Coupling free and porous-media flows:

modeling, analysis and numerical

approximation

Marco Discacciati

Special Semester on Multiscale Simulation & Analysisin Energy and the EnvironmentRICAM, Linz, October 5, 2011

Partial support of the Marie Curie Career Integration Grant 2011-294229

MOTIVATION

Modeling free and porous media flows requires to considercoupled differential models featuring Navier-Stokes equations inthe fluid domain and a filtration model in the porous domain, like

the Darcy equation.

⇒ Global coupled heterogeneous differential problem.

Environmental application Blood flow simulations

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Industrial applications: filters, porous foams, fuel cells...

PROBLEM SETTING

Fluid flow: Navier-Stokes equations

−divT(uf , pf ) + (uf · ∇)uf = f

divuf = 0in Ωf

where T(uf , pf ) = ν(∇uf +∇Tuf )− pf I is the Cauchy stresstensor.

Fluid through porous media: Darcy’s equations

K−1up +∇pp = 0

divup = 0in Ωp ⇔ −div (K∇pp) = 0 in Ωp

Free-fluid domain

Porous media domain

Γ

Ωf

Ωp

COUPLING (INTERFACE) CONDITIONSThe solution must satisfy three regularity conditions across Γ:

the continuity of the normal velocities

uf · n = up · n ⇔ uf · n = −K∇pp · n

a consequence of the incompressibility;

the continuity of the normal stresses

−n · T(uf , pf ) · n = pp

(pressures can be discontinuous across Γ);

a condition on the tangential component of the normal stress:Beavers–Joseph–Saffman equation

−τ · T(uf , pf ) · n = αuf · τ (−αup · τ )

[Miglio, Discacciati, Quarteroni (2002); Layton, Schieweck, Yotov (2003)]

COUPLING (INTERFACE) CONDITIONS

Experimental approach:

- 1967: Beavers and Joseph;- 1971: corrected by Saffman (removed up in the tangentialpart).

Mathematical approach:

- 1996, 2000, 2001: justification by Jager and Mikelic viahomogenization theory.

LITERATURE

A far-from-complete list of names:

Arbogast et al.; Badia, Codina;Becker et al; Bernardi et al.;Burman, Hansbo; Correa, Loula;D’Angelo, Zunino; Discacciati, Quarteroni;Iliev, Laptev; Kanschat;Layton; Fuhrmann et al.;Galvis, Sarkis; Gatica, Oyarzua;Girault; Gunzburger, Hua;Jager, Mikelic, Neuss; Mu, Xu, Zhu;Nassehi et al; Riviere;Urquiza et al.; Wolmuth, Helmig;Yotov; ...

MATHEMATICAL ANALYSIS OF THE COUPLEDPROBLEM

WEAK FORM OF THE COUPLED DARCY –

NAVIER-STOKES PROBLEM (I)

Find uf ∈ H1(Ωf ), pf ∈ L2(Ωf ), pp ∈ H1(Ωp):

∫

Ωf

ν∇uf · ∇v +

∫

Γα(uf · τ )(v · τ ) +

∫

Ωf

[(uf · ∇)uf ]v

−∫

Ωf

pf divv +

∫

Γpp(v · n) =

∫

Ωf

f · v∫

Ωf

qdivuf = 0

∫

Ωp

K∇pp · ∇ψ −∫

Γψ(uf · n) = 0

WEAK FORM OF THE COUPLED DARCY–NS

PROBLEM (II)

Find uf ∈ H1(Ωf ), pf ∈ L2(Ωf ), up ∈ L2(Ωp), pp ∈ H1(Ωp):

∫

Ωf

ν∇uf · ∇v +

∫

Γα(uf · τ )(v · τ ) +

∫

Ωf

[(uf · ∇)uf ]v

−∫

Ωf

pf divv +

∫

Γpp(v · n) =

∫

Ωf

f · v∫

Ωf

qdivuf = 0

∫

Ωp

K−1up ·w +

∫

Ωp

w · ∇pp = 0

∫

Ωp

up · ∇ψ +

∫

Γ(uf · n)ψ = 0

[Urquiza et al. (2008); Masud, Hughes (2005)]

ON THE WELL-POSEDNESS OF THE COUPLED

PROBLEMS

The Darcy-Stokes case: well-posedness can be easily provedusing the theory by Brezzi for saddle point problems.[Miglio et al. (2002); Layton et al. (2003); Urquiza et al. (2008)]

