs, ar in L 2 (ω); H 1 (ω) is th subst of H 1 (ω) consisting of functions whos boundary valus in th sns of tracs ar zro on ω. W rcall that th tracs of H 1 (ω) functions ar in H 1/2 ( ω), providd ω is smooth. For dtails on standard notation and th proprtis of Sobolv spacs, s [2]. Th spac H(div; ω) is th spac of vctor valud functions whos componnts, along with thir divrgnc, ar in L 2 (ω), and th associatd norm is q H(div;ω) := ω q qdx + ( q)2 dx. W rcall that normal tracs of lmnts of H(div; Ω) across any smooth curv ar continuous [17]. W shall also us anothr norm on H(div; ω). Lt us b givn a positiv dfinit uniformly boundd tnsor K = K(x), x ω, such as th on discussd blow. Thn on dfins (2) q K,ω := K 1 q qdx + ( q) 2 dx. ω This norm is quivalnt to th standard H(div; Ω) norm; th constants of quivalnc ar stablishd blow. Nxt, by Y (A) Y (B) w man th st of functions on A B whos rstrictions to sts A, B blong to th spacs Y (A) and Y (B), rspctivly. In contrast, Y + Z dnots a st of linar combinations of functions from Y and Z. Finally, th projctions to a subspac Y X ar dfind in th usual way: x X, π Y x Y and (x π Y x, y) Y =, y Y Statmnt of th problm. Considr a fixd opn boundd domain Ω R 2 and th flow of a singl phas incomprssibl fluid whos prssur is dnotd by p = p(x), x Ω. Th flow is govrnd by Darcy s law which dfins th vlocity (flux) u as proportional to th gradint of th prssur p (3) u = K p. Th (hydraulic conductivity) cofficint K combins th (absolut) prmability constant and th viscosity of th fluid. Hr w ignor th prsnc of gravity; gravity can b handld by rplacing p with a potntial variabl. Th conductivity tnsor K(x) is assumd to b symmtric, ssntially boundd and uniformly positiv dfinit. In fact, in this papr w assum that K is diagonal, that is, [ ] K1 (x) K(x) :=. K 2 (x) Dnot th uppr and lowr bounds on th ignvalus of this cofficint ovr a st ω by (4) (5) Obviously, κ 1 ω K ω := min x ω {K 1(x), K 2 (x)}, κ ω := min x ω {K 1 1 (x).k 1 2 (x)}. = max x ω {K 1 (x), K 2 (x)}} and K 1 ω = max x ω {K1 1 (x), K 1 2 (x)}}.

5 MORTAR ADAPTIVITY FOR FLOW 245 Whil Darcy s law (3) xprsss th consrvation of momntum, th consrvation of mass of th fluid (continuity quation) is xprssd by (6) u = f whr f rprsnts th sourcs and sinks of th fluid. Th systm (3),(6) can b rwrittn as an lliptic quation for th prssur (1) and solvd as such, or it can b solvd in th mixd formulation (3), (6). In ithr cas, boundary conditions on Ω nd to b spcifid. In thortical drivation of th rror stimator in Sctions 3, 4 w assum, for simplicity, that th prssur p satisfis th Dirichlt boundary condition (7) p Ω = g. In particular, a ho- In gnral, othr boundary conditions could b considrd. mognous Numann no-flow boundary condition (8) u n Ω =, is usful in applications; it is usd in our computational xampls. W sk a wak solution to th boundary-valu problm (3), (6), (7). St W := L 2 (Ω) and V := H(div; Ω). Th wak solution (u, p) V W satisfis (9) (1) (K 1 u, v) = (p, v) < g, v n >, v V ( u, w) = (f, w), w W This problm has a uniqu solution if f W, g H 1/2 ( Ω), K is uniformly boundd and lliptic and if Ω is boundd, s [17], IV.1.2. From now on shall assum Assumption 1. K is diagonal and thr xists K Ω > and κ Ω >. Assumption 2. Ω is opn, boundd, convx, with polygonal boundary Ω, f W. For convrgnc analysis, it is ncssary to assum that p is smoothr than p W, which in turn, by (9), and if K is smooth, incrass th smoothnss of u. In [7] it is assumd that p H 3/2+ε, ε >. In [56, 18] th problm (9) (1) is assumd 2-rgular that is, K, Ω, f, g ar smooth nough so that p H 2 (Ω). This is guarantd, for xampl, in addition to (A1) and (A2), if K is diffrntiabl and g H 3/2 ( Ω), s ([26], Thm 8.13), xhaustiv thory in [28], and ([15], Prop.2.2). W not that in gnral rsrvoir problms th picwis constant cofficints and point sourcs ar not smooth nough for th thory considrd hr. Nvrthlss, from now on w shall assum Assumption 3. Th data K, g, ar smooth nough so that p H 2 (Ω), u (H 1 (Ω)) d. (11) Finally, w not that with (A1) w hav κ Ω q 2 H(div;Ω) q 2 K,Ω K 1 Ω q 2 H(div;Ω) Multiblock dcomposition of Ω. Assum that thr is a (sufficintly smooth) multiblock dcomposition of Ω into n smooth nonovrlapping subdomains Ω i, i = 1,... n, ach of which satisfis (A2). Dnot by Γ ij := Ω i Ω j th (possibly mpty) intrfac btwn two subdomains Ω i and Ω j and for ach i, by Γ i := n j=1 Γ ij. Finally, dfin th intrfac Γ := i Γ i which can also b writtn as Γ = i,j Γ ij. From th point of viw of a subdomain, its boundary Ω i is mad of th subdomain intrfac Γ i := Ω i Γ and of parts Ω i \ Γ = Ω i Ω that ar part of Ω. Clarly, for som i, Ω i \ Γ = and for som i, j, Γ ij =.

