Schwarz waveform relaxation for nonlinear problems · 2011-10-17 · reef, wit h onl y a li mited...

Motivation for waveform relaxation Waveform relaxation Optimized Schwarz waveform relaxation algorithms for parabolic equations Nonlinear problems

Schwarz waveform relaxation for nonlinear problems

Laurence HALPERN

LAGA - Université Paris 13

Simulation of Flow in Porous Mediaand Applications in Waste Management and CO2 Sequestration

Radon Institute in LinzOktober 2011

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Outline

1 Motivation for waveform relaxation

2 Waveform relaxation

3 Optimized Schwarz waveform relaxation algorithms for parabolicequationsOptimized Schwarz algorithms for advection-diusion equationNumerical experiments

4 Nonlinear problemsA theoretical studyA modied approach

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Example : ocean-atmosphere coupling

couplers (OASIS for example), which can be difficult to implement and therefore goes against our principle of portability, a specific coupler has been devised for ROMSWRF (Lemarié et al., 2008). Actual 2way coupling algorithms for realistic highresolution regional applications are not numerous; examples can be found in Bao et al. (2000). The approach used for climate applications with low coupling frequency are not appropriate for generally fast modes regional applications, and may generate large inaccuracies in the numerical solutions. On the other hand, high frequency coupling (at the ocean time step) may also have its drawbacks as estimated turbulent fluxes can be uncertain on time scales lower than 10 minutes. The choice is either to implement more detailed physical processes relevant to high temporal frequency coupling, such as spray contributions to heat fluxes and the wave boundary layer (Bao et al., 2000), and/or to improve asynchronous methods. In this latter case, it may be useful to impose various coupling properties, in particular the convergence of oceanic and atmospheric fluxes

at the interface as well as flux conservation between the two systems. In our coupling algorithm, an iterative method based on domain decomposition (Globalintime nonoverlapping Schwarz methods) was implemented to obtain convergence between fluxes. A FORTRAN package manages the calls to oceanic and atmospheric models and their coupling. It has been applied successfully to the coupled simulation of real tropical cyclones showing the role of oceanic feedback in limiting the potential growth of tropical cyclones (Lemarié et al., 2008). The application of the coupling method to our nowcast/forecast system is in principle straightforward and will be implemented in the future.

6. The lagoon

Lagoons are very specific bodies of sea water, often largely isolated from oceanic waters by a barrier reef, with only a limited number of passes to make the connection. The Lagoon of New Caledonia is the

largest closed Lagoon in the world with large biological diversity and degrees of endemism. As such it is a valuable entity to be understood, surveyed and protected. The Lagoon is shallow and its circulation is dominated by barotropic tides and direct wind influences. Therefore, its modeling requires accurate tidal forcing and wind forcing. For tides, freely available data sets such as those from the global model of ocean tides TPXO (version 6 or 7),

8

coupled simulation (bottom) of Cyclone Erica moving over New Caledonia in March 2003 (the actual path is similar to the simulated one except that it crosses New Caledonia about 200km south of the simulated path). Note that cold waters are produced on the cyclone track due to strong surface mixing induced by the hurricaneforce winds; the cold water induces a negative feedback on the tropical cyclone which is weaker, hence and more realistic, as a results (orange rather than red color).

couplers (OASIS for example), which can be difficult to implement and therefore goes against our principle of portability, a specific coupler has been devised for ROMSWRF (Lemarié et al., 2008). Actual 2way coupling algorithms for realistic highresolution regional applications are not numerous; examples can be found in Bao et al. (2000). The approach used for climate applications with low coupling frequency are not appropriate for generally fast modes regional applications, and may generate large inaccuracies in the numerical solutions. On the other hand, high frequency coupling (at the ocean time step) may also have its drawbacks as estimated turbulent fluxes can be uncertain on time scales lower than 10 minutes. The choice is either to implement more detailed physical processes relevant to high temporal frequency coupling, such as spray contributions to heat fluxes and the wave boundary layer (Bao et al., 2000), and/or to improve asynchronous methods. In this latter case, it may be useful to impose various coupling properties, in particular the convergence of oceanic and atmospheric fluxes

at the interface as well as flux conservation between the two systems. In our coupling algorithm, an iterative method based on domain decomposition (Globalintime nonoverlapping Schwarz methods) was implemented to obtain convergence between fluxes. A FORTRAN package manages the calls to oceanic and atmospheric models and their coupling. It has been applied successfully to the coupled simulation of real tropical cyclones showing the role of oceanic feedback in limiting the potential growth of tropical cyclones (Lemarié et al., 2008). The application of the coupling method to our nowcast/forecast system is in principle straightforward and will be implemented in the future.

6. The lagoon

Lagoons are very specific bodies of sea water, often largely isolated from oceanic waters by a barrier reef, with only a limited number of passes to make the connection. The Lagoon of New Caledonia is the

largest closed Lagoon in the world with large biological diversity and degrees of endemism. As such it is a valuable entity to be understood, surveyed and protected. The Lagoon is shallow and its circulation is dominated by barotropic tides and direct wind influences. Therefore, its modeling requires accurate tidal forcing and wind forcing. For tides, freely available data sets such as those from the global model of ocean tides TPXO (version 6 or 7),

8

coupled simulation (bottom) of Cyclone Erica moving over New Caledonia in March 2003 (the actual path is similar to the simulated one except that it crosses New Caledonia about 200km south of the simulated path). Note that cold waters are produced on the cyclone track due to strong surface mixing induced by the hurricaneforce winds; the cold water induces a negative feedback on the tropical cyclone which is weaker, hence and more realistic, as a results (orange rather than red color).