The Darcy-Navier–Stokes case: if there holds

‖f‖L2(Ωf ) ≤ Cν2

the Darcy-Navier–Stokes problem has a solution which isunique if the normal velocity across the interface is ‘smallenough’:

uf · n ∈ Srm = η ∈ H1/200 (Γ) : ‖η‖

H1/200 (Γ)

≤ rm ⊂ H1/200 (Γ)

for a suitably defined radius rm.[Girault, Riviere (2009); Badea, Discacciati, Quarteroni (2010)]

FINITE ELEMENT APPROXIMATION

A conforming FE approximation of this problem would lead to solvea global nonlinear system, generally large, sparse andill-conditioned.

“Laplace” – Navier-Stokes problem:

Aff (uf ) DTf Af Γ(uf ) 0 0

Df 0 Df Γ 0 0

AΓf (uf ) DTf Γ A

fΓΓ(uf ) 0 MΓΓ

0 0 0 App ATΓp

0 0 −MTΓΓ AΓp A

pΓΓ

ufpfuΓfpppΓp

= F

with uΓf → nodal values of uhf · n on ΓpΓp → nodal values of php on Γ

Darcy – Navier-Stokes problem:

Aff (uf ) DTf Af Γ(uf ) 0 0 0

Df 0 Df Γ 0 0 0

AΓf (uf ) DTf Γ A

fΓΓ(uf ) 0 0 MΓΓ

0 0 0 Ap GTp G

TΓp

0 0 0 Gp Spp SpΓ

0 0 MTΓΓ GΓp SΓp S

pΓΓ

ufpfuΓfuppppΓp

= F

NUMERICAL ALGORITHMS

THE DARCY – NAVIER-STOKES CASE

Fixed-point (Picard) method

Newton methodConvergence result: if

• ‖f‖L2(Ωf ) ≤ Cν2

then

• the Navier-Stokes/Darcy problem has a unique solution

• the Newton method converges to this solution provided theinitial normal velocity u0

f · n on Γ is chosen ‘close enough’ tothe solution.

We have to solve a linearized coupled problem at each iteration.

[Badea, Discacciati, Quarteroni (2010)]

NUMERICAL RESULTS

We take Ωf = (0, 1) × (1, 2) and Ωp = (0, 1) × (0, 1).We use Taylor-Hood elements for the Navier-Stokes equations andquadratic Lagrangian elements for the Darcy equation.The exact solution is uf = ((y − 1)2 + (y − 1) +

√Kx(x − 1)),

pf = 2ν(x + y − 1), ϕ = K−1(x(1− x)(y − 1)+ (y − 1)3/3)+ 2νx .

Number of iterations with respect to the parameters ν and K:

ν K h = 0.1429 h = 0.0714 h = 0.0357FP N FP N FP N

1 1 7 4 7 4 7 41 10−4 5 4 5 4 5 4

10−1 10−1 10 5 10 5 10 510−2 10−1 15 6 15 6 15 610−2 10−3 13 6 13 6 13 6

AN APPLICATION: internal ventilation of motorcycle

helmets (collaboration with F. Cimolin, Politecnico di Torino)

The ventilation system is realized by means of a series of channelscrossing the helmet. The air enters the channels from the airintakes, and the objective is to extract as much heat as possible.

[Cimolin, Discacciati (2010)]

This simplified scheme shows how the heat is extracted by thefresh air, which flows above and through the porous comfort layersurrounding the head:

(Schematic frontal cross section of the helmet)

(Schematic longitudinal cross section of an air channel)

NUMERICAL RESULTS (I)

2D problem discretized using about 200,000 elements.Navier-Stokes/Darcy problem solved by the Newton method.

Velocity field

A “DECOUPLED” STRATEGY

We would like (in this case for software availability reasons) tosolve the coupled problem exploiting the “intrinsic” decoupledstructure of our physical problem ⇒ alternate the solution ofthe Navier-Stokes problem in Ωf and of Darcy equations inΩp:

1 Given a normal stress on the interface, solve the Navier-Stokesequations in Ωf and recover the corresponding normal velocityacross the interface;

2 Use the computed normal velocity across the interface asboundary condition for the Darcy equations in Ωp, solve themand recover the corresponding normal stress on the interface;

3 Iterate using a suitable convergence criterion and a relaxationprocedure to enhance convergence (if necessary).

Use a domain decomposition approach⇒ write the global problem as an interface problem,choosing suitable interface variables.