6 246 M. PESZYŃSKA For simplicity of th xposition blow and to rmov unncssary complications with notation, w shall assum in fact th following Assumption 4. Assum that ach Ω i is a rctangl (i.., a macrolmnt) and that ach Γ ij, for vry i, j, is a union of straight sgmnts. On ach subdomain Ω i w dfin locally th function spacs and rlat thm to th globally dfind spacs W, V. Th spac W = L 2 (Ω) can b straightforwardly dcomposd as W = n i=1 W i with th spacs of local tst functions W i = L 2 (Ω i ) and with a naturally dfind scalar product (, ) i bing a shorthand for (, ) Ωi and with th associatd norm,i. Th vlocity spac ovr Ω = n i=1 Ω i is constructd as follows n V 1 := with th local spacs V i := H(div; Ω i ). Not that V V 1 but V V 1 unlss n = 1, sinc th normal componnts of functions from V 1 ar not, in gnral, continuous across Γ. This is rflctd by th suprscript 1 in th notation V 1 as in [54, 53, 55]. In th squl w shall dfin an intrmdiat spac V in th sns V V V 1 whos mmbrs satisfy a wak continuity condition with rspct to th spac of discrt Lagrang multiplirs on Γ. Th local spacs ar rlatd to th global wak problm as follows: if (u, p) (V, W ) (V, H 1 (Ω)) satisfis th global problm (9)-(1), thn on can show, s [27], also ([17], IV.1.3), that th following local wak problm dfind ovr Ω i is satisfid: (12) (13) (K 1 u, v) i = (p, v) i < p, v n i > Ωi \ Ω < g, v n > Ωi Ω, v V i ( u, w) i = (f, w) i, w W i with p H 1 (Ω i ). In turn, if th local problms ar satisfid with u dfind locally ovr ach Ω i that is, u V 1, p W, thn, whn quations (12) ar addd ovr i = 1,... n, and if w tak v V 1, thn w s that th right hand sid of th summd quations (12) contains th trm (14) < p, v n i > Ωi \ Ω= < p, v n i > Γi = < p, v n i + v n j > Γij i i i,j Th global problm (9) is rcovrd if w rquir that u V and that th trm (14) vanishs. This is achivd if th tst functions v hav continuous normal componnts across Γ that is, v V, bcaus thn th jump (15) vanishs pointwis: (16) i=1 V i [v n] ij (s) := (v Ωi n i )(s) + (v Ωj n j )(s), s Γ ij [v n] ij (s) =, s Γ i,j, v V. In th discrt countrpart of ths quations, th rquirmnt that th jump trm vanishs pointwis will b rplacd by a rquirmnt that th discrt countrpart of th trm (14) vanishs. This condition will giv ris to th dfinition of th wakly continuous vlocitis from a spac V. W rfr to [27] or ([17], III.1.2) for mor dtails on this classical domain dcomposition stup.

7 MORTAR ADAPTIVITY FOR FLOW Discrt problm. Assum that ach Ω i is covrd by a rctangular partition T i so that Ω i = E T i Ē and ach E is a rctangl. This partition is a spcial cas of a quadrilatral partition (quadrangulation [13]). W can assum that th dgs E T i := E T i E ar alignd with coordinat axs. Also, ach E is an imag of th rfrnc lmnt Ê = [, 1]2 undr an affin rfrnc map F E : Ê E with E x = Fˆx = b + B( ˆx 1, ˆx 2 ) T with a diagonal matrix B. Dnot by h E = diam(e) which for rctangls is th maximum lngth of th dgs of an lmnt E and dfin h i := max E Ti h E. With this T i can b rfrrd to T hi. h Assum that thr is a uniform bound, E T i, E ρe CT i whr ρ E is th diamtr of a circl inscribd in E. This mans that for ach i th family T i is rgular in th sns of [22, 13]. In this multiblock (domain) dcomposition of Ω, thr is no global quadrangulation of Ω as in gnral th grids on Ω i do not hav to match. Howvr, sinc n is finit, on can naturally dfin h = max i h i and C T := max i CT i. Whn rfrring to th collction of quadrangulations T i, w shall us th notation T h or simply T. Nxt, w dnot by P k,l (ω) th st of polynomials of dgr k in th first and of dgr l in th scond variabl of x ω R 2 and by P k (ω) := P k,k (ω). In this papr w ar concrnd with on typ only of th mixd finit lmnt spacs associatd with ach T i. Ths ar th finit dimnsional spacs W hi W i and V h,i V i. Spcifically, th discrt prssur spac W hi is th spac of functions which ar P (E), E T i. In turn, th discrt vlocity spac V h,i is an RT [] spac, that is, on ach lmnt E T i, ach V h,i q, q E P 1, (E) P,1 (E). W rcall that this choic of spacs, for ach i, satisfis th Babuska-Brzzi condition ([17], III.3.2). W rfr to [17] for mor dtails on ths and othr mixd finit lmnt spacs. Th mortar mixd FE solution is dfind as in [7]. First w dfin V 1 h = n i=1 V h,i. W not that V 1 h V 1 but that, in gnral (if grids ar not matching), V 1 h V contains not much mor than th zro lmnt, as th mmbrs of th spacs V h,i and V h,j do not hav matching normal tracs on Γ i,j. To handl non-matching grids and to dfin som form of a coupling across th intrfac th mortar spacs wr introducd. Th goal is, on on hand, to rlax th continuity of th discrt solution across Γ which cannot b satisfid across Γ bcaus of non-matching grids, and on th othr hand, to coupl th subdomain solutions. In addition, introduction of th mortar unknowns (or Lagrang multiplirs) on th intrfac Γ allows on to formulat th discrt problm for (1) in trms of th intrfac unknowns and to ntirly dcoupl th subdomain (local) problms from on anothr; thir intraction is only through th mortar unknowns. For th conforming mortar mthods s,.g., [14, 12] and rfrncs thrin. In addition, s [3] and rfrncs thrin for discussion of itrativ mthods on th intrfac with optimal prconditionrs. For mixd mortar mthods th Lagrang multiplir mortar prssurs supply th Dirichlt boundary conditions on Γ applid to subdomain problms. In turn, th Numann data (valus of fluxs across Γ) nd to b matchd somhow to achiv th aformntiond global flow. In fact, avrags of thir projctions onto som spac ar matchd: this condition rlaxs th rquirmnt v h V which cannot b satisfid bcaus of nonmatching grids, and, on th othr hand, it rstricts th functions from V 1 h so that som global mass consrvation across Γ can b achivd.

