Weierstraß-Institut. für Angewandte Analysis und Stochastik. Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN

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1 Weierstraß-Institut ür Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN On iterative subdomain methods or the Stokes Darcy problem Alonso Caiazzo 1 Volker John 12 Ulrich Wilbrandt 2 submitted: July Weierstrass Institute Mohrenstr Berlin Germany Alonso.Caiazzo@wias-berlin.de Volker.John@wias-berlin.de 2 Free University o Berlin Department o Mathematics and Computer Science Arnimallee Berlin Germany uli-wilbrandt@gmx.de No Berlin 2013 Key words and phrases. Stokes Darcy problem iterative subdomain methods Robin boundary conditions inite element methods continuous and discontinuous updating strategy applications rom geosciences. The research o Ulrich Wilbrandt has been supported by the Helmholtz graduate research school GeoSim.

2 Edited by Weierstraß-Institut ür Angewandte Analysis und Stochastik (WIAS) Leibniz-Institut im Forschungsverbund Berlin e. V. Mohrenstraße Berlin Germany Fax: World Wide Web:

3 Abstract Iterative subdomain methods or the Stokes Darcy problem that use Robin boundary conditions on the interace are reviewed. Their common underlying structure and their main dierences are identiied. In particular it is clariied that there are dierent updating strategies or the interace conditions. For small values o luid viscosity and hydraulic permeability which are relevant in applications rom geosciences it is shown in numerical studies that only one o these updating strategies leads to an eicient numerical method i this strategy is used in combination with appropriate parameters in the Robin boundary conditions. In particular it is observed that the values o appropriate parameters are larger than those proposed so ar. Not only the size but also the ratio o appropriate Robin parameters depends on the coeicients o the problem. 1. INTRODUCTION Let us consider a bounded domain Ω R d d {2 3} and a decomposition Ω = Ω Γ I Ω p into two disjoint subdomains Ω and Ω p denoting a ree low domain and a porous medium respectively and possessing a common interace Γ I i.e. Ω Ω p = and Ω Ω p = Γ I see Figure 1. Assuming a moderate low velocity in the ree low domain the luid dynamics in Ω can be modeled with the incompressible Stokes equations or the velocity u : Ω R d [m/s] and the pressure P : Ω R [P a] ( T u P ) = in Ω (1) ρ u = 0 in Ω. (2) In (1) ρ [kg/m 3 ] represents the luid density and T(u p ) [m 2 /s 2 ] denotes the luid stress tensor T(u p ) : Ω R d d T = 2νD(u ) p I 1

4 2 A. CAIAZZO V. JOHN. U. WILBRANDT Figure 1. Sketch o the luid domain or the Stokes Darcy problems. where D(u ) = ( u+ u T )/2 : Ω R d d [1/s] is the velocity deormation tensor p = P /ρ [m 2 /s 2 ] and ν : Ω R [m 2 /s] denotes the kinematic viscosity o the luid. Outer orces acting on the ree low are modeled by : Ω R d [m/s 2 ]. The dynamics in the porous medium is described by the Darcy law K ϕ p +u p = 0 in Ω p (3) u p = p in Ω p (4) in terms o the unction ϕ p : Ω p R [m] called Darcy pressure or piezometric head and o the Darcy velocity u p : Ω p R d [m/s]. In (3) K : Ω p R d d [m/s] is the hydraulic conductivity tensor. Generally it will be assumed that K = K T K > 0. Here only the case K = KI K > 0 will be considered. In (4) the unction p : Ω p R [1/s] describes sinks and sources. System (3) (4) is called the mixed orm o the Darcy problem. An alternative ormulation is obtained by taking the divergence o equation (3) and using (4) (K ϕ p ) = p in Ω p (5) which is reerred to as the primal orm o the Darcy problem. The mixed orm is oten more important or applications as it allows to recover the Darcy velocity u p directly as an unknown o the problem. However in what ollows we will ocus on the primal ormulation which is generally simpler rom the analytical and numerical point o view. Having computed ϕ p with the primal ormulation the velocity u p can be computed in a post-processing step using (3). In the context o inite element methods the application o (3) results in loosing one approximation order such that the approximation estimate or u p is worse than or ϕ p. On the other hand velocity post-processing techniques based on gradient super convergence phenomena have been successully used to enorce mass conservation and improve global accuracy see e.g. [1]. Our motivation or considering the primal ormulation lies in the act that or the studied numerical methods there are already open questions or this ormulation. These questions should be studied beore proceeding to the more complicated mixed ormulation. Equations (1) (2) and (5) must be completed with appropriate boundary conditions and in particular with proper interace conditions at the Stokes Darcy interace Γ I. Denote by n and n p the unit outward normal vectors on Ω and Ω p respectively and by τ i i = 1...d 1 pairwise orthogonal unit tangential vectors on the interace Γ I. Note that n = n p on Γ I. Two standard coupling conditions on Γ I model the conservation o mass and the balance o normal stresses u n = K ϕ p n on Γ I (6) n T(u p ) n = gϕ p on Γ I (7) where g [m/s 2 ] is the gravitational acceleration. A classical third coupling condition is the so-called Beavers Joseph condition [2] which is based on experimental indings relating the jump o the tangential velocity along the interace to the luid stresses: (u u p ) τ i +ατ i T(u p ) n = 0 on Γ I i = 1...d 1 (8)

