Boundary Value Problems in Some Ramified Domains with a Fractal Boundary: Analysis and Numerical Methods. Part II: Non homogeneous Neumann Problems.

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Boundary Value Problems in Some Ramified Domains with a Fractal Boundary: Analysis and Numerical

Methods. Part II: Non homogeneous Neumann Problems.

Yves Achdou, Christophe Sabot, Nicoletta Tchou

To cite this version:

Yves Achdou, Christophe Sabot, Nicoletta Tchou. Boundary Value Problems in Some Ramified Do-

mains with a Fractal Boundary: Analysis and Numerical Methods. Part II: Non homogeneous Neu-

mann Problems.. 2004. �hal-00003632�

Boundary Value Problems in Some Ramified Domains with a Fractal Boundary: Analysis and Numerical Methods.

Part II: Non homogeneous Neumann Problems

Yves Achdou

^∗

, Christophe Sabot

^†

, Nicoletta Tchou

^‡

. December 16, 2004

Abstract

This paper is devoted to numerical methods for solving Poisson problems in self-similar ramified domains of R

²

with a fractal boundary. It is proved that a sequence of solutions to some nonhomogeneous Neumann problems posed on domains obtained by interrupting the fractal construction after a finite number of generations, converges to the solution of a Neumann problem posed in the whole domain. To define the Neumann problem on the infinitely ramified domain and for proving the above mentioned convergence, extension and trace results are given. Then, a method for computing the solution is proposed an analyzed.

In particular, it is shown that the small scales of the Neumann data are damped exponentially fast away from the boundary. A self similar finite element method is developed and tested.

1 Introduction

In this paper, we deal with the numerical simulation of diffusion phenomena in a self-similar ramified domain of R

²

with a fractal boundary. This work was motivated by a wider and very challenging project aiming at simulating the diffusion of medical sprays in the lungs. Our ambi- tions here are more modest, since the geometry of the problems (two dimensions only) and the underlying physical phenomena are much simpler, but we hope that giving rigorous results and methods will prove useful. The geometry under consideration is that of a self-similar ramified bidimensional domain, see Figure 1 below. It can be seen as a simple model for a tree or for lungs. This domain can be obtained by glueing together dilated/translated copies of a simple polygonal domain of R

²

, called ω

⁰

below.

Partial differential equations in domain with fractal boundaries or fractal interfaces is a rela- tively new topic: variational techniques have been developed, involving new results on fonctional analysis, see [9, 7, 8]. A very nice theory on variational problems in fractal media is given in [10].

The difficulty of solving boundary value problems with partial differential equations in this kind of domains comes essentially from the multiscale character of the boundary. Yet, when the equation is homogeneous, it is possible to make use of the geometric self-similarity in order to compute very accurately the restrictions of the solutions to subdomains obtained by interrupting

∗

UFR Math´ ematiques, Universit´ e Paris 7, Case 7012, 75251 Paris Cedex 05, France and Laboratoire Jacques- Louis Lions, Universit´ e Paris 6, 75252 Paris Cedex 05. [email protected]

†

CNRS, UMPA, UMR 5669, 46, Allee d’Italie, F-69364 Lyon Cedex 07, France. [email protected]

‡

IRMAR, Universit´ e de Rennes 1, Rennes, France, [email protected]

the fractal construction after a finite number of generations.

In a previous work [1], we have considered a Poisson problem with homogeneous Neumann con- ditions on the whole boundary, except on the bottom part of the boundary (see Figure 1), noted Γ

⁰

below, where a Dirichlet condition was imposed. For that, it is possible to solve an equivalent boundary value problem in a subdomain obtained by interrupting the fractal construction after a finite number of generations: this equivalent problem involves a nonlocal Dirichlet to Neu- mann operator T

⁰

, which maps a function defined on Γ

⁰

to the normal derivative of its harmonic lifting in the whole domain. It turns out that the Dirichlet to Neumann operator on Γ

⁰

can be computed very accurately by making use of the geometric self-similarity. The Dirichlet to Neumann operator is approximated as the limit of an inductive sequence, see (37) (38) (39) below. When discretizing the problem with finite elements with self similar meshes, the same procedure can be implemented. The numerical method developed in [1] is reminiscent of some of the techniques involved in the theoretical analysis of finitely ramified fractals (see [12],[15], [14], [13], and [2, 11, 6] for numerical simulations). The simple structure of these sets allows to do an explicit analysis of the spectral properties. This involves the dynamics of a renormalization map which acts on the Dirichlet to Neumann operator on the boundary (which for finitely ramified fractal consists only on a finite number of points). Here, the natural boundary is not so simple, but the numerical method is based on a similar strategy.

In the present paper, we are interested in solving the same kind of problem, except that the Neumann data is nonzero on the top part of the boundary, called Γ

^∞

below. The first thing to do is to give a meaning to this kind of problem. For that, it is necessary to prove nonstandard extension and trace theorems, which are, in our opinion, interesting by themselves. This is done in § 3.

Next, an interesting problem is to design a method which permits to approximate numerically the restriction of the solution to a subdomain obtained by interrupting the fractal construction after a finite number of generations. This will be done by expanding the Neumann data on Γ

^∞

on the basis of Haar wavelets. hen the Neumann data is a Haar wavelet, the solution of the boundary problem can be computed by using the operator T

⁰

mentioned above, thanks to self-similarity. The program described above is carried out at a continuous level in § 4, and at a discrete level in § 5, where finite element are used with special self-similar meshes. Finally, numerical examples are given in § 6, with reults in very good agremment with the theory.

