A parallel Robin-Robin domain decomposition method for the Stokes-Darcy system

METHOD FOR THE STOKES-DARCY SYSTEM

WENBIN CHEN^†, MAX GUNZBURGER^‡, FEI HUA ^§,AND XIAOMING WANG^¶

Abstract. We propose a new parallel Robin-Robin domain decomposition method for the cou- pled Stokes-Darcy system with Beavers-Joseph-Saffman-Jones interface boundary condition. In par- ticular, we prove that, with an appropriate choice of parameters, the scheme converges geometrically independent of the mesh size.

Key words. Stokes and Darcy equations, Robin-Robin domain decomposition, geometrical convergence

AMS subject classifications.65N55, 65N30, 35M20, 35Q35

1. Introduction. The Darcy equations are a well accepted model for flow in porous media such as that that is often found in the subsurface. Thus, discretized versions of these equations are often used to simulate both groundwater and petroleum flows. However, in these settings, one often finds that what the porous media does not completely cover subsurface regions of interest. For examples, in petroleum applica- tions, one often finds pockets of oil and in karst aquifers, one finds conduits in which free water flows. In such regions, the flow of the liquid cannot be accurately modeled by the Darcy equations, even though often, for expediency, that is exactly what is done in practice. A more accurate description of the flow of liquids in cavities and conduits is given by the Navier-Stokes equations. Due to the relative slow flows often encountered in such situations, one can simplify matters and use the linear Stokes equations instead. Of course, flows in conduits and cavities are coupled to the flow in the surrounding porous media so that conditions along the interfaces separating free flows and porous media flows must be imposed to effect the coupling. Several such in- terface conditions have been proposed; see, e.g., [1, 19]. Once a coupled Stokes-Darcy system has been defined, the remaining tasks are to define discrete systems whose solutions accurately approximate the exact solution of the continuous model and then to develop efficient methods for solving the discrete equations. These are the tasks that we address in this paper.

Here we consider a coupled Stokes-Darcy system on a bounded domain Ω = Ωp!

Ωf ⊂R^d, (d= 2,3). In the porous media region Ωp, the governing equations

∗Supported in part by the SCRMS program of the National Science Foundation under grant number DMS-0724273.

†School of Mathematical Science, Fudan University, Shanghai, P.R.China, 200433, [email protected]. Supported by the Chinese MOE under an 111 project grant, the National Basic Research Program under grant number 2005CB321701, and the Shanghai Natural Science Foundation under grant number 07JC14001.

‡Department of Scientific Computing, Florida State University, Tallahassee FL 32306-4120, [email protected]. Supported in part by the CMG program of the National Science Foundation under grant number DMS-0620035.

§Department of Mathematics, Florida State University, Tallahassee FL 32306-4120, [email protected]. Supported in part by the CMG program of the National Science Foundation under grant number DMS-0620035.

¶Department of Mathematics, Florida State University, Tallahassee FL 32306-4120, [email protected]. Supported in part by the CMG program of the National Science Foundation under grant number DMS-0620035.

1

are the Darcy equations

u_p=−K∇φp

∇ ·up= 0,

where u_p is the fluid velocity in the porous media,K is the hydraulic conductivity tensor, andφp is the hydraulic head. In the fluid region Ωf, the fluid flow is assumed to satisfy the Stokes equations

−∇ ·T(uf, pf) =f

∇ ·u_f = 0,

where u_f is the fluid velocity, pf is the kinematic pressure, f is the external body force, ν the kinematic viscosity of the fluid,T(u_f, pf) = 2νD(u_f)−pfIis the stress tensor, andD(u_f) = 1/2(∇u_f+∇^Tu_f) is the deformation tensor.

Porous media domain Ω

p

Fluid domain Ω

f

Interface Γ

Fig. 1.1.The fluid and porous media domainsΩf andΩp, respectively, an the interfaceΓ.

Let Γ = Ωp∩Ωf denote the interface between the fluid and porous media regions;

see Fig. 1.1. Along the interface Γ, we require

u_f·n_f =−u_p·n_p, (1.1)

−τ_j·(T(uf, pf)·n_f) =ατ_j·u_f, (1.2)

−n_f·(T(u_f, pf)·n_f) =gφp, (1.3) wheren_f and n_p denote the unit outer normal to the fluid and the porous media re- gions at the interface Γ, respectively;τ_j(j= 1,· · · , d−1) denote mutually orthogonal unit tangential vectors to the interface Γ; and the constant parameterαdepends onν andK. The second condition (1.2) is referred to as the Beavers-Joseph-Saffman-Jones (BJSJ) interface condition [12, 19] which is an approximation of the Beavers-Joseph interface boundary condition [1]. The BJSJ boundary condition is also related to the Navier’s slip boundary condition.

