An Introduction to Coupling Conditions Homogeneous Heterogeneous - - PowerPoint PPT Presentation

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

An Introduction to Heterogeneous Domain Decomposition with a New Approach based on Factorization

Martin J. Gander martin.gander@unige.ch

University of Geneva

September 2016 Joint work with Laurence Halpern and Veronique Martin

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Coupling Conditions for Homogeneous Problems

Lu := (η − ∆)u = f , in a domain Ω = ⇒ Solution u we want to compute is well defined !

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Coupling Conditions for Homogeneous Problems

Lu := (η − ∆)u = f , in a domain Ω = ⇒ Solution u we want to compute is well defined ! H.A. Schwarz 1869: Przemieniecki 1963:

¨ Uber einen Grenz¨ ubergang Matrix Structural Analysis durch alternierendes Verfahren

• f Substructures

Ω1 Ω2 Γ1 Γ2 ∂Ω Ω1 Ω2 Γ ∂Ω Lu1 = f , in Ω1 Lu2 = f , in Ω2 u1 = u2,

• n Γ1

u2 = u1,

• n Γ2

Lu1 = f , in Ω1 Lu2 = f , in Ω2 u1 = u2,

• n Γ

∂nu2 = ∂nu1,

• n Γ

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Coupling Conditions for Heterogeneous Problems

1) Physics are different in different regions:

• Structure

Fluid

Solution u we want to compute is well defined = ⇒ coupling conditions are given by the physics

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Coupling Conditions for Heterogeneous Problems

1) Physics are different in different regions:

• Structure

Fluid

Solution u we want to compute is well defined = ⇒ coupling conditions are given by the physics 2) Different models for computational savings:

◮ Using an expensive model only where it is necessary

airfoil

Ω1 Ω2 Γ

◮ Coupling across different dimensions

PDE Transistor Circuit ODE R C Ω2 Ω1 Γ

Blood Vessel 1d PDE model Heart 3d PDE model

Ω1 Ω2 Γ

= ⇒ coupling conditions are unknown !

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Coupling Conditions for Computational Savings

airfoil

Ω

LNSu = f in Ω = ⇒ Solution u we want to compute is well defined !

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Coupling Conditions for Computational Savings

airfoil

Ω

LNSu = f in Ω = ⇒ Solution u we want to compute is well defined !

airfoil

Ω1 Ω2 Γ

LNSu1 = f in Ω1 LEu2 = f in Ω2

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Coupling Conditions for Computational Savings

airfoil

Ω

LNSu = f in Ω = ⇒ Solution u we want to compute is well defined !

airfoil

Ω1 Ω2 Γ

LNSu1 = f in Ω1 LEu2 = f in Ω2

Dubach (1993): “L’objectif est alors d’essayer des conditions de transmission ad´ equates ` a la fronti` ere de fa¸ con ` a minimser l’erreur entre la solution du probl` eme de transmission et celle de Navier Stokes complet dans tout le domaine.”

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Coupling Conditions for Computational Savings

airfoil

Ω

LNSu = f in Ω = ⇒ Solution u we want to compute is well defined !

airfoil

Ω1 Ω2 Γ

LNSu1 = f in Ω1 LEu2 = f in Ω2

Dubach (1993): “L’objectif est alors d’essayer des conditions de transmission ad´ equates ` a la fronti` ere de fa¸ con ` a minimser l’erreur entre la solution du probl` eme de transmission et celle de Navier Stokes complet dans tout le domaine.”

Idea of best coupling conditions: find coupling conditions s.t. ||u − u1||Ω1 − → min!

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Coupling of Advection Diffusion with Advection

x t Ω1 Ω2 ∂tuad −ν∂xxuad +∂xauad +cuad = f ∂tua +∂xaua +cua = f New idea based on operator factorization in 1D: Lad := −ν∂xx + a∂x + c = (∂x − λ+)(−ν∂x + νλ−) where the characteristic roots are λ± = (a ±

• a2 + 4νc)/2ν.