NUMERICAL RESULTS (II)Steady-state flow field computed using a Navier-Stokes/Forchheimer model:

The normal component of the velocity through the interface andvelocity profile at the outlet:

10 20 30 40 50x

-0.1

0.0

0.1

0.2

uy

0.5 1.0 1.5 2.0 2.5 3.0 ux

-3

-2

-1

1

2

3

4

y

THE DOMAIN DECOMPOSITION FRAMEWORK

CHOICE OF THE INTERFACE VARIABLE

There are two possible strategies to choose the interface variable:

λ = uf · n on Γ; in that case we aim at satisfying

−K∇pp · n = λ on Γ

σ = pp on Γ; here, we aim at satisfying

−n · T(uf , pf ) · n = σ on Γ

Both choices are suitable from a mathematical standpoint sincethey yield well-posed subproblems in the fluid and the porous part.

INTERFACE EQUATION FOR DARCY–STOKESWe can equivalently express the Darcy-Stokes problem in terms ofthe solution λ (normal velocity across Γ) of the interface problem

Ssλ+ Sdλ = χ on Γ (1)

Ss continuous and coercive fluid operator:

Ss : λ (normal velocities on Γ)solve

−−−→Stokes

ξ = −n·T(uf , pf )·n (normal stresses on Γ).

Sd continuous, positive porous media operator:

Sd : λ (fluxes of pp on Γ)solve−−−→Darcy

ξ = pp|Γ.

Ss is spectrally equivalent to Ss + Sd : there exist two positiveconstants k1 and k2 (independent of η) such that

k1〈Ssη, η〉 ≤ 〈(Ss + Sd)η, η〉 ≤ k2〈Ssη, η〉 ∀η ∈ Λ0 ⊂ H1/200 (Γ).

There exists a unique solution λ ∈ H1/200 (Γ) for (1).

INTERFACE EQUATIONS AT THE DISCRETE LEVEL

The discrete counterpart of the interface equations are thesymmetric positive definite Schur complement systems:

Discrete interface equation for the normal velocity:

ΣsuΓf +Σdu

Γf = χs + χd

Discrete interface equation for the piezometric head:

Σf pΓp +Σpp

Γp = χf + χp

PRECONDITIONING TECHNIQUES (I)

Then, we can characterize the preconditioners:

for the interface equation involving uΓf :

P1 = (2α1)−1(Σs + α1I )(Σd + α1I ) α1 ≃

√ν

for the interface equation involving pΓp:

P2 = (2α2)−1(Σp + α2I )(Σf + α2I ) α2 ≃

√K

These preconditioners

have a multiplicative structure;

can be used within GMRES iterations;

generalize from the algebraic viewpoint the Robin-Robinmethod.

[see also Benzi (2009), Bai et al. (2003)]

NUMERICAL RESULTS (I)

Comparison between CG iterations without preconditioner andGMRES iterations...

with preconditioner P1 for the system involving uΓf

ν = 10−4, K = 10−3 ν = 10−6, K = 10−5 ν = 10−6, K = 10−8

CG GMRES + P1 CG GMRES + P1 CG GMRES + P1

h1 9 5 (α1 = 10−2) 9 4 (α1 = 10−3) 9 4 (α1 = 10−3)h2 20 7 (α1 = 10−2) 20 4 (α1 = 10−3) 20 4 (α1 = 10−3)h3 42 9 (α1 = 10−3) 42 4 (α1 = 10−3) 42 4 (α1 = 10−3)h4 64 9 (α1 = 10−3) 66 4 (α1 = 10−3) 66 4 (α1 = 10−3)

with preconditioner P2 for the system involving pΓp

ν = 10−4, K = 10−3 ν = 10−6, K = 10−5 ν = 10−6, K = 10−8

CG GMRES + P2 CG GMRES + P2 CG GMRES + P2

h1 11 8 (α2 = 10−2) 13 5 (α2 = 10−3) - 3 (α2 = 10−3)h2 22 9 (α2 = 10−2) 24 5 (α2 = 10−3) - 4 (α2 = 10−3)h3 47 10 (α2 = 10−2) 52 6 (α2 = 10−3) 57 4 (α2 = 10−3)h4 84 10 (α2 = 10−2) 108 6 (α2 = 10−3) 124 4 (α2 = 10−3)

