Application of the probing technique

In this subsection, we apply the probing technique introduced in Section 2.5 to the Stokes-Darcy system. To do so, we need to reformulate the Stokes-Stokes-Darcy system in a substruc-tured form, that is, as an equation involving a Steklov-Poincaré operator over an interface variable. Then we apply the ADI method to solve the Steklov-Poincaré system similarly to (1.3.16). To the best of author’s knowledge, a first substructured formulation of the cou-pled Stokes-Darcy system has been provided in [52]. We consider the geometrical setup described by Figure 5.1 and we setΓ^N_d = ;and²=0, that is, we impose a zero tangential velocity and we introduce the space

H_s^τ:=n

us∈(H¹(Ωs))^d:us=0 onΓs andus·τj=0,j=1, . . . ,d−1 onΓo .

The multidomain weak formulation of the Stokes-Darcy system, counterpart of (1.3.2) for the Laplace equation, is [52, Proposition 2.4.1]

Findus∈H_s^τ,ps∈Qs,p_d∈H_dsuch that:

ae(us,v)+bs(v,ps)=R

Ωsfs·v, ∀v∈H₀¹(Ωs), b_s(us,q_s)=0, ∀q_s∈Q_s, aed(pd,qd)=R

Ωdgd· ∇qd, ∀qd∈H₀¹(Ωd), R

Γpdη=R

Ωsfs·E^sη−ae(us,E^sη)−bs(E^sη,ps), ∀η∈Λ. R

Γ(us·n)η=aed(pd,E^dη)−R

Ωdgd· ∇E^dη, ∀η∈Λ,

(5.3.7)

whereE^s andE^d are general continuous extension operators fromΛto respectivelyH_s^τ andHd. Equations (5.3.7)4,5are the coupling conditions along the interface and they play the same role as the continuity of the traces and of the normal derivatives in (1.3.2)_3,4. To

obtain the Steklov-Poincaré equation for (1.3.2), we selected a primal interface function, the trace, so that we could decompose each subdomain solution as the sum of two func-tions, one which is the harmonic extension of the trace, and another one which takes into account the force term, that isui=Hi(u_|Γ)+Gi(fi). In order for theui to be solutions, they still need to satisfy the continuity of the dual variable, that is the normal derivative, and this is exactly what the Steklov-Poincaré equation imposes. We are now going to fol-low the same logical path for the Stokes-Darcy system. We remark it is possible to use two different interface variables,λ:=us·n=(−K∇pd+gd)·nandϕ:=pd= −n·T(us,ps)·n. In contrast to the Laplace case, the functionλcorresponds to the trace of the Stokes velocity and to the normal derivative of the Darcy pressure, whileϕcorresponds to the trace of the Darcy pressure and to the normal component of the normal stress for the Stokes domain.

We could choose eitherλorϕas primal variables, however choosingλleads to some tech-nical difficulties. First, we would need to solve a Dirichlet Stokes boundary value problem which, in order to have a unique solution, requires to deal with a quotient space for the pressure field. We would have then to compute the normal stress alongΓwhich depends on the pressure defined up to a constant. Moreover, for a complete Dirichlet Stokes prob-lem, the boundary condition needs to satisfy the compatibility condition (5.3.6) which, supposing homogeneous boundary conditions alongΓs, implies thatR

Γλ=0. Thus we have a further constraint on the interface variable. For these reasons, we choose as in-terface variable the Darcy pressureϕ, and we refer the reader to [55, Section 5.1] for a detailed discussion concerning the interface variableλ.

We define the operatorHs:Λ→H_s^τ×Q_ssuch thatHs(ϕ)=(Hs¹(ϕ),Hs²(ϕ)) satisfies aes(Hs¹(ϕ),v)+bs(v,Hs²(ϕ))= −

Z

Γϕ(v·n), ∀v∈H_s^τ, bs(Hs¹(ϕ),q)=0, ∀q∈Qs,

(5.3.8)

and the function (u⁰_s,p⁰_s) ∈H_s^τ×Qs solution of the boundary value problem with zero normal stress condition alongΓ³

aes(u⁰_s,v)+bs(v,p⁰_s)= Z

Ωs

f·v, ∀v∈H_s^τ, bs(u⁰_s,q)=0, ∀q∈Qs.

(5.3.9)

Concerning the Darcy domain, we defineHd:Λ→H_dsuch thatHd(ϕ) satisfies ae_d(Hd(ϕ),q_d)=0, ∀q_d∈H₀¹(Ωd),

Hd(ϕ)=ϕ, onΓ, (5.3.10)

and the functionp_d⁰∈H₀¹(Ωd) solution of

ae_d(p_d⁰,q_d)= Z

Ωd

gd· ∇q_d, ∀q_d∈H₀¹(Ωd). (5.3.11)

3Using the interface variableλ, we would have to solve a Dirichlet Stokes problem, introducing the quo-tient spaceQ⁰_s:=

n q∈Q_s:R

Ωsq=0o .