8 248 M. PESZYŃSKA Following [7], in th mixd mortar mthod th mortar prssurs ar dfind as mmbrs of a finit dimnsional subst Λ hm L 2 (Γ) associatd with th mortar grid T m which is dfind locally on ach Γ ij, with a discrtization paramtr h m. Rcall n m 1 h m is th numbr of mortar dgrs of frdom. W considr th spacs Λ hm,ij L 2 (Γ ij ) spannd by continuous picwis linars ovr th grid T m rstrictd to Γ ij, and thir dirct sum Λ hm := i,j Λ h m,ij. Th mortar grid paramtr h m may, in principl, b dfind indpndntly of th subdomain discrtization paramtr(s) h i as long as crtain approximation proprtis to b discussd in Sction 2.3 ar satisfid. Th spac Λ hm provids th mans to dfin a wak continuity condition for normal tracs which rplacs (16) so that not th jump (15) itslf but rathr its wightd avrag with rspct to th discrt wights from Λ hm vanishs (17) V = {v V 1 : < v n i, µ > Γi = < [v n] ij, µ > Γij =, i i,j µ Λ h } Not that V V V 1. Also, w dfin V h := V 1 h V. As statd in [7], V h is difficult to charactriz dirctly, but it is nonmpty, and it provids th appropriat hom for an optimally convrgnt approximation of u. Mor prcisly, in th discrt mixd mortar problm formulation on sks u h V h, p h W h, λ h Λ hm, which solv th discrt quivalnts of local problms (12), (13). (18) (K 1 u h, v h ) i = (p h, v h ) i < λ hm, v h ν i > Γi < g, v h ν > Ωi \Γ, v h V h,i, (19) ( u h, w h ) i = (f, w h ) i, w h W h,i, Ths local quations ar coupld by th mortar prssurs λ h in (18) and by th condition of wak continuity of fluxs u h V h. Equivalntly, th systm can b rwrittn by adding ovr all subdomains and by using tst functions from V h, whrby th rsulting jump trm involving mortar prssur λ hm arising on th rhs of (18) is liminatd by virtu of wak continuity of th tst functions. Th attractiv fatur of such a rwrit is that it taks apparanc of a global systm similar to (9), (1), with th coupling conditions imposd on th tst functions: u h V h, p h W h such that (2) (21) (K 1 u h, v h ) = (p h, v h ) < g, v h n > Ω v h V h, ( u h, w h ) = (f, w h ), w h W h. If w assum (A1) (A5) and additional assumptions on h m to b discussd blow, thn th schm (2) (21) has a uniqu solution as shown in ([7], Lmma 2.1) and is asymptotically convrgnt at an optimal rat O(h) in p and u as shown by a-priori stimats for th vlocity and prssur in ([7], Throms 4.2, 5.1), rspctivly. W discuss th assumptions on h m nxt Paramtrization of th mortar grid. Hr w considr th paramtrization of th mortar grid T m with rspct to th subdomain grids T h. Rcall that ths ar associatd, rspctivly, with paramtrs h m and h. Assum (22) h m = h m (h) whr h m is monoton function of h. Obviously, w ar intrstd in h m dpnding linarly or lss on h in ordr to kp n m from growing too much.

9 MORTAR ADAPTIVITY FOR FLOW 249 In [7] th following two assumptions on h m (h) ar mad. Rcall th dfinition of th L 2 projction Q h,i : L 2 (Γ i ) V h,i ν i Γi and its global countrpart Q h coming from th dirct sum ovr all subdomains. W also considr th projction P hm dfind from L 2 (Γ) onto Λ hm. Assumption 5. ([7], 3.18) Assum that for any µ Λ h thr is a constant C I indpndnt of h such that (23) µ,γi,j C I ( Q h,i µ,γi,j + Q h,j µ,γi,j ) on vry intrfac Γ i,j, i j. ([7], 3.1) Also assum that for ψ L 2 (Γ), (24) ψ P hm ψ s,γij C II ψ r,γij h r+s Ths two assumptions provid, rspctivly, a lowr bound for h m (h) for any fixd h, and an uppr bound for h m (h) as h. Th first assumption guarants uniqu solvability of (18) (19) and th scond guarants th optimal approximation proprtis of th mortar spac which in turn ar ncssary for th optimal convrgnc of th mortar algorithm. Rmark 2.1. Sinc th approximation proprtis of P hm dpnd only on th grid paramtr h m, condition (24) implicitly rquirs (25) h m = N M h whr N M const. Furthrmor, th computational xampls in [7] rcommnd N M 2 by stating that th mortar grid on ach intrfac is a coarsning by two in ach dirction of th trac of ithr on of th subdomain grids. Blow w show that that th choic of N M const > 2 implis (23) and by (25), naturally also (24). Nxt w show that a sublinar nonconstant dpndnc h m (h) may yild convrgnc of th mortar algorithm, albit at suboptimal rat. In summary, for a fixd h th mortar grid paramtr h m should b chosn adaptivly as shown in Sction Lowr bound on mortar paramtrization. Hr w xplor th dpndnc of C I in (23) on h m by way of a simpl xampl. For simplicity assum Γ ij = 1 or that all calculations ar don in th rfrnc domain (, 1). Also, assum matching grids h = h i = h j, and (25) with N M N. Considr µ Λ h. W hav µ(x) = k ψ k(x)µ k, whr ψ k ar th usual continuous picwis linar nodal basis functions associatd with nods of th mortar grid on Γ i,j and whr µ k ar th wights. W hav that th support of a basis function supp(ψ k ) = [(k 1)h M, kh m ]. On th othr hand, Q h,i µ(x) = k µ kq h,i ψ k (x) and Q h,i ψ k (x) is a linar combination of th charactristic functions ovr th subdomain grid χ m (x) =: χ [(m 1)h,mh] (x) with wights givn by midpoint valus of ψ k ovr th support of ach χ m. To driv C I w comput µ 2,Γ i,j = (26) µ 2 k (ψ k (x)) 2 dx, k supp(ψ k ) Q h,i µ 2,Γ i,j = (27) µ 2 k ((Q h,i ψ k (x)) 2 dx. k,m It is straighforward to comput (28) supp(ψ k ) supp(ψ k ) supp(χ m) (ψ k (x)) 2 dx = h m 3,

