Matching of asymptotic expansions for the wave propagation in media with thin slot

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Matching of asymptotic expansions for the wave

propagation in media with thin slot

Sébastien Tordeux, Patrick Joly

To cite this version:

Sébastien Tordeux, Patrick Joly. Matching of asymptotic expansions for the wave propagation in

media with thin slot. AG Analysis und Numerik, 2005, Baal, Switzerland. �inria-00528070�

Matching of asymptotic expansions

for the wave propagation in media

with thin slot

S ´ebastien Tordeux and Patrick Joly

AG Analysis und Numerik, January 2005

A typical application

How can we study the scattering in media with

thin slot

?

source

wave

reflected

transmitted

wave

A physical problem with two

caracteristical

lengthes

(

The

wavelength

λ

The

width

of the slot

ε

An

asymptotic

case:

A typical application

How can we study the scattering in media with

thin slot

?

source

wave

reflected

transmitted

wave

An

asymptotic

case:

ε

_λ

The numerical difficulty

A

mesh step

smaller than

ε

ε

Some references

- Thin slot:

Harrington

,

Auckland

(1980),

Tatout

(1996).

- Finite differences:

Taflove

(1995).

- Thin plates and junction theory,...

Ciarlet

,

Le Dret

,

Dauge

-

Costabel

.

- Matching of asymptotic expansions:

McIver

,

Rawlins

(1993),

Il’in

(1992).

- multiscale analysis

A simple problem

Ω

ε

ε

Scalar

wave equation:

∂

2

p

ε

∂t

2

− ∆

p

ε

_{= f}

Harmonic

solution:

p

ε

(x, y, t) = exp(

_−iωt)

u

ε

(x, y)

Helmholtz

Equation:

∆

u

ε

+ ω

2

u

ε

=

_−f

in

Ω

ε

A simple problem

Ω

ε

ε

Outgoing

solution at infinity:

∂

u

ε

∂n

− iω

u

ε

≤

_r

C

₂

,

for

r

large

,

Neumann

limit condition

(rigid wall)

∂

u

ε

∂n

= 0

on

∂Ω

A simple problem

Ω

ε

ε

Outgoing

solution at infinity:

∂

u

ε

∂n

− iω

u

ε

≤

_r

C

₂

,

for

r

large

,

Neumann

limit condition

(rigid wall)

∂

u

ε

∂n

= 0

on

∂Ω

ε

A numerical computation

Dirichlet

Neumann

Numerical computation done with the

high order finite

elements code

of (

M. Duruflé

, INRIA)

A numerical computation

Dirichlet

Neumann

ε

A numerical computation

Dirichlet

Neumann

ε

A numerical computation

Dirichlet

Neumann

ε

Objectives

Introduce

accurate

numerical methods

We need an

intermediate zone

A technique

the matching of asymptotic expansions

Define

new approximate models

to compute the

solution.

Use effectively “universal” technique of numerical

computation (mesh reffinement).

Objectives

Introduce

accurate

numerical methods

We need an

intermediate zone

A technique

the matching of asymptotic expansions

Define

new approximate models

to compute the

solution.

Use effectively “universal” technique of numerical

computation (mesh reffinement).

Objectives

Introduce

accurate

numerical methods

We need an

intermediate zone

A technique

the matching of asymptotic expansions

Define

new approximate models

to compute the

solution.

Contributions to the match. of as. exp.

A new presentation of the

matching principle

(not

allways clear) postulated by the english school.

The

mathematical justification

of this technique.

The proof are

inspirated

by the multiscale technique

Existence and unicity

of the terms of the expansions.

Specific technique:

error estimates

.

Contributions to the match. of as. exp.

A new presentation of the

matching principle

(not

allways clear) postulated by the english school.

The

mathematical justification

of this technique.

The proof are

inspirated

by the multiscale technique

Existence and unicity

of the terms of the expansions.

Specific technique:

error estimates

.

