Revue historique de m ethodes de couplage Boundary Conditions as - - PowerPoint PPT Presentation

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Revue historique de m´ ethodes de couplage

• V. Martin

veronique.martin@u-picardie.fr

LAMFA,Universit´ e de Picardie, Amiens.

En collaboration avec Martin Gander.

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Coupling Equations

Lu = f

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Coupling Equations

Lu = f

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Coupling Equations

• Lu = f

˜ Lu = f

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Coupling Equations

• Lu = f

˜ Lu = f The position of the interface is known a priori or is determined by the model itself. The domains can overlap or not.

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Outline

Introduction Overlapping Method Boundary Conditions as Coupling Conditions Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator χ-Formulation

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Outline

Introduction Overlapping Method Boundary Conditions as Coupling Conditions Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator χ-Formulation

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Overlapping Method

Dinh, Glowinski, Periaux, Terrasson (1988): On the Coupling

• f Viscous and Inviscid Models for Incompressible Fluid Flows Via

Domain Decomposition. “The main goal of this paper is to present a computational method for the coupling of two distinct mathematical models describing the same physical phenomenon, namely the flow of an incompressible viscous

• fluid. The basic idea is to replace the Navier-Stokes equations by

the potential one in those regions where we can neglect the viscous effects and where the vorticity is small”.

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Overlapping Method

Navier-Stokes equations:        ∂u ∂t − ν△u + (u · ∇)u + ∇p = 0 in Ω2 ∇ · u = 0 in Ω2 +Boundary Conditions Laplace equation (if the vorticity is small : u = ∇φ):

• △φ = 0 in Ω1

+Boundary Conditions on γ1

How to couple the two problems?

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Overlapping Coupling

Dinh, Glowinski, Periaux, Terrasson (1988): “To couple

the two models, we use a least squares approach in which we minimize over the overlapping region some distance between u and ∇φ” Navier-Stokes equations        ∂u ∂t − ν△u + (u · ∇)u + ∇p = 0 in Ω2 ∇ · u = 0 in Ω2 u = v on γ2 Laplace equations

• △φ = 0

in Ω1 φ = ψ

• n γ1

The coupling condition is : Inf(v,ψ) 1 2

• Ω1∩Ω2

|u − ∇φ|2.

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Illustration on a Model Problem

Advection diffusion equation Ladu := −νu′′ + au′ + cu = f

• n (−L1, L2)

u = g1

• n x = −L1

Bu =

• n x = L2

−L1 L2 L Advection diffusion equation Laduad = f

• n (−L1, L)

uad = g1 on x = −L1 uad = ψ on x = L Advection equation Laua := au′

a + cua = f on (0, L2)

ua = τ on x = 0 Bua = 0 on x = L How to determine ψ and τ ?

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

A Least Squares Approach

Idea in Dinh, Glowinski, Periaux, Terrasson (1988): ||uad(ψ) − ua(τ)||(0,L) − → min

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 x uad ua u

ν = 0.5, N = 1000, L = 100h

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Quality of this Coupling

We evaluate this coupling technique for ν small by comparing to the fully viscous solution on the entire domain. f = cos x +sin x, a = 1, c = 1, L1 = 1, L2 = 1, g1 = 1, B = ∂x Discretization with centered finite difference, N = 100000 points:

10

−4

10

−3

10

−2

10

−1

10 10

−10

10

−8

10

−6

10

−4

10

−2

10 10

2

Dirichlet, N=50000 de=10 nu error advection errad nu5/2 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

Dirichlet, N=50000 de=1000 nu error advection errad nu3/2

L = 10h, ||ead|| ∼ ν5/2 L = 1000h, ||ead|| ∼ ν3/2

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Quality of this Coupling, Boundary Layer

Experiments for ν small with a boundary layer f = cos x+sin x, a = −1, c = 0, L1 = 1, L2 = 1, g1 = 1, B = Id Discretization with centered finite difference, N = 100000 points:

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.5 1 1.5 x uad ua u 10

−4

10

−3

10

−2

10

−1

10 10

−3

10

−2

10

−1

10 10

1

10

2

Dirichlet, BL, N=50000 de=10 nu error advection errad nu

L = 10h, ||ead|| ∼ ν (independent of L).

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Literature

• Q. V. Dinh, R. Glowinski, J. Periaux, G. Terrasson (1988):

On the Coupling of Viscous and Inviscid Models for Incompressible Fluid Flows Via Domain Decomposition, DD1.

• R. Glowinski, J. Periaux, G. Terrasson (1988): On the

Coupling of Viscous and Inviscid Models for Compressible Fluid Flows Via Domain Decomposition. DD3.