10 25 M. PESZYŃSKA and supp(ψ k ) ((Q h,i ψ k (x)) 2 dx = N M m=1 supp(ψ k ) supp(χ m ) ((Q h,i ψ k (x)) 2 dx = h3 h 2 ( N M 3 m 3 N M 12 = h m 3 (1 1 4NM 2 ), From ths calculations, comparing Equations (26) and (27) w gt (29) C I (N M ) = (2(1 1 4NM 2 )) 1/2. This xampl shows that C I is a dcrasing function of N M. In ordr for (23) to b satisfid, on should choos N M N M,min ; this is consistnt with th rcommndation in ([7], Rmark 2.2) that mortar grid should not b too fin with rspct to th subdomain grids Uppr bound on paramtrization of h m. As w mntiond, th approximation proprtis of th projction oprator P h onto picwis linars Λ hm ovr grid T m dpnd on h m through th stimat, for smooth nough ψ, ψ P h ψ s,γi,j C ψ r,γij h r+s m If (25) holds and N M is fixd as h, thn (24) holds and for nonngativ r + s, C II in (24) is an non-dcrasing function of N M Sublinar paramtrization of h m (h). Assum first h with h m const. Thn N M = N M (h) blows up and so dos th constant C II. Now assum that h m is a monoton sublinar function of h (3) h m = O(h l ), with < l 1 (whn l = 1 w hav (25)). Considr th us of (24) with (3) instad of (25) in th proofs of th convrgnc rsults in [7]. W find that, for RT [] spacs, th assumption (3) instad of (24) lads to th ordr of convrgnc for vlocitis of ordr O(h 3/2 m h 1/2 ) = O(h 1 2 (3l 1) ) hnc, th mixd mortar schm convrgs as long as l > 1 3. W skip th dtails and rmark that obviously, th convrgnc of th schm is suboptimal for l < 1 asymptotically whn h. Rmark 2.2. From now on w shall assum that (23), (24) hold. In particular, w shall assum that (25) holds locally on ach part Γ ij of th union of intrfacs Γ. Th linar dpndnc of h m on h will b rflctd in notation Λ h Λ hm which w adopt from now on. In ordr to dtrmin optimal valu of n m or h m pr ach intrfac Γ ij w shall stablish a-postriori rror indicators. 3. Rsidual rror stimator for spac RT [] and for non-homognous Dirichlt conditions In this sction w formally dfin an a-postriori rror stimator for th vlocity in th mixd formulation for RT [] spacs and non-homognous boundary condition. Spcifically, w considr th following Dirichlt problm in a wak mixd form u V, p W such that (31) (32) (K 1 u, v) = (p, v) < g, v ν > Ω, v V, ( u, w) = (f, w), w W,

11 MORTAR ADAPTIVITY FOR FLOW 251 and its mixd FE solution, ovr a conforming partition T h of Ω into rctangular lmnts E, with V h, W h bing RT [] spacs: u h V h, p W h such that (33) (34) (K 1 u h, v h ) = (p h, v h ) < g h, v h ν > Ω, v h V h, ( u h, w h ) = (f, w h ), w h W h. This stup and th rsults of this sction can b applid in two diffrnt ways. First, (31)-(32) can b sn as th global wak problm (9) (1) with n = 1 posd ovr Ω, and (33)-(34) as its discrt countrpart (2)-(21), ach nhancd by nonhomognous boundary conditions g and g h, rspctivly, with g g h. Scond, this stup can b sn as (31)-(32) rprsnting th local continuous problm (12)-(13) posd ovr Ω i, drivd from th global problm, for som fixd i, with 1 i n, n > 1. Similarly, (33)-(34) is its discrt countrpart (18)-(19). In this contxt w assum that on th Γ i part of th boundary Ω i th valu g is givn as a trac of th global solution p Γi and is smooth nough. Likwis, in th discrt problm, w assum that a fixd λ h Γi which plays th rol of g h is givn, This intrprtation is convnint for our subsqunt analysis of th mortar formulation. Th rsults of this sction ar ssntially th sam for both applications thrfor thy ar combind togthr. W shall now assum that a gnric g and g h ar givn and procd to dvlop an a-postriori rror stimator and show that it provids an uppr bound for th rror. Hr w rcall that a-postriori rror stimators for Galrkin finit lmnts hav bn introducd first in [11] and hav bn a topic of many thortical and applid studis; s th rcnt monographs [48, 4] and rfrncs thrin. A-postriori stimators for mixd finit lmnts wr first dvlopd for th Laplac problm in [16] whr msh-dpndnt norms wr usd, and just rcntly for (1) in [56, 18]. Hr w follow ths two lattr works. Both paprs [56, 18] dvlop an stimator for vlocity for 2D problms with g g h. Th rsidual stimator in [56] is appropriat for triangular grids and RT spacs, and in [56] it is shown to b locally (and globally) quivalnt to othr stimators. Th stimator in [18] is also appropriat for triangular grids but is dfind for various highr ordr mixd spacs; it rducs to th on in [56] in th cas of RT spacs. Both of th stimator(s) of intrst to this papr ar of rsidual typ, and thir construction and analysis ar basd on th Hlmholtz dcomposition of th vctors in R 2 into thir solnoidal and (wakly) irrotational parts. Spcifically, th spac H(div; Ω) is dcomposd into (35) (36) (37) H(div; Ω) = (H(div; Ω) X ) (H(div; Ω) X K ), X = {q H(div; Ω) : q = }, X K = {r H(div; Ω) : K 1 r q =, q X }. This dcomposition is crucial for th drivation of th stimator and for th proof of th uppr bounds. Hr our main objctiv is to trat rctangular grids, RT [] spacs, and nonhomognous boundary trms g and g h. Whil xtnsion of th rsults in [56, 18] to our cas is in itslf not difficult, w supply all th dtails for th sak of compltnss. This allows, in particular, to s whr th gnralization of ths rsults to mor gnral cas(s) fails. Th structur of th rsults blow follows th on in similar works: first w dfin th rsidual r(v) as a linar functional on an lmnt v V which, whn valuatd Ω