2 Geometrical setting of the model problem

Consider the following T-shaped subset of R

²

Q

⁰

= ( − 1, 1) × (0, 2]

∪ (( − 2, 2) × (2, 3)) ∪ ((( − 2, − 1) ∪ (1, 2)) × { 3 } ) .

The fractal domain Ω

⁰

is constructed as an infinite union of subsets of R

²

obtained by translat- ing/dilating Q

⁰

; at a first stage, two copies of 1/2 · Q

⁰

are translated respectively on top-left and on top-right of Q

⁰

and are glued to Q

⁰

: more precisely, let F

₁

and F

₂

be the affine mappings

F

i

(x) = ξ

_i¹

+ 1

2 x, where ξ

₁¹

= ( − 3

2 , 3) and ξ

₂¹

= ( 3

2 , 3), (1)

and let Q

¹

be the set Q

¹

= F

₁

(Q

⁰

) ∪ F

₂

(Q

⁰

). Next, the construction is recursive: the points ξ

_iⁿ

for i = 1, . . . , 2

ⁿ

are defined by the relation: for j = 1, . . . , 2

ⁿ⁻¹

, ξ

ⁿ_2j−1

= ξ

_jⁿ⁻¹

+

₂n−1¹

ξ

₁¹

and ξ

_2jⁿ

= ξ

_jⁿ⁻¹

+

_2n−1¹

ξ

₂¹

, and the following sets are introduced:

Q

ⁿ

= ∪

²i=1ⁿ

Q

ⁿ_i

, with Q

ⁿ_i

= ξ

_iⁿ

+ 1

2

ⁿ

· Q

⁰

. (2)

For n ≥ 1, calling A

ⁿ

the set containing all the mappings from { 1, . . . , 2

ⁿ⁻¹

} to { 1, 2 } , and for σ ∈ A

n

, M

σ

(F

₁

, F

₂

) = F

_σ(1)

◦ F

_σ(2)

◦ · · · ◦ F

_σ(2ⁿ₎

, (2) can also be written

Q

ⁿ

= ∪

σ∈An

M

^σ

(F

₁

, F

₂

)(Q

⁰

).

It will sometimes be convenient to agree that A

0

= { 0 } and that M

0

(F

1

, F

2

) is the identity.

Finally, the fractal tree Ω

⁰

is defined by

Ω

⁰

= ∪

^∞n=0

Q

ⁿ

. (3)

The construction of Ω

⁰

is displayed on Figure 1. It is straightforward to see that Ω

⁰

⊂ ( − 3, 3) ×

Figure 1: Left: the first step of the construction. Right: the fractal tree (only a few generations are displayed)

(0, 6). Note that Ω

⁰

may also be obtained as a union of overlapping open subsets of R

²

, thus Ω

⁰

is an open set.

It will be useful to define the truncated fractal Ω

^N

:

Ω

^N

= ∪

^∞n=N

Q

ⁿ

. (4)

The following self-similarity property is true: Ω

^N

is the union of 2

^N

translated copies of

₂¹N

· Ω

⁰

, i.e.

Ω

^N

= ∪

σ∈AN

Ω

^σ

, (5)

where

Ω

^σ

= M

^σ

(F

₁

, F

₂

)(Ω

⁰

). (6)

Also, Ω

^N

\ Ω

^N+1

= Q

^N

for any N ≥ 0.

We define the bottom boundary of Ω

⁰

by Γ

⁰

= (( − 1; 1) × { 0 } ) and Σ

⁰

= ∂Ω

⁰

∩ { (x

1

, x

2

); x

1

∈ R , 0 < x

₂

< 6 } . We have

∂Ω

⁰

∩ { (x

1

, x

2

); x ∈ R , x

2

< 6 } = Γ

⁰

∪ Σ

⁰

. (7) Similarly, the bottom boundary of Ω

^N

is Γ

^N

= ∪

²_i=1^N

Γ

^N_i

, Γ

^N_i

= ξ

_i^N

+

₂¹N

· Γ

⁰

. In an equivalent manner,

Γ

^N

= ∪

σ∈AN

Γ

^σ

, (8)

where

Γ

^σ

= M

σ

(F

₁

, F

₂

)(Γ

⁰

). (9)

For N > 0, Γ

^N

is contained in the line x

₂

= 3 P

N−1

i=0

2

⁻ⁱ

. We define also Σ

^N

= ∂Ω

^N

∩ { (x

1

, x

2

); x

1

∈ R , 3 P

N−1

i=0

2

⁻ⁱ

< x

2

< 6 } . Calling Γ

^∞

= [ − 3 : 3] × { 6 } , one can check easily that

∂Ω

⁰

= Γ

⁰

∪ Σ

⁰

∪ Γ

^∞

. (10)

The aim of this paper is to study boundary value problems in Ω

⁰

with nonhomogeneous Neumann boundary condition on Γ

^∞

For what follows, it is also useful to introduce the open domains ω

^N

, for N ≥ 0:

ω

^N

= Int Ω

⁰

\ Ω

^N+1

. (11)

Remark 1 Note that it is also possible to construct very similar fractal trees using dilations with ratii α

ⁿ

with α ∈ ]0; 1/2]; here we have chosen α = 1/2.