PRECONDITIONING TECHNIQUES (II)

Effective preconditioners with additive structure can becharacterized as well, considering the following augmentedinterface systems:

Discrete augmented Dirichlet-Dirichlet (aDD) problem:

(

Σs −MΓ

MTΓ Σp

)(

uΓfpΓp

)

=

(

χs

χp

)

Discrete augmented Neumann-Neumann (aNN) problem:

(

Σd MΓ

−MTΓ Σf

)(

uΓfpΓp

)

=

(

χd

χf

)

In this case, we can characterize the preconditioners

for the aDD problem

P3 = (2α3)−1

(

Σs + α3I 00 Σp + α3I

)(

α3I −MΓ

MTΓ α3I

)

for the aNN problem

P4 = (2α4)−1

(

Σd + α4I 00 Σf + α4I

)(

α4I MΓ

−MTΓ α4I

)

These preconditioners

allow solving the fluid and the porous-media subproblemsindependently in a parallel fashion;

can be used within GMRES iterations.

NUMERICAL RESULTS (II)

Comparison between GMRES iterations without preconditioner forthe augmented systems...

with preconditioner P3 for the aDD problem

ν = 10−4, K = 10−3 ν = 10−6, K = 10−5 ν = 10−6, K = 10−8

GMRES GMRES + P3 GMRES GMRES + P3 GMRES GMRES + P3

h1 17 14 (α3 = 10−3) 17 7 (α3 = 10−3) 17 8 (α3 = 10−3)h2 33 17 (α3 = 10−3) 33 8 (α3 = 10−3) 33 10 (α3 = 10−3)h3 63 22 (α3 = 5 · 10−4) 65 8 (α3 = 5 · 10−4) 65 10 (α3 = 5 · 10−4)h4 67 23 (α3 = 5 · 10−4) 79 9 (α3 = 5 · 10−4) 101 11 (α3 = 5 · 10−4)

with preconditioner P4 for the aNN problem

ν = 10−4, K = 10−3 ν = 10−6, K = 10−5 ν = 10−6, K = 10−8

GMRES GMRES + P4 GMRES GMRES + P4 GMRES GMRES + P4

h1 17 16 (α4 = 0.1) 16 9 (α4 = 0.5) 9 8 (α4 = 1)h2 32 18 (α4 = 0.1) 32 8 (α4 = 0.5) 16 7 (α4 = 0.5)h3 59 20 (α4 = 5 · 10−2) 58 10 (α4 = 0.1) 30 5 (α4 = 0.8)h4 82 27 (α4 = 5 · 10−2) 81 8 (α4 = 0.1) 44 5 (α4 = 0.8)

[Discacciati (2011)]

THE NONLINEAR DARCY – NAVIER-STOKES CASE

An interface equation depending solely on the normal velocityλ across the interface can be characterized also for the Darcy– Navier-Stokes problem.

We can define a “fluid” non-linear pseudo-differential operatorSns analogous to Ss that associates to the normal velocity λon Γ the normal component of the corresponding Cauchystress tensor on Γ.

We can write the interface equation:

find λ ∈ Λ0 ⊂ H1/200 (Γ) : 〈Sns(λ) + Sdλ, µ〉 = 0 ∀µ ∈ Λ0

Preconditioning?

[Badea, Discacciati, Quarteroni (2010)]

SUMMARIZING...

Using domain decomposition techniques (without overlap)

we can reduce the global coupled problem to an equationdefined only on the interface, so that we get a linear systemwhose size coincides with the number of dofs on Γ;

we can characterize (optimal) preconditioners that we can usewithin iterative methods (CG, GMRES, ...);

we can solve the coupled problem considering separately each(simpler) subproblem.

This methodology relies strongly on the coupling conditions.We would like to study alternative approaches where the role ofcoupling conditions is not so strong.

A PENALIZATION APPROACH

Due to the difficulty of dealing with different type ofequations in the subdomains, a penalization approach(e.g., Iliev et al. 2004, 2007) is often adopted to model theflow over porous media.

This approach, similar to the fictitious domain approach ofAngot (1999), models the resistance induced by the porousmedium via penalization terms in the Navier-Stokes equations:

−divT(u, p) + (u · ∇)u+ν

Ku+

CF√K|u|u = f

divu = 0in Ωf ∪Ωp

CF is the inertial resistance coefficient.The penalization terms are set to zero in Ωf usingdiscontinous coefficients.

