The interface control domain decomposition (ICDD) method for the Stokes problem. (Received: 15 July Accepted: 13 September 2013)

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1 Journal of Coupled Systems Multscale Dynamcs Copyrght 2013 by Amercan Scentfc Publshers All rghts reserved. Prnted n the Unted States of Amerca do: /jcsmd J. Coupled Syst. Multscale Dyn. Vol. 1(3/ X/2013/001/021 The nterface control doman decomposton (ICDD method for the Stokes problem Marco Dscaccat 1,, Paola Gervaso 2, Alfo Quarteron 3, 4 1 Laborator de Càlcul Numèrc (LaCàN, Departament de Matemàtca Aplcada III (MA3, Unverstat Poltècnca de Catalunya (UPC BarcelonaTech, Campus Nord UPC C2, E Barcelona, Span 2 DICATAM, Unverstà d Bresca, va Branze 38, I Bresca, Italy 3 MOX, Poltecnco d Mlano, P.zza Leonardo da Vnc 32, I Mlano, Italy 4 MATHICSE, Char of Modellng Scentfc Computng, Ecole Polytechnque Fédérale de Lausanne, Staton 8, CH-1015 Lausanne, Swtzerl (Receved: 15 July Accepted: 13 September 2013 ABSTRACT We study the Interface Control Doman Decomposton (ICDD for the Stokes equaton. We reformulate ths problem ntroducng auxlary control varables that represent ether the traces of the flud velocty or the normal stress across subdoman nterfaces. Then, we characterze sutable cost functonals whose mnmzaton permts to recover the soluton of the orgnal problem. We analyze the well-posedness of the optmal control problems assocated to the dfferent choces of the cost functonals, we propose a dscretzaton of the problem based on hp fnte elements. The effectveness of the proposed methods s llustrated through several numercal tests. Keywords: Stokes Equatons, Doman Decomposton Methods, Optmal Control, hp-fnte Elements, ICDD. Secton: Mathematcal, Physcal & Engneerng Scences Research Artcle 1. INTRODUCTION The Interface Control Doman Decomposton (ICDD method was ntroduced n Refs. [4, 5] as a soluton strategy for boundary value problems governed by ellptc partal dfferental equatons. In ths paper we extend ths methodology to the Stokes equatons we study ts effectveness n computng the soluton of ths lnear model for lamnar ncompressble flows. The ICDD method, whch shares some smlartes wth the classc overlappng Schwarz method wth the Least Square Conjugate Gradent 10 the Vrtual Control 13 methods, s characterzed by a decomposton of the orgnal doman nto overlappng regons by the ntroducton of new auxlary varables on the subdoman nterfaces. In the case of the Stokes problem, these Author to whom correspondence should be addressed. Emal: marco.dscaccat@upc.edu varables may represent ether the trace of the flud velocty or the normal stress across the nterfaces. In ether case, they play the role of control varables that can be determned as soluton of an optmal control problem that mposes the mnmzaton of a sutably defned cost functonal nvolvng the solutons of well-posed local subproblem. The ICDD method can thus be regarded as a novel doman decomposton method whose nterest les n the fact that, at least n the case of two subdomans, t may show convergence rates ndependent of the computatonal grd, of the polynomal degree used for the numercal approxmaton, for a partcular choce of the cost functonal, also ndependence on the sze of the overlappng. The choce of the cost functonal s crucal to ensure the unqueness of the soluton on the overlappng area. In partcular, we show that, for the Stokes problem, the cost functonals must account for both the velocty the 1

2 Journal of Coupled Systems Multscale Dynamcs Research Artcle pressure across the nterfaces to ensure the matchng of these two varables n the overlappng regons. What makes the ICDD method even more attractve s also ts capablty of hlng dfferental problems of heterogeneous type,.e., governed by dfferent type of equatons n dfferent subregons of the computatonal doman. Some examples of such applcaton of the method were provded n Ref. [4, 5] n the case of advecton/advecton-dffuson problems. Another nterestng problem wth many sgnfcant applcatons s the couplng of Stokes Darcy equaton to model fltraton processes (see Ref. [3, 4, 6, 14]. The outlne of the paper s as follows. In Secton 2 we wrte the Stokes problem n a bounded doman we reformulate t n equvalent ways after splttng the orgnal doman nto two overlappng regons. In Secton 3, after ntroducng a dscretzaton of the problem usng hp fnte elements, we present the ICDD method consderng the cases of Drchlet, Neumann mxed control varables. In each case we wrte the correspondng optmalty system wth ts algebrac counterpart. In Secton 4 we present several numercal results amed at studyng the convergence behavor of the proposed ICDD methods wth respect to the grd sze, the polynomal degree, the sze of the overlappng regon. Fnally, Secton 5 s devoted to the theoretcal analyss of the dfferent methods. 2. PROBLEM SETTING Let d (d = 2 3 be an open bounded doman wth Lpschtz boundary. We assume that = D N wth D N = that D whle N mght be empty. We consder the Stokes problem: Problem : dv T u p = f n dv u = 0n u = D on D T u p n = N on N descrbng the moton of a steady, vscous, ncompressble flud confned n the regon. Here, T u p = 2 s u pi s the Cauchy stress tensor beng s u = 1/2 u + u T, >0 s the flud vscosty, u ts velocty p ts pressure n s the unt normal vector to drected outwards the doman. We assume that f L 2 d, D H 1/2 D d N H 1/2 N d are assgned functons. If D (.e., N =, the compatblty condton D n = 0 must hold, a further condton on p, e.g., p = 0 must be enforced to guarantee the well-posedness of problem (1. The weak form of problem (1 s: fnd u H 1 d, u = D on D, p L 2 such that, for all (1 v H 1 d, v = 0 on D, q L 2, a u v + b p v = f v + N v N b q u = 0 where a u v = u + u T v (2 b q v = q dv v (3 For smplcty of exposton, n the rest of the paper we wll often use the strong form of the Stokes problem, but t must be understood that the analyss s carred out n the weak settng. We consder an overlappng decomposton of the doman n two subdomans 1 2 = 1 2. We denote the overlappng regon by 12 = 1 2 let = \. Moreover, let D = D N = N (see Fg. 1. We reformulate the Stokes problem (1 on the splt doman n the followng possble ways. Problem t : dv T u p = f n = 1 2 dv u = 0n = 1 2 u = D on D = 1 2 T u p n = N on N = 1 2 u 1 = u 2 on 1 2 In case N = for some, we would supplement (4 wth the condton p = 0 to ensure the well-posedness of the correspondng local problem. Problem f : dv T u p = f n = 1 2 dv u = 0n = 1 2 u = D on D = 1 2 T u p n = N on N = 1 2 T u 1 p 1 n = T u 2 p 2 n on 1 2 n Γ 2 Ω 1 Ω 12 Fg. 1. Representaton of the computatonal doman of ts overlappng splttng. Ω 2 Γ 1 Ω n (4 (5 2 J. Coupled Syst. Multscale Dyn., Vol. 1(3, 1 21