• P. Gervasio, J.-L. Lions, A. Quarteroni (2001): Heterogeneous

Coupling by Virtual Control Methods.

• P. Gervasio, J.-L. Lions, A. Quarteroni (2001): Domain

Decomposition and Virtual Control for Fourth Order Problems.

• V. Agoshkov, P. Gervasio, and A. Quarteroni (2006).

Optimal control in heterogeneous domain decomposition methods for advection-diffusion equations.

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Outline

Introduction Overlapping Method Boundary Conditions as Coupling Conditions Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator χ-Formulation

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Outline

Introduction Overlapping Method Boundary Conditions as Coupling Conditions Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator χ-Formulation

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Non-Overlapping Coupling

Gastaldi, Quarteroni, Sacchi Landriani (1989): On the Coupling of Two Dimensional Hyperbolic and Elliptic Equations: Analytical and Numerical Approach.

“Physical evidence suggests that viscosity effects are negligible apart from a small region close to the rigid body. This is one instance where the mathematical model of the problem may lead to the use of equations of different character in separate regions, just by dropping the viscous terms when they are very small.”

Gastaldi, Quarteroni (1989): On the Coupling of Hyperbolic and Parabolic systems: Analytical and Numerical Approach.

”The justification of the interface conditions is based on a singular perturbation analysis, that is the hyperbolic system is rendered parabolic by adding a small artificial ‘viscosity’. As this goes to zero, the coupled parabolic-parabolic problem degenerates into the

• riginal one, yielding some conditions at the interface.”

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Non-Overlapping Coupling

Gastaldi, Quarteroni, Sacchi Landriani (1989): On the Coupling of Two Dimensional Hyperbolic and Elliptic Equations: Analytical and Numerical Approach.

“Physical evidence suggests that viscosity effects are negligible apart from a small region close to the rigid body. This is one instance where the mathematical model of the problem may lead to the use of equations of different character in separate regions, just by dropping the viscous terms when they are very small.”

Gastaldi, Quarteroni (1989): On the Coupling of Hyperbolic and Parabolic systems: Analytical and Numerical Approach.

”The justification of the interface conditions is based on a singular perturbation analysis, that is the hyperbolic system is rendered parabolic by adding a small artificial ‘viscosity’. As this goes to zero, the coupled parabolic-parabolic problem degenerates into the

• riginal one, yielding some conditions at the interface.”

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Coupling Hyperbolic and Parabolic Problems

Gastaldi, Quarteroni (1989): Start with two different problems to be coupled: −νu′′

ad + au′ ad + cuad

= f

• n (−L1, 0)

au′

a + cua

= f

• n (0, L2)

Introduce for regularization a small artificial viscosity ǫ: −νw′′

ǫ + aw′ ǫ + cwǫ

= f

• n (−L1, 0)

−ǫv ′′

ǫ + av ′ ǫ + cvǫ

= f

• n (0, L2)

Two types of boundary conditions are possible : Variational Conditions wǫ(0) = vǫ(0) νwǫ′(0) = ǫvǫ′(0) Non Variational Conditions wǫ(0) = vǫ(0) wǫ′(0) = vǫ′(0)

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Coupling Hyperbolic and Parabolic Problems

Gastaldi, Quarteroni (1989): Start with two different problems to be coupled: −νu′′

ad + au′ ad + cuad

= f

• n (−L1, 0)

au′

a + cua

= f

• n (0, L2)

Introduce for regularization a small artificial viscosity ǫ: −νw′′

ǫ + aw′ ǫ + cwǫ

= f

• n (−L1, 0)

−ǫv ′′

ǫ + av ′ ǫ + cvǫ

= f

• n (0, L2)

Two types of boundary conditions are possible : Variational Conditions wǫ(0) = vǫ(0) νwǫ′(0) = ǫvǫ′(0) Non Variational Conditions wǫ(0) = vǫ(0) wǫ′(0) = vǫ′(0)

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

What Happens in the Limit ǫ → 0 ?

Quarteroni et al. proved rigorously that as ǫ → 0, we have wǫ → uad in (−L1, 0) and vǫ → ua in (0, L2) with the B.C. : Variational coupling conditions : a > 0 a < 0 uad(0) = ua(0), u′

ad(0) = 0,

−νu′

ad(0) + auad(0) = aua(0).

Non-variational coupling conditions : uad(0) = ua(0), u′

ad(0) = u′ a(0),

uad(0) = ua(0), Gastaldi, Quarteroni, Sacchi Landriani (1989): “Among

all allowed choices, we make the most natural one, namely we take those interface conditions which are generated by a limit procedure on ’globally viscous problems”’.