5 ON ITERATIVE SUBDOMAIN METHODS FOR THE STOKES DARCY PROBLEM 3 where α = α 0 τi K τ i νg [s/m] and α 0 > 0 is a non-dimensional parameter depending on the properties o the porous medium which must be experimentally determined. The well-posedness o the Beavers Joseph interace condition (8) or the steady-state Stokes Darcy problem is established only or particular values o α 0 see e.g. [3 4 5]. Based on the observation that the Darcy velocity on Γ I is oten negligible compared with the Stokes velocity it was proposed [6 7] to simpliy (8) to u τ i +ατ i T(u p ) n = 0 on Γ I i = 1...d 1 (9) which is called Beavers Joseph Saman condition (or Beavers Joseph Saman Jones condition). A urther simpliication is based on the observation that both terms on the lethand side o (9) are small hence obtaining u τ i = 0 on Γ I i = 1...d 1. (10) The coupled problem deined by (1) (2) (5) with the conditions (6) (7) and (9) (or (10)) has been studied extensively in the literature both theoretically see among others [ ] and numerically e.g. [ ]. This report will consider the interace conditions(9) and(10) and dierent numerical methods to solve the resulting coupled problems in a inite element ramework. Since (9) and (10) do not couple the Stokes and the Darcy equations they yield ormulations simpler to analyze and to treat numerically than (8) see e.g. [10 16]. Nonetheless several questions are still open. The numerical methods or solving the coupled problem might be divided into two main classes. Direct (also called monolithic or single domain) methods aim at solving in a single step the whole coupled system. Proposals o inite element based approaches to solve the strongly coupled Stokes Darcy problem can be ound among others in [ ]. These techniques are in general stable and robust. However they might be computationally costly depending on the size o the problem and on the physics behind it. Preconditioned direct methods based on solving the problems on dierent grids are proposed in [24 25]. As an alternative decoupled(also called domain-decomposition or multidomain) approaches solve the coupled problem with a subdomain iterative procedure based at each iteration on the solution o the Stokes and Darcy problem separately. On the one hand an iterative method requires multiple solutions o the subproblems. On the other hand the advantage o these techniques is that they allow to use specialized solvers or Stokes and Darcy problems and to tailor the algorithms according to their mathematical and physical properties which might result in eicient procedures. Furthermore iterative approaches are generally preerred i eicient solvers or the subproblems are available. Our motivation or studying the coupled Stokes Darcy problem comes rom the numerical simulation o the luid dynamics in water basins and river beds. The discretization o these problems is characterized by complex geometries and might lead to large and ill-conditioned systems in three dimensions. For this reason and also since an eicient code or the simulation o incompressible ree lows and scalar elliptic problems is available [26] the use o iterative strategies is our preerred option. Surveying the literature on iterative methods one inds that depending on the splitting o the interace conditions dierent strategies were proposed. This splitting deines the boundary condition on Γ I or the individual Stokes and Darcy problems that have to be solved at each iteration. Principally it is possible to perorm the splitting such that essential natural or Robin (linear combination o essential and natural) boundary conditions on Γ I can be prescribed. A irst approach consists in using natural boundary conditions or the Stokes problems i.e. (7) (9) or (7) (10) which involve Dirichlet data rom the Darcy problem and also natural conditions or the Darcy problems i.e. (6) which use Dirichlet data rom the Stokes problems e.g. [27 12]. Since this strategy results in Neumann problems to be solved in both subdomains this approach will be called Neumann Neumann coupled ormulation. It is

6 4 A. CAIAZZO V. JOHN. U. WILBRANDT worth noting that rom the point o view o domain decomposition methods this strategy is oten considered as o Dirichlet-to-Neumann type [27 12]. However the Neumann Neumann coupling possesses some disadvantages. In particular it has been shown that it converges very slowly on ine meshes or small values o the viscosity or the hydraulic conductivity [12] a coniguration which is o utmost importance or applications in geosciences where e.g. ν = 10 6 [m 2 /s] (water) and K [10 9 I10 3 I] [m/s] (clay sand gravel) are values o interest. To overcome these problems it was irst proposed in[12] to use a Robin Robin coupling. The numerical results in [12] are very promising showing that the Robin Robin decomposition is robust and eicient in the parameter range that is o interest or geoscientiic applications. The idea behind the Robin Robin coupling or Stokes Darcy problems is to solve Robin problems rather than Neumann problems in both subdomains introducing two parameters γ and γ p which determine the linear combination o the essential and natural boundary condition or Stokes and Darcy problems respectively. As shown in [12] these additional parameters have to be tuned in order to achieve ast convergence o the iterative scheme. Moreover recent works showed that Robin boundary conditions can be seen in the context o the inite element method as a generalization o Neumann and Dirichlet conditions without signiicant loss o accuracy and stability see e.g. [28]. Ater the pioneering work o [12] more variants o Robin Robin iterative methods have been proposed and analyzed see e.g. [4 9 16]. For motivating our work the discussion o some details o the algorithms proposed in [12] and [16] is necessary. The sequential algorithm rom [12] solves at each iteration a Darcy and a Stokes problem sequentially i.e. updating ater each step the boundary conditions on Γ I using the latest solution available on the other subdomain. This approach corresponds to a Gauss Seidel strategy or the solution o the coupled problem. The presented numerical analysis in [12] suggests that the Robin Robin parameters should satisy γ > γ p at least or ν K 1. The numerical studies considered the choice γ = 0.3 and γ p = 0.1 showing robustness with respect to grid size viscosity and permeability but restricted to the simpliied interace condition (10) and ν < 10 4 K < In [9 16] a very similar algorithm was analyzed where the Stokes and the Darcy problem are solved simultaneously in each iteration updating simultaneously also the boundary condition on Γ I or both subproblems. This approach called parallel algorithm in [16] is the canonical modiication o the sequential (Gauss Seidel) algorithm proposed in [12] to the Jacobi-type. In act one has to solve the same subproblems and the updates o the Robin boundary data at the interace have the same structure see Section 3. The only dierence between both algorithms is that in the deinition o the Robin data or the Stokes problem the solution o the Darcy problem is either rom the current or rom the previous iterate. In [16] the convergence was proved theoretically or γ < γ p or the Beaver Joseph Saman condition (9) and any initial iterate. However the numerical studies have been restricted to the value γ p = 1 considering also unitary viscosity and permeability ν = 1 K = 1. Hence although the algorithms proposed in [12] and [16] dier only slightly the analysis and the results presented in these papers suggest completely dierent choices o appropriate Robin Robin parameters. This puzzling situation is o course unsatisactory and it makes the usage o Robin Robin coupling prohibitive in practical applications. In particular several questions remain to be addressed concerning the dependence o the numerical parameters on the coeicients o the problem. One o the main contributions o this report consists in a uniied presentation o the considered algorithms thereby identiying the common underlying structure and highlighting their dierences. In particular it will be clariied that dierent updating strategies or the Robin conditions on the interace can be applied. One o them is based on rewriting the Neumann Neumann ormulation as a Robin Robin ormulation whereas the other one directly uses a Robin Robin ormulation o the coupled problem. The irst approach leads or standard inite element methods to continuous data at the interace and the second one to discontinuous data. Another main contribution is the assessment o the algorithms or problems whose