To enable direct comparisons with the results of [8] and for simplicity, we assume that the hydraulic headφp and the fluid velocityu_f satisfies homogeneous Dirichlet boundary condition except on Γ , i.e.,φp = 0 on the boundary∂Ωp\Γ andu_f = 0 on the boundary∂Ωf\Γ.

The spaces that we utilize are

Xf ={v_f ∈[H¹(Ωf)]^d | v_f = 0 on∂Ωf\Γ} Qf =L²(Ωf)

Xp ={ψp∈H¹(Ωp) | ψp= 0 on∂Ωp\Γ}.

For the domainD(D= Ωf or Ωp), (·,·)Ddenotes theL²inner product on the domain D, and &·,·' denotes the L² inner product on the interface Γ or the duality pairing betweenH^−1/2(Γ) andH₀₀^1/2(Γ).

With these notations, the weak formulation of the coupled Stokes-Darcy problem is given as follows [2, 8]: find (uf, pf)∈Xf×Qf and φp ∈Xp such that

af(u_f,v_f) +bf(v_f, pf) +gap(φp, ψp) +&gφp,v_f·n_f' − &gu_f ·n_f, ψp' +α&Pτuf, Pτvf'= (f,vf)Ωf ∀vf ∈Xf, ψp∈Xp, (1.4)

bf(uf, qf) = 0, ∀qf ∈Qf, (1.5)

where the bilinear forms are defined as

ap(φp, ψp) = (K∇φp,∇ψp)Ωp

af(u_f,v_f) = 2ν(D(u_f),D(v_f))Ωf

bf(vf, q) =−(∇ ·v_f, q)Ωf

andPτ denoted the projection onto the tangent space on Γ, i.e.

Pτu=

d−1"

j=1

(u·τ_j)τ_j.

It is easy to see that the system (1.4) and (1.5) is well posed forf ∈[L²(Ωf)]^d [2, 8].

Since the governing equations are different for the fluid and the porous media region, it is natural to utilize a domain decomposition method so that off-the-shelf efficient solvers for the Darcy system and the Stokes system can be utilized [8]. The central issue is then to determine if the domain decomposition method convergence and its associated convergence rate. The main contribution of this paper is the devel- opment and analysis of anew parallel domain decomposition method based on Robin boundary conditions that converges with a rate that, for appropriate choices of accel- eration parameters, is independent of the mesh size .

For classical second-order elliptic problems, a Robin-type domain decomposition method (DDM) was introduced in [13] where it was also proved that the solution of the Robin DDM converges weakly to the solution of the elliptic problems with respect to theH¹ norm. In [4, 5], new update techniques for the Robin data were introduced and it was proved that the weak convergence inH¹could induce strong convergence.

In [9], it was pointed out that a convergence rate 1−O(h^1/2) can be achieved in the case of two subdomains. Recently, a rigorous analysis for the case of many subdomains was given in [16,17] where it was proved, in certain cases such as for a small number of subdomains, that the convergence rate could be 1−O(h^1/2H^−1/2), whereH is the size

of the subdomain andhis the size of the finite element grid. In particular, the new term “winding number” was proposed in [17] to describe the depth of subdomains, and in the case of many subdomains and it was shown that the convergence rate not only depends on the mesh sizehand the sizeH of subdomains, but also on the winding number. Furthermore, Robin-type DDMs for parabolic problems was also studied in [18].

In [8], two Robin DDMs for the Stokes-Darcy equations, one a serial version(sRR) and the other a parallel version(pRR), are considered and compared with the Dirichlet- Neumann DDMs [6, 7]; mesh-independent convergence rates were observed for serial Robin DDMs numerically but not proved rigorously. In addition to providing a rig- orous analysis, in this paper, we treat the more general case of the Beavers-Joseph- Saffman-Jones interface boundary condition instead of the further simplified interface boundary conditions considered in [8]. However, the full Beavers-Joseph interface boundary condition is not treated here since well posedness in the steady state case is established only for particular choices of parameters [2]. Other algorithms that combine ideas from multi-grid and DDMs can be found in [14], where the authors proposed to solve the coupled problem directly on a coarse grid (with mesh size hcoarse) and then use the coarse solution to provide boundary conditions for the Stokes and Darcy systems at the interface so that they may be solved separately on a finer mesh (with mesh size of the order ofh

3

coarse2 ). For DDMs for other settings, and especially for the parallel Robin-Robin domain decomposition methods, one may refer to [4–8, 13, 15–17, 20, 21] and the references cited therein.