Florian Lemarié thèse 2009http://www.atmos.ucla.edu/ florian/Research.html

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The issues

One couples dierent models in dierent zones, which live ondierent time-scales.

The coupling must be very ecient : one or two iterations.

The data of one model are not available at each time-step of theneighboring model.

One must avoid the communication time to overrule thecomputational time.

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The issues





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The issues





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The issues





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Contents





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Outline





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The ancestor : Émile Picard

Journ. de Math.(4ème série), tomeVI.-Fasc II, 1890Traité d'Ana-lyse, tome 2,1891,Gauthier-Villars.

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The MOS

Exemple original de 1982 pour les méthodes de relaxation d'ondes

Circuit complet Circuit partitionné

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Waveform relaxationReview : Burrage et al, Appl. Num. Math. 1996.

y1 = f1(t, y1, y2, · · · , yp),y2 = f2(t, y1, y2, · · · , yp)yj = fj (t, y1, y2, · · · , yp)yp = fp(t, y1, y2, · · · , yp)

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Approximations successives

y(k+1)1

= f1(t, y(k)1, y

(k)2, · · · , y (k)

p ),

y(k+1)2

= f2(t, y(k)1, y

(k)2, · · · , y (k)

p )

y(k+1)j

= fj (t, y(k)1, y

(k)2, · · · , y (k)

p )

y(k+1)p = fp(t, y

(k)1, y

(k)2, · · · , y (k)

p )

‖y (k+1)−y‖∞ ≤Lk(T − t0)k

k!‖y (0)−y‖∞

Lindelöf, Journ. de Math. (4ème sé-rie), tome X.-Fasc II, 1894

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Jacobi

y(k+1)1

= f1(t, y(k+1)1

, y(k)2, y

(k)j, · · · , y (k)

p ),

y(k+1)2

= f2(t, y(k)1, y

(k+1)2

, y(k)j, · · · , y (k)

p )

y(k+1)j

= fj (t, y(k)1, y

(k)2, · · · , y (k+1)

j, · · · , y (k)

p )

y(k+1)p = fp(t, y

(k)1, y

(k)2, y

(k)j, · · · , y (k+1)

p )

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Gauss-Seidel

y(k+1)1

= f1(t, y(k+1)1

, y(k)2, · · · , y (k)

j, y

(k)p ),

y(k+1)2

= f2(t, y(k+1)1

, y(k+1)2

, · · · , y (k)j, y

(k)p )

y(k+1)j

= fj (t, y(k+1)1

, y(k+1)2

, · · · , y (k+1)j

, · · · , y (k)p )

y(k+1)p = fp(t, y

(k+1)1

, y(k+1)2

, · · · , y (k+1)j

, y(k+1)p )

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Theoretical study : O. Nevanlinna, Remarks on Picard-Lindelöf iterations.BIT 1989.

There can be a slow convergence in the case of strong coupling

between subcircuits. Fortunately for integrated problems, this

strong coupling occurs only on short time intervals.

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Outline





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The Schwarz waveform relaxation algorithm

Ω1 Ω2Γ1Γ2

t

(∂t −∆)uk+1

1= f in Ω1 × (0,T )

uk+1

1(·, 0) = u0 in Ω1

uk+1

1= uk2 on Γ1 × (0,T )

(∂t −∆)uk+1

2= f in Ω2 × (0,T )

uk+1

2(·, 0) = u0 in Ω2

uk+1

2= uk1 on Γ2 × (0,T )

Gander 1997 (thesis), Giladi-Keller 1997 (Icase report).

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Properties

Gander-Stuart 1997, Gander-Zhao 2002. Giladi-Keller 2002(advection-diusion)

792 M. J. GANDER AND H. ZHAO

0 2 4 6 8 10 12 14 16 18 2010

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

iteration k

max

imum

err

or

0.1 overlap 0.06 overlap

Figure 5.5: Two dimensional model problem with four subdomains, algorithm in thesuperlinear convergence regime for T = 0.05.

Table 5.1: Number of iterations needed to reach a certain tolerance for different numbersof subdomains and two different time intervals. While the algorithm does not scale inthe linear convergence regime with respect to the number of subdomains, it does scale inthe superlinear convergence regime.

2 × 2 3 × 3 4 × 4 6 × 6T = 5 12 15 19 28T = 0.01 5 5 5 6

added to the overlapping Schwarz waveform relaxation algorithm. This is howevernot necessary when the algorithm is in the superlinear convergence regime. Herethe algorithm converges independently of the number of subdomains, as predictedby Theorem 4.3 and shown in Table 5.1 experimentally. This can be understoodintuitively by noting that it is the initial condition which determines over shorttime intervals mainly the solution of a parabolic evolution problem and thus wehave error decay away from the interior boundaries even if they are far away fromthe real boundary, since the initial conditions are known for all subdomains. Thisis illustrated in Figure 5.6 which shows the error of three consecutive iterates for4 × 4 subdomains at the end of the time interval on the left for T = 5 where thealgorithm is in the linear convergence regime and on the right for T = 0.01 wherethe algorithm is in the superlinear convergence regime. One can clearly see howthe error is diminished in all subdomains on the right due to the initial condition,whereas it has to be eliminated from the original boundaries on the left.