7 ON ITERATIVE SUBDOMAIN METHODS FOR THE STOKES DARCY PROBLEM 5 coeicients are o the magnitude which is relevant or applications rom geosciences. It will turn out that only one o the updating strategies will lead to eicient methods i it is used with appropriately chosen parameters in the Robin boundary conditions. The size o appropriate parameters is considerably larger at least in the studied examples than proposed so ar in the literature. The numerical studies also show that not only the size but also the ratio o appropriately chosen Robin parameters depends on the coeicients o the problem. The remainder o the report is organized as ollows. Section 2 introduces the variational and the inite element ormulations o the coupled problems while Section 3 ocuses on the two iterative strategies considered in the report. Computational results are presented in Section 4 and Section 5 gives a summary and an outlook. 2. WEAK FORMULATION OF THE COUPLED STOKES DARCY PROBLEM The scope o this section is to introduce the basic notation and the weak ormulations o the coupled Stokes Darcy problem considering dierent strategies to include the interace conditions Weak ormulations The irst step o the numerical simulation o any physical model is its non-dimensionalization using characteristic scales. It will be assumed without loss o generality that the equations (1) (2) and (5) and the interace conditions (6) (7) and (9) (or (10)) are non-dimensionalized with unity characteristic scales such that the non-dimensional equations have exactly the same orm. Moreover as usual the Stokes pressure is deined such that it incorporates the density. For simplicity o presentation our notation will not distinguish between dimensional and nondimensional quantities. Besides interace conditions on Γ I also boundary conditions on the external boundaries ( Ω Ω p )\Γ I have to be speciied. In what ollows only homogeneous conditions will be considered. However the presented arguments can be extended also to the general nonhomogeneous case. For the Stokes problem it is assumed that u = 0 on Γ e (no-low essential boundary conditions) T(u p ) n = 0 on Γ n (outlow natural boundary conditions) where Γ e Γ n = Ω \Γ I and Γ e Γ n =. For the Darcy problem the boundary conditions are deined by ϕ p = 0 on Γ pe (essential/dirichlet) K ϕ p n = 0 on Γ pn (natural/neumann) with Γ pe Γ pn = Ω p \Γ I and Γ pe Γ pn =. Furthermore the boundary parts where essential boundary conditions are prescribed are assumed to have positive measure Γ e > 0 Γ pe > 0. For an open set ω R d let H m (ω) denote the standard Sobolev spaces and let L 2 (ω) = H 0 (ω). In addition ( ) ω denotes the L 2 (ω) inner product and ΓI stands or the inner product in L 2 (Γ I ). For the derivation o a weak ormulation the essential boundary conditions are incorporated into the unction spaces. Hence the space or the Stokes velocity is given by V = {v ( H 1 (Ω ) ) } d : v = 0 on Γe. The correct choice o the pressure space depends on the boundary conditions. I Γ n = then the pressure is sought in the space { } Q = L 2 0(Ω ) = q L 2 (Ω ) : q dx = 0 Ω

8 6 A. CAIAZZO V. JOHN. U. WILBRANDT otherwise in Q = L 2 (Ω ). For the Darcy problem the space or the piezometric head is deined by Q p = { ψ H 1 (Ω p ) : ψ = 0 on Γ pe }. The weak ormulations o the Stokes and the Darcy equations are derived in a standard way by multiplying (1) (2) and (5) with appropriate test unctions and applying integration by parts. Due to the choice o the boundary conditions all integrals on ( Ω Ω p )\Γ I vanish. One obtains the ollowing weak orm o the Stokes Darcy problem: Find (u p ϕ p ) V Q Q p such that or all (vqψ) V Q Q p (T(u p )D(v)) Ω +( u q) Ω T(u p ) n v ΓI = ( v) Ω (11) (K ϕ p ψ) Ωp + K ϕ p n ψ ΓI = ( p ψ) Ωp. (12) The Beavers Joseph Saman condition (9) can be naturally included into the weak ormulation (11) decomposing the integral over Γ I in (11) into normal and tangential components yielding d 1 T(u p ) n v ΓI = n T(u p ) n v n ΓI τ i T(u p ) n v τ i ΓI. }{{} i=1 = 1 α u τ iv τ i ΓI Rearranging terms leads to a (u v)+b (vp ) b (u q) n T(u p ) n v n ΓI = ( v) Ω (13) a p (ϕ p ψ)+ K ϕ p n ψ ΓI = ( p ψ) Ωp (14) with a : V V R b : V Q R and a p : Q p Q p R deined by d 1 1 a (uv) = (2νD(u)D(v)) Ω + α u τ iv τ i ΓI b (vp) = ( vp) Ω a p (ϕψ) = (K ϕ ψ) Ωp. In the case o the simpliied interace condition (10) one obtains equations o the same orm as (13) (14) with the ollowing bilinear orm d 1 a (uv) = (2νD(u)D(v)) Ω τ i T(u p ) n v τ i ΓI where the second term does not vanish in the case o weakly imposed boundary conditions see Section Neumann Neumann coupled ormulation For completeness o presentation and or highlighting the dierences to the Robin Robin ormulation irst the Neumann Neumann coupled ormulation is reviewed. Inserting the interace conditions (7) and (6) into (13) (14) yields the coupled Stokes Darcy weak ormulation i=1 i=1 a (u v)+b (vp ) b (u q)+ gϕ p v n ΓI = ( v) Ω (15) a p (ϕ p ψ) u n ψ ΓI = ( p ψ) Ωp. (16) Here the coupling is based on Neumann interace conditions or both the Stokes and the Darcy domain and (15) (16) will be thereore called a Neumann Neumann coupled problem.

9 ON ITERATIVE SUBDOMAIN METHODS FOR THE STOKES DARCY PROBLEM 7 In order to describe better the dierent coupled ormulations arising rom the Stokes Darcy problem it is helpul to rewrite (15) (16) in a more abstract orm. Let V = (V Q ) be the unction space in which the solution (velocity and pressure) to the Stokes problem is sought and let V denote its dual space. Similarly let Q p denote the dual o Q p. Then deine the operators S : V V and D : Q p Q p such that (Dϕψ) := a p (ϕψ) (Sst) := a (uv)+b (vp) b (uq) or all s = (up) t = (vq) V and ϕ ψ Q p and the coupling operator C : V Q p by (Csψ) := u n ψ ΓI. Then the Neumann Neumann coupled problem (15) (16) can be equivalently written as: Find ϕ Q p and s = (up) V such that ( )( ) ( ) D C ϕ Fp gc =. (17) S s F The operator C : Q p C is the adjoint o C and the right-hand sides F V and F p Q p are deined by (F p ψ) := ( p ψ) Ωp (F t) := ( v) Ω or all t = (vq) V and all ψ Q p. An equivalent orm o the coupled Stokes Darcy system can be obtained using Lagrange multipliers on the interace Γ I. This approach allows to ormally decouple the Stokes and the Darcy subproblems introducing additional interace variables η η p H 1/2 (Γ I ) and considering the system D R p I E S R E p ϕ η s I where I is the identity operator on H 1/2 (Γ I ) η p = F p 0 F 0 E : H 1/2 (Γ I ) V (E η t) = η v n ΓI E p : H 1/2 (Γ I ) Q p (E p η p ψ) = η p ψ ΓI Q p H 1/2 (Γ I ) V (18) H 1/2 (Γ I ) correspond to extension operators into the Stokes and Darcy domain respectively and R p : Q p H 1/2 (Γ I ) ψ gψ ΓI R : V H 1/2 (Γ I ) t v ΓI n are restriction operators on H 1/2 (Γ I ) rom the two subdomains. The ormulation (18) is equivalent to (17) since by deinition 2.3. Robin Robin coupled ormulation gc = E R p and C = E p R. Instead o the interace conditions (6) and (7) one can as well consider two linear combinations γ u n +n T(u p ) n = γ K ϕ p n gϕ p on Γ I (19) γ p K ϕ p n gϕ p = γ p u n +n T(u p ) n on Γ I (20)