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Idea Explained in the Stationary Case on R

Integrate the factored advection diffusion equation on (x, +∞): (∂x − λ+)(−νu′ + νλ−u) = f

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Idea Explained in the Stationary Case on R

Integrate the factored advection diffusion equation on (x, +∞): (∂x − λ+)(−νu′ + νλ−u) = f = ⇒ (∂x − λ+)˜ ua = f

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Idea Explained in the Stationary Case on R

Integrate the factored advection diffusion equation on (x, +∞): (∂x − λ+)(−νu′ + νλ−u) = f = ⇒ (∂x − λ+)˜ ua = f = ⇒ ˜ ua(x) = − ∞

x

f (σ)eλ+(x−σ) dσ

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Idea Explained in the Stationary Case on R

Integrate the factored advection diffusion equation on (x, +∞): (∂x − λ+)(−νu′ + νλ−u) = f = ⇒ (∂x − λ+)˜ ua = f = ⇒ ˜ ua(x) = − ∞

x

f (σ)eλ+(x−σ) dσ = ⇒ −νu′(x) + νλ−u(x) = − ∞

x

f (σ)eλ+(x−σ) dσ

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Idea Explained in the Stationary Case on R

Integrate the factored advection diffusion equation on (x, +∞): (∂x − λ+)(−νu′ + νλ−u) = f = ⇒ (∂x − λ+)˜ ua = f = ⇒ ˜ ua(x) = − ∞

x

f (σ)eλ+(x−σ) dσ = ⇒ −νu′(x) + νλ−u(x) = − ∞

x

f (σ)eλ+(x−σ) dσ To have the exact solution in Ω1, we thus need to impose −νu′

ad(0) + νλ−uad(0) = −

∞ f (σ)eλ+(x−σ) dσ

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Idea Explained in the Stationary Case on R

Integrate the factored advection diffusion equation on (x, +∞): (∂x − λ+)(−νu′ + νλ−u) = f = ⇒ (∂x − λ+)˜ ua = f = ⇒ ˜ ua(x) = − ∞

x

f (σ)eλ+(x−σ) dσ = ⇒ −νu′(x) + νλ−u(x) = − ∞

x

f (σ)eλ+(x−σ) dσ To have the exact solution in Ω1, we thus need to impose −νu′

ad(0) + νλ−uad(0) = −

∞ f (σ)eλ+(x−σ) dσ Hence with the modified advection equation

• La˜

ua := ˜ u′

a − λ+˜

ua = f in Ω2 := (0, ∞) the optimal (exact!) coupling condition is −νu′

ad(0) + νλ−uad(0) = ˜

ua(0).

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Stationary Case on a Bounded Domain

Integration of the factored equation on (x, L2) gives

−νu′(x)+νλ−u(x)=( − νu′+ νλ−u)(L2)eλ+(x−L2)− L2

x

f (σ)eλ+(x−σ)dσ

Hence, with the modified advection equation

• La˜

ua := ˜ u′

a − λ+˜

ua = f sur (0, L2), i.e. ˜ ua(x) = ˜ ua(L2)eλ+(x−L2) − L2

x

f (σ)eλ+(x−σ) dσ. the optimal (exact!) coupling condition is

−νu′(0)+νλ−u(0)=(−νu′(L2)+νλ−u(L2)− ˜ ua(L2))e−λ+L2 + ˜ ua(0)

Remark: we have

◮ If a > 0 λ+ = a ν + O(1) ◮ Otherwise ˜

ua(L2) = α1ν + α2ν2 + · · · + αmνm

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Coupling Conditions from the Literature

◮ Var: Variational coupling conditions (Gastaldi and

Quarteroni 1988)

◮ NVar: Non-variational coupling conditions (Gastaldi

and Quarteroni 1988)

◮ ABC: Coupling conditions based on absorbing boundary

conditions (Dubach 1993)