T=0.05 : superlinear convergenceregime

‖e2nj‖ ≤ erfc( nδ√

T)‖e0

j‖

L∞(Γj × (0,T ))

OVERLAPPING SCHWARZ WAVEFORM RELAXATION 791

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

100

Iteration

max

imum

err

or


Figure 5.4: Two dimensional model problem with four subdomains, algorithm in thelinear convergence regime for T = 3.

analysis. We decompose the unit square into four smaller squares which formoverlapping subdomains of equal size. We run the overlapping Schwarz waveformrelaxation algorithm in its additive version. For the overlap parameter δ we chosetwo values, δ ∈ 0.1, 0.06. We solve the subdomain problems using a centeredfinite difference scheme with ∆x = 0.02 and integrate in time using backwardEuler. Figure 5.4 shows the algorithm in the linear convergence regime, integratingup to T = 3.

Figure 5.5 shows the algorithm in the superlinear convergence regime, integratingover a shorter time interval, T = 0.05. Note the different scale which shows howmuch faster the superlinear convergence is compared to the linear convergence inthe previous experiment.

5.3 Scaling in the number of subdomains.

We finally analyze the scaling behavior of overlapping Schwarz waveform re-laxation numerically. We solve again the heat equation in two dimensions (5.2)but now vary the number of subdomains. We use a discretization in space with∆x = 1/30. Table 5.1 shows the number of iterations needed to decrease the errorbelow a given tolerance for different numbers of subdomains and the two timeintervals [0, 5] and [0, 0.01] with fixed overlap parameter δ = 1/30.

The table shows that over long time intervals, the overlapping Schwarz waveformrelaxation algorithm does not scale with respect to the number of subdomains.This is due to the fact that the convergence rate is limited by the steady statesolution and corresponds to the m factor in Theorem 3.2. For elliptic problems itis well known that overlapping Schwarz needs a coarse mesh to exhibit convergenceindependent of the number of subdomains [9]. Such a coarse mesh could also be

T=3 : linear convergence regime

‖e2nj‖ ≤ (φ(δ))n‖e0

j‖

L∞(Γj × R+)

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Properties

Gander-Stuart 1997, Gander-Zhao 2002. Giladi-Keller 2002(advection-diusion)

792 M. J. GANDER AND H. ZHAO

0 2 4 6 8 10 12 14 16 18 2010

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

iteration k

max

imum

err

or


Figure 5.5: Two dimensional model problem with four subdomains, algorithm in thesuperlinear convergence regime for T = 0.05.

Table 5.1: Number of iterations needed to reach a certain tolerance for different numbersof subdomains and two different time intervals. While the algorithm does not scale inthe linear convergence regime with respect to the number of subdomains, it does scale inthe superlinear convergence regime.

2 × 2 3 × 3 4 × 4 6 × 6T = 5 12 15 19 28T = 0.01 5 5 5 6

added to the overlapping Schwarz waveform relaxation algorithm. This is howevernot necessary when the algorithm is in the superlinear convergence regime. Herethe algorithm converges independently of the number of subdomains, as predictedby Theorem 4.3 and shown in Table 5.1 experimentally. This can be understoodintuitively by noting that it is the initial condition which determines over shorttime intervals mainly the solution of a parabolic evolution problem and thus wehave error decay away from the interior boundaries even if they are far away fromthe real boundary, since the initial conditions are known for all subdomains. Thisis illustrated in Figure 5.6 which shows the error of three consecutive iterates for4 × 4 subdomains at the end of the time interval on the left for T = 5 where thealgorithm is in the linear convergence regime and on the right for T = 0.01 wherethe algorithm is in the superlinear convergence regime. One can clearly see howthe error is diminished in all subdomains on the right due to the initial condition,whereas it has to be eliminated from the original boundaries on the left.

T=0.05 : superlinear convergenceregime

‖e2nj‖ ≤ erfc( nδ√

T)‖e0

j‖

L∞(Γj × (0,T ))

OVERLAPPING SCHWARZ WAVEFORM RELAXATION 791

0 2 4 6 8 10 12 14 16 18 2010

−3

10−2

10−1

100

Iteration

max

imum

err

or


Figure 5.4: Two dimensional model problem with four subdomains, algorithm in thelinear convergence regime for T = 3.

analysis. We decompose the unit square into four smaller squares which formoverlapping subdomains of equal size. We run the overlapping Schwarz waveformrelaxation algorithm in its additive version. For the overlap parameter δ we chosetwo values, δ ∈ 0.1, 0.06. We solve the subdomain problems using a centeredfinite difference scheme with ∆x = 0.02 and integrate in time using backwardEuler. Figure 5.4 shows the algorithm in the linear convergence regime, integratingup to T = 3.

Figure 5.5 shows the algorithm in the superlinear convergence regime, integratingover a shorter time interval, T = 0.05. Note the different scale which shows howmuch faster the superlinear convergence is compared to the linear convergence inthe previous experiment.

5.3 Scaling in the number of subdomains.

We finally analyze the scaling behavior of overlapping Schwarz waveform re-laxation numerically. We solve again the heat equation in two dimensions (5.2)but now vary the number of subdomains. We use a discretization in space with∆x = 1/30. Table 5.1 shows the number of iterations needed to decrease the errorbelow a given tolerance for different numbers of subdomains and the two timeintervals [0, 5] and [0, 0.01] with fixed overlap parameter δ = 1/30.