10 8 A. CAIAZZO V. JOHN. U. WILBRANDT where γ 0 and γ p > 0 are constant. The conditions (19) and (20) correspond to Robin boundary conditions or the Stokes and Darcy subproblems respectively. Inserting them into the weak ormulations (13) (14) leads to a (u v)+b (vp ) b (u q) + γ u n v n ΓI + gϕ p v n ΓI + γ K ϕ p n v n ΓI = ( v) Ω (21) 1 1 a p (ϕ p ψ)+ gϕ p ψ + n T(u p ) n ψ u n ψ γ p Γ I γ ΓI = ( p ψ) Ωp (22) p Γ I which will be denoted as the Robin Robin coupled ormulation. The restrictions γ 0 and γ p > 0 guarantee positiveness o the bilinear orms a (u v)+ γ u n v n ΓI and 1 a p (ϕ p ψ)+ γ p gϕ p ψ. Introducing the operators S ΓI : V V D Γ I : Q p Q p C p : V Γ I Q p and C : Q p V deined by (S ΓI st) = u n v n ΓI (D ΓI ϕψ) = gϕ p ψ ΓI ( C ϕt ) = K ϕ n v n ΓI (C p sψ) = n T(up) n ψ ΓI or all s = (up) V t = (vq) V ϕ ψ Q p one can rewrite (21) (22) as ( )( ) (( ) ( Drob C rob ϕ D C γ 1 := S rob s gc p + S C rob C γ )( DΓI ))( ) C p ϕ = S ΓI s ( Fp F ). (23) The second term represents the additional operators in the Robin Robin problem (21) (22) in comparison with the Neumann Neumann ormulation (15) (16) compare also with (17). Remark 2.1 The resulting coupled weak ormulations (15) (16) and (21) (22) are equivalent i and only i V n ΓI = Q p ΓI. This equality is satisied i Γ e Γ I = Γ pe Γ I i.e. i the types o boundary conditions (essential/natural) imposed on the intersection between interace and outer boundary coincide. Similarly to the Neumann Neumann case one introduces two interace variables η η p H 1/2 (Γ I ) and considers the system D rob R prob I E S rob R rob E p ϕ F p Q p η 0 s = F H 1/2 (Γ I ) V (24) I 0 H 1/2 (Γ I ) where I is the identity operator on H 1/2 (Γ I ). Then one deines or all t = (vq) V and all ψ Q p : R prob : Q p H 1/2 (Γ I ) ψ (gψ +γ K ψ n ) ΓI R rob : V H 1/2 (Γ I ) η p t ( γ 1 p n T(vq) v ) ΓI n. In (24) the operators E p and E are deined on H 1/2 (Γ I ). By construction (24) is equivalent to (23) since C rob = E R prob and C rob = E p R rob.

11 ON ITERATIVE SUBDOMAIN METHODS FOR THE STOKES DARCY PROBLEM A Robin Robin ormulation or the Neumann Neumann problem In the context o inite element methods computing derivatives o discrete unctions in general leads to loss o approximation orders. Hence the Robin Robin ormulation might result in a suboptimal accuracy. On the other hand Robin boundary conditions allow or additional lexibility at the interace and they can give eicient algorithms as reported in [12]. An alternative weak ormulation equivalent to the Neumann Neumann ormulation (17) but basedonthesolutionotworobinproblemsinthesubdomainsω andω p whichdonotrequire the computation o derivatives has been proposed in [12 16] in the context o subdomain iterative methods. Using the ramework introduced above this coupled ormulation can be written in the orm D +γp 1 D ΓI γ 1 br p I ai E S +γ S ΓI ci dr I p E p ϕ η s η p = F p 0 F (25) 0 where the actors a b c and d have to be determined in order to obtain a ormulation equivalent to (17) (and to (18)). The choice o the orm (25) is motivated by two reasons. Firstly it prevents the subproblems to be coupled directly (irst and third row). An indirect coupling is enorced only through the interace unctions η and η p (or the Stokes and Darcy problem respectively). Secondly each interace unction can be computed rom the solution in a single subdomain (empty blocks in the second and ourth row). The signs o the operators and the positions o γ and γ p are chosen such that the resulting scheme will coincide with the ones presented in [12 16]. System (25) is equivalent to (17) i D +γp 1 D ΓI γ 1 p E p ϕ D C br p I ai η E S +γ S ΓI s = C S ci dr I η p γp 1 D ΓI C γ 1 br p I ai C E γ S ΓI ci dr I p E p ϕ F p 0 0 s = F 0 0 ϕ η s = 0 η p which yields the conditions η p = γ p u n +gϕ in Λ p η = bgϕ+aη p in H 1/2 (Γ I ) η = γ u n gϕ in Λ η p = du n +cη in H 1/2 (Γ I ) (26) with the interace (trace) spaces Λ := V n ΓI = {v ΓI n : v V } and Λ p := Q p ΓI = {ψ ΓI : ψ Q p }. Remark 2.2 I the interace spaces Λ and Λ p coincide one can solve (26) or a b c d obtaining a = γ γ p b = 1 a c = 1 d = γ +γ p as done in [16]. This is the same condition as in Remark 2.1. I the interace spaces are not equal we still use these parameters in the numerical simulations.