3 Some functions spaces

Let q be a real number such that q ≥ 1. Consider the function space W

^1,q

(Ω

ⁿ

) = { v ∈ L

^q

(Ω

ⁿ

) s.t. ∇ v ∈ (L

^q

(Ω

ⁿ

))

²

} . Similarly, for all positive integer p, it is possible to define W

^p,q

(Ω

ⁿ

) as the space of functions whose partial derivatives up to order p belong to L

^q

(Ω), and for all positive real number s 6∈ N , W

^s,q

(Ω

ⁿ

) is defined by interpolation between W

^p,q

(Ω

ⁿ

) and W

^p+1,q

(Ω

ⁿ

), where p is the integer such that p ≤ s < p + 1. Likewise, it is possible to define the Sobolev spaces W

^s,q

(ω

ⁿ

) for all nonnegative integers n. All the spaces introduced below endowed with their natural norms are Banach spaces. For general results on Sobolev spaces for domains with Lipschitz regular boundaries (which is not the case here), see [3, 4]. In the case q = 2, the spaces are Hilbert spaces, and we use the special notation H

^s

(Ω

⁰

) = W

^s,2

(Ω

⁰

).

Of course, for all n ≥ 0, the restriction of a function v ∈ H

¹

(Ω

⁰

) to ω

ⁿ

belongs to H

¹

(ω

ⁿ

), so it is possible to define the trace of v on Γ

ⁿ

. The trace operator on Γ

ⁿ

is bounded from H

¹

(Ω

⁰

) to L

²

(Γ

ⁿ

), so one can define the closed subspace of H

¹

(Ω

ⁿ

):

V (Ω

ⁿ

) = { v ∈ H

¹

(Ω

ⁿ

) s.t. v |

Γⁿ

= 0 } . (12) In what follows, for a function u integrable on Γ

^σ

, the notation h u i

Γ^σ

will be used for the mean value of u on Γ

^σ

.

We will also use the notation . to indicate that there may arise constants in the estimates, which are independent of the index n in Ω

ⁿ

or ω

ⁿ

or on the mesh size when dealing with finite elements.

3.1 Poincar´ e’s inequality and consequences By generalizing slightly the results proved in [1], we have the

Theorem 1 For p ∈ R , 1 ≤ p and any function u ∈ W

^1,p

(Ω

⁰

) whose trace on Γ

⁰

is zero,

k u k

L^p(Ω⁰)

≤ 8p

^−1p

k∇ u k

L^p(Ω⁰)

. (13)

There exists a positive constant C such that

• for all n ≥ 0 and for all u ∈ W

^1,p

(Ω

⁰

), k u k

^p_L^p_(Ωⁿ₎

≤ C

2

^−np

k∇ u k

^p_L^p_(Ωⁿ₎

+ 2

⁻ⁿ

k u |

Γⁿ

k

^p_L^p_(Γⁿ₎

, (14)

• for all integers n, q, n > q ≥ 0 and for all v ∈ W

^1,p

(Ω

⁰

),

k v |

Γ^q

k

^p_L^p_(Γ^q₎

− k v |

Γⁿ

k

^p_L^p_(Γⁿ₎

≤ C2

^(1−p)q

k∇ v k

^p_L^p_(ωⁿ_\ω^q₎

. (15)

• for all u ∈ W

^1,p

(Ω

⁰

), for all N ≥ 0, k u k

^p_Lp(Ω^N)

≤ C2

^−N

k∇ u k

^p_L^p_(Ω0)

+ k u |

Γ⁰

k

^p_L^p_(Γ0)

.

The imbedding from W

^1,p

(Ω

⁰

) in L

^p

(Ω

⁰

) is compact.

3.2 Extensions and traces 3.2.1 Orientation

We aim at constructing an extension operator mapping a function of W

^1,q

(Ω

⁰

), q ≥ 1, to a function defined in a simple polygonal domain of R

²

. Call ˜ Q

⁰

of R

²

the convex hull of the points

0000 0000 0000 0000 00

1111 1111 1111 1111 11 00 00 0 11 11 1

00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111

00 00 00 00 0

11 11 11 11 0 1 00 00 11 11 1

0000 0000 0000 0000 0000 0000 0000 0000 00

1111 1111 1111 1111 1111 1111 1111 1111 11

00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111

00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 0000

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11110000000000000000000000000000000000

1111 1111 1111 1111 1111 1111 1111 1111 11000000000

11 11 11 11 1

000000 000000 000000 000000 000

111111 111111 111111 111111 1110 00 00 00 00

11 11 11 11 1

Figure 2: The extension is performed in two steps ( −

³₂

, 0), (

³₂

, 0), ( −

⁵₂

, 3), (

⁵₂

, 3), and ˜ Ω

⁰

the new fractal domain

Ω ˜

⁰

= [

n∈N

[

σ∈An

M

^σ

(F

₁

, F

₂

) ˜ Q

⁰

.

It is represented on the right part of Figure 2 with full lines. It is clear that Ω

⁰

⊂ Ω ˜

⁰

.

We call also Ω b

⁰

the convex hull of the points ( −

³₂

, 0), (

³₂

, 0), ( − 3, 6), (3, 6). Note that ˜ Ω

⁰

is

obtained by removing an infinite family of nonoverlapping triangles from Ω b

⁰

. More precisely,

consider the triangle T = conv((0, 6), (1, 3), ( − 1, 3)) ⊂ Ω b

⁰

\ Ω ˜

⁰

, which is visible on the right part of Figure 2 and call G

₁

, G

₂

the affine mappings G

₁

(x) = (1+

^x₂¹

, 3+

^x₂²

), G

₂

(x) = ( − 1+

^x₂¹

, 3+

^x₂²

).