12 252 M. PESZYŃSKA for v = with th rror in vlocity := u u h givs ris to th natural norm K of th rror. Thn w prform intgration by parts, which allows to xtract, in ach trm of th stimator, a part which can b calculatd dirctly from th finit lmnt solution and which givs ris to th dfinition of th stimator η, and a part which can b stimatd using som norm of th rror. Th stimats ar achivd with hlp of a local rgularization oprator. Rcall that in [56, 18] th Clémnt oprator P C appropriat for triangular mshs is usd. In this papr w us th local rgularization P BG dfind by Brnardi and Girault in [13] which works for triangular and quadrilatral mshs. Sinc th rsidual is akin to th squar of this natural norm of th rror, this procdur lads to an uppr bound for th rror (38) K Cη Dfinition of rsidual and rsidual calculations. Hr w follow th notation and assumptions introducd in Sction and dfin th rsidual r(v), for v H(div; Ω), as th following linar functional (39) r(v) := (K 1 u h, v) + (p h, v) < g h, v n > Ω. Using th quadrangulation T h of Ω, dnoting by th dgs of E T h and writing Ω = Ω w can writ r(v) = (K 1 u h, v) E + (p h, v) E < g h, v n > E E Ω By (33) w not that r(v h ) =, v h V h. W procd to dfin local (lmntwis) contributions to th rsidual. Th ky is th abov mntiond dcomposition of th spac H(div; Ω). For any v H(div; Ω), w hav v = v +v K whr v H(div; Ω) X and whr v K H(div; Ω) X K. In addition, a solnoidal vctor v = Curlφ for som φ = φ(v ) H 1 (Ω), with φ dtrmind up to a constant. In fact, w fix φ to b such that (φ, 1) = that is, to hav zro avrag ovr Ω. This dcomposition of H(div; Ω) is discussd in [56] and [18]. Now on calculats by intgration by parts ovr an lmnt E, for any vctor q = (q 1, q 2 ), and a tst function v H(div; Ω) X, and an associatd φ = φ(v ) q v dx = q Curlφdx = (q 1,2 φ q 2,1 φ)dx (q 1 n 2 φ q 2 n 1 φ)ds E E E E = curlqφdx q tφds. E E As it is don in [56], for q = K 1 u h RT, th intgral curlqφdx vanishs. E Indd, RT lmnts hav th gnral form of (P (E)) 2 +xp (E). In othr words, w hav q E = (K1,1 1 (α 1 + βx 1 ), K2,2 1 (α 2 + βx 2 )) for som lmnt-wis chosn constants α 1, α 2, β and trivially q 1,2 =, q 2,1 =. If K is nondiagonal but symmtric, thn still q 1,2 q 2,1 =. Howvr, if K is non-symmtric and/or whn spacs of highr ordr than RT ar usd, thn th trms involving curl(k 1 u h ) E may not vanish (s [18]). For q RT [] which is th spac of intrst in this papr, w hav that lmntwis q E = (K1,1 1 (α 1 + β 1 x 1 ), K2,2 1 (α 2 + β 2 x 2 )), and by rasoning as abov, if K

13 MORTAR ADAPTIVITY FOR FLOW 253 is diagonal, curl(q) = curl(k 1 u h ) =. In summary, w gt K 1 u h v dx = (K1,1 1 (u h) 1 n 2 φ K2,2 1 (u h) 2 n 1 φ)ds E E = E K 1 u h tφds. Now w tak a sum ovr all lmnts E in th calculation of th rsidual valu r(v ) by (39), with v H(div; Ω) X and not that th scond trm in (39) vanishs by virtu of v =, and w gt r(v ) = (4) K 1 u h tφds < g h, v n > E E Ω = [K 1 u h t]φds + (K 1 u h tφ g h v n)ds \ Ω Ω whr th jump [ ] is dfind as in (15) on th dgs \ Ω of lmnts E which ar intrior to Ω Calculation of K. Dcompos th rror = + K and not that w hav, by th dfinition of th norms (2) and proprtis of and K, (41) 2 K= (K 1 ( + K ), + K ) + ( K, K ) = (K 1, ) + (K 1 K, K ) + ( K, K ) = 2 K + K 2 K. Nxt, by subtracting (34) from (32) which, from th proprtis of th L 2 (Ω) projction Π : W W h can b rwrittn as ( u h, w) = ( u h, Π w) = (f, Π w) = (Π f, w), w gt (, w) = ( (u u h ), w) = (f Π f, w), w W. By applying Cauchy-Schwarz inquality twic w gt = f Π f which is also tru on any lmnt E so that,e = f Π f,e. Finally w s that K = = f Π f. In th squl w shall us th fact that on H(div; Ω) X K th natural norm in H(div; Ω) is quivalnt to th norm K, hnc, K H(div;Ω) C q K, and by (11) w hav (42) K K max(1, K 1 Ω )C q K C K f Π f whr w hav st C K := max(1, K 1 Ω )C q. Now w comput th norm of K using th dfinition of th rsidual r(v). By adding r(v) to both sids of (31), w s that th rror satisfis (K 1 (u u h ), v) (p p h, v)+ < g g h, v n > Ω = r(v), v H(div; Ω) hnc, rstricting th tst functions to X w gt (K 1 (u u h ), v ) = < g g h, v n > Ω +r(v ), v H(div; Ω) X. Now us v = to gt, by (2) and th abov calculations (43) 2 K = (K 1, v ) = (K 1 (u u h ), v ) = < g g h, v n > Ω +r(v ).

14 254 M. PESZYŃSKA Upon insrting th xprssion (4) into (43) and cancling trms w obtain, with a φ = φ(v ) 2 K = (44) [K 1 u h t]φds \ Ω + (K 1 u h tφ(v ) gv n)ds Ω Now w nd to dcompos ths xprssions as products of a computabl quantity and of on that can b stimatd by K Local rgularization oprator. To achiv this aim, on considrs a suitabl finit dimnsional projction vh of v. In this contxt, in [56, 18], th Clémnt rgularization oprator (quasi-intrpolant) P C φ of φ is usd which, locally P C φ E P 1 (E), so that Curlφ h RT. In othr words, P C is compatibl with th mixd spac RT. Also, sinc in [56, 18] g g h, on has r(curlφ h ) =, and th trm r(curlφ h ) can b subtractd from r(v ) without changing u u h 2 K. In our cas, for RT [] spacs, w us φ h := P BG φ whr P BG is th local rgularization oprator on quadrilatral grids dfind in [13]. In cas of rctangular lmnts, φ h := P BG φ is th uniqu lmnt of Q 1 (E) constructd, roughly spaking, by avraging th valus of φ ovr th nighborhood E of th lmnt E. S [13] for dtails which in fact ar carrid through in th rfrnc domain. Th approximation φ h := P BG φ of φ satisfis th local stimats (rspctivly, as ([13], q. 4.2, 4.1), for k = 1) (45) φ φ h,e c BG h E φ 1, E (46) φ φ h, C BG he φ 1, E, E Thr xists a natural compatibility btwn P BG and RT [] spacs, at last on rctangular mshs. Indd, for any φ h Q 1 (E) if ach E is a rctangl, w hav Curlφ h RT []. To s that, considr a basis function ψ h Q 1 (E). Rcall ths ar dfind as ψ h = ˆψ h F 1 E whr ˆψ h ( ˆx 1, ˆx 2 ) is dfind on th rfrnc lmnt Ê and is on of bilinar polynomials (1 ˆx 1 )(1 ˆx 2 ), ˆx 1 (1 ˆx 2 ), ˆx 1ˆx 2, (1 ˆx 1 )ˆx 2. To show Curlφ h RT [] it is nough to show that Curl ˆψ h RT [] holds on Ê. This can b don by computing ˆψ h from which dirctly w s that Curl ˆφ h = T ˆφ h RT []. S also [17], Lmma III.3.3. With this compatibility stablishd, for a givn v and its associatd φ(v ), w lt φ h = P BG (φ), and dnot vh := Curlφ h. Sinc vh RT [], by (33), w hav r(vh ) =. Thus, w can subtract r(v h ) from th rhs of (43) to find (47) 2 K = < g g h, v n > Ω +r(v ) r(v h) = < g g h, v n > Ω +r(v vh) = [K 1 u h t](φ φ h )ds \ Ω + (K 1 u h t(φ φ h ) Ω g h (v vh) n (g g h )v n)ds := (I int )ds + (B I + B II + B III )ds \ Ω Ω