Three zones

η

_S

η

_H

_x

y

Far field

(2D field)

Three zones

η

_S

η

_H

_x

y

The

asymptotic assumptions:

Three zones

x

y

The

asymptotic assumptions:

Three zones

x

y

Far field

The

asymptotic assumptions:

Three zones

x

y

Near field

The

asymptotic assumptions:

Three zones

x

y

Slot field

The

asymptotic assumptions:

Three zones

x

y

Far and near

The

asymptotic assumptions:

Three zones

x

y

Slot and near

The

asymptotic assumptions:

The different steps of the method

Derivate

the asymptotic expansions:

Formal

part

The different steps of the method

Derivate

the asymptotic expansions:

Formal

part

Several presentations are possible

Describe

the asymptotic expansions

Rigorous

part

The different steps of the method

Derivate

the asymptotic expansions:

Formal

part

Several presentations are possible

Describe

the asymptotic expansions

Rigorous

part

Definition

of the terms of the asymptotic expansions

Mathematical validation

of the asymptotic expansions

Rigorous

part

The different steps of the method

2

Derivate

the asymptotic expansions:

Formal

part

Several presentations are possible

1

Describe

the asymptotic expansions

Rigorous

part

Definition

of the terms of the asymptotic expansions

3

Mathematical validation

of the asymptotic expansions

Rigorous

part

Far field

Asymptotic context:

ε

_η

_H

_λ

.

η

_S

η

_H

Far field

Asymptotic context:

ε

_η

_H

_λ

.

(x, y)

η

_H

No

normalization

:

X = x,

Y = y.

Far field

Asymptotic context:

ε

_η

_H

_λ

.

(x, y)

η

_H

_ε

_{→ 0}

(x, y)

Ω

A

No

normalization

:

X = x,

Y = y.

Far field

Asymptotic context:

ε

_η

_H

_λ

.

(x, y)

η

_H

_ε

_{→ 0}

(x, y)

Ω

A

u

ε

=

u

0

+

+

_∞

X

i=1

i

₋₁

X

k=0

ε

i

(log

ε

)

k

u

k

_i

+ o(

ε

∞

),

in

Ω.

Far field

Asymptotic context:

ε

_η

_H

_λ

.

(x, y)

η

_H

_ε

_{→ 0}

(x, y)

Ω

A

where the

u

k

_i

satisfy the

homogeneous Helmholtz

equation

Slot field

u

ε

(x, y) =

U

ε

(x,

y

Slot field

ε

η

_S

x

y

The

asymptotic

context:

ε

_η

_S

_λ

.

The

normalization:

X = x,

Y =

y

Slot field

ε

η

_S

x

y

scaling

1

η

_S

x

Y

The

asymptotic

context:

ε

_η

_S

_λ

.

The

normalization:

X = x,

Y =

y

Slot field

ε

η

_S

x

y

scaling

1

η

_S

x

Y

ε

_{→ 0}

1

x

Y

The

asymptotic

context:

ε

_η

_S

_λ

.

The

normalization:

X = x,

Y =

y

Slot field

ε

η

_S

x

y

scaling

1

η

_S

x

Y

ε

_{→ 0}

1

x

Y

u

ε

(x, Y ε) =

U

ε

(x, Y ) =

+

_∞

X

i=0

i

X

k=0

ε

i

(log

ε

)

k

U

_i

k

(x, Y ) + o(

ε

∞

),

in

_O

₁

.

Slot field

ε

η

_S

x

y

scaling

1

η

_S

x

Y

ε

_{→ 0}

1

x

Y

where the

U

k

_i

satisfy the

1D Helmholtz

equation:

d

2

U

k

_i

dx

2

+ ω

2

_U

k

Near field

u

ε

(x, y) =

u

ε

_p

(

x

ε

,

y

Near field

The

Asymptotic

context:

ε

_η

_H

_λ,

ε

_η

_S

_λ.

The

normalization

:

X =

x

ε

,

Y =

y

Near field

(x, y)

ε

η

_S

η

_H

The

Asymptotic

context:

ε

_η

_H

_λ,

ε

_η

_S

_λ.

The

normalization

:

X =

x

ε

,

Y =

y

Near field

(x, y)

ε

η

_S

η

_H

1

η

H

ε

η

S

ε

1

(X, Y )

scaling

The

Asymptotic

context:

ε

_η

_H

_λ,

ε

_η

_S

_λ.

The

normalization

:

X =

x

ε

,

Y =

y

Near field

(x, y)

ε

η

_S

η

_H

1

η

H

ε

η

S

ε

1

(X, Y )

scaling

ε

_{→ 0}

(X, Y )

B

1

The

Asymptotic

context:

ε

_η

_H

_λ,

ε

_η

_S

_λ.