We have that

Ω ˜

⁰

= Ω b

⁰

\ [

n∈N

[

σ∈An

M

σ

(G

1

, G

2

)T

! .

3.2.2 Bounded extension from W

^1,q

(Ω

⁰

) to W

^1,q

( ˜ Ω

⁰

), q ≤ 2

Our first goal is to construct an extension operator, bounded from W

^1,q

(Ω

⁰

) to W

^1,q

( ˜ Ω

⁰

), q, 1 ≤ q ≤ 2,. This is done in two steps.

First step Call D

L

the L-shaped compact set D

L

= conv ( − 1, − 2), ( −

¹₂

, − 2), ( −

¹₂

, 2), ( − 1, 2)

∪ conv ( −

¹₂

, 2), ( − 2, 2), ( − 2,

⁵₂

), ( −

¹₂

,

⁵₂

)

and D

_R

the image of D

_L

by the symmetry of axis x

₁

= 0.

Call also T

L

the triangle conv (( − 1, − 2), ( − 1, 2), ( − 2, 2)) and T

R

the image of T

L

by the sym- metry of axis x

₁

= 0.

It is possible to construct an extension operator E

L

, which maps continuously W

^1,q

(D

_L

) to W

^1,q

(D

L

∪ T

L

), for all q ∈ [1, 2]: for all q ∈ [1, 2], there is a positive constant c such that, for all u ∈ W

^1,q

(D

L

), kE

^L

u k

W^1,q(TL)

≤ c k u k

W^1,q(DL)

.

Call Ω

⁰_L

= Ω

⁰

∩ S

n∈N

F

₁ⁿ

(D

L

) and ˜ Ω

⁰_L

= ˜ Ω

⁰

∩ S

n∈N

F

₁ⁿ

(D

L

∪ T

L

). Similarly Ω

⁰_R

= Ω

⁰

∩ S

n∈N

F

₂ⁿ

(D

_R

) and ˜ Ω

⁰_R

= ˜ Ω

⁰

∩ S

n∈N

F

₂ⁿ

(D

_R

∪ T

_R

). The previous observation and the facts that

• | F

₁ⁿ

(T

L

) ∩ F

₁^m

(T

L

) | = 0 if n 6 = m.

• for any point x in Ω

⁰_L

, there exist at least one and at most two integers n such that x ∈ F

₁ⁿ

(D

L

).

enable to construct an extension operator ˜ E

L

, bounded from W

^1,q

(Ω

⁰_L

) to W

^1,q

( ˜ Ω

⁰_L

), 1 ≤ q ≤ 2, by:

if x ∈ F

₁ⁿ

(T

_L

), n ≥ 1, then ( ˜ E

L

u)(x) = E

L

((u ◦ F

₁ⁿ

) |

DL

) (F

₁ⁿ

)

⁻¹

(x) , if x ∈ Ω ˜

⁰

∩ T

L

, then ( ˜ E

L

u)(x) = E

L,0

(u |

DL

)(x),

where E

L,0

is any extension operator which maps continuously W

^1,q

(Ω

⁰

∩ D

_L

) to W

^1,q

( ˜ Ω

⁰

∩ (D

L

∪ T

L

)), 1 ≤ q ≤ 2.

By symmetry, it possible to construct an extension operator, bounded from ˜ E

R

from W

^1,q

(Ω

⁰_R

) to W

^1,q

( ˜ Ω

⁰_R

), 1 ≤ q ≤ 2.

Second step Let G

^L

= { F

₂

◦ M

^σ

(F

₁

, F

₂

), σ ∈ A

ⁿ

, n ∈ N } and G

^R

= { F

₁

◦ M

^σ

(F

₁

, F

₂

), σ ∈ A

n

, n ∈ N } . We observe that for any point x in ˜ Ω

⁰

\ Ω

⁰

, one and only one of the four successive items is true:

1. either x ∈ Ω ˜

⁰_L

\ Ω

⁰

, 2. or x ∈ Ω ˜

⁰_R

\ Ω

⁰

,

3. or there exists a unique tranformation τ in G

^L

such that x ∈ τ ( ˜ Ω

⁰_L

\ Ω

⁰

), 4. or there exists a unique tranformation τ in G

R

such that x ∈ τ ( ˜ Ω

⁰_R

\ Ω

⁰

),

From this observation, it is possible to construct an extension operator E , bounded from

W

^1,q

(Ω

⁰

) to W

^1,q

( ˜ Ω

⁰

), 1 ≤ q ≤ 2, by: for x ∈ Ω ˜

⁰

\ Ω

⁰

,

• if item 1 is true: ( E u)(x) = ( ˜ E

^L

u)(x),

• if item 2 is true: ( E u)(x) = ( ˜ E

R

u)(x),

• if item 3 is true: ( E u)(x) = ( ˜ E

L

(u ◦ τ ))(τ

⁻¹

(x)),

• if item 4 is true: ( E u)(x) = ( ˜ E

R

(u ◦ τ ))(τ

⁻¹

(x)),

3.2.3 Bounded extension from W

^1,q

( ˜ Ω

⁰

) to W

^1,q

( Ω b

⁰

), 1 ≤ q < 2

Consider the triangle T b = conv((0, 6), (

³₂

, 2), ( −

³₂

, 2)) which is displayed on the left of Figure 2, with interrupted lines. The key observation is that