The rest of the paper is organized as follows. In Section 2, we propose the Robin boundary conditions at the interface for the Stokes and Darcy systems. A necessary and sufficient condition on the equivalence of the Stokes-Darcy system with Beavers- Joseph-Saffman-Jones interface boundary condition and the new decoupled Stokes and Darcy systems with Robin boundary conditions is derived. In Section 3, we propose our new parallel Robin-Robin domain decomposition method. We establish the convergence of the new scheme for the case of equal acceleration parameters γf =γp and the case of γf < γp, with a convergence rate for appropriate choices of the acceleration parameters. Finite element approximations are considered in Section 4. In particular, we derive a convergence rate (depending on the mesh size h) for the case of equal acceleration parameters and the case of γp < γf. Although the convergence rate is derived for globally regular triangulations only, we may easily generalize the result to mortar elements so that solvers with different mesh size may be utilized for the fluid and the porous media regions. We present our results of some computational experiments in Section 5. These results are in accordance with our analyses.

2. Robin boundary conditions. In order to solve the coupled Stokes-Darcy problem utilizing the domain decomposition idea, we naturally consider (partial) Robin boundary conditions for the Stokes and the Darcy equations because Robin boundary conditions are more general and embody both the Neumann and Dirichlet type conditions in (1.1)–(1.3).

Let us consider the following Robin condition for the Darcy system: for a given constantγp>0 and a given functionηp defined on Γ,

γpK∇φ#p·n_p+gφ#p=ηp on Γ. (2.1) Then, the corresponding weak formulation for the Darcy system is given by: for

ηp ∈L²(Γ), findφ#p∈Xp such that

γpap(φ#p, ψp) +&gφ#p, ψp'=&ηp, ψp' ∀ψp∈Xp.

Similarly, we propose the following Robin type condition for the Stokes equations:

for a given constantγf >0 and a given functionηf defined on Γ,

nf·(T(#uf,p#f)·nf) +γf#uf·nf =ηf on Γ. (2.2) Then, the corresponding weak formulation for the Stokes system is given by: for ηf ∈L²(Γ), findu#_f ∈Xf andp#f∈Qf such that

af(u#_f,v_f) +bf(vf,#pf) +γf&#u_f·n_f,v_f·n_f'

+α&Pτ#uf, Pτvf'= (f,vf)Ωf +&ηf,vf·nf' ∀vf ∈Xf (2.3) bf(#u_f, qf) = 0 ∀qf ∈Qf.

We can combine the Stokes and Darcy systems with Robin boundary conditions into one system. Indeed, for any positive constant ω, it is easy to see that ifηp ∈ L²(Γ) and ηf ∈ L²(Γ) are given, then, there exists a unique solution (φ#p,#uf,p#f)

∈Xp×Xf×Qf such that

af(#u_f,v_f) +bf(v_f,p#f) +ωγpap(φ#p, ψp) +ω&gφ#p, ψp'+γf&#u_f·n_f,v_f·n_f'

+α&Pτ#u_f, Pτv_f'= (f,v_f)Ωf +&ηf,v_f·n_f'+ω&ηp, ψp' ∀ψp∈Xp,v_f ∈Xf

bf(#u_f, qf) = 0 ∀qf ∈Qf.

(2.4) Remark. 2.1. Note that the solution (φ#p,#u_f,p#f) is independent of the parameter ω.

Our next goal is to show that, for appropriate choices of γf, γp, ηf, and ηp, (smooth) solutions of the Stokes-Darcy system are equivalent to solutions of (2.4), and hence we may solve the latter system instead of the former.

Lemma 2.2. Let (φp,u_f, pf) be the solution of the coupled Stokes-Darcy sys- tem (1.4)–(1.5)and let(φ#p,#uf,#pf)be the solution of the decoupled Stokes and Darcy system with Robin boundary conditions at the interface (2.4). Then, (φ#p,u#_f,p#f) = (φp,u_f, pf)if and only if γf,γp,ηf, andηp satisfy the following compatibility condi- tions:

ηp=γp#uf·nf+gφ#p (2.5) ηf =γfu#f·nf−gφ#p. (2.6) Proof. For the necessity, we setψp = 0 in the Stokes-Darcy system (1.4)–(1.5) and deduce that (φp,u_f, pf) solves (2.3) if

< ηf−γfu_f·n_f+gφp,v_f·n_f >= 0 ∀v_f ∈Xf (2.7) which implies (2.6). The necessity of (2.5) can be derived in a similar fashion.

As for the sufficiency, by setting ω = g/γp in (2.4) and substituting the com- patibility conditions (2.5)–(2.6), we easily see that (φ#p,#u_f,#pf) solves the coupled Stokes-Darcy system (1.4)–(1.5).

Since the solution to the Stokes-Darcy system is unique, we have (φ#p,#u_f,p#f) = (φp,u_f, pf).

3. Robin-Robin domain decomposition methods.

3.1. The Robin-Robin domain decomposition algorithm. Now we pro- pose the following parallel Robin-Robin domain decomposition method for solving the coupled Stokes-Darcy system.