The table shows that over long time intervals, the overlapping Schwarz waveformrelaxation algorithm does not scale with respect to the number of subdomains.This is due to the fact that the convergence rate is limited by the steady statesolution and corresponds to the m factor in Theorem 3.2. For elliptic problems itis well known that overlapping Schwarz needs a coarse mesh to exhibit convergenceindependent of the number of subdomains [9]. Such a coarse mesh could also be

T=3 : linear convergence regime

‖e2nj‖ ≤ (φ(δ))n‖e0

j‖

L∞(Γj × R+)

Mathematical tools : maximum principle and Laplace-Fourier transformin time/transverse space variables. 12 / 49


The Modied Schwarz algorithm

Jacobi or Gauss-Seidel way :

Lu := ∂tu + (a · ∇)u − ν∆u + cu in Ω× (0,T )Luk+1

1= f in Ω1 × (0,T )

uk+1

1(·, 0) = u0 in Ω1

B1uk+1

1= B1uk2 on Γ1 × (0,T )

Luk+1

2= f in Ω2 × (0,T )

uk+1

2(·, 0) = u0 in Ω2

B2uk+1

2= B2uk+1

1on Γ2 × (0,T )

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Generalisation

Optimal Schwarz Waveform relaxation WITH OR WITHOUT overlap.Equation

Lu := ∂tu + (a · ∇)u − ν∆u + cu in Ω× (0,T )

Boundary operators

Bju := (ν∇u − a) · nj + pu + q(∂t + a · ∇u − ν∆Su + cu)

Theorems

Under some conditions on p, q, the algorithm is well-posed in suitedSobolev spaces and converges with and without overlap, includingvariable coecients and curved boundaries (Martin, Halpern, Japhet,Szeftel, Gander, without overlap, Minh Binh Tran with overlap).

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Choice of the coecients

♣ lLu := ∂tu + (a · ∇)u − ν∆u + cu in Ω× (0,T )

Bju := (ν∇u − a) · n + pu + q(∂t + a · ∇u − ν∆Su + cu)

Best approximation problem,

p∗, q∗(a, ν, c,∆t,∆x).

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Asymptotic formulae ∆t ∼ C ′h

Dirichlet or Neumann transmission conditions

ρ∗ ∼ 1− Ch overlap L ≈ Ch

Robin transmission conditionsρ∗ ∼ 1− Ch1/2 without overlap

ρ∗ ∼ 1− Ch1/3 with overlap

q 6= 0 ρ∗ ∼ 1− Ch1/4 without overlap

ρ∗ ∼ 1− Ch1/5 with overlap

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Test case : porosity + rotating velocity+ nonconformal grids

Lu = ϕ∂u

∂t+∇ · (a(x)u − ν(x)∇u) + cu = f

a1 = (−sin(π2

(y − 1))cos(π(x − 1

2)), 3cos(π

2(y − 1))sin(π(x − 1

2))),

ν1 = 0.003, ϕ1 = 0.1 , a2 = a1, ν2 = 0.01 , ϕ2 = 1

Computational domain (0, 1)× (0, 2), nal time T = 1.5

−0.2 0 0.2 0.4 0.6 0.8 1 1.2−0.5

0

0.5

1

1.5

2

2.5

x

y

Velocity field

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2 DG−OSWR Solution, At time t=T=0

x

y

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

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Convergence

mean value of the (p∗, q∗) obtained by optimization of the convergencefactor.

0 2 4 6 8 10 12 14 16 18 2010−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100Error between momodomain (variational) and DGOSWR solutions versus the iterations

Iterations

log 10

(Err

or)

Order2Robin

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Conclusion for linear parabolic problems

Robin transmission conditions are better than Dirichlet, but secondorder transmission conditions improve signicantly.

overlap is better if possible, but nonoverlapping with second ordershould be considered if not.

The convergence rate is almost independent of the discretizationparameters.

Very robust when applied to variable coecients.

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Outline





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Frame

Frame : ANR SHP CO2. Collaborators : Filipa Caetano, A. Michel (IFP),Florian Haeberlein (doctorant).Martin Gander, , Jérémie Szeftel, Minh Binh Tran (doctorant).

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A theoretical study

L. Halpern and J. Szeftel. Nonlinear Schwarz Waveform Relaxation for SemilinearWave Propagation. Math. Comp. 2009,

F. Caetano, M. Gander, L. Halpern and J. Szeftel. Schwarz waveform relaxation

algorithms for semilinear reaction-diusion. September 2010

ut − ν∆u + f (u) = 0 dans Rd × (0,T ),u(·, 0) = u0 dans Rd ,

f ∈ C2(R), f (0) = 0.

Theorem

If u0 ∈ H2(R2), then there exists T > 0 such that the Cauchy problem

has a unique weak solution u ∈ L2(0,T ;H1(R2))∩ C([0,T ]; L2(R2)). Inaddition u ∈ L∞(0,T ;H2(R2)).