◮ Fac: Based on operator factorization (G, Halpern and

Martin, 2009) a > 0 a < 0 Var uad = ua −νu′

ad + auad = aua

−νu′

ad + auad = aua

NVar uad = ua u′

ad = u′ a

uad = ua ABC uad = ua u′

ad = u′ a

−νu′

ad + auad = aua

Fac −νu′

ad + νλ−uad = a˜

ua uad = ua −νu′

ad + νλ−uad = a˜

ua

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Stationary Problem: Rigorous Error Estimates

Theorem (G, Halpern, Martin 2009)

For the stationary advection reaction diffusion problem in 1d, the following error estimates hold: a > 0 Var NVar Fac eadΩ1 O(ν3/2) O(ν5/2) O(e− a

ν )

eaΩ2 O(ν) O(ν) O(ν) a < 0 eadΩ1 O(ν) O(ν) O(νm), m = 1, 2, . . . eaΩ2 O(ν) O(ν) O(ν) Viscous Problems with a Vanishing Viscosity Approximation in Subregions: a New Approach Based on Operator Factorization, G, Halpern, Japhet, Martin, ESAIM, 2009.

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Numerical Experiment Stationary Problem: a > 0

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.8 1 1.2 x u uad Factorization ua Factorization uad Variational ua Variational uad Non−Variational ua Non−Variational

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Numerical Experiment Stationary Problem: a < 0

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 x u uad Factorization ua Factorization uad Variational ua Variational uad Non−Variational ua Non−Variational

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Asymptotics Stationary Problem: a > 0

10

−4

10

−3

10

−2

10

−1

10 10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

10

2

ead Factorization ea Factorization ead Variational ea Variational ead Non−Variational ea Non−Variational ν ν ν1.5 ν2.5

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Asymptotics Stationary Problem: a < 0

10

−4

10

−3

10

−2

10

−1

10 10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

10

2

ead Factorization m=1 ead Factorization m=2 ead Factorization m=3 ea Factorization ead Variational ea Variational ead Non−Variational ea Non−Variational ˜ ea ν ν ν2 ν3

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Algorithm for Time Dependent Case a > 0

First solve a transport problem in Ω2 into the positive x direction Lauk

a

= f in Ω2 × (0, T), uk

a (0, ·)

= uk−1

ad (0, ·)

• n (0, T),

uk

a (·, 0)

= h in Ω2, followed by a modified transport problem into the negative x direction in Ω2 with adapted source Lmauk

ma

=

a2 ν f + R uk a

in Ω2 × (0, T), uk

ma(L2, ·)

=

• n (0, T),

uk

ma(·, 0)

= f (·, 0) + νd2

x h

in Ω2, and finally an advection reaction diffusion problem in Ω1, Laduk

ad

= f in Ω1 × (0, T), uk

ad(−L1, ·)

= g1

• n (0, T),

Lauk

ad(0, ·)

= uk

ma(0, ·)

• n (0, T),

uk

ad(·, 0)

= h in Ω1.

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Algorithm for Time Dependent Case a < 0

Start with an advection reaction problem in Ω2, Lau1

a

= f in Ω2 × (0, T), u1

a(L2, ·)

= g2

• n (0, T),

u1

a(·, 0)

= h in Ω2, followed by another advection reaction problem in the same domain, Lau2

a

=

a2 ν f + R u1 a

in Ω2 × (0, T), u2

a(L2, ·)

= Lmau1

a(L2, ·)

• n (0, T),

u2

a(·, 0)

= Lmau1

a(·, 0)

in Ω2, and finally an advection reaction diffusion problem in Ω1, Laduad = f in Ω1 × (0, T), uad(−L1, ·) = g1

• n (0, T),

Lmauad(0, ·) = u2

a(0, ·)

• n (0, T),

uad(·, 0) = h in Ω1.