3 Journal of Coupled Systems Multscale Dynamcs Condton (5 5 on 1 should be understood as follows. The normal vector n on 1 s drected outward of 1 the normal component of the tensor T u 2 p 2 s computed upon restrctng t to 12. On the other h, on 2 the normal vector n s drected outward of 2 the normal component of the tensor T u 1 p 1 s taken upon restrctng t to 12. Moreover, we consder the problem: Problem tf : dv T u p = f n = 1 2 dv u = 0n = 1 2 u = D on D = 1 2 T u p n = N on N = 1 2 u 1 = u 2 on 1 T u 1 p 1 n = T u 2 p 2 n on 2 (6 Proposton 2.1 (Equvalence Between t. The Stokes problems t are equvalent f Assumpton 2 1 holds. Equvalence holds n the sense that f u p u p ( = 1 2 are the unque solutons of t, respectvely, there exst two unquely determned constants C 1 C 2, possbly null, such that, for = 1 2, u = u p = p + C. Proof. We treat the dfferent cases separately. (1 Assume frst that N 12. Then, problem (1 s well-posed n u p V D Q the restrctons of ts soluton to satsfy (4 by constructon. Vceversa, for = 1 2, let u p V D Q ( = 1 2 be the solutons of the well-posed local problems dv T u p = f n dv u = 0n u = D on D If N 1 =, we should mpose p 1 = 0 1 to guarantee the well-posedness of the Stokes problem n 1. Let us ntroduce the followng spaces V= H 1 d V = H 1 d =1 2 Q =L 2 Q 0 = q Q q =0 Q =L 2 Q 0 = q Q q =0 =1 2 the followng affne manfolds V D = v H 1 d v = D on D (8 V D = v H 1 d v = D on D = 1 2 Fnally, we set (7 V 0 = v H 1 d v = 0 on D = 1 2 (9 To prove that the Stokes problem (1 s equvalent to ether (4, or (5, or (6, we wll denote w = u 1 12 u 2 12 q = p 1 12 p 2 12 the dfference n 12 between the local solutons. Note that w q satsfes the Stokes equatons: dv T w q = 0 n 12 (10 dv w = 0n 12 The boundary condtons fulflled by w q on 12,as well as the spaces to whch these functons belong wll be specfed case by case. Assumpton 2.1. We suppose that one of the followng assumptons s verfed: N = ; N N 12 ; N 12 = wth N connected. T u p n = N on N u = u j on j = 3 By constructon, the functons w q satsfy problem (10 wth boundary condtons T w q n = 0 on 12 N w = 0 on 12 \ N Ths problem s well-posed admts the unque soluton w = 0 q = 0, hence u 1 = u 2 p 1 = p 2 n 12. Thus, we can set u 1 n 1 \ 12 u = u 1 = u 2 n 12 (11 u 2 n 2 \ 12 p 1 n 1 \ 12 p = p 1 = p 2 n 12 (12 p 2 n 2 \ 12 By constructon, functons u p belong to V D Q they satsfy problem (1. In ths case C 1 = C 2 = 0. (2 Let now N 12 = assume that N s connected. In ths case, ether N 1 = or N 2 =. We consder the latter case; the former can be treated analogously. If u p V D Q s the soluton of,fweset u = u ( = 1 2, p 1 = p 1, p 2 = p 2 1 p we can mmedately verfy that u p V D Q ( = 1 2 are solutons of t wth 2 p 2 = 0. Thus, C 1 = 0 C 2 = 1/ 2 2 p 2. Vceversa, let u 1 p 1 V 1 D Q 1, u 2 p 2 V 2 D Q 2 0 be the solutons of t. Research Artcle J. Coupled Syst. Multscale Dyn., Vol. 1(3,