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Lu := ∂tu − ν∆u + f (u)Luk+1

1= 0 in Ω1 × (0,T k+1

1)

uk+1

1(·, 0) = u0 in Ω1

B1uk+1

1= B1uk2 on Γ1 × (0,T k+1

1)

Luk+1

2= 0 in Ω2 × (0,T k+1

2)

uk+1

2(·, 0) = u0 in Ω2

B2uk+1

2= B2uk+1

1on Γ2 × (0,T k+1

2)

Bi (u) = ν∂u

∂ni+ pu + q

(∂u∂t− ν ∂

2u

∂y2

), p > 0, q ≥ 0

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Existence for the iterates

Theorem

Let g01and g0

2in H1(0,T ; L2(Γ)) ∩ L∞(0,T ;H

1

2 (Γ)), u0 ∈ H2(R2),p > 0 and q ≥ 0 be given. Suppose that (ν∂ni u0 + pu0)|Γ = g0

i (0, ·), ifq = 0. Then, the algorithm denes a unique sequence of iterates (uk

1, uk

2)

q = 0 : uki ∈W 1,∞(0,T ki ; L2(Ωi )) ∩ L∞(0,T k

i ;H2(Ωi )) ∩ H1(0,T ki ;H1(Ωi )),

q > 0 : uki ∈W 1,∞(0,T ki ; L2(Ωi )) ∩ L∞(0,T k

i ;H2

2 (Ωi )) ∩ H1(0,T ki ;H1(Ωi )),

for some T ki ≤ T

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Existence for the iterates

Theorem

Let g01and g0

2in H1(0,T ; L2(Γ)) ∩ L∞(0,T ;H

1

2 (Γ)), u0 ∈ H2(R2),p > 0 and q ≥ 0 be given. Suppose that (ν∂ni u0 + pu0)|Γ = g0

i (0, ·), ifq = 0. Then, the algorithm denes a unique sequence of iterates (uk

1, uk

2)

q = 0 : uki ∈W 1,∞(0,T ki ; L2(Ωi )) ∩ L∞(0,T k

i ;H2(Ωi )) ∩ H1(0,T ki ;H1(Ωi )),

q > 0 : uki ∈W 1,∞(0,T ki ; L2(Ωi )) ∩ L∞(0,T k

i ;H2

2 (Ωi )) ∩ H1(0,T ki ;H1(Ωi )),

for some T ki ≤ T

T k1 ≤ T k−1

2, T k

2 ≤ T k−11

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Proof

q = 0. Banach's xed point theorem in a ball in

H(T ) := W 1,∞(0,T ; L2(Ωi )) ∩ L∞(0,T ;H2(Ωi )) ∩ H1(0,T ;H1(Ωi )).

Lemma

Let T > 0. Let u0 ∈ H2(Ωi ), g ∈ H1(0,T ; L2(Γ)) ∩ L∞(0,T ;H1

2 (Γ)).For any v ∈ H(T ), the linear problem

wt − ν∆w = −f (v), in Ωi × (0,T ),

w(·, ·, 0) = u0|Ωi, in Ωi ,

ν ∂w∂ni + pw = g , over Γ× (0,T ).

(1)

has a unique solution in H(T ), hence dening an application w = T (v)in H(T ), with

‖T (v)‖2H(T ) ≤ CeT(‖u0‖2H2(Ωi )

+ ‖g‖2H1(0,T ;L2(Γ))∩L∞(0,T ;H

1

2 (Γ))

+T (ϕ(‖v‖L∞((0,T )×Ωi )))2‖v‖2W 1,∞(0,T ;L2(Ωi ))

)(2)

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continue

Let M be such that

M2 ≥ 4C (‖u0‖2H2(Ωi )+ ‖g‖2

H1(0,T ;L2(Γ))∩L∞(0,T ;H1

2 (Γ))), (3)

where C is the universal constant of estimate (2), and dene the time

T0(M) = supT ′ ≤ T , max(eT ′

2, 2CeT

′(ϕ(M))2T ′, 2e

T′2

√T ′ϕ(M)

)≤ 1.

(4)

Lemma

Dene

BM := w ∈ H(T0) : ‖w‖H(T0) ≤ M.Then T (BM) ⊆ BM , BM is a closed metric subspace of

L∞(0,T0; L2(Ωi )), and T is a contraction in BM .

26 / 49


A common time of existence

Theorem

There exists M and T such that, if

‖u0‖2H2(Ωi )+ ‖g0

i ‖2

H1(0,T ;L2(Γ))∩L∞(0,T ;H1

2 (Γ))≤ M 2, (5)

(uk1, uk

2) is dened in the interval [0,T ] for all positive k.

EKS =K∑k=0

Ek , UK (t) = sup0≤k≤K

‖uk1 (t)‖∞ + sup0≤k≤K

‖uk2 (t)‖∞.EKS (t) +

∫ t

0

GK (s)ds ≤ 2

∫ t

0

ϕ(UK (s)

)EKS (s)ds +

∫ t

0

G0(s)ds,

max0≤k≤K

∫ t

0

Gk(s)ds ≤ 2

∫ t

0

ϕ(UK (s)

)EKS (s)ds +

∫ t

0

G0(s)ds.

(6)

UK (t) ≤ 4C4(1+EKS (t)+(ϕ′(UK (t)))2EKS (t)+ max0≤k≤K

∫ t

0

Gk(s) ds). (7)

From this get T on which all the quantities are bounded.27 / 49


Convergence

Theorem

With the notations of Theorem 5, the sequence (uk1, uk

2) converges, as

k →∞, to (u|Ω1, u|Ω2

), in L∞(0,T ;H1(Ωi )).