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

What Happens in the Limit ǫ → 0 ?

Quarteroni et al. proved rigorously that as ǫ → 0, we have wǫ → uad in (−L1, 0) and vǫ → ua in (0, L2) with the B.C. : Variational coupling conditions : a > 0 a < 0 uad(0) = ua(0), u′

ad(0) = 0,

−νu′

ad(0) + auad(0) = aua(0).

Non-variational coupling conditions : uad(0) = ua(0), u′

ad(0) = u′ a(0),

uad(0) = ua(0), Gastaldi, Quarteroni, Sacchi Landriani (1989): “Among

all allowed choices, we make the most natural one, namely we take those interface conditions which are generated by a limit procedure on ’globally viscous problems”’.

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Rigorous Error Estimates

Theorem (Gander, Halpern, Japhet, Martin (2008))

For a > 0, the error between the fully viscous solution u and the coupled solution satisfies for the variational coupling conditions ||u − uad||2 = O(ν3/2), ||u − ua||2 = O(ν), and for the non-variational coupling conditions ||u − uad||2 = O(ν5/2), ||u − ua||2 = O(ν). For a < 0, both choices give ||u − uad||2 = O(ν), ||u − ua||2 = O(ν).

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Literature

• F. Gastaldi, A. Quarteroni (1989): On the Coupling of

Hyperbolic and Parabolic Systems: Analytical and Numerical Approach.

• F. Gastaldi, A. Quarteroni, G. Sacchi Landriani (1990):

On the Coupling of Two Dimensional Hyperbolic and Elliptic Equations: Analytical and Numerical Approach.

• A. Quarteroni and F. Pasquarelli and A. Valli (1992):

Heterogeneous domain decomposition principles, algorithms, applications. Lie, Bourgat, Le Tallec Tidriri Qiu, Schenk Hebeker, Alonso Valli . . .

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Outline

Introduction Overlapping Method Boundary Conditions as Coupling Conditions Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator χ-Formulation

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

A Posteriori Correction Technique

Case a < 0

Coclici, Morosanu, Wendland (2000): The Coupling of Hyperbolic and Elliptic Boundary Value Problems with Variable Coefficients. Consider the coupling model proposed by Quarteroni et al. :

−νu′′

ad + au′ ad + cuad = f on (−L1, 0)

au′

a + cua = f on (−L1, 0)

−νu′

ad(0) + auad(0) = aua(0)

Coclici et al: “But this transmission condition implies that

solutions of the coupled hyperbolic-elliptic problem exhibit jumps at the interface. . . ”

Different point of view :the original problem is

−νw ′′

ǫ + aw ′ ǫ + cwǫ = f

−ǫv ′′

ǫ + av ′ ǫ + cvǫ = f

ǫv ′

ǫ(0) = νw ′ ǫ(0)

vǫ(0) = wǫ(0)

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Asymptotic Correction

Coclici et al:“(...) we can affirm that the approximate solution

to the heterogeneous coupled Navier-Stokes/Euler problem is a first approximation of exterior viscous flows taking into account viscosity as well as far field behaviour. This coupled solution needs to be corrected by special terms (...)”

The coupled solution obtained represents only the first term in an asymptotic expansion: wǫ(x) = uad(x) + rǫ(x) vǫ(x) = ua(x) + lǫ(x) + sǫ(x) where lǫ represent the boundary layer term missing for continuity, and rǫ and sǫ are small for ǫ small. The correction term is computed lǫ analytically.

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Numerical Experiment, Coclici et al.

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Outline

Introduction Overlapping Method Boundary Conditions as Coupling Conditions Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator χ-Formulation

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Idea of the method on the stationary case

Gander, Halpern, Japhet, Martin (2008): Viscous

Problems with Inviscid Approximations in Subregions: a New Approach Based on Operator Factorization.

֒ → Can we have a better error estimate than O(ν5/2)? Factorization of the operator : Lu = (∂x − λ+)L−u = f . We can prove that if (∂x − λ+)˜ ua = f , then L−u(0) = ˜ ua(0) + (L−u(L2) − ˜ ua(L2))e−λ+L2, So that the exact transmission condition is : L−uad(0) = ˜ ua(0) + (L−u(L2) − ˜ ua(L2))e−λ+L2.

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

New approach : the procedure

• 1. We solve the modified advection equation
• La˜

ua := ˜ u′

a − λ+˜

ua = f on (0, L2), ˜ ua(L2) = α1ν + α2ν2 + · · · + αmνm.

• 2. We solve the advection-diffusion equation

Laduad := −νu′′

ad + au′ ad + cuad = f on (−L1, 0),

−νu′

ad(0) + νλ−uad(0) = ˜

ua(0).