15 MORTAR ADAPTIVITY FOR FLOW 255 If g g h, thn all th boundary trms B I, B II, B III vanish and th intrior trms ar stimatd as in [56, 18] Intrior stimats. Us v = u u h, by Cauchy-Schwarz inquality and (46) I int ds = (48) [K 1 u h t](φ φ h )ds \ Ω \ Ω \ Ω \ Ω [K 1 u h t], (φ φ h ), [K 1 u h t], C BG he φ 1, E Now, by Poincaré inquality (rcall that th avrag of φ ovr Ω is zro) w hav φ,ω C P φ 1,Ω = C P v,ω C P κω v K,Ω. With this, and by Cauchy- Scharz inquality for sums, and noting that thr can b at most four lmnts in E for vry E, th trms in (48) can b stimatd as follows 1 (49) I int ds C int h E [K 1 u h t] 2, v K,Ω, κω \ Ω \ Ω whr C int := C BG 4(1 + C 2 P ) and which is ssntially th sam as appropriat trms in [56, 18]. W not in passing that, if th norm on th rhs of (46) can b rplacd by th 1-sminorm, (as in [23] or in [56]), on can improv th abov bound by rplacing κ Ω by a local stimat of κ E and th rror stimt, instad of (49) would rad as follows h E I int ds C int [K κ 1 u h t] 2, v K,Ω. E \ Ω \ Ω At this tim howvr w ar unabl to dtrmin whthr it can b don without furthr loss of gnrality Boundary trms. Hr w considr th gnral cas of g g h and procd to stimat th boundary trms B I, B II, B III. W only considr intgration ovr thos dgs which li on Ω and w drop th notation Ω for brvity. First w considr th following calculation on an dg which blongs to Ω. Dnot th two ndpoints of by P 1, P 2. Lt ρ and ψ b two functions dfind on, smooth in th intrior of and dfind maningfully pointwis at its ndpoints. W hav by intgration by parts, ρcurlψ nds = ρψ P 2 P Curlρ nψds = ρψ P 2 1 (5) P ( ρ t)ψds. 1 If ρ and ψ ar continuous on Ω, w sum ovr all dgs Ω, th pointwis valus cancl and w gt ρcurlψ nds = Curlρ nψds = (51) ρ tψds. This calculation suggsts how to handl trms arising in B I, B II, B III and how to distinguish diffrnt cass dpnding on th dfinition and smoothnss of g h. W considr two cass: (i) g h is th picwis linar intrpolant of g, and (ii) g h is th picwis constant L 2 ( Ω) projction of g. In ach cas th discrt valus of g h ar dfind with rspct to th trac of th triangulation T h of Ω on Ω.

16 256 M. PESZYŃSKA Cas (i). In th first cas, w s that g h and φ φ h ar smooth nough for calculation (51) to work. Hnc, w obtain B II ds = g h Curl(φ φ h ) nds = Curlg h n(φ φ h )ds and, aftr it is combind with th trm B I, w gt (K 1 u h t(φ φ h ) g h (v v h) n)ds = (K 1 u h + g h ) t(φ φ h )ds which can b stimatd in a mannr similar to th intrior trms in (48), and w gt 1 (B I + B II )ds C I h (K 1 u h + g h ) t 2, v K,Ω, κω whr now C I := C BG 2(1 + C 2 P ) and whr h dnots th maximum of h E for lmnts E adjacnt to th lmnt containing. Nxt, to dal with th trm B III, w us th avrag φ of φ ovr ach lmnt E adjacnt to th dg and xtndd by to th rst of Ω. Clarly Curl φ n = and w subtract this from th trm B III and us (5) in th following (g g h )v nds = (g g h )(v Curl φ) nds = (g g h )(φ φ) P 2 P Curl(g g 1 h ) n(φ φ)ds whr ach of th trms (g g h )(φ φ) P 1 vanishs by virtu of g h intrpolating g at th ndpoints of. Hnc w gt, for th trm B III, B III ds = g h )v (g nds = Curl(g g h ) n(φ φ)ds = (g g h ) t(φ φ)ds (g g h ) t, φ φ, which follows from th Cauchy-Schwarz inquality. proprty of φ Now, by th approximating φ φ, C ap 1/2 φ t,e C ap he Curlφ n,e = C ap he v,e whr w hav usd th fact that th lngth h E, w gt, (g g h )v nds C ap Ω h E κ E (g g h ) t 2, v K. Combining all ths stimats togthr w gt 2 (52) (B I + B II + B III )ds max{c I, C ap } κω ( h E (K 1 u h + g h ) t 2, + (g g h) t,) 2 v K,Ω.