4 Journal of Coupled Systems Multscale Dynamcs Research Artcle The functons w q satsfy (10 wth w = 0 on 12. Then, w = 0 q = const n 12. The functon q s unquely determned by 12 q = 12 p 1 p 2 whch mples q = 1 p 12 1 p 2 12 If we take u as n (11 p 1 n 1 \ 12 p = p 1 = p 2 + q n 12 p 2 + q n 2 \ 12 then u p satsfy the thess follows wth C 1 = 0 C 2 = q. (3 Let u p V D Q 0 be the soluton of. Then, for = 1 2, the functons u = u p = p 1 p belong to V D Q 0 they satsfy t. Thus, C 1 = 0 C 2 = 1/ p. Vceversa, let u p V D Q 0 be solutons of t. Then, the functons w q satsfy (10 wth boundary condton w = 0 on 12. Then, w = 0 n 12 q = const n 12. The constant q s unquely determned by q = p 1 p that s q = 1 p 12 1 p 2 12 If we defne the constants C 1 = 1 ( p 2 \ 1 q 12 C 2 = 1 ( p q 12 snce C 2 C 1 = q, then p 1 + C 1 = p 2 + C 2 n 12. Thus, we can easly verfy that the functons u p defned respectvely as n (11 as p 1 + C 1 n 1 \ 12 p = p 1 + C 1 = p 2 + C 2 n 12 (13 p 2 + C 2 n 2 \ 12 are solutons of wth p = 0. Remark 2.1. If 12 N = N = 1 2, problems t are not equvalent. In fact, f u p are the solutons of t, the functons w q satsfy (10 wth boundary condton w = 0 on 12. Then, w = 0 q = const n 12 wth q unquely gven by q = 1 p 12 1 p 2 12 Then, proceedng smlarly to the thrd case of the proof of Proposton 2.1, there exst two unque constants C 1 C 2 wth q = C 2 C 1 so that we can defne u p as n (11 (13, respectvely. The Neumann boundary condtons n t mply T u p n = N on N, by defnton of u p, wehave T u p n = N + C n on N Thus, u p satsfy problem f only f C 1 = C 2 = 0, but we cannot guarantee that ths condton s fulflled. Proposton 2.2 (Equvalence Between f. If 12 D, the Stokes problems f are equvalent n the sense that there exst unque constants C 1 C 2 such that u = u p = p + C, u p u p = 1 2 beng, respectvely, the unque solutons of f. Proof. The proof goes along the same arguments used for Proposton 2.1 so that we only defne the constants n the cases N or N =. In the frst case t s straghtforward to see that the equvalence holds wth C 1 = C 2 = 0. On the other h, f N =, the functons w q satsfy the problem (10 wth boundary condtons w = 0 on 12 T w q n = 0 on 1 2 Ths problem s well-posed ts soluton s w = 0 q = 0. Thus, u 1 = u 2 p 1 = p 2 n 12 we can defne velocty u a pressure p analogously to (11 (12. However, the functon p would belong to Q but not to Q 0, so that we defne C 1 = C 2 = 1 p p = p + C 1 to recover the null average. Remark 2.2. Problems f are not equvalent f 12 D =. In fact, n ths case problem (10 n 12 would be supplemented wth the boundary condton T w q n = 0 on 12 whch has nfnte non-trval solutons that may dffer one from another not only by a constant. Proposton 2.3 (Equvalence Between tf. The Stokes problems tf are equvalent f ether =, or N 12, or N 12 = N N 1. Equvalence holds n the sense that f u p u p ( = 1 2 are the unque solutons of t, respectvely, there exst two unquely determned constants C 1 C 2, possbly null, such that, for = 1 2, u = u p = p + C. 4 J. Coupled Syst. Multscale Dyn., Vol. 1(3, 1 21

5 Journal of Coupled Systems Multscale Dynamcs Proof. The proof develops along the lnes of the prevous propostons. Let us only pont out that the equvalence holds wth C 1 = C 2 = 0f N. Otherwse, f N =, f u p V D Q 0 s the soluton of, then u = u, p 2 = p 2 p 1 = p 1 1 p are the solutons of tf. Vceversa, f u 1 p 1 V 1 D Q 1 0 u 2 p 2 V 2 D Q 2 are the solutons of tf, then we need to set C 1 = C 2 = 1 p 2 2 \ 12 Remark 2.3. Problems tf are not equvalent f 12 N =, N 1 =, N 2. In fact, f u 1 p 1 V 1 D Q 1 0 u 2 p 2 V 2 D Q 2 are the solutons of tf, then w q satsfy problem (10 n 12 wth boundary condton T w q n = 0 on 2 w = 0 on 12 \ 2. The soluton of ths problem n 12 s dentcally null. However, snce 12 q = 12 p 1 p 2 wth p 1 Q 1 0 p 2 unquely determned by the Neumann boundary condton on N 2, we cannot guarantee that q = 0. Notce that a result smlar to Proposton 2.3 could be obtaned by swtchng the role of the nterface condtons (6 5 (6 6,.e., consderng Problem ft : dv T u p = f n = 1 2 dv u = 0 n = 1 2 u = D on D = 1 2 T u p n = N on N = 1 2 T u 1 p 1 n = T u 2 p 2 n on 1 (14 concde n 12 that both nterfaces 1 2 do not cross any element of 1 or 2. We dscretze both prmal dual problems n each subdoman by hp fnte element methods (hp-fem. Because of the dffculty to compute ntegrals exactly for large p, typcally when quadrlaterals are used, Legendre Gauss Lobatto quadrature formulas are employed to approxmate the blnear forms a b (see (2 (3 as well as the L 2 -nner products n on the nterfaces. Ths leads to the so called 1, 2 Galerkn approach wth Numercal Integraton (G-NI to the Spectral Element Method wth Numercal Integraton (SEM-NI. In partcular, we consder ether nf-sup stable fnte dmensonal spaces or stablzed couples of spaces of the same degree (see Refs. [7, 8, 11, 15] to approxmate the velocty the pressure we assume that the polynomals used for the pressure are contnuous (see, e.g., Refs. [9, 16]. More precsely, gven an nteger p 1, let p be the space of polynomals whose global degree s less than or equal to p n the varables x 1 x d p be the space of polynomals that are of degree less than or equal to p wth respect to each varable x 1 x d. The space p s assocated wth smplcal parttons, whle p to quadrlateral ones. We ntroduce the fnte dmensonal space on defned by X p h = v C0 v T P p T n the smplcal case, by X p h = v C0 v T F T p T for quadrlaterals. Then, the fnte dmensonal spaces for velocty pressure are, respectvely, V h = X p h d V 0 Q h = X r h (15 for sutable polynomal degrees p r. Research Artcle u 1 = u 2 on 2 3. FORMULATION OF THE ICDD METHOD FOR THE STOKES PROBLEM For the sake of smplcty we wll consder homogeneous boundary condtons,.e., we wll set D = 0 on D N = 0 on N. Moreover, snce we wll be nterested n computng a fnte dmensonal approxmaton of the soluton of the Stokes problem, we ntroduce the ICDD method drectly at the dscrete level hp-fem Dscretzaton We ntroduce two regular computatonal grds 1 2 n 1 2 made by ether smplces or quadrlaterals/hexahedra. We suppose that each element T s obtaned by a C 1 dffeomorphsm F T of the reference element ˆT we suppose that two adjacent elements of share ether a common vertex or a complete edge or a complete face (when d = 3. Moreover, we assume that they 3.2. ICDD Method wth Drchlet Controls Assume, for smplcty, that 12 N D. (We wll dscuss ths ssue more n detal n Secton 5. We defne the space of dscrete Drchlet controls as D h = h C 0 v h V h wth h = v h let D h = D 1 h D 2 h For = 1 2, we consder two control functons h D h the state problems: fnd u h p h V h Q h such that, for all v h q h V h Q, v h = 0 on, a u h v h + b p h v h = f v h b q h u h = 0 (16 u h = h on where a b denote the restrcton of the blnear forms (2 (3 to. In fact, u h p h depends on both h J. Coupled Syst. Multscale Dyn., Vol. 1(3,