T b \ T ⊂ Ω ˜

⁰

, (16)

and that

∀ σ

₁

, σ

₂

∈ [

n∈N

A

ⁿ

such that σ

₁

6 = σ

₂

, M

^σ1

(G

₁

, G

₂

)( T b ) ∩ M

^σ2

(G

₁

, G

₂

)( T b ) = ∅ . (17) It is well known that there exists a extension operator ˜ F , bounded from W

^1,q

( T b \ T) to W

^1,q

( T), b for any q, 1 ≤ q < 2. Note that the operator cannot be bounded in from W

^1,2

( T b \ T ) to W

^1,2

( T), b because its construction involves polar coordinates around the common vertex of T and T b . From this and (17), we can construct an extension operator F , bounded from W

^1,q

( ˜ Ω

⁰

) to W

^1,q

( Ω b

⁰

), 1 ≤ q < 2 by

if x ∈ M

^σ

(G

₁

, G

₂

)(T ), F (u)(x) = ˜ F

(u ◦ M

^σ

(G

₁

, G

₂

)) |

Tb\T

◦ ( M

^σ

(G

₁

, G

₂

))

⁻¹

(x).

3.2.4 Bounded extension from W

^1,q

(Ω

⁰

) to W

^1,q

( Ω b

⁰

), 1 ≤ q < 2

By composing the extension operators E and F constructed above, we have proved the

Theorem 2 There exists an extension operator J bounded from W

^1,q

(Ω

⁰

) to W

^1,q

( ˆ Ω

⁰

), for all q, 1 ≤ q < 2,

Remark 2 Of course, since Ω

⁰

is a bounded domain, we have also that for q, 1 ≤ q < 2, the extension operator J is bounded from H

¹

(Ω

⁰

) to W

^1,q

( ˆ Ω

⁰

).

As a consequence of Theorem 2, we have the Sobolev imbeddings:

Proposition 1 (Sobolev imbeddings) Let q be a real number such that 1 ≤ q < 2: we have the continuous imbeddings:

W

^1,q

(Ω

⁰

) ⊂ L

^p

(Ω

⁰

), ∀ p, 1 ≤ p ≤ q

^∗

, q

^∗

= 2q 2 − q , and the imbedding is compact if p < q

^∗

.

Furthermore, for all q, p, 1 ≤ q < 2, 1 ≤ p ≤ q

^∗

, there exists a constant C such that for all N ≥ 0,

k u k

^p_Lp(Ω^N)

≤ C

2

2N(p−q)−qpN

q

k∇ u k

^p_Lq(Ω^N)

+ 2

^N(p−2q)q

k u k

^p_Lq(Γ^N)

. (18)

For all real number p ≥ 1, H

¹

(Ω

⁰

) ⊂ L

^p

(Ω

⁰

) with continuous and compact imbedding.

3.2.5 Density results

Theorem 3 For q, 1 ≤ q < 2, the space C

^∞

(Ω

⁰

) is dense in W

^1,q

(Ω

⁰

), and there exists a sequence of linear operators ( S

n

)

_n∈^N

from W

^1,q

(Ω

⁰

) to C

^∞

( Ω b

⁰

) such that

∀ u ∈ W

^1,q

(Ω

⁰

), lim

n→∞

k u − ( S

n

u) |

Ω⁰

k

W^1,q(Ω⁰)

= 0, (19) and for a constant c,

∀ u ∈ W

^1,q

(Ω

⁰

), kS

n

u k

_W1,q(Ωb⁰)

≤ c k u k

W^1,q(Ω⁰)

. (20) Proof. We know that C

^∞

( Ω b

⁰

) is dense in W

^1,q

(b Ω

⁰

) and that there exists a sequence of linear operators ( S b

n

)

_n∈^N

from W

^1,q

(b Ω

⁰

) to C

^∞

( Ω b

⁰

) such that

∀ u ∈ W

^1,q

(b Ω

⁰

), lim

n→∞

k u − S b

ⁿ

u k

_W^1,q(bΩ⁰)

= 0, and for a constant c,

∀ u ∈ W

^1,q

( Ω b

⁰

), k S b

n

u k

_W^1,q₍Ωb⁰)

≤ c k u k

_W^1,q₍Ωb⁰)

.

Consider an extension operator J as in Theorem 2. The operators S

ⁿ

= S b

ⁿ

◦ J answer the question.

3.2.6 Traces

Take q, 1 < q ≤ 2 and call N

^pq

the mapping

N

p^q

: W

^1,q

(Ω

⁰

) → R

₊

, N

p^q

(v) = k v |

Γ^p

k

^q_L^q_(Γ^p₎

. (21) Estimate (15) tells us that for any v ∈ W

^1,q

(Ω

⁰

), ( N

p^q

(v))

_p∈^N

is a Cauchy sequence in R

₊

and that it converges to a real number N

∞^q

(v). It is an easy matter to prove the following

Lemma 1 Let q be a real number such that 1 < q ≤ 2. The mapping v 7→ N

∞^q

(v) is homogeneous of degree q.