The

normalization

:

X =

x

ε

,

Y =

y

Near field

(x, y)

ε

η

_S

η

_H

1

η

H

ε

η

S

ε

1

(X, Y )

scaling

ε

_{→ 0}

(X, Y )

B

1

u

ε

(

ε

X,

ε

Y ) =

u

ε

_p

(X, Y ) =

+

_∞

X

i=0

i

X

k=0

ε

i

(log

ε

)

k

(

u

_p

)

k

_i

(X, Y ) + o(

ε

∞

).

Near field

(x, y)

ε

η

_S

η

_H

1

η

H

ε

η

S

ε

1

(X, Y )

scaling

ε

_{→ 0}

(X, Y )

B

1

where the

(

u

_p

)

k

_i

satisfy the

(in)-homogeneous Laplace

equation

.



Order 0 :

u

0

,

(

u

_p

)

0

₀

,

U

0

₀

A

Far field:



























Find

u

0

_{∈ H}

_loc

1

(Ω)

such that

:

−∆

u

0

_{− ω}

2

u

0

= f,

in

Ω,

∂

u

0

Order 0 :

u

0

,

(

u

_p

)

0

₀

,

U

0

₀

B

1

Near field:

Order 0 :

u

0

,

(

u

_p

)

0

₀

,

U

0

₀

O

1

Slot field:

Order 1 :

u

0

₁

,

(

u

_p

)

0

₁

,

(

u

_p

)

1

₁

,

U

0

₁

,

U

1

₁

A

r

θ

Approximation

of the exact Solution:

Order 1 :

u

0

₁

,

(

u

_p

)

0

₁

,

(

u

_p

)

1

₁

,

U

0

₁

,

U

1

₁

A

r

θ

explicit form of

u

0

₁

u

0

₁

(r, θ) =

₋

ω

2

u

0

_{(A) H}

(1)

0

(ωr).

Order 1 :

u

0

₁

,

(

u

_p

)

0

₁

,

(

u

_p

)

1

₁

,

U

0

₁

,

U

1

₁

X

Y

B

1

ρ

θ

Approximation

of the exact solution:







u

ε

(

ε

X,

ε

Y ) =

u

ε

_p

(X, Y ),

Order 1 :

u

0

₁

,

(

u

_p

)

0

₁

,

(

u

_p

)

1

₁

,

U

0

₁

,

U

1

₁

X

Y

B

1

ρ

θ

Near field:

Find

(

u

_p

)

0

₁

_{∈ H}

_loc

1

(

_B

₁

)

such that:











∆(

u

_p

)

0

₁

= 0,

in

_B

₁

∂(

u

_p

)

0

₁

B

Order 1 :

u

0

₁

,

(

u

_p

)

0

₁

,

(

u

_p

)

1

₁

,

U

0

₁

,

U

1

₁

X

Y

B

1

ρ

θ

Behavior at infinity

in the half-space:

(

u

_p

)

0

₁

(ρ, θ)

₋

∂

u

0

∂y

(A) ρ cos θ +

ω

2

u

0

_(A)

h

_{1 +}

2i

π

log ρ + γ

i

= O(

1

ρ

).

Order 1 :

u

0

₁

,

(

u

_p

)

0

₁

,

(

u

_p

)

1

₁

,

U

0

₁

,

U

1

₁

X

Y

B

1

ρ

θ

Behavior at infinity

in the half-space:

(

u

_p

)

0

₁

(ρ, θ)

₋

∂

u

0

∂y

(A) ρ cos θ +

ω

2

u

0

_(A)

h

_{1 +}

2i

π

log ρ + γ

i

= O(

1

ρ

).

Order 1 :

u

0

₁

,

(

u

_p

)

0

₁

,

(

u

_p

)

1

₁

,

U

0

₁

,

U

1

₁

X

Y

B

1

ρ

θ

(

u

_p

)

1

₁

=

₋

iω

π

u

0

_(A)

Order 1 :

u

0

₁

,

(

u

_p

)

0

₁

,

(

u

_p

)

1

₁

,

U

0

₁

,

U

1

₁

O

1

x

Approximation

of the exact solution:







u

ε

(x,

ε

Y ) =

U

ε

(x, Y ),

Order 1 :

u

0

₁

,

(

u

_p

)

0

₁

,

(

u

_p

)

1

₁

,

U

0

₁

,

U

1

₁

O

1

x

The slot field:

U

0

₁

(x) =

Z

1

0

U

0

Order 1 :

u

0

₁

,

(

u

_p

)

0

₁

,

(

u

_p

)