6 Journal of Coupled Systems Multscale Dynamcs Research Artcle f, however such dependence wll be understood for the sake of notaton. The unknown controls on the nterface are obtaned by solvng a mnmzaton problem for a cost functonal sutably dependng on the dfference between u 1 h u 2 h on the nterfaces 1 2. More precsely, nspred by (4 5, we look for [ nf J t h = 1 ] u h = 1 h 2 h 2 1 h u 2 h 2 L 2 (17 =1 To the mnmzaton problem (17 we can assocate the followng optmalty system: fnd h = 1 h 2 h D h, for = 1 2, u h p h V h Q h, w h q h V h Q h such that, for all v h h V h Q h wth v h = 0 on, a u h v h + b p h v h = f v h b h u h = 0 (18 u h = h on a w h v h + b q h v h = 0 b h w h = 0 (19 w h = 1 +1 u 1 h u 2 h on, for all 1 h 2 h D h, u 1 h u 2 h + w 2 h 1 h d + 1 u 1 h u 2 h + w 1 h 2 h d = 0 2 ( Algebrac Formulaton of ICDD wth Drchlet Controls To the Stokes problem n subdoman ( = 1 2 we can assocate the matrx ( A B T S = B 0 where A corresponds to the fnte dmensonal approxmaton of the blnear form a (see (2, whle B corresponds to the dscretzaton of b (see (3. When stablzaton s used, the matrces S take the form ( ( A B S = T Ã B + T B 0 B C where Ã, B C are assembled locally, element by element, they take nto account the ntegraton of the dfferental Stokes operator. In the followng we wll denote by the ndex I the degrees of freedom for the velocty the pressure belongng to \, whle the ndex wll refer to the degrees of freedom on the nterface. For the sake of exposton, we wll reorder the nodes n puttng those assocated wth \ frst followed by those on the nterfaces. Correspondngly, wth obvous choce of notaton, we can rewrte the Stokes matrx S as A I I BI T I A I BI T S = B I I 0 B I 0 A I B T I A B T B I 0 B 0 ( SI I = S I S I S Moreover, we wll ndcate by M the mass matrx on the nterface. Fnally, n the rest of the secton, we wll denote by F the rght-h sde for the state problems n, whle U W wll be the vectors of unknown velocty pressure n for the state the adjont problems, respectvely. s the vector of the unknown Drchlet controls on : = ( 1 N j = h x j j where s the set of the N ndces correspondng to the velocty degrees of freedom on the nterface x j s a node on ( j s the nodal value of the dscrete control functon h at the node x j. We consder now the optmalty system assocated wth the functonal J t wth Drchlet controls that we ntroduced n Secton If R j denotes the algebrac restrcton operator of the velocty unknowns n j to the nterface ( j = 1 2, the algebrac counterpart of (18 (20 reads: S t y t = b t (21 where y t = U I1 U I2 W I1 W I2 1 2 T, b t = F 1 F T the matrx S t s defned as S I1 I S I S I2 I S I2 2 0 S I1 1 R 12 S I1 I 1 0 S I1 1 0 S I2 2 R S I2 I 2 0 S I2 2 0 M 1 R 12 0 M 1 R 12 M 1 0 M 2 R 21 0 M 2 R M 2 For the numercal soluton of the lnear systems (21, we compute the Schur complement system wth respect to the control varables 1 2 solve them through an teratve method lke, e.g., B-CGstab. 20 The Schur complement system reads ( 1 t = t (22 2 where ( M 1 I 1 R 12 SI 1 1 I 1 S I1 1 2 t = M 2 I 2 R 21 SI 1 2 I 2 S I J. Coupled Syst. Multscale Dyn., Vol. 1(3, 1 21

7 Journal of Coupled Systems Multscale Dynamcs t = ( M 1 R 12 I 1 S 1 I 1 I 1 S I1 1 R 12 S 1 I 1 I 1 F 1 M 2 R 21 I 2 S 1 I 2 I 2 S I2 2 R 21 S 1 I 2 I 2 F 2 I s the dentty matrx on the nterface ICDD Method wth Neumann Mxed Controls Let N h denote the space of dscrete Neumann controls on. We requre that N h L2. For = 1 2, gven the control functons h N h, consder the dscrete state problems: fnd u h p h V h Q h, such that, for all v h q h V h Q h, a u h v h +b p h v h = h v h + f v h (23 b q h u h =0 Let T k be a generc element n ; we ntroduce the set E = k meas T k >0, for any k E, the edges e k = T k. Thanks to the defnton of X p h d, for any v h X p h d q h X r h, t holds v h T k C 1 T k d q h T k C 0 T k, then we defne the dscrete normal stress ˆ h = T u h p h n on Ths defnton makes sense n classc way on each e k, so that ˆ h L 2 d. We are nterested n evaluatng the dscrete normal stress assocated wth u h p h also on the nterface j (j = 3, whch s nternal to. Wth ths am we frst restrct u h p h to 12 then extend t to j n such a way that such extenson ũ h p h belongs to V j h Q j h. Then we defne ˆ j h = T ũ h p h n on j t holds ˆ j h L 2 j d. Followng (5 5, the dscrete Neumann controls h on the nterface are obtaned as soluton of the followng mnmzaton problem nf 1 h 2 h [J f 1 h 2 h = 1 2 =1 ] ˆ 1 h ˆ 2 h 2 L 2 (24 In practce, the dscrete normal stresses on the nterfaces are obtaned as resduals of the frst equaton n (23, as we are gong to show. I p Let I u be the sets of ndces of the nodes of the meshes n for the velocty the pressure, respectvely. Moreover, let u I u be the subsets of ndces of the nodes lyng on. We consder matchng meshes on the overlap 12.In X p h d we take the bass B u of the characterstc Lagrange polynomals l wth l I u. Smlarly, n Q h we consder the bass B p of the characterstc Lagrange polynomals k, wth k I p. Now, let u h p h be the soluton of (23. For any l u, we defne the vectors l R d of the weak dscrete normal stresses on assocated wth u h p h as l = a u h l + b p h l f l (25 Smlarly, for any l j u j l B u j we defne the vectors j l d of the weak dscrete normal stresses on j assocated wth u h p h as j l = a j ũ h j l + b j p h j l f j l (26 j It holds j l = ˆ j h j l l u j j 1 2 j To the mnmzaton of problem (24 we can assocate the followng optmalty system: fnd h = 1 h 2 h N h, for = 1 2, u h p h w h q h V h Q h such that a u h l +b p h l = h l =1 b k u h =0 + f l l I u k I p a w h l + b q h l = l j l l I u b k w h = 0 k I p (27 (28 [ ] l j l + j l = 0 l u (29 where j = 3 j l = a j w h j l + b j q h j l s the weak representaton of the dscrete normal stress on j assocated wth the dual state soluton w h q h. An alternatve strategy conssts n choosng mxed controls, e.g., a dscrete Drchlet control 1 h D 1 h on 1 a Neumann control 2 h N 2 h on 2 to mnmze the dfference between both nterface veloctes nterface normal stresses. Followng (6 5 (6 6, the correspondng mnmzaton problems would read: nf [J tf 1 h 2 h = 1 1 h 2 h 2 u 1 h u 2 h 2 L ] 2 ˆ 1 2 h ˆ 2 2 h 2 L 2 2 (30 Research Artcle J. Coupled Syst. Multscale Dyn., Vol. 1(3,