1. Initial values ofη_p⁰ andη_f⁰ are guessed. They may be taken to be zero.

2. For k = 1,2, . . ., independently solve the Stokes and Darcy systems with Robin boundary conditions. More precisely,φ^k_p ∈Xp is computed from

γpap(φ^k_p, ψp) +&gφ^k_p, ψp'=&η_p^k, ψp' ∀ψp∈Xp; (3.1) andu^k_f ∈Xf andp^k_f ∈Qf are computed from

af(u^k_f,v_f) +bf(v_f, p^k_f) +γf&u^k_f·n_f,v_f ·n_f'+α&Pτu^k_f, Pτv_f'

=&η_f^k,v_f·n_f'+ (f,v_f)Ωf ∀v_f ∈Xf, (3.2)

bf(u^k_f, qf) = 0 ∀qf ∈Qf. (3.3)

3. η_p^k+1 andη^k+1_f are updated in the following manner:

η_f^k+1 =aη^k_p+bgφ^k_p (3.4) η_p^k+1 =cη_f^k+du^k_f·n_f (3.5) where the coefficientsa, b, c, dare chosen as follows:

a=γf

γp

b=−1−a (3.6)

c=−1 d=γf+γp. (3.7)

In the special case for whichγf =γp =γ, we have

a= 1 b=−2 c=−1 d= 2γ.

The relations (3.6)–(3.7) are necessary to ensure the convergence of the scheme.

Indeed, suppose thatη_f^k and η_p^k converge toη_f^∗ and η_p^∗, respectively, andφ^k_p,u^k_f also converge to the true solutionφ^∗_pandu^∗_f, respectively. Then, by (3.4)–(3.5) and Lemma 2.2, we see that the following relationships hold:

η_f^∗=aη_p^∗+bgφ^∗_p=γfu^∗_f·nf−gφ^∗_p, (3.8) η^∗_p=cη^∗_f+du^∗_f·n_f =γpu^∗_f·n_f+gφ^∗_p. (3.9) This leads to

$ bgφ^∗_p du^∗_f·n_f

%

=

$ b 0 0 d

% $ gφ^∗_p u^∗_f·n_f

%

=

$ 1 −a

−c 1

% $ η_f^∗ η_p^∗

%

=

$ 1 −a

−c 1

% $ −1 γf

1 γp

% $ gφ^∗_p u^∗_f ·n_f

%

(3.10) which implies the consistency equations (3.6)–(3.7) on the coefficients a, b, c, d and γf, γp.

These relationships (3.6)–(3.7) among the parameters are used in the convergence analysis of the Robin-Robin domain decomposition method.

Remark. 3.1. If the updating strategy (3.4)–(3.5) is changed to η^k+1_f =a1η^k_p+b1gφ^k_p+c1u^k_f·n_f

η^k+1_p =a2η^k_f+b2gφ^k_p+c2u^k_f·n_f, then, the “consistency” conditions change to

$ −a1 1 1 a2

% $ 1 γp

−1 γf

%

=

$ b1 c1

b2 c2

% .

In this case, we have more flexibility. However, the convergence analysis is somewhat more complicated, and will be addressed elsewhere.

The parallel Robin-Robin domain decomposition algorithm proposed here is re- lated to the serial version (sRR algorithm) of [8]. As a matter of fact, the sRR al- gorithm can be obtained by implementing our algorithm serially as follows. Initialize η_p⁰, and, fork= 0,1, . . . ,

1. Findφ^k_p by solving the Darcy system (3.1).

2. Setη_f^k=aη_p^k+bgφ^k_p and findu^k_f andp^k_f by solving the Stokes system (3.2)–

(3.3).

3. Setη_p^k+1=cη_f^k+du^k_f ·nf.

In [8], it is proved, for γf = γp = γ, that the solution of the sRR algorithm converges to the solution of the Darcy-Stokes systems. Here, we are able to prove a similar convergence result. Moreover, we are able to demonstrate that the convergence could be geometric for appropriate choice ofγf < γp.

3.2. Convergence of the parallel Robin-Robin DDM. We follow the ele- gant energy method proposed in [13] to demonstrate the convergence of the parallel Robin-Robin domain decomposition method for appropriate choice of parametersγp

andγf.

To this end, let (φp,u_f, pf) denote the solution of the coupled Stokes-Darcy sys- tem (1.4)–(1.5). Then, we have that (φp,u_f, pf) solves the equivalent decoupled sys- tem (2.4) with γf, γp, ηp, ηf satisfying the compatibility conditions (2.5)–(2.6) with the hats removed.