1

₁

,

U

0

₁

,

U

1

₁

O

1

x

The slot field:

U

1

₁

(x) =

₋

iω

π

u

The far fiel of order

i > 1

The field

u

k

_i

are defined in the

half space

:

The far fiel of order

i > 1

The field

u

k

_i

are defined in the

half space

:

A

The far fields

u

k

_i

satisfy the

homogeneous Helmholtz

equation

are

singular

at the neighborhood of the origin

The far fiel of order

i > 1

The field

u

k

_i

are defined in the

half space

:

A

u

k

_i

=

+

_∞

X

p=0

a

_p

H

_p

(1)

(ωr) cos pθ

The far fiel of order

i > 1

The field

u

k

_i

are defined in the

half space

:

A

u

k

_i

=

i

_−k−1

_X

p=0

The far fiel of order

i > 1

The field

u

k

_i

are defined in the

half space

:

A

u

k

_i

=

i

_−k−1

_X

p=0

The far fiel of order

i > 1

The field

u

k

_i

are defined in the

half space

:

A

Im

(H

₀

(1)

(ωr))

u

k

_i

=

i

_−k−1

_X

p=0

a

_p

H

_p

(1)

(ωr) cos pθ

The far fiel of order

i > 1

The field

u

k

_i

are defined in the

half space

:

A

Im

(H

₁

(1)

(ωr) cos θ)

u

k

_i

=

i

_−k−1

_X

p=0

a

_p

H

_p

(1)

(ωr) cos pθ

The near fields of order

i > 1

The near fields of order

i > 1

The

(

u

_p

)

k

_i

(X, Y )

are defined in the

canonical

domain:

by

Laplace

equation:

∆(

u

_p

)

k

_i

= 0,

(i = k

ou

k + 1),

∆(

u

_p

)

k

_i

=

_−ω

2

(

u

_p

)

k

_i

₋₂

,

(i

> k + 2),

The near fields of order

i > 1

The

(

u

_p

)

k

_i

(X, Y )

are defined in the

canonical

domain:

by

Laplace

equation:

by polynomial

growings

at infinity:

The

growings

in the half space are functions of

far

field of lower (or equal) order

The near fields of order

i > 1

The

(

u

_p

)

k

_i

(X, Y )

are defined in the

canonical

domain:

Proof of the

existence-unicity

:

with truncature functions, we subtract the growing

behavior at infinity of the

(

u

_p

)

k

_i

The slot field of order

i > 1

The

U

k

_i

are defined on the

canonical

domain:

O

1

The slot field of order

i > 1

The

U

k

_i

are defined on the

canonical

domain:

O

1

x

The slot field of order

i > 1

The

U

k

_i

are defined on the

canonical

domain:

O

1

x

The

U

k

_i

does not depend on

y

.

U

k

_i

(x) =

Z

1

0

Some properties

We see that:

More

i

_{− k}

is

large

more

u

k

_i

is

singular

at the origin:

Some properties

We see that:

More

i

_{− k}

is

large

more

u

k

_i

is

singular

at the origin:

r

−p

terms,

p = 0, ..., i

_{− k − 1}

More

i

_{− k}

is

large

more

(

u

_p

)

k

_i

is

growing

:

(

ρ

p

terms,

p = 0, ..., i

_{− k,}

X

p

termss,

p = 0, ..., i

_{− k,}

Some properties

We see that:

More

i

_{− k}

is

large

more

u

k

_i

is

singular

at the origin:

r

−p

terms,

p = 0, ..., i

_{− k − 1}

More

i

_{− k}

is

large

more

(

u

_p

)

k

_i

is

growing

:

(

ρ

p

terms,

p = 0, ..., i

_{− k,}

X

p

termss,

p = 0, ..., i

_{− k,}

Dependance diagram of the asymp. terms

4

3

2

1

0

i =

0

1

2

3

4

k=

ε

i

log

k

ε

Dependance diagram of the asymp. terms

4

3

2

1

0

i =

0

1

2

3

4

k=

ε

i

log

k

ε

Natural scheduling of the computations

4

3

2

1

0

i =

0

1

2

3

4

k=

ε

i

log

k

ε

Devirvate the terms of the as. exp.

We search for solutions of the form:

X

i

_∈Z

X

k

_∈Z

ε

i

(log

ε

)

k

u

k

_i

(far field)

X

i

_∈Z

X

k

_∈Z

ε

i

(log

ε

)

k

(

u

_p

)

k

_i

(near field)

X

i

_∈Z

X

k

_∈Z

Devirvate the terms of the as. exp.