17 MORTAR ADAPTIVITY FOR FLOW Cas (ii): Hr w assum that g h is picwis constant on Ω in fact, that it is th L 2 projction of g. In addition, w assum that g is dfind pointwis and is continuous on Ω. Sinc g h is th avrag of g, and sinc vh n is constant on ach, w hav (g g h)vh nds = and this lattr trm can b addd to th trm B III. Togthr thy can b rwrittn as (53) B III ds = B III ds + (g g h )vh nds = (g g h )(v vh) nds = (g g h )(φ φ h ) P 2 P + 1 Curl(g g h ) n(φ φ h )ds. Considr now th trm B II. Whn intgratd by parts, B II givs ris to th sam intgral Curlg h n(φ φ h )ds as in Cas (i) and to pointwis trms g h (φ φ h ) P 2 P 1 which cannot b assumd to vanish whn summd ovr Ω. But, ths pointwis valus will cancl with thos arising from th trm B III as appars from (53). Thn thr rmains th sum of pointwis trms g(φ φ h ) P 2 P which vanishs by virtu of 1 smoothnss of g and of φ φ h. In summary, w ar lft with = = (B I + B II + B III )ds (K 1 u h t(φ φ h ) g h (v vh) n (g g h )v n)ds (K 1 u h + g h ) t(φ φ h )ds + Curl(g g h ) n(φ φ h )ds whos individual pics can b stimatd in th sam way as it was don in Cas (i) and on gts (54) 1 (B I + B II + B III )ds C I κω h E ( K 1 u h + g h ) t 2, + (g g h) t 2, ) v K,Ω, Summary. Combining (41), (42), (43), (47), (48), and (52) or (54) w obtain th following main rsult of this sction. Thorm 3.1. Assum that T h is a rgular partition of Ω and that Assumptions 1, 2, 3, 4 ar satisfid. Thn th following uppr bound for th mixd finit lmnt solution of (33) (34) for RT [] spacs holds (55) K,Ω C(K, Ω)η Ω,

18 258 M. PESZYŃSKA whr th rsidual stimator η Ω is dfind as (56) η 2 Ω := η 2 f,ω + η 2 int,ω + η 2 II, Ω + η 2 III, Ω ηf,ω 2 := f Π f 2,E, E T ηint,ω 2 := h [K 1 u h t] 2, T,\ Ω ηii, Ω 2 := h (K 1 u h + g h ) t 2, Ω ηiii, Ω 2 := h (g g h ) t 2, Ω and whr th constant C(K, Ω) dos not dpnd on h and whr h dnots th maximum lngth of th dgs of any rctangl E adjacnt to th rctangl E such that th dg E. Hr w do not attmpt to stablish a lowr bound for th rror; this has bn don for RT spacs and homognous boundary conditions in [56]. Th proof of th lowr bound in [18] is not don and th radr is rfrrd to [48]. Rmark 3.1. On can also driv a bound for th total rror which, in addition to rror in vlocity includs th rror p p h in th prssur variabl. This is don in [56] by dcomposing p p h p Π o p + Π o p p h. Th first part is stimatd on ach lmnt from approximation proprtis of Π o by h E p,e which lads to a computabl part and an rror part by way of p = K 1 u h + K 1 (u u h ). Th scond part is stimatd in trms of by duality, s ([17], II.2.7) or [1]. In summary, additional intrior trms involving K 1 u h,e nd to b includd in η but this dirction will not b pursud hr. Rmark 3.2. It is intrsting to compar th structur of th parts of th stimator to th on for conforming mthods. In particular, whil th η f is standard and appars similarly in th Galrkin a-postriori stimats, th countrpart of η int for conforming mthods is th masur of th jump of th normal componnt of th gradint of th solution. On th othr hand, th trm η II masurs how badly th discrt gradint of th boundary data approximats th gradint of th discrt prssur. In this sns, this trm is similar to th prvious on. Finally, th last trm in th stimator is rathr standard as it masurs th consistncy rror btwn g and g h ( data rror in [3]). Most important for our subsqunt analysis of mortar formulation ar th η II and η III parts of th stimator. 4. Rsidual rror stimator for mortar formulation In this sction w dfin an a-postriori rror stimator η(n) for th problm (18)-(19) with n subdomains and thn w prov an uppr bound for th rror using η(n). W ar mainly intrstd in handling trms arising on th intrfac Γ and spcifically, in th part of th rror stimator rlatd to th jump of th numrically computd flux across Γ. If n = 1, th stimator η(n) drivd blow rducs, of cours, to th on discussd in Thorm 3.1. For n > 1, η(n) includs th subdomain intrior trms akin to th trms η f,i, η int,i in (56) and which ar calculatd on vry subdomain Ω i, th

19 MORTAR ADAPTIVITY FOR FLOW 259 boundary trms dfind on Ω i Ω, if nonmpty, and th intrfac trms dfind on th intrfac Γ. Th lattr combin th trms η II,i, η III,i in (56). All th trms aris naturally from th dfinition of th rsidual. To prov th uppr bound for th vlocity rror in th (brokn) norm on V 1 (57) 2 K,i= 2 K,i + K 2 K,i, i i whr similarly as in (42) on can show (58) K 2 K,i C K,i f Π,i f i,, i i w follow som of th idas dvlopd in [54] for th mortar formulations for standard Galrkin mthod. Thr, as a distinct fatur, th various hirarchical and rsidual stimators considrd includ trms which stimat th rror on subdomains as wll as trms which prtain to th intrfac problms and provid a masur of nonconformity arising from th us of non-matching grids and mortar spacs. To handl ths, an important tchnical assumption calld th saturation assumption is usd; it is basd on asymptotic a-priori stimats drivd for mortar formulations. Th quasi-intrpolant usd in th proofs of uppr bounds is similar to on by Clémnt, and it is adaptd to various choics of basis functions on th intrfac. In this papr, th proof of th uppr bound in our cas is analogous to [54] but also vry diffrnt sinc th undrlying mixd spacs and mortar mthods hav an ntirly diffrnt structur Dfinition of rsidual and calculations. First w dfin, for vry v V 1, that is, for v i locally in V i, th rsidual, (59) r(v) := i r i (v i ) (6) r i (v i ) := (K 1 u h, v i ) i + (p h, v i ) i < g h, v i ν > Ωi \Γ i < λ h, v i ν > Γi. Notic that th rsidual r i (v i ) has bn dfind analogously to (39). On ach subdomain w can dcompos v i V i in such a way that v i = vi + vk i with th dcomposition bing local to Ω i. Also, th corrsponding φ i = φ i (vi ) H 1 (Ω i ) but bcaus of lack of continuity of its normal flux across Γ, φ(v ) H 1 (Ω). Nxt, v K is K-wakly orthogonal to v only locally, that is, (K 1 v K, q) i =, for any q H(div; Ω i ) X. Finally, its rgularization φ i,h = P BG φ i is local to Ω i. Now w procd with rsidual calculations similar to thos that gav (44). W us v = u u h and its rstriction v i on Ω i, to gt v 2 K,Ω i = (61) [K 1 u h t]φ i ds + \ Ω i ( Ω i\γ i) (K 1 u h tφ i (vi ) gvi n)ds + (K 1 u h tφ i (vi ) pvi n)ds Γ i whr w hav rplacd g by p on th intrfac part Γ i of Ω i. Nxt, as it was don to gt (47), w tak into account th discrt problm (18), subtract th Curl of th