8 Journal of Coupled Systems Multscale Dynamcs Research Artcle Alternatvely, followng (14 5 (14 6, we could consder a dscrete Neumann control on 1 a dscrete Drchlet control on 2 the correspondng mnmzaton problem: nf [J ft 1 h 2 h = 1 1 h 2 h 2 ˆ 1 1 h ˆ 2 1 h 2 L ] 2 u 1 h u 2 h 2 L 2 2 (31 To the mnmzaton problem (30 we assocate the followng optmalty system: fnd h = 1 h 2 h D 1 h N 2 h, for = 1 2, u h p h V h Q h, w q V h Q h such that a 1 u 1 h 1 l +b 1 p 1 h 1 l = 1 f 1 l l I u 1 b 1 1 k u 1 h =0 k I p 1 u 1 h = 1 h on 1 (32 a 2 u 2 h 2 l +b 2 p 2 h 2 l = 2 2 h 2 l + 2 f 2 l l I2 u b 2 2 k u 2 h =0 k I p 2 a 1 w 1 h 1 l + b 1 q 1 h 1 l = 0 l I u 1 (33 b 1 1 k w 1 h = 0 k I p 1 (34 w 1 h = u 1 h u 2 h on 1 a 2 w 2 h 2 l + b 2 q 2 h 2 l = 2 2 l 1 2 l l I2 u b 2 2 k w 2 h = 0 k I p 2 u 1 h 1 l u 2 h 1 l + w 2 h 1 l j j ( j = 0 l u 1 j u 2 (36 To the mnmzaton problem (31 we now assocate the optmalty system: fnd h = 1 h 2 h N 1 h D 2 h, for = 1 2, u h p h V h Q h, w q V h Q h such that a 1 u 1 h 1 l +b 1 p 1 h 1 l = 1 1 h 1 l + 1 f 1 l l I1 u (37 b 1 1 k u 1 h =0 k I p 1 a 2 u 2 h 2 l +b 2 p 2 h 2 l = 2 f 2 l l I u 2 b 2 2 k u 2 h =0 k I p 2 u 2 h = 2 h on 2 a 1 w 1 h 1 l + b 1 q 1 h 1 l = 1 1 l 2 1 l l I1 u (38 (39 a 2 w 2 h 2 l + b 2 q 2 h 2 l = 0 l I u 2 b 2 2 k w 2 h = 0 k I p 2 w 2 h = u 1 h u 2 h on 2 ( j 2 1 j j + u 1 h 2 l u 2 h 2 l + w 1 h 2 l j u 1 l u 2 ( Algebrac Formulaton of ICDD wth Neumann Mxed Controls Usng the prevous notatons, the dscrete values of the Neumann controls are gven by h l = k E e k h l l u Denotng by T j the fnte dmensonal counterpart of the operator that assocates to the velocty pressure n the correspondng normal stress tensor on the nterface j (j = 1 2 (as n (25, after dscretzaton the optmalty system (27 (29 for the functonal J f wth Neumann controls yelds the followng matrx: S I S I 2 0 T 12 S 1 0 I 1 0 (42 T S 2 0 I 2 0 T 12 0 T 12 I 1 0 T 21 0 T I 2 The correspondng Schur complement system becomes ( 1 f = f (43 where f = f = 2 ( I 1 T 12 S I 2 T 21 S2 1 2 ( T12 S1 1 I 1 + T 12 S1 1 F 1 T 21 S2 1 I 2 + T 21 S2 1 F 2 Fnally, the matrx assocated wth the optmalty system (32 (36 for the functonal J tf wth mxed controls s: S I1 I S I S I 2 0 S I1 1 R 12 S I1 I 1 0 S I1 1 0 T S 2 0 I 2 0 M 1 R 12 0 M 1 R 12 M 1 0 b 1 1 k w 1 h = 0 k I p 1 T 21 0 T I J. Coupled Syst. Multscale Dyn., Vol. 1(3, 1 21