We search for solutions of the form:

X

i

_∈Z

X

k

_∈Z

ε

i

(log

ε

)

k

u

k

_i

(far field)

X

i

_∈Z

X

k

_∈Z

ε

i

(log

ε

)

k

(

u

_p

)

k

_i

(near field)

X

i

_∈Z

X

k

_∈Z

ε

i

(log

ε

)

k

U

k

_i

(slot field)

Devirvate the terms of the as. exp.

We search for solutions of the form:

X

i

_∈Z

X

k

_∈Z

ε

i

(log

ε

)

k

u

k

_i

(far field)

X

i

_∈Z

X

k

_∈Z

ε

i

(log

ε

)

k

(

u

_p

)

k

_i

(near field)

X

i

_∈Z

X

k

_∈Z

ε

i

(log

ε

)

k

U

k

_i

(slot field)

We

inject

the equations (Helmholtz, Neumann)

We obtain the

coupling

conditions: (

the difficulty

)

Far-Near coupling

λ

η

_H

ε

η

_H

In a

thick zone

:

ε

_η

_H

_λ.

We write the coupling condition:

u

ε

(η

_H

, θ) = (

u

_p

)

ε

(

η

H

ε

, θ).

+

_∞

X

X

Far-Near coupling

X

i

_∈Z

X

k

_∈Z

ε

i

(log

ε

)

k

u

k

_i

(η

_H

, θ)

_'

X

i

_∈Z

X

k

_∈Z

ε

i

(log

ε

)

k

(

u

_p

)

k

_i

(

η

H

ε

, θ)

η

_H

_{→ 0}

η

H

ε

→ +∞

We

expand

the left serie according to

η

_H

near

0

The right serie according to

η

_H

/

ε

tending ot

infinity

The conclusion of the coupling

The

far

field-

near

field coupling:

The

singular

behavior of the

far field

is coupled with the

none growing

behavior of the

near field at infinity

.

The conclusion of the coupling

The

far

field-

near

field coupling:

The

growing

behavior of the

near field at infinity

is coupled

with the

none singular

behavior of the

far field

.

The conclusion of the coupling

The

far

field-

near

field coupling:

The

near

field-

slot

field coupling:

The

growing

behavior of the

near field

is coupled with the

none growing

behavior of the

slot field

(derivative values)

Mathematical analysis

Ω

R,R

0

R

0

R

u

ε

₋

u

0

₋

n

X

i=1

i

₋₁

X

k=0

ε

i

(log

ε

)

k

u

k

_i

H

1

_(Ω

R,R0

₎

≤ C

ε

n+1

_(log

_ε

₎

n

kfk

L

2

_(Ω)

.

Mathematical analysis

Ω

R,R

0

R

0

R

B

₁

L,L

0

L

L

0

u

ε

₋

u

0

₋

n

X

i=1

i

₋₁

X

k=0

ε

i

(log

ε

)

k

u

k

_i

H

1

_(Ω

R,R0

₎

≤ C

ε

n+1

_(log

_ε

₎

n

kfk

L

2

_(Ω)

.

u

ε

_p

₋

n

X

i=0

i

X

k=0

ε

i

(log

ε

)

k

(

u

p

)

k

_i

H

1

₍

_B

L,L0

1

)

≤ C

ε

n+1

(log

ε

)

n+1

_kfk

_L

2

_(Ω)

.

Mathematical analysis

Ω

R,R

0

R

0

R

B

₁

L,L

0

L

L

0

l

0

l

O

₁

l,l

0

u

ε

₋

u

0

₋

n

X

i=1

i

₋₁

X

k=0

ε

i

(log

ε

)

k

u

k

_i

H

1

_(Ω

R,R0

₎

≤ C

ε

n+1

_(log

_ε

₎

n

kfk

L

2

_(Ω)

.

u

ε

_p

₋

n

X

i=0

i

X

k=0

ε

i

(log

ε

)

k

(

u

p

)

k

_i

H

1

₍

_B

L,L0

1

)

≤ C

ε

n+1

(log

ε

)

n+1

_kfk

_L

2

_(Ω)

.