12 10 A. CAIAZZO V. JOHN. U. WILBRANDT Remark 2.3 The dierence between the two Robin Robin ormulations is that (23) is derived starting rom two Robin subproblems in strong orm i.e. (19) and (20) while (25) is equivalent to two Robin problems obtained rom the Neumann Neumann strong orm (15) (16). Remark 2.4 Unlike the Neumann Neumann approach(18) and the Robin Robin approach(24) the Robin Robin ormulation (26) can only be written in operator orm using Lagrange multipliers as the interace variables η and η p are coupled to each other. 3. ROBIN ROBIN SUBDOMAIN ITERATIVE METHODS In the literature one can ind so-called sequential and parallel Robin Robin subdomain iterative methods. This section starts by presenting the principal orm o these algorithms. Then the important issue o the deinition o the boundary data at the interace will be considered. Finally some details concerning the implementation o the algorithms will be presented and possible stopping criteria o the iterative process will be discussed Sequential and parallel approaches The considered sequential orm o the iterative method was proposed and studied e.g. in [10 12]. Algorithm S. (Sequential or Gauss Seidel-type iteration) 0. Given η 0 pη 0 L 2 (Γ I ) γ 0 γ p > 0 θ (01]. Set k = Solve a Darcy problem with Robin boundary data η k p giving the solution ϕ k+1 2. Set η k+1 as a linear combination o η k ηk p and the solution o the Darcy problem by using the damping actor θ. 3. Solve a Stokes problem with Robin boundary data η k+1 giving the solution (u k+1 p k+1 ). 4. Set ηp k+1 as a linear combination o ηp k η k+1 and the solution o the Stokes problem by using the damping actor θ. 5. I not converged: Increase k by 1 and go to step 1. A straightorward modiication o Algorithm S into a version where the subproblems are solved in parallel was studied in [16]. Algorithm P. (Parallel or Jacobi-type iteration) 0. Given η 0 η0 p L 2 (Γ I ) γ 0 γ p > 0 θ (01]. Set k = Do in parallel: a) Solve a Darcy problem with Robin boundary data ηp k giving the solution ϕ k+1 p. b) Solve a Stokes problem with Robin boundary data η k giving the solution (u k+1 p k+1 ). 2. Do in parallel: a) Set η k+1 as a linear combination o ηp k η k and the solution o the Darcy problem by using the damping actor θ. b) Set ηp k+1 as a linear combination o ηp k η k and the solution o the Stokes problem by using the damping actor θ. 3. I not converged: Increase k by 1 and go to step 1. Note that in [ ] the undamped versions i.e. θ = 1 were considered. The Robin problems to be solved in Algorithms S and P have the orm p.

13 ON ITERATIVE SUBDOMAIN METHODS FOR THE STOKES DARCY PROBLEM 11 Darcy: Find ϕ k+1 p Stokes: Find (u k+1 Q p such that or all ψ Q p 1 gϕ k+1 p ψ γ p a p ( ϕ k+1 p ψ ) + p k+1 Γ I = ( p ψ) Ωp + 1 η k γ pψ p Γ I ) V Q such that or all (vq) V Q ( a u k+1 v ) ( ) ( +b vp k+1 b u k+1 q ) + γ u k+1 n v n Γ I η k = ( v) Ω + v n where in Algorithm S it is k = k +1 and in Algorithm P it is k = k. The matrix-vector representation o the two Robin Robin ormulations (24) and (25) possesses the structure ( ( ) ϕ Fp ) ( ) A11 A 12 η ( ) A 21 A 22 s = η p 0 ). It can be seen that Algorithm S is a block Gauss Seidel method (with a orward solve) or this system whereas Algorithm P is a block Jacobi method both methods are not damped. Moreover both algorithms solve the subproblems involving the diagonal blocks A ii i {12} with one Gauss Seidel step. Note that the subblocks A ii i {12} are lower triangular twoby-two block matrices themselves with one diagonal block typically much bigger than the other one. Hence Algorithm P is strictly speaking a nested Jacobi and Gauss Seidel algorithm The boundary conditions at the interace A main issue o the algorithms is updating the Robin data η k+1 and ηp k+1 steps 2 and 4 o Algorithm S and step 2 o Algorithm P. In [ ] the ollowing approach was considered with θ = 1 ( ) γ (27) η k+1 = (1 θ)η k +θ ( ηp k+1 = (1 θ)ηp k +θ ( F 0 ηp k γ p +γ gϕ k+1 p γ p γ p Γ I η k +(γ p +γ )u k+1 n ). (28) This updating strategy is a damped version o applying the second and ourth line in (26) i.e. it originates rom a Robin Robin ormulation or the Neumann Neumann coupled problem. Using conorming inite element spaces then ϕ k+1 p and u k+1 are continuous unctions. Thus i η 0 and ηp 0 are also continuous then the updating strategy (27) (28) gives also continuous Robin boundary data at the interace. For this reason it will be called C-RR (continuous Robin Robin). For θ = 1 Algorithm S with the updating strategy C-RR is the Robin Robin method that was analyzed in [12]. This analysis covers the convergence o the method in the setting o ininite dimensional spaces where the Stokes Darcy problem with the interace conditions (6) (7) and (10) was considered. The updating strategy C-RR (with θ = 1) with Algorithm P was proposed and analyzed in [16]. In this paper the Stokes Darcy problem with the coupling conditions (6) (7) and the Beavers Joseph Saman condition (9) was studied. Although both approaches are rather similar the corresponding algorithms rely on completely dierent choices o the Robin parameters: γ > γ p or small ν and K in [12] and γ < γ p or unitary ν and K in [16]. In both cases it was reported that the opposite choice would not lead to an eicient strategy. On the one hand both papers [12] and [16] study dierent regimes o the coeicients o the Stokes Darcy problem but on the other hand a dependence o the Robin parameters on these coeicients is not considered. Altogether it remains still unclear how to choose the Robin Robin parameters or given ν and K in general situations.