Next, we define the error functions )^k_p=ηp−η_p^k )^k_f =ηf−η^k_f

e^k_φ=φp−φ^k_p e^k_u=u_f−u^k_f e^k_p =pf−p^k_f. Then, the error functions satisfy the following error equations:

γpap(e^k_φ, ψp) +&ge^k_φ, ψp'=&)^k_p, ψp' ∀ψp∈Xp, (3.11) af(e^k_u,v_f) +bf(vf, e^k_p) +γf&e^k_u·n_f,v_f·n_f'+α&Pτe^k_u, Pτv_f'

=&)^k_f,v_f ·n_f' ∀v_f ∈Xf, (3.12)

bf(e^k_u, qf) = 0 ∀qf ∈Qf, (3.13)

and, along the interface Γ,

)^k+1_f =a)^k_p+bge^k_φ (3.14)

)^k+1_p =c)^k_f+de^k_u·n_f. (3.15)

Equation (3.15) leads to

*)^k+1_p *²Γ=c²*)^k_f*²Γ+d²*e^k_u·n_f*²Γ+ 2cd&)^k_f,e^k_u·n_f'. Settingvf =e^k_u in (3.12), we deduce

&)^k_f,e^k_u·nf'=af(e^k_u,e^k_u) +γf*e^k_u·nf*²Γ+α*Pτe^k_u*²Γ

and, hence, combining the last two equations, we have

*)^k+1_p *²Γ=c²*)^k_f*²Γ+ (d²+ 2cdγf)*e^k_u·n_f*²Γ+ 2cd af(e^k_u,e^k_u) + 2cd α*Pτe^k_u*²Γ. (3.16) Similarly, (3.14) implies

*)^k+1_f *²Γ=a²*)^k_p*²Γ+b²*ge^k_φ*²Γ+ 2ab&)^k_p, ge^k_φ'. Settingψp=ge^k_φ in (3.11), we have

&)^k_p, ge^k_φ'=γpap(e^k_φ, ge^k_φ) +&ge^k_φ, ge^k_φ'. Combining the last two equations, we deduce

*)^k+1_f *²Γ=a²*)^k_p*²Γ+ (b²+ 2ab)*ge^k_φ*²Γ+ 2abγpg ap(e^k_φ, e^k_φ). (3.17) Substituting (3.6)–(3.7) into (3.16) and (3.17), we have the following result.

Lemma 3.2. The error functions satisfy

*)^k+1_p *²Γ=*)^k_f*²Γ+ (γ_p²−γ_f²)*e^k_u·n_f*²Γ

−2(γf+γp)af(e^k_u,e^k_u)−2(γf+γp)α*Pτe^k_u*²Γ

*)^k+1_f *²Γ=

$γf

γp

%2

*)^k_p*²Γ+

&

1−

$γf

γp

%2'

*ge^k_φ*²Γ−2γf

$ 1 + γf

γp

%

g ap(e^k_φ, e^k_φ).

We are now ready to demonstrate the convergence of our parallel Robin-Robin domain decomposition method. The convergence analysis forγf =γp and γf +=γp

are different and will be treated separately.

Case 1: γf =γp=γ. In this case, we have

*)^k+1_p *²Γ=*)^k_f*²Γ−4γaf(e^k_u,e^k_u)−4γα*Pτe^k_u*²Γ

*)^k+1_f *²Γ =*)^k_p*²Γ−4γgap(e^k_φ, e^k_φ).

Adding the two equations and summing overkfromk= 0 toN, we deduce

*)^N+1_p *²Γ+*)^N_f⁺¹*²Γ=*)⁰_p*²Γ+*)⁰_f*²Γ−4γ

"N

k=0

(af(e^k_u,e^k_u) +g ap(e^k_φ, e^k_φ) +α*Pτe^k_u*²Γ).

This implies that*)^N+1_p *²Γ+*)^N_f⁺¹*²Γ is bounded from above by*)⁰_p*²Γ+*)⁰_f*²Γ and thate^k_uande^k_φtend to zero in (H¹(Ωf))^d andH¹(Ωp), respectively. The convergence ofe^k_φtogether with the error equation (3.11) implies the convergence of)^k_p inH⁻¹²(Γ).

Combining the convergence of)^k_pande^k_φand the error equation on the interface (3.14), we deduce the convergence of)^k_f in H⁻¹²(Γ). The convergence of the pressure then

follows from the inf-sup condition and (3.12)–(3.13). Note that we have no rate of convergence here. Hence, we have proved the following result.

Theorem 3.3. If γp=γf =γ, then

φ^k_p−→^Xp φp u^k_f −→^Xf u_f p^k_f −→^Qf pf, and

η^k_p ^H

− 1 2(Γ)

−→ γuf·nf+gφp =−γK∇φp·np+gφp

η^k_f ^H

− 1 2(Γ)

−→ γuf·n_f−gφp=n_f·(T(uf, pf)·n_f) +γuf·n_f.

Case 2: γf < γp. In this case, because e^k_φ∈Xp, we deduce that, thanks to the trace theorem and the Poincar´e inequality, there exists a constant Cp (independent ofK) such that

*e^k_φ*²Γ≤Cp*K⁻¹*ap(e^k_φ, e^k_φ). (3.18) Thus, ifγf < γp and

1 γf − 1

γp ≤ 2

gCp*K⁻¹*, (3.19)

we have

&

1−

$γf

γp

%2'

*ge^k_φ*²Γ−2γf

$ 1 +γf

γp

%

gap(e^k_φ, e^k_φ)

≤γfg

$ 1 + γf

γp

% $$ 1 γf − 1

γp

%

gCp*K⁻¹* −2

%

ap(e^k_φ, e^k_φ)

≤0.