_X

n

_X

i

Idea of the proof

We want to define an

approximation

_e

u

ε

_n

of the exact solution

which

coincide

with:

the

truncated

expansion of the

far field

away from the

slot in the half space.

u

H,

_n

ε

(x, y) =

u

0

(x, y) +

n

X

i=1

i

₋₁

X

k=0

ε

i

(log

ε

)

k

u

k

_i

(x, y)

Idea of the proof

We want to define an

approximation

_e

u

ε

_n

of the exact solution

which

coincide

with:

The

truncated

expansion of the

near field

in the

neighbourhood of the end of the slot

u

N,

_n

ε

(x, y) =

n

X

i=0

i

X

k=0

ε

i

(log

ε

)

k

(

u

_p

)

k

_i

(

x

ε

,

y

ε

)

Idea of the proof

We want to define an

approximation

_e

u

ε

_n

of the exact solution

which

coincide

with:

the

truncated

expansion of the

slot field

far away in the

slot

u

S,

_n

ε

(x, y) =

n

X

i=0

i

X

k=0

ε

i

(log

ε

)

k

U

k

_i

(x,

y

ε

)

Idea of the proof

Introduce a partition of unity

e

u

ε

_n

(r, θ) = χ

ε

_H

u

H,

_n

ε

+ χ

_N

ε

u

N,

_n

ε

+ χ

ε

_S

u

S,

_n

ε

with

χ

ε

_H

+ χ

ε

_N

+ χ

ε

_S

= 1.

1

0

η

(

ε

)

χ

ε

_H

0

1

η

(

ε

)

η

_S

(

ε

)

χ

ε

_N

η

_S

(

ε

)

χ

ε

_S

Idea of the proof

The

error

equation:

e

ε

_n

=

_e

u

ε

_n

₋

u

ε



























∆

e

ε

_n

+ ω

2

e

_n

ε

= (

δ

_N

)

ε

_n

+ (

δ

_H

_−N

)

ε

_n

+ (

δ

_S

_−N

)

ε

_n

,

in

Ω

_ε

,

∂

e

ε

_n

∂n

= 0,

on

∂Ω

ε

,

e

ε

_n

is outgoing.

(

δ

_N

)

ε

_n

is related to the

approximation

of the

Helmholtz

equation by the

near

field

Idea of the proof

The

error

equation

e

ε

_n

=

_e

u

ε

_n

₋

u

ε



























∆

e

ε

_n

+ ω

2

e

_n

ε

= (

δ

_N

)

ε

_n

+ (

δ

_H

_−N

)

ε

_n

+ (

δ

_S

_−N

)

ε

_n

,

dans

Ω

_ε

,

∂

e

ε

_n

∂n

= 0,

on

∂Ω

ε

,

e

ε

_n

is outgoing.

(

δ

_H

_−N

)

ε

_n

is related to the

matching error

between the

far

field ans the

near

field

Idea of the proof

The

error

equation

e

ε

_n

=

_e

u

ε

_n

₋

u

ε



























∆

e

ε

_n

+ ω

2

e

_n

ε

= (

δ

_N

)

ε

_n

+ (

δ

_H

_−N

)

ε

_n

+ (

δ

_S

_−N

)

ε

_n

,

dans

Ω

_ε

,

∂

e

ε

_n

∂n

= 0,

sur

∂Ω

ε

,

e

ε

_n

is outgoing.

(

δ

_S

_−N

)

ε

_n

is related to the

matching error

between the

slot

field and the

near

champ

Idea of the proof

The

error

equation:

e

ε

_n

=

_e

u

ε

_n

₋

u

ε



























∆

e

ε

_n

+ ω

2

e

_n

ε

= (

δ

_N

)

ε

_n

+ (

δ

_H

_−N

)

ε

_n

+ (

δ

_S

_−N

)

ε

_n

,

in

Ω

_ε

,

∂

e

ε

_n

∂n

= 0,

on

∂Ω

ε

,

e

ε

_n

os outgoing.

classical asymptotic techniques:

Stability

: proof by

contradiction

(Helmholtz)

Idea of the proof

Global error estimates











_u

ε

_{− e}

_u

ε

n

H

1

_(Ω

R,δ

ε

)

6

C

h

η

_H

(

ε

)



n

+



ε

η

_H

(

ε

)



n

i

+ C

h

η

_S

(

ε

)



n

+



ε

η

_S

(

ε

)



n

i

.