Solvers comparison

The normal component of the velocity through the interfaceand velocity profile at the outlet:

10 20 30 40 50

-0.2

-0.1

0.0

0.1

0.2

NS-Darcy – NS-Forchheimer – Penalization

THE VIRTUAL CONTROL APPROACH(Joint work with P. Gervasio and A. Quarteroni)

A TOY PROBLEM

Consider the boundary value problem

Lu = f in Ωu = φD on ΓD∂nLu = φN on ΓN

where L is the linear elliptic second-order operator

Lu = −div(K∇u) + b0u

We split Ω into two overlapping subdomains:

The problem can be equivalently rewritten as

Lu1 = f in Ω1 Lu1 = f in Ω1

Lu2 = f in Ω2 or Lu2 = f in Ω2

u1 = u2 in Ω12 Ψ(u1) = Ψ(u2) on Γ1 ∪ Γ2b.c . on ∂Ωi \ Γi b.c . on ∂Ωi \ Γi

where

Ψ(ui) =

uior βui + ∂nLui

on Γ1 ∪ Γ2

so that in the second problem we impose either

u1 = u2 on Γ1 ∪ Γ2

orβu1 + ∂nLu1 = βu2 + ∂nLu2 on Γ1 ∪ Γ2

VIRTUAL CONTROL APPROACH (with overlap)

Ω1, Ω2 ⊂ Ω, Ω12 = Ω1 ∩ Ω2 6= ∅, Γk = ∂Ωk \ ∂Ω, k = 1, 2.

Ω1

Ω1

Ω2

Ω2

Γ1Γ1 Γ2Γ2λ1 λ2

Lu1 = f in Ω1

Ψ(u1) = λ1 on Γ1b.c . on ∂Ω1 \ Γ1

Lu2 = f in Ω2

Ψ(u2) = λ2 on Γ2b.c . on ∂Ω2 \ Γ2

λ1, λ2 are solutions of a suitable minimization problem

infλ1,λ2

J(λ1, λ2)

VIRTUAL CONTROL APPROACH (with overlap)

It represents the formal mathematical justification ofengineering practice.

Origin and mathematical foundations: theory of optimalcontrol [J.-L. Lions (1971); Glowinski, Dinh, Periaux (1983); Glowinski,

Periaux, Terrasson (1990); Lions, Pironneau (1998, 1999)]

It is more “indifferent” w.r.t. interface conditions (no a-prioriinfo required), contrary to the domain decompositionapproach without overlap.

FUNCTIONAL SETTING

λi are the virtual controls that, depending on the choice of Ψ, maybe either

admissible Dirichlet controls

λi ∈ ΛDi ⊂ H1/2(Γi )

or admissible Robin controls

λi ∈ ΛRi ⊂ H−1/2(Γi )

We have to solve a control problem with boundary controls and...

... either distributed or boundary (interface) observation dependingon the choice of the cost functional J:

Minimization in the norm L2(Ω12)

J0(λ1, λ2) =12‖u1(λ1)− u2(λ2)‖2L2(Ω12)

Minimization in the norm H1(Ω12)

J1(λ1, λ2) =12‖u1(λ1)− u2(λ2)‖2H1(Ω12)

Minimization in the norm H1/2(Γ1 ∪ Γ2)

J1/2(λ1, λ2) =12‖u1(λ1)− u2(λ2)‖2H1/2(Γ1∪Γ2)

Minimization in the norm H−1/2(Γ1 ∪ Γ2)

J−1/2(λ1, λ2) =12‖∂nLu1(λ1)− ∂nLu2(λ2)‖2H−1/2(Γ1∪Γ2)

Theorem. For all the choices of functionals and for eitherDirichlet or Robin controls, the minimization probleminfλ1,λ2

J(λ1, λ2) has a unique solution.The solution of the minimization problemcoincides with the solution of the original problem.

Moreover, we can prove that the functionals J1/2 and J−1/2 are

equivalent to J1.