28 / 49


Algorithme de Schwarz waveform relaxation pour uneéquation non linéaire

∂tw

k+1

i + Lwk+1

i + F(wk+1

i ) = fw dans Ωi × (0,T )

wi (·, 0) = w0 dans Ωi

Gwi = g dans (∂Ωi \ Γ)× ((0,T )

Biwk+1

i = Biwk3−i dans Γ× (0,T )

Biu := ∂ni u + pu

29 / 49


Numerics

The numerical approximation

Discretization in each sub-domain:

⊲ Finite elements in space, finite differences in time.⊲ Semi-implicit Euler method.

Discretization of the iterative algorithm:

⊲ Construction of the interface operator (u1, u2) −→ (B1(u2), B2(u1)).

Numerical tests:

⊲ Computational domain: Ω×]0, T[, where Ω =]− 1, 1[×]0, 1[.

⊲ Ω1 =]− 1, 0[×]0, T[, Ω2 =]0, 1[×]0, T[.

⊲ Nonlinear reactions: f (u) = u3, f (u) = eu − 1, . . .

⊲ Numerical solutions: we compare

⊲ The mono-domain solution in Ω;⊲ The domain decomposition solution.

Filipa Caetano – p. 11

30 / 49


Numerics

Numerical results - Robin transmission conditions

Error, in the L∞ norm, between the DD and the M solutions.Robin conditions: different values of p.

1 2 3 4 5 6 7 8 9 1010

−5

10−4

10−3

10−2

10−1

100

Number of iterations

Linf

err

or

Linf error between DD and monodomain solutions after 10 iterations

Robin p = 9.5Robin p = 40


31 / 49


Numerics

Optimized parameters - nonlinear transmission conditions

Motivation: there exists optimized parameters popt and (popt, qopt) for the linearreaction-diffusion equation

∂tu− ν∆u+ bu = 0,

which satisfy

popt ∼ popt(b, ν,∆t), qopt ∼ qopt(b, ν,∆t)

([Bennequin, Gander, Halpern]).

−→ Replace b by f ′(u): p(u) = popt( f ′(u), ν,∆t), q(u) = qopt( f ′(u), ν,∆t).

Nonlinear transmission conditions:

Robin: Bi(u) =∂u∂ni

+ p(u)u.

Order 2: Bi(u) =∂u∂ni

+ p(u)u+ q(u)(∂tu− ν∆yu).


32 / 49


Numerics

Nonlinear Robin and nonlinear Order 2

Difference, in the L∞ norm, between the DD solution and the M solutions.Nonlinear Robin and order 2 conditions. f (u) = u3. T = 1. ν = 1. h = 0.125

and h = 0.0625.

1 2 3 4 5 6 7 8 9 1010−15

10−10

10−5

100


Lin

f e

rro

r


Nonlinear RobinPopt−num=8.5Nonlinear Order 2

1 2 3 4 5 6 7 8 9 1010−12

10−10

10−8

10−6

10−4

10−2

100


Lin

f e

rro

r




33 / 49


Numerics

Nonlinear Robin and nonlinear Order 2

Difference, in the L∞ norm, between the DD solution and the M solutions.Nonlinear Robin and order 2 conditions. f (u) = u3. T = 1. ν = 0.2. h = 0.125

and h = 0.0625.

1 2 3 4 5 6 7 8 9 1010

−12

10−10

10−8

10−6

10−4

10−2

100


Lin

f e

rro

r



1 2 3 4 5 6 7 8 9 1010−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100


Lin

f e

rro

r




34 / 49


Numerics

A simple model in geological CO2 storage modelling

The model: reactive chemical system with two types of materials, evolvingthrough equilibrium values u

eq1

and ueq2 .

Heterogeneous distribution of the materials in the spatial domain.

f (x, y, u) = k1S1(x, y)(u− ueq1)3 + k2S2(x, y)(u− u

eq2 )3.

k1, k2: reaction speeds.

S1, S2: distribution of the materials in the spatial domain.


35 / 49


Numerics


−1 −0.5 0 0.5 100.511.52

0.49

0.495

0.5

0.505

DD solution

0.494

0.496

0.498

0.5

0.502

−1 −0.5 0 0.5 10

12

0.49

0.495

0.5

0.505

Monodomain solution

0.494

0.496

0.498

0.5

0.502

−1 −0.5 0 0.5 10

0.5

1

1.5

2 Error at time t=1

0

2

4

6

8

x 10−5


36 / 49


Numerics


Difference, in the L∞ norm, between the DD solution and the M solution.Nonlinear Robin conditions: different values of h.

1 2 3 4 5 6 7 8 9 1010

−6

10−5

10−4

10−3

10−2

10−1

100


Linf

err

or


h=0.125h=0.0625h=0.03125


37 / 49


Schwarz waveform relaxation for a nonlinear equation :interface problem

∂tw

k+1

i + Lwk+1

i + F(wk+1



Gwi = g dans (∂Ωi \ Γ)× ((0,T )

Biwk+1


Biu := ∂ni u + pu

38 / 49



∂tw

k+1

i + Lwk+1

i + F(wk+1



Gwi = g dans (∂Ωi \ Γ)× ((0,T )

Biwk+1


Mi : ( λ , f ) 7→ (wi ) solution de

∂twi + Lwi + F(wi ) = fw dans Ωi × (0,T )


Gwi = g dans (∂Ωi \ Γ)× (0,T )

Biwk+1

i = λ dans Γ× (0,T )

λk1 = −λk−12

+ 2pM2(λk−12

, f )

λk2 = −λk−11

+ 2pM1(λk−11

, f ).