δ

R

Idea of the proof

Global error estimate











_u

ε

_{− e}

_u

ε

n

H

1

_(Ω

R,δ

ε

)

6

C

h

η

_H

(

ε

)



n

+



ε

η

_H

(

ε

)



n

i

+ C

h

η

_S

(

ε

)



n

+



ε

η

_S

(

ε

)



n

i

.

One can choose

η

_H

(

ε

)

and

η

_S

(

ε

)

to

optimize

this relation

η

_H

(

ε

) = η

_S

(

ε

) =

√

ε

This leads to

Idea of the proof

_u

ε

−e

u

ε

_n

_H

₁

_(Ω

R,δ

ε

)

6 C

ε

n

2

=

_⇒

u

ε

_{− e}

u

ε

_n

_H

₁

_(Ω

_R,R0

₎

6 C

ε

n

2

δ

R

Idea of the proof

_u

ε

−e

u

ε

_n

_H

₁

_(Ω

R,δ

ε

)

6 C

ε

n

2

=

⇒

u

ε

−e

u

ε

n

H

1

_(Ω

R,R0

₎

6 C

ε

n

2

R

R

0

Idea of the proof

_u

ε

−e

u

ε

_n

_H

₁

_(Ω

R,δ

ε

)

6 C

ε

n

2

=

⇒

u

ε

−e

u

ε

n

H

1

_(Ω

R,R0

₎

6 C

ε

n

2

In the

far field

zone:

e

u

ε

_n

=

u

H,

_n

ε

=

u

0

+

n

X

i=1

i

₋₁

X

k=0

ε

i

(log

ε

)

k

u

k

_i

Idea of the proof

_u

ε

−e

u

ε

_n

_H

₁

_(Ω

R,δ

ε

)

6 C

ε

n

2

=

⇒

u

ε

−e

u

ε

n

H

1

_(Ω

R,R0

₎

6 C

ε

n

2

In the

far field

zone:

e

u

ε

_n

=

u

H,

_n

ε

=

u

0

+

n

X

i=1

i

₋₁

X

k=0

ε

i

(log

ε

)

k

u

k

_i







_u

ε

−

u

H,

_3n

ε

_H

₁

_(Ω

_R,R0

₎

6 C

ε

3n

2

_u

H,

ε

3n

−

u

H,

n

ε

H

1

_(Ω

R,R0

₎

6 C

ε

n+1

_log

n

_ε

Idea of the proof

_u

ε

−e

u

ε

_n

_H

₁

_(Ω

R,δ

ε

)

6 C

ε

n

2

=

⇒

u

ε

−e

u

ε

n

H

1

_(Ω

R,R0

₎

6 C

ε

n

2

In the

far field

zone:

e

u

ε

_n

=

u

H,

_n

ε

=

u

0

+

n

X

i=1

i

₋₁

X

k=0

ε

i

(log

ε

)

k

u

k

_i







_u

ε

−

u

H,

_3n

ε

_H

₁

_(Ω

_R,R0

₎

6 C

ε

3n

2

_u

H,

ε

3n

−

u

H,

n

ε

H

1

_(Ω

R,R0

₎

6 C

ε

n+1

_log

n

_ε

Mathematical analysis

Ω

R,R

0

R

0

R

B

₁

L,L

0

L

L

0

l

0

l

O

₁

l,l

0

u

ε

₋

u

0

₋

n

X

i=1

i

₋₁

X

k=0

ε

i

(log

ε

)

k

u

k

_i

H

1

_(Ω

R,R0

₎

≤ C

ε

n+1

_(log

_ε

₎

n

kfk

L

2

_(Ω)

.

u

ε

_p

₋

n

X

i=0

i

X

k=0

ε

i

(log

ε

)

k

(

u

p

)

k

_i

H

1

₍

_B

L,L0

1

)

≤ C

ε

n+1

(log

ε

)

n+1

_kfk

_L

2

_(Ω)

.

_X

n

_X

i

Perspectives

1. Mathematical analysis of the finite slot (

resonance

phenomena)

x

y

Perspectives

1. Mathematical analysis of the finite slot (

resonance

phenomena)

Perspectives

1. Mathematical analysis of the finite slot (

resonance

phenomena)

2. Comparison with the

multi-scale

technique

3. The 3D

Maxwell

equation

Perspectives

1. Mathematical analysis of the finite slot (

resonance

phenomena)

2. Comparison with the

multi-scale

technique

3. The 3D

Maxwell

equation

4. The

time domain

(evolution equation)

∂

2

u

∂t

2

− c