14 12 A. CAIAZZO V. JOHN. U. WILBRANDT A dierent updating strategy consists in applying η k+1 = (1 θ)η k +θ ( gϕ k+1 p γ K ϕ k+1 p n ) (29) η k+1 p = (1 θ)η k p +θ ( γ p u k+1 n n T(u k+1 p k+1 ) n ). (30) This strategy corresponds to using the second and ourth row o (24) together with a damping i. e. it comes rom a Robin Robin ormulation. Note that or continuous inite element spaces ϕ k+1 p and T(u k+1 p k+1 ) are in general discontinuous unctions such that (29) and (30) lead to discontinuous Robin boundary data. Thus this strategy will be called D-RR (discontinuous Robin Robin). In the literature we could ind the use o the D-RR updating strategy only in [29] within Algorithm S. Remark 3.1 (Possible variations) In Algorithm S one could as well solve a Stokes problem irst and then a Darcy problem in every iteration step. Furthermore the damping actor θ could be dierent or the update o the two interace variables η k+1 and ηp k+1. Another modiication o the D-RR updating strategy could be to use the solution rom the previous iteration step instead o the previous interace variable i. e. η k+1 = (1 θ) ( gϕ k p γ K ϕ k p n ) +θ ( gϕ k+1 p γ K ϕ k+1 p n ) η k+1 p = (1 θ) ( γ p u k n n T(u k p k ) n ) +θ ( γp u k+1 n n T(u k+1 p k+1 ) n ). In our experience so ar none o the proposed variations yields a qualitative dierence to the results presented in Section Implementation aspects The matrices corresponding to the Darcy and the Stokes problem with Robin boundary conditions do not change during the iterative processes in both Algorithms S and P. Thereore one assembly prior to the iteration suices and a actorization can be computed. Furthermore v n (corresponding to E Γ in (24) and (25)) which is added I to the right-hand side o the Stokes system is linear in η and can be computed by a matrixvector multiplication. Similar ideas apply to the update o the right-hand side in the Darcy problem i.e. to the operator E p and to the restriction operators R p R R prob R rob in (24) and (25) respectively. Hence no urther assembling is needed during the iteration. The simpliied condition (10) is implemented weakly penalizing the tangential component o the velocity on the interace by a Nitsche technique [30 31] rather than imposing it directly in the velocity unction spaces. It is assumed that the Stokes subdomain is triangulated with a regular inite element mesh which induces a partition on the (conormal) interace Γ I. Denoting by h E the diameter o a ace (edge) E Γ I then or imposing the interace condition (10) the ollowing term is added to the ormulation the interace contribution η k+1 d 1 τ i T(u p ) n v τ i ΓI + d 1 i=1 i=1 E Γ I γ h E u τ i v τ i E where γ > 0 is a constant penalty parameter whose value depends in general on the particular problem [30]. We used γ = 10 in all simulations. The irst term resulting rom integration by parts is not replaced but it enters the matrix Stopping criteria Standard stopping criteria e.g. the ones used in [10 12] depend on the absolute or relative dierences between successive iterates computed on pressure or velocity solution vectors or

15 ON ITERATIVE SUBDOMAIN METHODS FOR THE STOKES DARCY PROBLEM 13 on interace variables i.e. based on terms o the orm s k+1 s k+1 h s k h s k h h l 2 or s k l h 2 where s k h stands or uk h pk h ϕk ph ηk ph or ηk h. In [10 12] the relative increment o the discrete normal velocity on the interace u h n ΓI was used. Another possibility consists in checking the accuracy o the coupling conditions (6) and (7) i.e. monitoring the term e k := u k h +K ϕ k ph n 2 + n T(u k L 2 (Γ I) hp k h) n +gϕ k 2 ph. L 2 (Γ I) However these quantities depend on the considered inite element spaces and in general they might not vanish even or the exact solution o the discrete coupled linear system. On the other hand the relative dierence E k := ek e k+1 e k = 1 ek+1 e k provides a measure o the progress made in the iteration. All these criteria can be used on their own or in some (linear) combination. However these stopping criteria only measure the progress o the iteration but not the quality o the solution i.e. they are not able to detect whether the computed iterate is indeed close to the solution o the discrete problem or i the iteration sticks at an early stage. Thereore we considered a urther stopping criterion based on the residual o the (discrete versions o the) equations (17) or (23) i.e. ( )( R k := Dh C h ϕ k+1) ( ) Ch ph Fh S h s k+1. F h ph l 2 I not stated otherwise in the simulations presented in Section 4 the iterative procedure has been stopped when the ollowing conditions were satisied R k < eps and u k+1 h uk h l 2 u k h l 2 + p k+1 h pk h l 2 p k h l 2 l 2 + ϕ k+1 ph ϕk ph l 2 ϕ k ph l 2 < eps (31) or a prescribed threshold eps. In the second condition we omitted the denominator whenever it was smaller than one. 4. COMPUTATIONAL RESULTS The goal o the numerical studies consists in assessing Algorithms S and P in combination with the updating strategies C-RR and D-RR with respect to their eiciency. Since in all methods the numerical costs per iteration are very similar the eiciency will be measured in terms o the number o iterations or reaching convergence. Because o our motivation to study applications in computational geosciences the algorithms will be assessed especially or physical parameters which are relevant in this context. For all numerical experiments the Taylor Hood P 2 /P 1 pair o inite element spaces was used in the Stokes subdomain while P 2 elements have been used or the piezometric head. This choice o spaces is the same as e.g. in [ ]. The Robin Robin updating strategies were applied with θ = 1. All simulations were perormed with the code MooNMD [26]. The linear systems o equations were solved with the sparse direct solver UMFPACK [32]. To veriy the implementation the solver was benchmarked considering a solution belonging to the inite element spaces. In these studies the inal errors or ν = 1 and K = I were o the order o the machine accuracy but they increased with ν 1 and K 1. This observation relects

16 14 A. CAIAZZO V. JOHN. U. WILBRANDT the act that or small ν and K the condition numbers o the inite element linear systems increase. As already mentioned above a motivation or the present study is the unclear situation concerning the choice o the Robin parameters γ and γ p. To address this question we irst validated our implementation against two examples taken rom [ ] to reproduce the published results using the algorithms considered in the respective paper. In addition all other algorithms were also assessed or these examples. Finally we studied an example related to a geoscientiic application [33 34] Example 1 Thisexamplewasusedin[16]toillustratethebehavioroAlgorithmP withthec-rrupdating strategy or the viscosity ν = 1 and the hydraulic conductivity K = I. Let Ω p = (0π) ( 10) Ω = (0π) (01) and Γ I = (0π) {0}. The hydraulic conductivity has the orm K = KI and the solution o the coupled problem (1) (2) (5) is given by u (xy) = ( ) v (y)cos(x) p v(y) sin(x) (xy) = 0 ϕ p (xy) = e y sin(x) where v(y) = K gy ( 2ν + αg 4ν 2 + K ) y 2. 2 On the outer boundaries i.e. Γ e = Ω \Γ I and Γ pe = Ω p \Γ I essential boundary conditions were prescribed. This example considers the Beavers Joseph Saman condition i.e. the coupling conditions at the interace are given by(6)(7) and(9). Numerical simulations were perormed on meshes obtained by uniormly red reining an initial coarse grid (level 0) consisting o eight triangles see Figure 2. Note that no inormation about the mesh is available in [16] such that our setup might be dierent in this respect. The initial iterate was always set to be zero. The iterative algorithm was stopped according to the criterion (31) with a tolerance eps = The use o this stopping criterion is another dierence to [16]. Figure 2.Example 1: initial grid (level 0) with Ω in blue (top) and Ω p in red (bottom). Since the main goal o this irst example is the validation o our results against the results rom [16] only the case ν = 1 and K = I will be considered. Figure 3 shows the evolution o the relative discrete errors o the iterates o the Stokes velocity or dierent choices o the Robin parameters. In all cases the curves are in agreement with the results reported in [16 Fig. 6.1]. Details concerning the needed iterations are provided in Table I. For the updating strategy D-RR Algorithm S converged aster than Algorithm P. Moreover or all methods the number o iterations is independent o the level. As stated also in [16] one can see in Figure 3 a very ast convergence i γ < γ p a slow convergence i γ = γ p and divergence in the case γ > γ p. The speed o convergence depends essentially on the ratio γ /γ p. In this example the smallest number o iterations were needed or γ p = 1. We could observe the same convergence behavior or the Stokes pressure and the piezometric head. In summary the results rom [16] could be reproduced very well.