Similarly, thanks to the trace theorem and Korn’s inequality, there exists a con- stantCf such that

*e^k_u·n_f*²Γ≤Cf

(

Ωf

|D(e^k_u)|²dx. Therefore, under the additional constraint

γp−γf ≤ 4ν Cf

,

we have

(γ_p²−γ_f²)*e^k_u·n_f*²Γ≤2(γf+γp)af(e^k_u,e^k_u). (3.20) Hence we have, when combined with Lemma 3.2,

*)^k+1_p *²Γ≤ *)^k_f*²Γ, *)^k+1_f *²Γ≤

$γf

γp

%2

*)^k_p*²Γ

which leads to

*)^k+1_p *^Γ≤ γf

γp*)^k−1_p *^Γ, *)^k+1_f *^Γ≤ γf

γp*)^k−1_f *^Γ.

This implies the convergence of the η which further implies the convergence of the velocity e^k_u, the pressuree^k_p, and the hydraulic heade^k_φ through the error equations (3.11) and (3.12)–(3.13).

Hence we have derived the following geometric convergence result.

Theorem 3.4. Assume that the parameters γp andγf are chosen so that

0< γp−γf≤ 4ν Cf

1 γf − 1

γp ≤ 2

gCp*K⁻¹*. (3.21) Then, the solutions of the parallel Robin-Robin domain decomposition method converge to the solution of the Stokes-Darcy system. Moreover,

ap(e^kφ, e^kφ) +af(e^k_u,e^k_u) +*e^kp*²Γ+*)^kp*²Γ+*)^kf*²Γ≤C

$γf

γp

%#^k2$)

*)⁰p*²Γ+*)⁰f*²Γ

*.

The last inequality follows from the geometric convergence of )^k_p, )^k_f, the error relationship at the interface and the error equations.

4. Finite element approximations. We next consider finite element discretiza- tion of the Robin domain decomposition method that was proposed in the previous section. One of the advantages of considering finite element approximations is that we can then derive explicit convergence rates, even for the case γf = γp = γ. Of course, the rate of convergence will depend on the size of the element h. This is different from the case ofγf < γp where the rate of convergence is independent ofh.

We will also demonstrate the convergence of the finite element approximation even in the parameter region ofγp < γf which may seem unlikely in view of Lemma 3.2.

One of the key advantages of the finite element setting is the availability of an inverse Poincar´e type inequality [3] that allows us to control various terms.

We consider a regular triangulation T^h of the global domain Ωp!

Ωf which is assumed to be regular and quasi-uniform. We also assume that the triangulation T^f,h,T^p,h induced on the subdomains Ωf and Ωp are compatible on Γ and the mesh on the interface Γ is quasi-uniform. The induced triangulation on Γ will be denoted B^h. Non-matching grid or mortar cases will be considered elsewhere. We denote by Xp,h ⊂ Xp a finite element space on the porous media domain Ωp and denote by Xf,h ⊂ Xf and Qf,h ⊂ Qf finite element spaces on the fluid domain Ωf. We use these spaces to approximate the hydraulic head in the porous media and the fluid velocity and pressure.

Specifically, we choose Xp,h={ψp,h∈C⁰(Ωp)

++

++ ψp,h|^T ∈P2(T), ∀T ∈ T^p,h, ψp,h

++

∂Ωp\Γ = 0} Xf,h={v_f,h∈(C⁰(Ωf))^d

++

++ v_f,h|^T ∈(P2(T))^d, ∀T ∈ T^f,h, v_f,h+

+∂Ωf\Γ= 0} Qf,h={qp,h∈C⁰(Ωf)

++

++ qf,h|^T ∈P1(T),∀T ∈ T^f,h}.

The spacesXf,handQf,hare assumed to satisfy the discrete LBB or inf-sup condition [10, 11].

We also define the discrete trace space on the interface Zh={ηh∈C⁰(Γ)

++

++ ηh|^τ ∈P2(τ), ∀τ∈ B^h, ηh|^∂Γ = 0}. It is easy to see thatZh is the trace space in the sense that

Yp,h: =Xp,h

++

Γ=Zh, Yf,h: =Xf,h

++

Γ·n_f =Zh.

The discrete weak formulation of the coupled Stokes-Darcy problem is then given by: find (uf,h, pf,h)∈Xf,h×Qf,handφp,h∈Xp,h such that

af(uf,h,v_f) +bf(vf, pf,h) +gap(φp,h, ψp) +&gφp,h,v_f·n_f' − &guf,h·n_f, ψp' +α&Pτu_f,h, Pτv_f'= (f,v_f)Ωf ∀v_f ∈Xf,h, ψp∈Xp,h

bf(uf,h, qf) = 0, ∀qf ∈Qf,h.