There exists a constant C such that forall v ∈ W

^1,q

(Ω

⁰

),

N

∞^q

(v) ≤ C k v k

^q_W1,q(Ω⁰)

. (22) For v ∈ C

^∞

( Ω b

⁰

),

N

∞^q

(v |

Ω⁰

) = 1

3 k v k

^q_L^q_(Γ^∞₎

. (23)

Theorem 4 Let q be a real number such that 1 < q < 2. There is a continuous trace operator γ from W

^1,q

(Ω

⁰

) onto W

^1−1q,q

(Γ

^∞

), such that for all u ∈ C

^∞

( Ω b

⁰

),

γ(u |

Ω⁰

) = u |

Γ^∞

.

Proof. Consider a sequence of operators ( S

n

)

_n∈^N

as in Theorem 3. Let u be a function in W

^1,q

(Ω

⁰

). The function S

n

(u) has a trace in W

^1−1q,q

(Γ

^∞

), and we have from (20) that for a constant c (independent of n)

kS

ⁿ

(u) k

_W1−1

q ,q(Γ^∞)

≤ c k u k

W^1,q(Ω⁰)

.

Therefore, one can extract a subsequence S

φ(n)

such that S

φ(n)

(u) |

Γ^∞

converges weakly in W

^1−1q,q

(Γ

^∞

) and strongly in L

^q

(Γ

^∞

) to some function w ∈ W

^1−1q,q

(Γ

^∞

). There remains to prove that w is unique (i.e. the whole sequence converges), and that w depends only on u and not on the sequence S

n

.

Assume that there exist two subsequences ( S

φ(n)

(u))

_n∈N

and ( S

ψ(n)

(u))

_n∈N

whose traces on Γ

^∞

converges respectively to w and w

^′

weakly in W

^1−1q,q

(Γ

^∞

) and strongly in L

^q

(Γ

^∞

). We have from (23) that

k w − w

^′

k

^q_L^q_(Γ^∞₎

= lim

n→∞

kS

φ(n)

(u) − S

ψ(n)

(u) k

^q_L^q_(Γ^∞₎

= 3 lim

n→∞

N

∞^q

( S

φ(n)

(u) |

Ω⁰

− S

ψ(n)

(u) |

Ω⁰

) From (22),

k w − w

^′

k

^q_L^q_(Γ^∞₎

≤ 3C lim sup

n→∞

kS

φ(n)

(u) |

Ω⁰

− S

ψ(n)

(u) |

Ω⁰

k

^q_W1,q(Ω⁰)

, and we conclude from (19) that

k w − w

^′

k

^q_L^q_(Γ^∞₎

= 0, and therefore w = w

^′

.

The same argument leads to the fact that w does not depend on the sequence of operators ( S

n

)

_n∈N

.

Therefore the mapping γ : u 7→ w is a continuous linear operator from W

^1,q

(Ω

⁰

) to W

^1−1q,q

(Γ

^∞

).

It can also be checked by reproducing the argument above that if u ∈ C

^∞

(b Ω

⁰

) then γ(u |

Ω⁰

) = u |

Γ^∞

. Therefore γ is a trace operator from W

^1,q

(Ω

⁰

) to W

^1−1q,q

(Γ

^∞

).

It is surjective because the trace operator from W

^1,q

( Ω b

⁰

) to W

^1−1q,q

(Γ

^∞

) is surjective.

From the continuous imbedding H

¹

(Ω

⁰

) ⊂ W

^1,q

(Ω

⁰

), and from the Sobolev imbedding theorems in dimension one, we have the

Corollary 1 The trace operator is continuous from H

¹

(Ω

⁰

) to W

^1−1q,q

(Γ

^∞

), for all real numbers q with 1 < q < 2. The trace operator is continuous from H

¹

(Ω

⁰

) to L

^p

(Γ

^∞

), for all real number p with 1 ≤ p < ∞ .

Corollary 2 For all u ∈ H

¹

(Ω

⁰

), for all real number p, p > 1, N

∞^p

(u) = 1

3 k γ(v) k

^p_Lp(Γ^∞)

. (24)

For all u, v ∈ H

¹

(Ω

⁰

),

n→∞

lim Z

Γⁿ

u |

Γⁿ

v |

Γⁿ

= 1 3

Z

Γ^∞

γ(u)γ (v). (25)

4 Poisson problems with nonzero Neumann data on Γ ^∞

4.1 The boundary value problem

Let p be a real number greater than one, take g ∈ L

^p

(Γ

^∞

). We are interested in the variational problem

find u ∈ V (Ω

⁰

) such that and for all v ∈ V (Ω

⁰

), Z

Ω⁰

∇ u · ∇ v = 1 3

Z

Γ^∞

gγ(v). (26)

From Corollary 1, the linear form v 7→ R

Γ^∞

gγ(v) is bounded on H

¹

(Ω

⁰

), and from Theorem 1, problem (26) has a unique solution.

The next result says that the solution to (26) can be approximated by solving boundary value problems in ω

ⁿ

, n → ∞ :

Proposition 2 Let p be a real number such that 1 < p ≤ 2. If g is the trace on Γ

^∞

of a function

˜

g belonging to W

^1,p

(b Ω

⁰

), then

n→∞

lim k u |

^ωn

− u

n

k

H¹(ωⁿ)