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Evoution Problem: Rigorous Error Estimates

Theorem (G, Halpern, Martin 2016)

For the time dependent advection reaction diffusion problem in 1d, the following error estimates hold: a > 0 Var NVar Fac eadΩ1 O(ν3/2) O(ν5/2) O(ν9/2) eaΩ2 O(ν) O(ν) O(ν) a < 0 eadΩ1 O(ν) O(ν) O(ν2) eaΩ2 O(ν) O(ν) O(ν)

Multiscale analysis of heterogeneous domain decomposition methods for time-dependent advection reaction diffusion problems, G, Halpern, Martin, submitted, 2016. A new Algorithm Based on Factorization for Heterogeneous Domain Decomposition, G, Halpern, Martin, Numerical Algorithms, 2016.

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Numerical Experiment Evolution Problem: a > 0

−1 −0.5 0.5 1 0.2 0.4 0.6 0.8 1

• Visc. Sol.

uad ua

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Numerical Experiment Evolution Problem: a > 0

−1 −0.5 0.5 1 0.2 0.4 0.6 0.8 1

• Visc. Sol.

uad ua

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Numerical Experiment Evolution Problem: a > 0

−1 −0.5 0.5 1 0.2 0.4 0.6 0.8 1

• Visc. Sol.

uad ua

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

− 1 − 9 −8 −7 − 6 −5 − 5 − 4 − 3

Variational Meth. Error, t=0.49751 −0.1 −0.08 −0.06 −0.04 −0.02 0.2 0.4 0.6 0.8

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

−9 −8 − 7 − 6 −6 −5 − 4

Non Variational Meth. Error, t=0.49751 −0.1 −0.08 −0.06 −0.04 −0.02 0.2 0.4 0.6 0.8

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

−12 − 1 1 −10 − 9 − 8 − 7 −6 −6 − 6

Factorization Error Iter 2 , t=0.49751 −0.1 −0.08 −0.06 −0.04 −0.02 0.2 0.4 0.6 0.8

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Asymptotics Evolution Problem: a > 0

10

−4

10

−3

10

−2

10

−8

10

−6

10

−4

10

−2

• Fact. iter1
• Fact. iter2

Var Non Var ν1.5 ν2.5 ν4

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Asymptotics Evolution Problem: a < 0

10

−4

10

−3

10

−2

10

−5

10

−4

10

−3

10

−2

10

−1

10 Fact. Var Non Var ν ν2

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

Summary

◮ Two types of Heterogeneous DD methods:

• 1. Coupling different physical phenomena
• Structure

Fluid

• 2. Coupling the same physical phenomenon with cheaper

models to save computation

airfoil

Ω1 Ω2 Γ

◮ New idea in case 2: ||u − u1|| −

→ min = ⇒ coupling conditions based on operator factorization.

◮ For advection reaction diffusion problems:

◮ A modified advection problem needs to be solved. ◮ Rigorous error estimates in ν.

Heterogeneous DD Martin J. Gander Coupling Conditions

Homogeneous Problems Heterogeneous Problems Best Coupling Conditions

Advection- Diffusion

Model Problem Factorization

Stationary Case

Algorithm Error Estimates Numerical Experiments

Evolution Case

Algorithm Error Estimates Numerical Experiments

Conclusions

Summary References

References for This New Approach

• 1. Multiscale analysis of heterogeneous domain decomposition

methods for time-dependent advection reaction diffusion problems, G, Halpern, Martin, submitted, 2016.

• 2. A new Algorithm Based on Factorization for Heterogeneous

Domain Decomposition, G, Halpern, Martin, Numerical Algorithms, 2016.

• 3. An Asymptotic Approach to Compare Coupling Mechanisms

for Different Partial Differential Equations, G, Martin, DD 20, 2014.

• 4. How Close to the Fully Viscous Solution can one get When

Inviscid Approximations are Used in Subregions? G, Halpern, Martin, DD 19, 2010.

• 5. Viscous Problems with a Vanishing Viscosity Approximation

in Subregions: a New Approach Based on Operator Factorization, G, Halpern, Japhet, Martin, ESAIM, 2009.

• 6. Advection Diffusion Problems with Pure Advection

Approximation in Subregions, G, Halpern, Japhet, Martin, DD 16, 2006.