(4.1) The FEM approximation of the decoupled Stokes-Darcy system with Robin bound- ary conditions (2.1)–(2.2) can be formulated in the following way: for given ηp,h ∈ L²(Γ) andηf,h∈L²(Γ), find (φ#p,h,#u_f,h,p#f,h)∈Xp,h×Xf,h×Qf,h such that

af(#u_f,h,v_f) +bf(vf,p#f,h) +ωγpap(#φp,h, ψp) +ω&gφ#p,h, ψp'+γf&#u_f,h·n_f,v_f ·n_f' +α&Pτ#u_f,h, Pτv_f'= (f,v_f)Ωf +&ηf,h,v_f·n_f'+ω&ηp,h, ψp', ∀ψp∈Xp,h,v_f ∈Xf,h

bf(u#_f,h, qf) = 0, ∀qf ∈Qf,h.

(4.2) Similar to the continuous case, finite element approximations of the coupled Stokes-Darcy system, defined by (4.1), and of the revised Robin approximation, de- fined by (4.2), are related in the following fashion:

Lemma 4.1. For ηp,h∈Yp,h and ηf ∈Yf,h, (φ#p,h,u#_f,h,p#f,h) = (φp,h,u_f,h, pf,h) if and only if ηp,h andηf,h satisfy

ηp,h=Pp,h(γpu_f,h·n_f+gφp,h) ηf,h=Pf,h(γfu_f,h·n_f−gφp,h), wherePp,h andPf,h are L²(Γ)-projections onto the spacesYp,handYf,h respectively, i.e., forv∈L²(Γ),

&Pp,hv, wp'=&v, wp' ∀wp∈Yp,h &Pf,hv, wf'=&v, wf' ∀wf ∈Yf,h.

Remark. 4.2. Comparing with the Robin conditions (2.1) and (2.2), we see that we have heuristically used

Pp,h(γpK∇φp,h·n_p+gφp,h) =ηp,h

Pf,h(nf·(T(uf,h, pf,h)·n_f) +γfu_f,h·n_f) =ηf,h.

The choices of the spacesYp,handYf,hare not unique; other choices are also possible.

The parallel Robin-Robin domain decomposition finite element method is defined as follows.

1. The initial values of η⁰_p,h ∈ Yp,h and η_f,h⁰ ∈ Yf,h are guessed; they may be taken to be zero.

2. For k = 1,2, . . ., solve the discrete Stokes and Darcy systems with Robin conditions independently, i.e.,φ^k_p,h∈Xp,h is determined from

γpap(φ^k_p,h, ψp) +&gφ^k_p,h, ψp'=&η_p,h^k , ψp' ∀ψp∈Xp,h; andu^k_f,h∈Xf,h andp^k_f,h∈Qf,h are determined from

af(u^k_f,h,v_f) +bf(v_f, p^kf,h) +γf&u^k_f,h·n_f,v_f·n_f'+α&Pτu^k_f,h, Pτv_f'

=&η^k_f,h,v_f·n_f'+ (f,v_f)Ωf ∀v_f ∈Xf,h

bf(u^k_f,h, qf) = 0 ∀qf ∈Qf,h. 3. η_p,h^k+1 andη^k+1_f,h are updated by

η_f,h^k+1=Pf,h(aη^k_p,h+bgφ^k_p,h) η^k+1_p,h =Pp,h(cη_f,h^k +du^k_f·n_f).

One important observation is thatη^k_p,h, η_f,h^k , φ^k_p,h+

+Γ,u^k_f·n_f+

+Γ∈Zh for allk, pro- vided that the initial guesses belong toZh. Therefore, the projectionsPp,h andPf,h

in the implementation of the algorithm are identity operators.

We now consider the error functions for finite element approximations, just as in the continuous case studied in the previous section. Let

)^k_p,h=ηp,h−η_p,h^k )^k_f,h=ηf,h−η^k_f,h,

e^k_φ,h=φp,h−φ^k_p,h e^k_u,h=u_f,h−u^k_f,h e^k_p,h=pf,h−p^k_f,h. It is straightforward to verify that

)^k_p,h∈Zh )^k_f,h∈Zh e^k_φ,h|^Γ∈Zh e^k_u,h·nf∈Zh. It is also easy to see that the error functions satisfy the error equations

γpap(e^k_φ,h, ψp) +&ge^k_φ,h, ψp'=&)^k_p,h, ψp' ∀ψp∈Xp,h, (4.3)

af(e^k_u,h,v_f) +bf(v_f, e^k_p,h) +γf&e^k_u,h·n_f,v_f·n_f'+α&Pτe^k_f,h, Pτv_f'

=&)^k_f,h,vf·nf' ∀vf ∈Xf,h (4.4)

bf(e^k_u,h, qf) = 0 ∀qf ∈Qf,h, (4.5)

and, along the interface,

)^k+1_f,h =Pf,h(a)^k_p,h+bge^k_φ,h) )^k+1_p,h =Pp,h(c)^k_f,h+de^k_u,h·n_f).