38 / 49







Biwk+1


λk1 = −λk−12

+ 2pM2(λk−12

, f )

λk2 = −λk−11

+ 2pM1(λk−11

, f ).

Linear case : optimized Schwarz = Jacobi for the interface problem(I I − 2pM2(·, 0)

I − 2pM1(·, 0) I

)·(λ1λ2

)=

(2pM2(0, f )2pM1(0, f )

)

39 / 49







Biwk+1


λk1 = −λk−12

+ 2pM2(λk−12

, f )

λk2 = −λk−11

+ 2pM1(λk−11

, f ).

Linear case : optimized Schwarz = Jacobi for the interface problem(I I − 2pM2(·, 0)

I − 2pM1(·, 0) I

)·(λ1λ2

)=

(2pM2(0, f )2pM1(0, f )

)

39 / 49


Newton/Schwarz/Krylov or Schwarz/Newton/Krylov ou ... ?

1 classical approach : the interface problem is solved by a xed pointalgorithm

Ψ(λ, f ) :=

(−λ2 + 2pM2(λ2, f )−λ1 + 2pM1(λ1, f )

)−(λ1λ2

)= 0.

2 Nested approach

3 Common Newton approach

40 / 49




Ψ(λ, f ) :=

(−λ2 + 2pM2(λ2, f )−λ1 + 2pM1(λ1, f )

)−(λ1λ2

)= 0.

One iteration = resolution of a non linear problem. cf above

2 Nested approach


40 / 49




Ψ(λ, f ) :=

(−λ2 + 2pM2(λ2, f )−λ1 + 2pM1(λ1, f )

)−(λ1λ2

)= 0.

2 Nested approachFixed point → Newton : Ψ′(λn) · (λn+1 − λn) = −Ψ(λn)

Mlin

i : ( A , λ , f ) 7→ wi

solution de

∂twi + Lwi + A wi = fw Ωi × (0,T )

wi (·, 0) = w0 Ωi

Jwi = g (∂Ωi \ Γ)× (0,T )

Biwi = λ Γ× (0,T )

Ψ′(λ) =

(−I −I + 2pMlin

2(F ′(M2(λ2, 0)), ·, 0)

−I + 2pMlin1

(F ′(M2(λ1, 0)), ·, 0) −I

).

Krylov resolution of the linear system


40 / 49




Ψ(λ, f ) :=

(−λ2 + 2pM2(λ2, f )−λ1 + 2pM1(λ1, f )

)−(λ1λ2

)= 0.

2 Nested approach


40 / 49




Ψ(λ, f ) :=

(−λ2 + 2pM2(λ2, f )−λ1 + 2pM1(λ1, f )

)−(λ1λ2

)= 0.

2 Nested approach3 Common Newton approach

1 Newton from the beginning

(∂t + L+ F ′(wn))wn+1 = F ′(wn)wn −F(wn) + fw on Ω× (0,T )

wn+1(x , 0) = w0(x) on Ω

G(wn+1) = g(x , t) on ∂Ω× (0,T )

2 One get a linear interface problem, with matrix Ψ′(λn) :(Id Id−2pMlin

2 (F ′(wn2 ), ·, 0)

Id−2pMlin1 (F ′(wn

1 ), ·, 0) Id

)·(λn+1

1

λn+1

2

)=

=

(2pMlin

2 (F ′(wn2 ), 0, (F ′(wn

2 )wn2 −F(wn

2 ) + q,w0, g))

2pMlin1 (F ′(wn

1 ), 0, (F ′(wn1 )wn

1 −F(wn1 ) + q,w0, g))

).

3 Resolution with Krylov40 / 49




Ψ(λ, f ) :=

(−λ2 + 2pM2(λ2, f )−λ1 + 2pM1(λ1, f )

)−(λ1λ2

)= 0.




wn+1(x , 0) = w0(x) on Ω

G(wn+1) = g(x , t) on ∂Ω× (0,T )


2 (F ′(wn2 ), ·, 0)


1 ), ·, 0) Id

)·(λn+1

1

λn+1

2

)=

=

(2pMlin

2 (F ′(wn2 ), 0, (F ′(wn

2 )wn2 −F(wn

2 ) + q,w0, g))

2pMlin1 (F ′(wn

1 ), 0, (F ′(wn1 )wn

1 −F(wn1 ) + q,w0, g))

).





Ψ(λ, f ) :=

(−λ2 + 2pM2(λ2, f )−λ1 + 2pM1(λ1, f )

)−(λ1λ2

)= 0.




wn+1(x , 0) = w0(x) on Ω

G(wn+1) = g(x , t) on ∂Ω× (0,T )


2 (F ′(wn2 ), ·, 0)


1 ), ·, 0) Id

)·(λn+1

1

λn+1

2

)=

=

(2pMlin

2 (F ′(wn2 ), 0, (F ′(wn

2 )wn2 −F(wn

2 ) + q,w0, g))

2pMlin1 (F ′(wn

1 ), 0, (F ′(wn1 )wn

1 −F(wn1 ) + q,w0, g))

).



Comparaison

101 102

103

Number of grid cells per dimension, Nx = Ny

Num

ber

mat

rix

inve

rsio

ns

Classical Approach

Nested Iteration Approach

Common Iteration Approach

O(N1/2.75)

O(N1/7)

Implementation by Florian Haeberlein, nite volumes, q 6= 0,nonconformal meshes, to some reactive system, with chemistry.