9 Journal of Coupled Systems Multscale Dynamcs Its correspondng Schur complement system becomes ( 1 tf = tf (44 2 Table I. Test case wth analytc soluton. Results for the functonal J t wth Taylor-Hood elements wth respect to dfferent values of h. Fxed overlap wth = 0 2. h #ter nf J t e u 1 where tf = ( M 1 I 1 R 12 SI 1 1 I 1 S I1 1 2 tf = I 2 T 21 S2 1 2 ( M 1 R 12 SI 1 1 I 1 I 1 S I1 1 R 12 SI 1 1 I 1 F 1 T 21 S 1 2 I 2 + T 21 S 1 2 F e e e e e e e e e e e e e e e 05 Table II. Test case wth analytc soluton. Results for the functonal J t wth stablzed p p elements wth respect to dfferent polynomal degrees p. Fxed overlap wth = 0 2. p #ter nf J t e u 1 4. NUMERICAL RESULTS 4.1. Test Cases wth Respect to an Analytc Soluton We consder the doman = wth 1 = /2 2 2 = /2, >0 beng a sutable parameter characterzng the wdth of the overlappng regon. The vscosty s set to 1, whle the force f the boundary condtons are chosen such that the Stokes problem admts the soluton u = exp y exp x T p = exp x sn y. Concernng the boundary condtons, we mpose Neumann condtons on the boundary whle Drchlet boundary condtons are mposed on the remanng boundares. We compute the soluton of the optmalty system usng the B-CGStab method on the Schur complement (22 settng the tolerance to Frst, we consder the case of an overlap wth fxed wdth = 0 2. We use both Taylor-Hood elements wth three computatonal meshes characterzed by h = , stablzed hp-fem p p. 8 In the latter case, we consder 4 5 quad elements n each subdoman,4 1 elements n 12 each quad element has sdes of length h = 2 2. In Tables I II we report the number of teratons requred to converge, the computed nfmum of the cost functonal J t the errors e u 1 = u 1 u 1 h 2 H u 2 u 2 h 2 H 1 2 1/2 e p 0 = p 1 p 1 h 2 L p 2 p 2 h 2 L 2 2 1/2 e12 0 u = u 1 h u 2 h L 2 12, = p 1 h p 2 h L 2 12, where u h V h p h Q h are the solutons of (18 (20. The number of teratons s ndependent of both the grd sze h the polynomal degree p. Notce that the convergence order for the errors e1 u ep 0 n Table I agrees wth the expected optmal accuracy for the Taylor-Hood elements. Next, we study the case where the wdth of the overlap tends to zero on a fxed computatonal mesh. When usng the Taylor-Hood elements, we set h = 0 04 = 5h h; the subdomans are defned e e e e e e e e e e e e e e e e e e e e 08 as follows: for = 5h, 1 = = ; for = 4h, 1 = = ; for = 3h, 1 = = ; for = 2h, 1 = = ; for = h, 1 = = For stablzed p p approxmatons, we take p = 4 we partton each subdoman n 4 5 quad elements; \ 12 s parttoned nto 4 4 equal quad elements of sze h x h y, h x = 0 25 h y = 1 /2 /4; 12 s parttoned n 1 5 quads of sze h x ; the value of ranges from 0.2 to Results reported n Tables III IV show that the requred number of teratons ncreases when decreases. Fnally, we carry out a convergence test wth Taylor- Hood elements settng = h lettng h 0. Also n Table III. Test case wth analytc soluton. Results for the functonal J t wth Taylor-Hood elements wth h = #ter nf J t e u 1 5h e e e e e 05 4h e e e e e 05 3h e e e e e 05 2h e e e e e 04 h e e e e e 04 Table IV. Test case wth analytc soluton. Results for the functonal J t wth stablzed p p elements wth respect to dfferent polynomal degrees p for 0. By we denote that the method dd not converge wthn 250 teratons. #ter nf J t e u e e e e e e e e e e e e e e e e e e e e e e e e e+00 Research Artcle J. Coupled Syst. Multscale Dyn., Vol. 1(3,

10 Journal of Coupled Systems Multscale Dynamcs Table V. Test case wth analytc soluton. Results for the functonal J t wth Taylor-Hood elements wth = h 0. = h #ter nf J t e u 1 1/ e e e e e 02 1/ e e e e e 03 2/ e e e e e 04 1/ e e e e e 04 Table VIII. Test case wth analytc soluton. Results for the functonal J tf wth Taylor-Hood elements wth respect to dfferent values of h. Fxed overlap wth = 0 2. h #ter nf J tf e u e e e e e e e e e e e e e e e 05 Research Artcle ths case we can see that the number of teratons requred to converge grows when h decreases. Results are reported n Table V. These numercal results show that the ICDD method s not very effectve especally when consderng small overlappng regons. Ths behavor may due to the fact that the functonal J t nvolves no nformaton on the pressure felds n the overlap, snce t mposes only the contnuty of veloctes on the nterfaces. The number of teratons s ndependent of the mesh sze h of the polynomal degree p. However, a dependence on the sze of the overlap can be estmated as #ter C 1 for a sutable postve constant C>0. We consder now the case of Neumann mxed controls. Frst, we consder the case of an overlap wth fxed wdth = 0 2. The settng the dscretzaton are the same used before. In Tables VI VII we report the number of teratons the computed errors for the case of the functonal J f usng Taylor-Hood stablzed p p approxmatons, respectvely, whle n Tables VIII IX we report the results obtaned for the functonal J tf. Then, we consder the case where the wdth of the overlap tends to zero on a fxed computatonal mesh. Results Table VI. Test case wth analytc soluton. Results for the functonal J f wth Taylor-Hood elements wth respect to dfferent values of h. Fxed overlap wth = 0 2. h #ter nf J f e u e e e e e e e e e e e e e e e 05 Table IX. Test case wth analytc soluton. Results for the functonal J tf wth stablzed p p elements wth respect to dfferent polynomal degrees p. Fxed overlap wth = 0 2. p #ter nf J tf e u e e e e e e e e e e e e e e e e e e e e 09 are shown n Tables X XII for the Taylor-Hood elements wth h = 0 04 n Tables XI XIII for the stablzed p p elements wth p = 4. Both functonals J f J tf are used. Table X. Test case wth analytc soluton. Results for the functonal J f wth Taylor-Hood elements wth h = #ter nf J f e u 1 5h e e e e e 05 4h e e e e e 05 3h e e e e e 05 2h e e e e e 05 h e e e e e 06 Table XI. Test case wth analytc soluton. Results for the functonal J f wth stablzed p p elements wth respect to dfferent polynomal degrees p for 0. By we denote that the method dd not converge wthn 250 teratons. #ter nf J f e u e e e e e e e e e e e e e e e e e e e e e e e e e 03 Table VII. Test case wth analytc soluton. Results for the functonal J f wth stablzed p p elements wth respect to dfferent polynomal degrees p. Fxed overlap wth = 0 2. p #ter nf J f e u e e e e e e e e e e e e e e e e e e e e 09 Table XII. Test case wth analytc soluton. Results for the functonal J tf wth Taylor-Hood elements wth h = #ter nf J tf e u 1 5h e e e e e 05 4h e e e e e 05 3h e e e e e 05 2h e e e e e 05 h e e e e e J. Coupled Syst. Multscale Dyn., Vol. 1(3, 1 21