• 3. We solve the advection equation (optional)

Laua := au′

a + cua = f on (0, L2),

ua(0) = uad(0).

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Error Estimates

a > 0 Factorization Variational Non-Variational eadΩ− O(e− a

ν )

O(ν3/2) O(ν5/2) eaΩ+ O(ν) O(ν) O(ν) a < 0 Factorization Variational Non-Variational eadΩ− O(νm), m = 1, 2, . . . O(ν) O(ν) eaΩ+ O(ν) O(ν) O(ν)

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Outline

Introduction Overlapping Method Boundary Conditions as Coupling Conditions Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator χ-Formulation

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

The χ-Formulation

Brezzi, Canuto, Russo (1989): A Self- Adaptative Formulation for the Euler/Navier-Stokes Coupling

“This means, assuming again that ν is constant and small everywhere, that the diffusion effects are negligible in the region where ∆u is not too large. We, therefore, propose to introduce an additional nonlinearity, by considering, instead of ∆u, a function χ(∆u) which coincides with ∆u when |∆u| ≥ δ (δ to be chosen) and vanishes otherwise”

−νχ(u′′) + au′ + cu = f

• n (−L1, L2)

u = g1

• n x = −L1

Bu =

• n x = L2

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

The χ-Formulation

Brezzi, Canuto, Russo (1989): A Self- Adaptative Formulation for the Euler/Navier-Stokes Coupling

“This means, assuming again that ν is constant and small everywhere, that the diffusion effects are negligible in the region where ∆u is not too large. We, therefore, propose to introduce an additional nonlinearity, by considering, instead of ∆u, a function χ(∆u) which coincides with ∆u when |∆u| ≥ δ (δ to be chosen) and vanishes otherwise”

−νχ(u′′) + au′ + cu = f

• n (−L1, L2)

u = g1

• n x = −L1

Bu =

• n x = L2

−0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2

χ(s) =    0, 0 ≤ s < δ − σ (s − δ + σ) δ

σ, δ − σ ≤ s ≤ δ

s, s > δ

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

The χ-solution

Advective solution Intermediate solution Viscous solution

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4

ν = 0.1

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

The χ-solution

Advective solution Intermediate solution Viscous solution

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4

ν = 0.01

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

The χ-solution

Advective solution Intermediate solution Viscous solution

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4

ν = 0.001

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Theoretical Results

For the model problem (in 2d) : Brezzi, Canuto and Russo (1989)

◮ proved for each (δ, σ) pair existence of a solution in C 1. ◮ gave the following estimate :

√νu − uχH1 + u − uχL2 ≤ C νδ α .

Canuto and Russo (1993)

◮ proved uniqueness of the solution. ◮ proved convergence of the iterative procedure.

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Literature

• F. Brezzi, C. Canuto, A. Russo (1989): A Self-

Adaptative Formulation for the Euler/Navier-Stokes Coupling.

• C. Canuto and A. Russo (1993): On the

Elliptic-Hyperbolic Coupling I: the advection-diffusion equation via the χ-formulation.

• Y. Achdou, O. Pironneau (1993): The χ-Method for the

Navier-Stokes Equations.

• R. Arina, C. Canuto (1994): A chi-formulation of the

viscous-inviscid domain decomposition for the Euler/Navier-Stokes equations. C.-H. Lai, A. M. Cuffe and K. A. Pericleous (1998): A defect equation approach for the coupling of subdomains in domain decomposition methods

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Summary

Overlapping Method L(ν△u, u) = f L(0, u) = f Singular Perturbation Strategy L(ν△u, u) = f L(0, u) = f Boundary Layer Correction L(ν△u, u) = f L(ǫ△u, u) = f Factorization L(ν△u, u) = f ˜ L(0, u) = f χ-Formulation L(νχ(△u), u) = f L(0, u) = f

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Summary : error estimates

Overlapping Method O(ν5/2) − −O(ν3/2) O(ν) Singular Perturbation Strategy O(ν5/2) O(ν) Boundary Layer Correction ? O(ν) Factorization O(e−a/ν) O(ν) χ-Formulation O(ν5/2) O(ν)

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Convergence of the Newton Algorithm for the χ-formulation

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Convergence of the Newton Algorithm for the χ-formulation

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Convergence of the Newton Algorithm for the χ-formulation

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Convergence of the Newton Algorithm for the χ-formulation

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Coupling Methods Martin V´ eronique Introduction Overlapping Method Boundary Conditions as Coupling Conditions

Singular Perturbation Method Boundary Layer Correction Method Based on the Factorization of the Operator

χ-Formulation

Convergence of the Newton Algorithm for the χ-formulation