= 0, where u is defined by (26) and u

n

∈ V (ω

ⁿ

) is the solution to:

for all v ∈ V (ω

ⁿ

), Z

ωⁿ

∇ u

n

· ∇ v = Z

Γⁿ⁺¹

˜

g |

Γⁿ⁺¹

v |

Γⁿ⁺¹

. (27) Proof. Calling e

n

the error e

n

= u |

^ωn

− u

n

∈ V (ω

ⁿ

), we see that

Z

ωⁿ

∇ e

n

· ∇ v = 1

3 Z

Γ^∞

gγ(v) − Z

Γⁿ⁺¹

˜

g |

Γⁿ⁺¹

v |

Γⁿ⁺¹

− Z

Ωⁿ⁺¹

∇ u · ∇ v. (28) It can be proved that for all v ∈ V (Ω

⁰

),

¹₃

R

Γ^∞

gγ(v) = lim

_n→∞

R

Γⁿ⁺¹

˜ g |

Γⁿ⁺¹

v |

Γⁿ⁺¹

, so the first term in the right hand side tends to zero. More precisely, Proposition 1 tells us that for all r, 1 < r < p, the function ˜ gv belongs to W

^1,r

(Ω

⁰

). It is easy to check by a scaling argument that, for all m < m

^′

,

Z

Γ^m+1

˜

g |

Γ^m+1

v |

Γ^m+1

− Z

Γ^m′+1

˜

g |

_Γ^m′+1

v |

_Γ^m′+1

. 2

^m1−rr

k∇ (˜ gv) k

L^r(ω^m′\ω^m)

. Since ˜ g is fixed, this implies from Proposition 1 that

Z

Γ^m+1

˜

g |

Γ^m+1

v |

Γ^m+1

− Z

Γ^m′+1

˜

g |

_Γ^m′+1

v |

_Γ^m′+1

. 2

^m1−rr

k v k

H¹(Ω⁰)

. Thus, the sequence of continuous linear forms on V (Ω

⁰

): v 7→ R

Γⁿ⁺¹

˜ g |

Γⁿ⁺¹

v |

Γⁿ⁺¹

is a Cauchy sequence in the dual of V (Ω

⁰

), and the limit can be identified by using Theorem 3: therefore,

n→∞

lim sup

v∈V(Ω⁰),v6=0

¹₃

R

Γ^∞

gγ(v) − R

Γⁿ⁺¹

g ˜ |

Γⁿ⁺¹

v |

Γⁿ⁺¹

k v k

H¹(Ω⁰)

= 0. (29)

We also have that

n→∞

lim sup

v∈V(Ω⁰),v6=0

R

Ωⁿ⁺¹

∇ u · ∇ v k v k

H¹(Ω⁰)

= 0. (30)

From (28) (29) and (30), we deduce the desired result.

We will try to solve (26) numerically. Of course, it is not possible to represent completely the

domain Ω

⁰

in numerical simulations, because this would imply an infinite memory and computing

time. Rather, for some n ∈ N , we aim at computing as well as possible the restriction of u in

(26) to ω

ⁿ

. This turns out to be possible, but for that, we need to use nonlocal operators on

Γ

^σ

, σ ∈ A

n+1

. These operators have been studied theoretically and numerically in [1].

4.2 The Dirichlet to Neumann operator

Here we give results which where proved in [1]. For an integer n ≥ 0, and for σ ∈ A

ⁿ

, one can define the harmonic lifting operator H

^σ

from H

¹²

(Γ

^σ

) to H

¹

(Ω

^σ

): for all u ∈ H

¹²

(Γ

^σ

), the trace of H

^σ

(u) on Γ

^σ

is u and for all v ∈ V (Ω

^σ

), R

Ω^σ

∇H

^σ

(u) · ∇ v = 0. Since A

0

= { 0 } , we denote by H

⁰

the harmonic lifting in Ω

⁰

. It is easy to check that, for all v ∈ H

¹²

(Γ

^σ

),

H

^σ

(v) ◦ M

σ

(F

₁

, F

₂

) = H

⁰

(v ◦ M

σ

(F

₁

, F

₂

)). (31) Theorem 5 There exists a positive constant C such that, for all u ∈ H

¹²

(Γ

⁰

),

k∇H

⁰

(u) k

L²(ω⁰)

≥ C k∇H

⁰

(u) k

L²(Ω⁰)

. (32) There exists a real number ρ, 0 < ρ < 1 such that for all u ∈ H

¹²

(Γ

⁰

),

Z

Ω^N

|∇H

⁰

(u) |

²

≤ ρ

^N

Z

Ω⁰

|∇H

⁰

(u) |

²

. (33)

For σ ∈ A

ⁿ

, one can define the operators T

^σ

, from H

¹²

(Γ

^σ

) to their respective duals by h T

^σ

u, v i = R

Ω^σ

∇H

^σ

(u) ·∇H

^σ

(v) = R

Ω^σ

∇H

^σ

(u) ·∇ v, for any function ˜ ˜ v ∈ H

¹

(Ω

^σ

) such that ˜ v |

Γ^σ

= v. From the self-similarity of Ω

⁰

, we have that

∀ u, v ∈ H

¹²

(Γ

^σ

), h T

^σ

u, v i = h T

⁰

(u ◦ M

σ

(F

1

, F

2

)) , (v ◦ M

σ

(F

1

, F

2

)) i , (34) where the duality pairing in left (resp., right) hand side of (34) is the duality

H

¹²

(Γ

^σ

)

′

- H

¹²

(Γ

^σ

) (resp.,

H

¹²

(Γ

⁰

)

′

- H

¹²

(Γ

⁰

)).