41 / 49


Conclusion

Extension of optimized Schwarz waveform relaxation to nonlinearsystems.

Used to rene locally the mesh in time and space ♣

Possible to follow the reactive front

Implementation in Arcane

See the real stu in Anthony Michel's talk.

42 / 49


Conclusion






42 / 49


Conclusion






42 / 49


Conclusion






42 / 49


Conclusion






42 / 49


A few numbers

∂u

∂t+∇ · (a(x , z)u −

(νx(z) 00 νz(z)

)∇u) = 0.

Ocean : (0, L)× (−hO , 0), atmosphere :(0, L)× (0, hA)hO = 5km, hA = 30km, L = 5000km.

a(x , z) =

( 506sin( 2πx

L) cos(πz

hA),−0.1 cos( 2πx

L) sin(πz

hA)), −hO ≤ z ≤ 0

(0.5 sin( 2πxL

) cos( πzhO

),−0.001 cos( 2πxL

) sin( πzhO

)), 0 ≤ z ≤ hA

νx(z) =

100, z ≤ 0

10000, z ≥ 0, νz(z) =

10−5 + (10−2 − 10−5)e

−(z+100)2

8000 , z ≤ 0

10−3 + (0.1− 10−3)e−(z−30)2

1000 , z ≥ 0

Initial state

u0 =

14e−ln(7)

105(z+495)2) + 2, −hO ≤ z ≤ −495,

16, −495 ≤ z ≤ 0,16− 41z

30000, 0 ≤ z ≤ hA.

Total time of computation : 5 days.Homogeneous Neumann boundary conditions. 43 / 49


Simulation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 106

−0.5

0

0.5

1

1.5

2

2.5

3x 10

4

x

y

Mesh

NA,x = 60,NA,z = 70,NA,t = 90 Discontinuous Galerkin P1 in timeNO,x = 240,NO,z = 50,NO,t = 12 P1 nite elements in space.

Nw = 20 time windows of 6 hour each

3 iterations are sucient to reach the global scheme accuracy.

44 / 49


Complexity

Nx = max(N0,x ,NA,x) = 240, Nz = NO,z + NA,z = 120,Nt = max(NO,t ,NA,t) = 90.

1 N1 : number of elementary operations for the monodomain solution,

2 N2(p) number of elementary operations for the Schwarz waveformrelaxation algorithm with p iterations.

N1 ∼ 2N3

zNx + 4N2

zNxNtNw

N2(p) ∼ 2max(N3

A,zNA,x ,N3

O,zNO,x) + 4p max1≤i≤2

(N2

i,yNi,xNi,t)Nw

45 / 49


Complexity

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

10

number of iterations p

com

plex

ity

N

2(3)

N2(p)

N1

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9x 10

10

number of iterations p

com

plex

ity

N

2(p) with N

1,t=12, N

2,t=90

N2(p) with N

1,t=N

2,t=90

N1

full ne computation/ conformal in time/domain decomposition non conformal in time

with non conformal mesh renementComputations : Caroline Japhet.LH& C. Japhet & J. Szeftel. Space-Time Nonconforming Optimized Schwarz

Waveform Relaxation for Heterogeneous Problems and General Geometries. Domain

Decomposition Methods in Science and Engineering XIX, Lect. Notes Comput. Sci.

Eng., 78, Springer, Berlin 2011. Huang, Y. ; Kornhuber, R. ; Widlund, O. ; Xu, J. eds.

pp 75-86. ♣

46 / 49


Convergence factor

The case of half-spaces and constant coecients : Fourier transform intime and transverse space

δ(z) = a2 + 4νc + 4νz , z = i(ω + b · k) + ν|k |2,

Convergence factor

ρ(ω, k,P, L) =

(P − δ1/2P + δ1/2

)2

e−2δ1/2L/ν , P(z) = p + qz .

47 / 49



φ(z) = a2 + 4νc + 4νz , z = i(ω + b · k) + ν|k|2

ρ(z ,P, L) =

(P(z)− φ1/2(z)

P(z) + φ1/2(z)

)2

e−2δ1/2L

Taylor expansion,P(z) =√φ(0) + 2νz/

√φ(0),

Best approximation

infP∈Pn

supz∈K|ρ(z ,P, L)|, K = (

π

T,π

∆t), kj ∈ (

π

Xj,π

∆xj)

♣

48 / 49



φ(z) = a2 + 4νc + 4νz , z = i(ω + b · k) + ν|k|2

ρ(z ,P, L) =

(P(z)− φ1/2(z)

P(z) + φ1/2(z)

)2

e−2δ1/2L


√φ(0),

Best approximation

infP∈Pn


π

T,π

∆t), kj ∈ (

π

Xj,π

∆xj)

♣

48 / 49



φ(z) = a2 + 4νc + 4νz , z = i(ω + b · k) + ν|k|2

ρ(z ,P, L) =

(P(z)− φ1/2(z)

P(z) + φ1/2(z)

)2

e−2δ1/2L


√φ(0),

Best approximation

infP∈Pn


π

T,π

∆t), kj ∈ (

π

Xj,π

∆xj)

♣

48 / 49


Convergence

mean value of the (p∗, q∗) obtained by optimization of the convergencefactor.

0 2 4 6 8 10 12 14 16 18 2010−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100Error between momodomain (variational) and DGOSWR solutions versus the iterations

Iterations

log 10

(Err

or)

Order2Robin

49 / 49

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