11 Journal of Coupled Systems Multscale Dynamcs Table XIII. Test case wth analytc soluton. Results for the functonal J tf wth stablzed p p elements wth respect to dfferent polynomal degrees p for 0. l 1 #ter nf J tf e u 1 l 10 Ω 1 l e e e e e e e e e e e e e e e e e e e e e e e e e 08 l 9 Γ 2 Ω 12 l 6 Γ 1 Fnally, we study the behavor of the ICDD method wth functonals J f J tf usng Taylor-Hood elements settng = h lettng h 0. Results are reported n Tables XIV XV. Dfferently from the case of Drchlet controls wth functonal J t, we can see that both functonals J f J tf requre a much lower number of teratons to converge. Ths shows that controllng the pressure not only the velocty on the nterfaces s crucal for the Stokes problem. Moreover, we can see that the best convergence results are obtaned wth mxed controls functonal J tf :as a matter of fact, n ths case the number of teratons s ndependent from the mesh sze h, from the degree p of polynomal used, from the measure of the overlap. Neumann controls wth functonal J f also provde a number of teratons ndependent of the mesh sze h of the polynomal degree p. However, a dependence on the sze of the overlap can be notced as #ter C 1/2 for a sutable postve constant C> A Test Case Wthout Analytc Soluton We consder the computatonal doman = wth 1 = /2 2 2 = /2, as represented schematcally n Fgure 2. The force Table XIV. Test case wth analytc soluton. Results for the functonal J f wth Taylor-Hood elements wth = h 0. = h #ter nf J f e u 1 1/ e e e e e 03 1/ e e e e e 04 2/ e e e e e 05 1/ e e e e e 06 Table XV. Test case wth analytc soluton. Results for the functonal J tf wth Taylor-Hood elements wth = h 0. = h #ter nf J tf e u 1 1/ e e e e e 03 1/ e e e e e 04 2/ e e e e e 04 1/ e e e e e 05 Fg. 2. l 8 l 5 Ω 2 l 4 Schematc representaton of the computatonal doman. s set to f = 0 the vscosty s = 2 e 3. We mpose homogeneous Neumann boundary condtons for the flud normal stress on the edges l 4 l 7. On the remanng boundares, apart from the edge l 6, we mpose homogeneous Drchlet boundary condtons for the flud velocty unless on where we set a parabolc profle wth maxmum equal to 1. On the edge l 6 we may mpose ether homogeneous Neumann or Drchlet boundary condtons to compare the behavor of the dfferent methods that we have studed. In partcular, we want to show that the functonal J t wth Drchlet controls wll not provde a correct soluton when l 6 s set as a Drchlet boundary, snce ths case volates Assumpton 2.1. For ths problem, besdes the errors e12 0 u ep 12 0 on the overlap, we also compute e u 1 = U 1 h u 1 h 2 H U 2 h u 2 h 2 H 1 2 1/2 e p 0 = P 1 h p 1 h 2 L P 2 h p 2 h 2 L 2 2 1/2 where U h P h s the restrcton to the subdoman of the soluton computed on the same mesh but consderng the doman as a whole wthout any splttng solvng (1. Frst, we mpose homogeneous Drchlet boundary condtons on l 6. The results obtaned n correspondence of the dfferent functonals J t, J f J tf are reported n Table XVI for Taylor-Hood elements n table XVII for stablzed p p elements wth p = 6. As expected, the mnmzaton of the functonal J t does not allow to recover the correct soluton, whereas both J f J tf converge to the correct soluton. In Fgure 3 we show the orgnal soluton, whle n Fgures 4 5 we show, respectvely, the solutons obtaned through mnmzaton of the functonal J t J tf. We can see that the functonal J t has no control on the pressure, whch therefore does not match on the overlap. Now, we mpose homogeneous Neumann boundary condtons on l 6. In ths case, accordng to the theory, all Research Artcle J. Coupled Syst. Multscale Dyn., Vol. 1(3,

12 Journal of Coupled Systems Multscale Dynamcs Table XVI. Test case wthout analytc soluton. Drchlet boundary condton on l 6. Results for the functonals J t (top, J f (md J tf (bottom wth Taylor-Hood elements wth fxed h = #ter nf J t e u 1 5h e e e e e 04 4h e e e e e 03 3h e e e e e 03 2h e e e e e 04 h e e e e e 03 #ter nf J f e u 1 5h e e e e e 04 4h e e e e e 05 3h e e e e e 05 2h e e e e e 05 h e e e e e 05 #ter nf J tf e u 1 Research Artcle 5h e e e e e 04 4h e e e e e 05 3h e e e e e 05 2h e e e e e 05 h e e e e e 04 functonals allow to correctly compute the sngle-doman soluton ther behavors are smlar to those observed n the prevous tests wth analytc soluton. The functonal J tf assocated wth mxed controls s the one that converges n the lowest number of teratons wth a slght dependence on. Results are reported n Table XVIII for Taylor-Hood elements n Table XIX for stablzed p p elements wth p = 6. In Fgure 6 we show the sngle-doman soluton, whle n Fgures 7 8 we show, respectvely, the solutons Table XVII. Test case wthout analytc soluton. Drchlet boundary condton on l 6. Results for the functonals J t (top, J f (md J tf (bottom wth stablzed p p elements wth fxed p = 6 0. #ter nf J t e u e e e e e e e e e e e e e e e e e e e e e e e e e 03 #ter nf J f e u e e e e e e e e e e e e e e e e e e e e e e e e e 05 #ter nf J tf e u e e e e e e e e e e e e e e e e e e e e e e e e e 05 Fg. 3. Test case wthout analytc soluton. Drchlet boundary condton on l 6. Reference monodoman soluton computed usng Taylor-Hood fnte elements. obtaned through mnmzaton of the functonal J t J tf. We can see that, although the functonal J t has no control on the pressure, the Neumann boundary condton on the edge l 6 allows the pressure to match almost perfectly n 12 J. Coupled Syst. Multscale Dyn., Vol. 1(3, 1 21