17 ON ITERATIVE SUBDOMAIN METHODS FOR THE STOKES DARCY PROBLEM 15 Updating strategy Algorithm Level 1 Level 2 Level 3 Level 4 C-RR Algorithm P C-RR Algorithm S D-RR Algorithm P D-RR Algorithm S Table I.Example 1: Number o iterations or γ p = 1 and γ = γ p/ Example 2 This example was used in [10 12] or assessing Algorithm S with the C-RR updating strategy or dierent values o the kinematic viscosity ν and the hydraulic conductivity K. Let Ω p = (01) 2 and Ω = (01) (12) with the interace Γ I = Ω p Ω = (01) {1} the hydraulic conductivity o the orm K = KI and the solution given by ( ) y u s (xy) = 2 2y +1 x 2 x p (xy) = 2ν(x+y 1)+ g 3K (32) ϕ p (xy) = 1 ) (x(1 x)(y 1)+ y3 K 3 y2 +y + 2ν g x. Dirichlet boundary conditions were imposed on Ω \Γ I and on the bottom boundary (0 1) {0}. On the remaining parts Neumann boundary conditions were prescribed. In this example the simpliied Beavers Joseph Saman condition (6) (7) and (10) is considered. We used unstructured grids with 98 (mesh 1) 470 (mesh 2) 1914 (mesh 3) and 8216 (mesh 4) cells with a total o 406 (mesh 1) 1690 (mesh 2) 6553 (mesh 3) and (mesh 4) degrees o reedom. The initial iterate was always chosen to be zero. Using values or the parameters ν and K relevant or geoscientiic applications which are typically very small the pressure and the piezometric head in (32) consist o two parts. A very small contribution scaled with ν and a very large part due to the scaling with K 1. This second part leads to values o p and ϕ p which are unrealistic in applications. In this respect this example has some deiciencies. To reproduce the results rom [10 12] we irst used as stopping criterion only u k+1 u k u k < 10 6 (33) similarly as it has been applied in [10 12] where the relative increment o the discrete normal velocity on the interace and a somewhat smaller tolerance were used. Note that u is the only part o the solution that does not scale with K 1. Because o the large condition number o the linear system o equations or small ν and K which was already mentioned above one generally has to relax the tolerances o stopping criteria in this case compared with the case ν = 1 K = 1. Table II reports the results with Robin parameters γ p = 1 γ = γp 3 which was the most eicient choice in Example 1 in combination with the stopping criterion (33). Clearly all algorithms ailed or those parameters in the case o small ν and small K. Indeed ollowing [10 12] one would expect eicient simulations or a dierent choice o the Robin parameters namely γ p = 0.1 γ = 3γ p. In this case one obtains the results presented in Table III. One can see that the algorithms with the updating strategy D-RR converged or all choices o physical parameters. We observed a similar behavior o the algorithms with the D- RR updating strategy also or γ = 3γ p with γ p { }. The results are similar

18 16 A. CAIAZZO V. JOHN. U. WILBRANDT Relative discrete errors or the Stokes velocity γ p = number o iterations γ p = number o iterations γ p = γ p = number o iterations number o iterations γ p = number o iterations γ =γ p /3 γ =γ p /2 γ =γ p γ =2γ p γ =3γ p u k+1 h Figure 3. Example 1: Relative discrete errors u h l 2 o Algorithm S with ν = 1 and K = 1 u h l 2 or dierent values o γ p and γ on reinement level 4.

19 ON ITERATIVE SUBDOMAIN METHODS FOR THE STOKES DARCY PROBLEM 17 Updating strategy Algorithm ν K mesh 1 mesh 2 mesh 3 mesh 4 C-RR D-RR all Algorithm P Algorithm S Algorithm P Algorithm S both Table II.Example 2 Number o iterations or γ p = 1 and γ = γ p/3 stopping criterion (33) +++ means did not converge within 100 iterations and means diverged. type algorithm ν K mesh 1 mesh 2 mesh 3 mesh 4 C-RR D-RR Algorithm P Algorithm S Algorithm P Algorithm S Table III.Example 2: Number o iterations or the stopping criterion (33) γ p = 0.1 and γ = 0.3 means diverged. to those reported in [10 12] or the updating strategy C-RR. In our simulations however the C-RR updating strategy was less successul or all studied choices o the Robin parameters and we could not reproduce the results rom [10 12]. Consulting the Ph.D. thesis [29] one gets the impression that in [10 12] the C-RR updating strategy is presented and analyzed but the numerical studies in these papers were perormed with the D-RR updating strategy. In Table III one can see that Algorithm P was (or the D-RR updating strategy) more eicient than Algorithm S or very small values o the physical coeicients. For both algorithms the number o iterations is independent o the mesh. Next the convergence properties o the algorithms using the harder stopping criterion (31) alsowitheps = 10 6 wereinvestigated.onecanobserveinfigure4thatorγ p = 0.1γ = 3γ p