20 26 M. PESZYŃSKA local rgularization φ i,h = P BG φ i, and calculat individual parts of th rsidual, for ach i, as follows: (62) 2 K,Ω i = < p λ h, vi n > Γi < g g h, vi n > Ωi\Γ i + r(vi vi,h) = [K 1 u h t](φ i φ i,h )ds \ Ω i + (K 1 u h t(φ i φ i,h ) ( Ω i\γ i) g h (vi vi,h) n (g g h )vi n)ds + (K 1 u h t(φ i φ i,h ) Γ i := + + λ h (vi vi,h) n (p λ h )vi n)ds I int,i ds T i, Ω i T i, Ω i \Γ i T i, Γ i (B i,i + B i,ii + B i,iii )ds (D i,i + D i,ii + D i,iii )ds whr th dfinitions of th trms B I,i,... D III,i follow naturally as in (47) xcpt that B trms apply on th xtrnal parts of th boundary Ω i Ω and D apply on its intrfac part Γ i. Whn adding (62) ovr all subdomains i = 1,... n, w find that i 2 K,Ω i is mad of parts intrior to subdomains I int,i, th xtrnal boundary parts B I,i... B III,i and of th intrfac parts D I,i... D III,i. Th handling of th formr is xactly th sam as in Sctions 3.3 and 3.4. W focus thrfor on th lattr which involvs trms on th intrfac Γ. In fact, w show blow that th trms D I,i and D II,i ar combind togthr for ach i and tratd as boundary trms in th sns of calculations don in Sction 3.4 and that thy act along Γ, s Rmark 3.2. On th othr hand, th trms D III,i whn addd ovr all i giv ris to th part of th stimator daling with th jump across ach Γ ij which is most intrsting to us as it provids th natural masur of inconsistncy or dfct btwn discrt vlocitis dfind on on both sids of Γ ij which as w rcall ar only wakly continuous Intrfac trms. In what follows it will b convnint to considr an intrsction of grids. Rcall that ach Ω i is covrd by quadrangulation T i whos trac T i Ω i provids a grid on ach Γ i. Also, ach intrfac Γ ij has its own associatd mortar grid T m. In th calculations blow it will b convnint to us a rfinmnt of ths grids, or in othr words, a partition T i m and T i j of Γ ij so that and T i m := {ξ : ξ = i m, i E i Γ ij, E i T i, m T m } T i j := {χ : χ = i j, i E i Γ ij, E i T i, j E j Γ ij, E j T j, }

21 MORTAR ADAPTIVITY FOR FLOW 261 Not that ths nw partitions ar nothing but rfinmnts of th original grids. By abus of notation, w could writ T i j = T i T j, and T i m = T i T m Intrfac trms D i,i + D i,ii along Γ. Hr w considr calculations for ach subdomain i. Rcall that λ h is picwis linar on th grid T m on Γ ij and it is thrfor also smooth on T i m. Using th sam calculations as in Sction 3.4.2, w rwrit th sum ovr as a sum ovr ξ T i m and considr th contributions from D i,i + D i,ii as (63) (K 1 u h + λ h t(φ i φ i,h ) ξ Γ i whr w undrstand by λ h t th dirctional drivativ alongsid Γ of λ h. In this calculation which follows from intgration by parts, w ar lft, as in (53), with additional pointwis trms λ h (φ φ h ) at thos ndpoints P Ω i := Γ i ( Ω) of Γ i which may li on Ω. W ignor ths pointwis valus as wll as thos in th cornr points whr som Γ ij and Γ kl intrsctas thir handling is implmntation dpndnt. Mor prcisly, whn summing ovr th dgs ξ w hav ξ ξ λ h(vi v i,h ) nds = ξ ξ λ h t(φ φ h )ds + P Ω λ h (φ φ h ) P Ω. Th pointwis valus at P Ω i must b combind with B i,ii. In practic, handling of ths trms which ntr th η Ω part of th stimator dpnds upon th typ of approximation of boundary condition imposd on Ω and its dtails will not b discussd. To handl th trms arising from (63), w procd as in Sction by noticing that on ach ξ th norm φ φ h,ξ can b stimatd from abov by φ φ h,, for som containing ξ. Thrfor w gt th sam typ of trms with th part of th stimator η 2 Γ,,ξ := h E (K 1 u h + λ h ) t 2,ξ akin to th trms η II,Ω in (56). Naturally η 2 Γ, := i ξ Γ i η 2 Γ,,ξ and w conclud by stating th rsult which can b provn using th sam tchniqu as th on applid in Sction 3.4.2: (64) (D I,i + D II,i )ds Cη Γ, K, i ξ Γ i whr th constant C dpnds on Ω and K but not on h. Rmark 4.1. In implmntation, th valus of λ h usd as a boundary condition for th local discrt problm (18),(19) ar rplacd by its picwis constant projction onto th grid of Ω i xprssd by Q h,i λ h. Thn th calculation as abov is no mor valid as Q h,i λ h is picwis constant. In such a cas, w propos to considr, on a union of two dgs 1 2, instad of th quantity λ h, its discrt gradint h λ h 1 2 := Q h i λ h 2 Q hi λ h 1 h 1. It is not hard to s that, a similar stimat as, 2 abov will still b valid. Also thn, th norm ovr ξ in th dfinition of η Γ, can b rplacd by th norm ovr i which is much asir to comput in implmntation. W not in passing that th us of th discrt gradint is rlatd to th dfinition of an intrpolation oprator I i which is discussd in Sction 4.3.