13 Journal of Coupled Systems Multscale Dynamcs Research Artcle Fg. 4. Test case wthout analytc soluton. Drchlet boundary condton on l 6. Soluton computed by mnmzng the functonal J t usng Taylor- Hood fnte elements. the overlappng regon. Notce that the dfference shown n Fgure 7 s of the same order of the errors reported n Tables XVIII XIX. Fnally, let us consder a test case n whch the nterface s a pecewse lnear curve (dentfed by element edges, as shown n Fgure 9. We compute the soluton by mposng a Neumann boundary condton on the bound- Fg. 5. Test case wthout analytc soluton. Drchlet boundary condton on l 6. Soluton computed by mnmzng the functonal J tf usng Taylor- Hood fnte elements. ary l 6 consderng stablzed p p elements wth p = 6 0. The teratons numbers shown n Table XX behave smlarly to those presented n the thrd block of Table XIX: the algorthm s not strongly nfluenced by the shape of the nterface. J. Coupled Syst. Multscale Dyn., Vol. 1(3,

14 Journal of Coupled Systems Multscale Dynamcs Table XVIII. Test case wthout analytc soluton. Neumann boundary condton on l 6. Results for the functonals J t (top, J f (md J tf (bottom wth Taylor-Hood elements wth fxed h = #ter nf J t e u 1 5h e e e e e 04 4h e e e e e 04 3h e e e e e 04 2h e e e e e 05 h e e e e e 04 #ter nf J f e u 1 5h e e e e e 05 4h e e e e e 05 3h e e e e e 05 2h e e e e e 05 h e e e e e 05 #ter nf J tf e u 1 Research Artcle 5h e e e e e 05 4h e e e e e 05 3h e e e e e 05 2h e e e e e 05 h e e e e e 05 To assess the robustness of the method wth respect to the vscosty coeffcent, we compute the soluton of the problem wth Neumann boundary condton on l 6 usng the ICDD method assocated wth the functonal J tf, the one that provded the best results n the prevous tests. We consder a dscretzaton by Taylor-Hood elements on a mesh wth fxed h = we set the vscosty = Numercal results are reported n Tables XXI, XXII; clearly they show that the method s robust wth respect to varatons of the parameter. Table XIX. Test case wthout analytc soluton. Neumann boundary condton on l 6. Results for the functonals J t (top, J f (md J tf (bottom wth stablzed p p elements wth fxed p = 6 0. By we denote that the method dd not converge wthn 250 teratons. #ter nf J t e u e e e e e e e e e e e e e e e e e e e e e e e e e 03 #ter nf J f e u e e e e e e e e e e e e e e e e e e e e e e e e e 06 #ter nf J tf e u e e e e e e e e e e e e e e e e e e e e e e e e e 05 Fg. 6. Test case wthout analytc soluton. Neumann boundary condton on l 6. Reference monodoman soluton computed usng Taylor-Hood fnte elements. 5. ANALYSIS OF THE ICDD METHOD FOR THE STOKES PROBLEM In ths secton we analyze the ICDD method that we have presented n the prevous sectons wth the am of guaranteeng the well-posedness of the mnmzaton problem. We begn wth the analyss n the contnuous case wth Drchlet controls J. Coupled Syst. Multscale Dyn., Vol. 1(3, 1 21

15 Journal of Coupled Systems Multscale Dynamcs Research Artcle Fg. 7. Test case wthout analytc soluton. Neumann boundary condton on l 6. Soluton computed by mnmzng the functonal J t usng Taylor-Hood fnte elements Analyss of the Optmal Control Problem wth Drchlet Controls For = 1 2, we ntroduce the followng spaces: = H 1/2 d v H 1 d (45 v = on v = 0 on D (46 Fg. 8. Test case wthout analytc soluton. Neumann boundary condton on l 6. Soluton computed by mnmzng the functonal J tf usng Taylor-Hood fnte elements. 0 = n = 0 (47 We wll denote by f N = 1 2 D = (48 0 f N = J. Coupled Syst. Multscale Dyn., Vol. 1(3,

16 Journal of Coupled Systems Multscale Dynamcs 2 Table XXI. Test case wthout analytc soluton. Neumann boundary condton on l 6. Results obtaned for the functonal J tf wth Taylor-Hood elements wth fxed h = The vscosty s = 10 2 (top, = 10 4 (md, = 10 6 (bottom. = #ter nf J tf e u 1 5h e e e e e 04 4h e e e e e 04 3h e e e e e 04 2h e e e e e 05 h e e e e e 04 = 10 4 #ter nf J tf e u 1 5h e e e e e 06 4h e e e e e 06 3h e e e e e 06 2h e e e e e 07 h e e e e e 06 Research Artcle Fg. 9. Computatonal mesh for stablzed p p elements n the case of pecewse lnear nterfaces. In the fgure = the spaces of admssble Drchlet controls. Moreover, we wll denote D = D 1 D 2 (49 For = 1 2, we consder two unknown control functons D the assocated state problems dv T u f dv u f u f p f = f n = 0n = on (50 wth sutable homogeneous boundary condtons on \.If N =, we add the constrant p f = 0. The unknown controls on the nterface are obtaned by solvng the mnmzaton problem [ nf J t = 1 ] u 1 f = 1 2 D 1 u 2 f 2 2 L 2 2 (51 =1 where, for the sake of smplcty, we adopt the same notaton as n the dscrete case. Table XX. Test case wthout analytc soluton. Pecewse lnear nterfaces. Neumann boundary condton on l 6. Results for the functonals J tf wth stablzed p p elements wth fxed p = 6 0. = 10 6 #ter nf J tf e u 1 5h e e e e e 08 4h e e e e e 08 3h e e e e e 08 2h e e e e e 09 h e e e e e 08 Table XXII. Test case wthout analytc soluton. Neumann boundary condton on l 6. Results obtaned for the functonal J tf wth stablzed p p elements wth fxed p = 6 0. The vscosty s = 10 2 (top, = 10 4 (md, = 10 6 (bottom. = 10 2 #ter nf J tf e u e e e e e e e e e e e e e e e e e e e e e e e e e 04 = 10 4 #ter nf J tf e u e e e e e e e e e e e e e e e e e e e e e e e e e 06 = 10 6 #ter nf J tf e u e e e e e e e e e e e e e e e e e e e e e e e e e 05 #ter nf J tf e u e e e e e e e e e e e e e e e e e e e e e e e e e J. Coupled Syst. Multscale Dyn., Vol. 1(3, 1 21