20 18 A. CAIAZZO V. JOHN. U. WILBRANDT (weobtainedsimilarresultsorγ p { })evenithecriterion(33)issatisiedtheresidual o the complete coupled problem is ar rom being small. Figure 4 shows also that relative changes o the pressure and the piezometric head are still large. For urther comparisons we computed the solution o the monolithic discrete Stokes Darcy problem by applying a direct solver to the coupled inite element problem (an approach which is not easible in many applications). Then one can see that the iterates or the pressure and the piezometric head still dier considerably rom the discrete solution. These observations conirm that the stopping criterion (33) is not suicient to assess the properties o the iterative algorithms and the number o iterations given in Table III might provide a wrong impression o their eiciency. Further numerical studies revealed that a considerable speed-up o the algorithms with the D-RR updating strategy could be achieved by increasing the Robin parameters thereby keeping the relation γ = 3γ p see Figure 5 and Table IV. Similar results were obtained or γ p {50200}. We think that the slow convergence in the case K = 10 7 is due to the unrealistic large values or the pressure and the hydraulic head. One can see in Table IV that Algorithm S perormed better than Algorithm P. The number o iterations is independent o the mesh residuals o coupled system R k 10 0 discrete errors Stokes velocity Stokes pressure piezometric head number o iterations number o iterations relative dierences o the iterates Stokes velocity Stokes pressure piezometric head sum o the three number o iterations Figure 4. Example 2: Evolution o the residual discrete errors and relative dierences or Algorithm S mesh 4 using the D-RR updating strategy and γ p = 0.1 γ = 3γ p ν = 10 4 and K = In summary despite the shortcomings o the used example it was clariied that or the case o small viscosity and small hydraulic conductivity the use o the D-RR updating strategy and the choice γ > γ p e.g. γ = 3γ p o the Robin parameters are important to obtain an eicient subdomain iteration Example 3: Water low over a porous river bed As a inal example a model o a unidirectional steady water low over a porous bed is considered separated by a non-straight interace. In the context o computational geosciences

21 ON ITERATIVE SUBDOMAIN METHODS FOR THE STOKES DARCY PROBLEM residuals o coupled system R k discrete errors Stokes velocity Stokes pressure piezometric head number o iterations number o iterations 10 6 relative dierences o the iterates 10 4 Stokes velocity Stokes pressure 10 2 piezometric head sum o the three number o iterations Figure 5.Example 2: Evolution o the residual discrete errors and relative dierences versus the number o iterations or Algorithm S mesh 4 using the D-RR updating strategy and γ p = 100 γ = 3γ p ν = 10 4 and K = type algorithm ν K mesh 1 mesh 2 mesh 3 mesh 4 C-RR D-RR Algorithm P Algorithm S Algorithm P Algorithm S Table IV.Example 2: Number o iterations or the stopping criterion (31) γ p = 100 and γ = means did not converge within 1000 iterations means diverged. this model has been proposed or the study o the hydrodynamic interactions between the water low and the underlying river bed [33 34]. In order to simulate the water low over two triangular dunes consider a rectangular domain Ω = [02L] [0H +H p ]

22 20 A. CAIAZZO V. JOHN. U. WILBRANDT where H and H p denote the heights o the water low domain and the porous river bed at x {0L2L}. The interace representing the dune is composed o two triangles whose highest points are located at x {l D L+l D } while the maximum height o the dunes with respect to the entrance porous bed is denoted by h D see Figure 6. l D {}}{ H }h D H p 2L Figure 6.Computational domain or the water low over a porous river bed model mesh 1. The boundary conditions and the values o the physical parameters have been chosen as in the numerical simulations presented in [33 34]. In particular the lower boundary is considered impermeable (no low boundary condition) while no-slip conditions are imposed at the upper boundary. At inlet and outlet boundaries the ollowing inhomogeneous periodic boundary conditions or Stokes and Darcy problems have been imposed: u inlet = u outlet T(u inlet p inlet ) n = T(u outlet p outlet ) n +p 0 ϕ inlet = ϕ outlet +p 0. Note that the pressure is unique only up to an additive constant in this example. We ixed this constant by orcing one pressure node to be zero. Furthermore the simulation parameters are L = 1 H = 0.5 H p = 1.5 l D = 0.9 h D = 0.1 p 0 = 10 3 = 0 p = 0. Figure 7 depicts the numerical solution or ν = 10 6 and K = The prescribed pressure drop p 0 induces a low rom let to right which partially penetrates the porous bed due to the inclination o the interace. However the impact o the water low onto the porous media (and vice versa) remains local i. e. the low remains unperturbed and unidirectional away rom the interace. Note the pressure drop arising behind the dunes which locally causes a low in the opposite direction underneath the dunes. For assessing the perormance o the iterative algorithms three dierent unstructured meshes were considered see Figure 6 or an example and Table V or detailed inormation. From the results obtained in Example 2 it can be expected that one has to use the updating strategy D-RR. In act we could observe in this example that the subdomain iteration did not converge i the updating strategy C-RR was employed. The obtained results are presented in Tables VI and VII where the threshold o the stopping criterion (31) was set to be eps = Dierent values o the Robin parameters γ and γ p were chosen always satisying the same ratio as in Example 2 and in [10 12]. One can see that both algorithms perormed well on all meshes or the Robin parameters γ p {10100} and γ = 3γ p. The number o iterations or Algorithm S was in the most cases independent o the mesh. For some physical parameters in particular or very small ν and K Algorithm P converged aster than Algorithm S.

23 ON ITERATIVE SUBDOMAIN METHODS FOR THE STOKES DARCY PROBLEM 21 Figure 7.Example 3: Numerical solution or ν = 10 6 and K = 10 7 showing the velocity ield u and ϕ p (unscaled arrows top let) the velocity streamlines (top right) and a pressure elevation plot which is semi-transparent in the Darcy subdomain (bottom). All three plots are colored according to the pressure ield scaled so that the minimum value is zero. mesh interace edges cells degrees o reedom Stokes Darcy Stokes Darcy Table V. Example 3: Number o interace edges Stokes and Darcy cells and degrees o reedom or the three dierent meshes used. 5. SUMMARY AND OUTLOOK This paper reviewed iterative subdomain methods or solving the Stokes Darcy problem that use Robin boundary conditions at the interace. In particular it was clariied that there are dierent updating strategies or the Robin boundary conditions. For coeicients in the Stokes Darcy problem that are relevant or applications rom geosciences the use o the updating strategy D-RR in combination with an appropriate choice o the Robin parameters turned out to be crucial or designing an eicient numerical method. Concerning the Robin parameters the choice γ = 3γ p resulted in eicient methods i γ p was chosen appropriately in the considered examples γ p [10100]. These values are considerably larger than those proposed so ar in the literature. Finally it was observed that the serial update o the interace conditions Algorithm S needed oten less iterations than the parallel update Algorithm P. Altogether the main goal o our studies was achieved: the identiication o an eicient iterative subdomain method or the Stokes Darcy problem with coeicients that are relevant in geosciences.