Knowing a priori (u^ex_s ,p^ex_s ,p^ex_d ) solution of (5.3.7) and definingϕ=p^ex_d,|Γ, we would have that (u^ex_s ,p^ex_s )=(Hs¹(ϕ)+u⁰s,Hs²(ϕ)+p⁰s) andp_d^ex=Hd(ϕ)+pd. However we do not know the solution of (5.3.7) a priori. Nevertheless, we have that (Hs¹(ϕ)+u⁰_s,Hs²(ϕ)+p⁰_s) and Hd(ϕ)+p_dsatisfy already the first four equations of (5.3.7). We use the fifth to obtain an equation for the unknown interface variableϕ. Replacing (Hs¹(ϕ)+u⁰_s,Hs²(ϕ)+p⁰_s) and Hd(ϕ)+pdinto (5.3.7)5, we get

〈Sϕ,η〉 = 〈χ,η〉, (5.3.12) whereS =Ss+Sdare defined as

〈Ssϕ,η〉:= − Z

Γ(Hs¹(ϕ)·n)η, 〈Sdϕ,η〉:=aed(Hd(ϕ),Hd(η)), (5.3.13) andχ∈Λ⁰is such that

〈χ,η〉 = Z

Γ(u⁰_s·n)η+ Z

Ωd

gd· ∇Hd(η)−aed(p⁰_d,Hd(η)).

We emphasise that we set the extension operatorEd=Hd. Solving equation (5.3.12) per-mits to obtain the exact trace pressureϕ, from which we can then recover the exact lo-cal solutions performing subdomain solves. For a proof of existence and uniquess of the solution to (5.3.12) we refer to [52, Proposition 2.6.1]. Repeating the same calculations presented at the end of Section 1.3.4 for the Laplace equation, we can rewrite the trans-mission conditions of (5.1.7) at the weak level as

〈(s2Ss+I)λⁿ,η〉 = 〈(I−s2Sd)λⁿ⁻¹,η〉 + 〈s2χ,η〉, ∀η∈Λ,

〈(s₁Sd+I)λⁿ,η〉 = 〈(I−s₁Ss)λⁿ⁻¹,η〉 + 〈s₁χ,η〉, ∀η∈Λ, (5.3.14) which leads to the fixed point iteration for the error equation

λⁿ⁺¹=(s₂Ss+I)⁻¹(I−s₂Sd)(s₁Sd+I)⁻¹(I−s₁Ss)λⁿ⁻¹=T(s₁,s₂)λⁿ⁻¹.

We are now perfectly in the framework discussed in Section 2.5. We introduce finite di-mensional approximationsΣs,Σd of the operatorsSs andSd. We thus choose a set of probing vectors©

xj

ªwith j in some index setK, and we aim to minimize numerically the ratio

smin1,s2∈Rmax

j∈I

ks2Σsxj−M_Γxjk ks1Σdxj+M_Γxjk

ks1Σdxj−M_Γxjk

ks2Σsxj+M_Γxjk, (5.3.15) whereM_Γis the mass matrix over the interface defined as (M_Γ)i,j:=R

Γφiφj, and© φkª

is a basis for the finite element approximation ofΛ.

We set the physical parameters equal toµ=0.1,η1=η2=1. We consider the probing vectorsxjsuch thatxjis an approximation of sin(πkx), andkbelongs to a index setK. The optimized parameters are equal tos2=^p_η¹

1η2p ands1=2µp, withp∈R. On the left of Figure 5.8, we consider Dirichlet boundary conditions all along∂Ω. We consider two different sets,K1:=©

1,p N,Nª

andK2:={1, 2,N}, whereN is the number of degrees of

0 20 40 60 80 100 p

0 0.5 1 1.5 2

ρ(T(s1, s2)) Fourier analysis P robing−K₁ P robing−K₂

0 20 40 60 80 100

p 0

0.5 1 1.5 2

ρ(T(s1, s2)) Fourier analysis P robing−K₁

Figure 5.8: Comparison between the spectral radius of the iteration operatorT(s1,s2) and several estimated parameters through the probing technique.

freedom alongΓ. As we discussed, the Fourier analysis does not provide good estimates.

However, also probing with the set indexK1does not provide good results. Guided by the left panel of Figure 5.4 where we saw that the initialisation with the first even frequency lead to the slowest numerical convergence factor, and since the first odd frequency does not satisfy (5.3.6), we decided to probe with the setK2. We remark that the addition of the first even frequency in the index set leads to a very precise estimate of the optimal pa-rameter. The choice of the set of probing vector is thus not trivial for the case of Dirichlet boundary conditions. We report that we also tried the power method approach described in Section 2.5, but we did not observe any significant improvement with respect toK1. We then impose a zero normal stress condition on the upper horizontal edge ofΩs. Due to this conditions, the velocity field does not need to satisfy (5.3.6). We remark that both the Fourier analysis and the probing technique with index setK1permit to get excellent optimized parameters. This experiment corroborates our statement that the compatibil-ity condition is the key element for the failure of the Fourier analysis for the Stokes-Darcy coupling.