It is easy to verify the following relationship for the error functions, just as in the continuous case.

Lemma 4.3. The error functions satisfy

*)^k+1_p,h*²Γ=*)^k_f,h*²Γ+ (γ_p²−γ_f²)*e^k_u,h·n*²Γ−2(γf+γp)af(e^k_u,h,e^k_u,h)−2(γf+γp)*Pτe^k_u,h*²Γ

*)^k+1_f,h*²Γ=

$γf

γp

%2

*)^k_p,h*²Γ+

&

1−

$γf

γp

%2'

*ge^k_φ,h*²Γ−2γf

$ 1 + γf

γp

%

gap(e^k_φ,h, e^k_φ,h).

The key ingredient in deriving an explicit convergence rate for the case ofγf =γp

is the following estimate.

Lemma 4.4.

*)^k_p,h*²Γ≤C(*K⁻¹*^1/2+γph^−1/2*K*^1/2)²ap(e^k_φ,h, e^k_φ,h).

Proof. The key in the proof of this lemma is an extension operator. LetN^p,hbe the set of nodes for the finite element triangulation on Ωp, and letN^p,Γ =N^p,h|^Γ. DenoteEp,h the zero extension operator fromYp,h=Zh toXp,h:

Ep,h)^k_p,h(P) =

, )^k_p,h(P) ifP ∈ N^p,Γ 0 ifP ∈ N^p,h\N^p,Γ. Then, we have

*Ep,h)^k_p,h*²L²(Ωp)≈h^d "

P∈Np,h

(Ep,h)^k_p,h(P))²≈h^d "

P∈Np,Γ

()^k_p,h(P))²≈h*)^k_p,h*²Γ. (4.6) Note thatEp,h)^k_p,h∈Xp,hdue to the definition of the extension operator and the fact that)^k_p,h∈Zh. Hence, we may setψp=Ep,h)^k_p,hin (4.3) and utilize Cauchy-Schwarz inequality to deduce

*)^k_p,h*²Γ=γpap(e^k_φ,h, Ep,h)^k_p,h) +&ge^k_φ,h, )^k_p,h'

≤γpap(e^k_φ,h, e^k_φ,h)^1/2ap(Ep,h)^k_p,h, Ep,h)^k_p,h)^1/2+*ge^k_φ,h*^Γ*)^k_p,h*^Γ. (4.7) On the other hand, thanks to the inverse inequality for finite element spaces and (4.6), we deduce

ap(Ep,h)^k_p,h, Ep,h)^k_p,h)≤ *K**∇Ep,h)^k_p,h*²L²(Ωp)

≤Ch⁻²*K**Ep,h)^k_p,h*²L²(Ωp)≤Ch⁻¹*K**)^k_p,h*²Γ. (4.8) Combining the inequality (3.18) with (4.7)–(4.8), we obtain

*)^k_p,h*²Γ≤Cγph^−1/2*K*^1/2ap(e^k_φ,h, e^k_φ,h)^1/2*)^k_p,h*^Γ+C_p^1/2*K⁻¹*^1/2ap(e^k_φ,h, e^k_φ,h)^1/2*)^k_p,h*^Γ which implies the lemma.

Remark. 4.5. Similar results are available forP1conforming and nonconforming elements for classical second-order elliptic problems( [16, 17]).

Now forγf =γp=γ, by Lemma 4.3, we have

*)^k+1_p,h*²Γ =*)^k_f,h*²Γ−4γaf(e^k_u,h,e^k_u,h)−4γ*Pτe^k_u,h*²Γ

*)^k+1_f,h*²Γ =*)^k_p,h*²Γ−4γgap(e^k_φ,h, e^k_φ,h).

Combining the above equalities with Lemma 4.4, we deduce

*)^k+1_p,h*²Γ≤ *)^k_f,h*²Γ, *)^k+1_f,h*²Γ≤(1−Cγ(*K⁻¹*^1/2+γh⁻¹²*K*^1/2)⁻²)*)^k_p,h*²Γ. Therefore, we have proved the following lemma.

Lemma 4.6. Forγf =γp=γ, we have

*)^k+1_p,h*²Γ≤(1−Cγ(*K⁻¹*^1/2+γh^−1/2*K*^1/2)⁻²)*)^k−1_p,h*²Γ

*)^k+1_f,h*²Γ≤(1−Cγ(*K⁻¹*^1/2+γh^−1/2*K*^1/2)⁻²)*)^k−1_f,h*²Γ.