[Discacciati, Gervasio, Quarteroni (2011, in preparation)]

The solution of the virtual control approach satisfies (in weaksense) the optimality system.Considering, e.g., Dirichlet controls and minimization in the normL2(Ω12), we obtain:

State equations:

Lui = f in Ωi

ui = λi on Γib.c . on ∂Ωi \ Γi

Adjoint equations:

L∗pi = (−1)i+1(u1 − u2)χ12 in Ωi

pi = 0 on Γib.c . on ∂Ωi \ Γi

Euler equations:∂nLpi = 0 on Γi

ASSOCIATED ALGEBRAIC SYSTEM

Considering a conforming finite element approximation withmatching grids on Ω12, we obtain the linear system

A1 0 0 0 DTΓ10

0 A2 0 0 0 DTΓ2

M121 −M12

2 A1 0 0 0−M12

1 M122 0 A2 0 0

0 0 DΓ1 0 0 00 0 0 DΓ2 0 0

u1u2p1p2λ1

λ2

=

F1

F2

0

0

0

0

We can use two possible strategies to solve this system:

compute the Schur complement wrt the controls λi

solve the system at once for all variables

In both cases suitable preconditioners should be designed.

NUMERICAL RESULTS

We consider a test case with b0 = 1, Q1 elements.

Virtual control method: cost functional J0h Hδ K = 10−4 K = 1 K = 104

10−1 10−1 28 25 255 · 10−2 10−1 35 37 36

2.5 · 10−2 10−1 32 30 311.25 · 10−2 10−1 32 33 316.25 · 10−3 10−1 36 35 37

h Hδ K = 10−4 K = 1 K = 104

2 · 10−2 10−1 30 30 302 · 10−2 8 · 10−2 50 43 452 · 10−2 6 · 10−2 70 68 812 · 10−2 4 · 10−2 127 115 1232 · 10−2 2 · 10−2 236 213 266

Virtual control method: cost functional J1h Hδ K = 10−4 K = 1 K = 104

10−1 10−1 13 28 255 · 10−2 10−1 15 15 15

2.5 · 10−2 10−1 15 15 151.25 · 10−2 10−1 14 14 146.25 · 10−3 10−1 14 14 14

h Hδ K = 10−4 K = 1 K = 104

2 · 10−2 10−1 14 14 142 · 10−2 8 · 10−2 15 15 152 · 10−2 6 · 10−2 18 18 182 · 10−2 4 · 10−2 22 22 222 · 10−2 2 · 10−2 30 156 145

BiCGStab iterations on the Schur complement system;tol = 10−14.

Additive Schwarz methodh Hδ K = 10−4 K = 1 K = 104

10−1 10−1 95 95 945 · 10−2 10−1 94 94 94

2.5 · 10−2 10−1 94 94 951.25 · 10−2 10−1 95 94 946.25 · 10−3 10−1 94 95 94

h Hδ K = 10−4 K = 1 K = 104

2 · 10−2 10−1 95 94 942 · 10−2 8 · 10−2 117 117 1172 · 10−2 6 · 10−2 153 153 1532 · 10−2 4 · 10−2 226 226 2262 · 10−2 2 · 10−2 441 441 441

Multiplicative Schwarz methodh Hδ K = 10−4 K = 1 K = 104

10−1 10−1 49 49 495 · 10−2 10−1 48 48 48

2.5 · 10−2 10−1 48 48 491.25 · 10−2 10−1 49 49 486.25 · 10−3 10−1 49 48 49

h Hδ K = 10−4 K = 1 K = 104

2 · 10−2 10−1 48 48 482 · 10−2 8 · 10−2 60 60 602 · 10−2 6 · 10−2 78 78 782 · 10−2 4 · 10−2 116 116 1162 · 10−2 2 · 10−2 224 224 224

Tol = 10−14.

ONGOING WORK: VIRTUAL CONTROLS FOR

DARCY–STOKES

Ωp

ΩfΓp

ΓfΩfp

Stokesequations

−divT(uf , pf ) = f, div uf = 0 in Ωf

uf = λ1 on Γf

Darcyequations

−div (K∇pp) = 0 in Ωp

pp = λ2 on Γp

λ1, λ2 virtual controls, solutions of the MINIMUM PROBLEM

infλ1,λ2

J(λ1, λ2)

A possible choice for J may be: J(λ1, λ2) =∫

Ωfp(K∇pp − uf )

2

[Discacciati, Gervasio, Quarteroni (2010)]

OVERALL FRAMEWORK

Domain decomposition

With overlap Without overlap

Schwarz methods Steklov-Poincare equation- Schur complement system- DN/RR preconditioners

Virtual controls Virtual controls- boundary observation - boundary observation- distributed observation