Lemma 2 For all u ∈ H

¹²

(Γ

⁰

), for n ≥ 1, the restriction of H

⁰

(u) to ω

ⁿ⁻¹

is the solution to the following boundary value problem: find u ˆ ∈ H

¹

(ω

ⁿ⁻¹

) such that u ˆ |

Γ⁰

= u and ∀ v ∈ V (ω

ⁿ⁻¹

),

Z

ωⁿ⁻¹

∇ u ˆ · ∇ v + X

σ∈An

T

⁰

(ˆ u |

Γ^σ

◦ M

^σ

(F

₁

, F

₂

)) , v |

Γ^σ

◦ M

^σ

(F

₁

, F

₂

)

= 0. (35)

Furthermore, ∀ v ∈ H

¹

(ω

ⁿ⁻¹

), h T

⁰

u, v |

Γ⁰

i =

Z

ωⁿ⁻¹

∇ u ˆ · ∇ v + X

σ∈An

h T

^σ

u ˆ |

Γ^σ

, v |

Γ^σ

i

= Z

ωⁿ⁻¹

∇ u ˆ · ∇ v + X

σ∈An

T

⁰

(ˆ u |

Γ^σ

◦ M

σ

(F

₁

, F

₂

)), v |

Γ^σ

◦ M

σ

(F

₁

, F

₂

) .

(36)

Lemma 2, in the case n = 1, leads us to introduce the cone O of self adjoint, positive semi- definite, bounded linear operators from H

¹²

(Γ

⁰

) to its dual, vanishing on the constants, and the mapping M : O 7→ O defined as follows: for Z ∈ O , define M (Z ) by

∀ u ∈ H

¹²

(Γ

⁰

), ∀ v ∈ H

¹

(ω

⁰

), h M (Z)u, v |

Γ⁰

i = Z

ω⁰

∇ u ˆ · ∇ v + X

2

i=1

D Z(ˆ u |

Γ¹_i

◦ F

i

), v |

Γ¹_i

◦ F

i

E ,

(37) where ˆ u ∈ H

¹

(ω

⁰

) is such that ˆ u |

Γ⁰

= u and

∀ v ∈ V (ω

⁰

), Z

ω⁰

∇ u ˆ · ∇ v + X

2 i=1

D Z (ˆ u |

Γ¹_i

◦ F

_i

), v |

Γ¹_i

◦ F

_i

E

= 0. (38)

Lemma 2 tells that T

⁰

is a fixed point of M . In fact, we have the

Theorem 6 The operator T

⁰

is the unique fixed point of M . Moreover, for all Z ∈ O , there exists a positive constant C independent of n such that, for all n ≥ 0,

k M

ⁿ

(Z) − T

⁰

k ≤ Cρ

ⁿ⁴

, (39) where ρ, 0 < ρ < 1 is the constant appearing in Theorem 5.

4.3 Solving (26) with g = 1

Let u

F

∈ V (Ω

⁰

) be such that for all v ∈ V (Ω

⁰

), Z

Ω⁰

∇ u

F

· ∇ v = 1 3

Z

Γ^∞

v. (40)

Let y ∈ (H

¹²

(Γ

⁰

))

^′

be the normal derivative of u

F

on Γ

⁰

, defined by : for all v ∈ H

¹

(Ω

⁰

), h y, v i =

Z

Ω⁰

∇ u

F

· ∇ v − 1 3

Z

Γ^∞

v. (41)

It can be checked that for all n > 0, for σ ∈ A

ⁿ

, u

^σ_F

= u

F

◦ ( M

^σ

(F

₁

, F

₂

))

⁻¹

satisfies: u

^σ_F

∈ V (Ω

^σ

) and for all v ∈ V (Ω

^σ

), Z

Ω^σ

∇ u

^σ_F

· ∇ v = 1 3

Z

Γ^∞

v ◦ M

σ

(F

₁

, F

₂

). (42) Call now ˜ u

^σ_F

∈ V (Ω

^σ

), the solution to the following problem:

for all v ∈ V (Ω

^σ

), Z

Ω^σ

∇ u ˜

^σ_F

· ∇ v = 1 3

Z

Γ^∞∩Ω^σ

v. (43)

It is easy to prove that

˜ u

^σ_F

= 1

2

ⁿ

u

^σ_F

. (44)

Let us call ˜ u

F

the function defined by

˜

u

_F

|

ω⁰

= 0,

˜

u

F

|

Ω^σ

= ˜ u

^σ_F

, σ ∈ A

1

. (45) From the definition of ˜ u

^σ_F

, we see that ˜ u

_F

∈ V (Ω

⁰

), and we can check that for all v ∈ V (Ω

⁰

),

Z

Ω⁰

∇ u ˜

F

· ∇ v = 1 3

Z

Γ^∞

v + 1 2

X

2 i=1

y, v |

Fi(Γ⁰)

◦ F

i

. (46)

Calling e the error e = u

F

− u ˜

F

, we have: for all v ∈ V (Ω

⁰

), Z

Ω⁰

∇ e · ∇ v = − 1 2

X

2 i=1

y, v |

Fi(Γ⁰)

◦ F

i

. (47)

Since e |

Ω¹

is harmonic, and e coincides with u

_F

in ω

⁰

, we have that for all i = 1, 2 and v ∈ H

¹

(F

i

(Ω

⁰

)), Z

Fi(Ω⁰)

∇ e · ∇ v =

T

⁰

(u

F

◦ F

i

) , v |

Fi(Γ⁰)

◦ F

i

. (48)

From the fact that e and u

F

coincide in ω

⁰

, from (47) and (48), we obtain that u

F

|

ω⁰

is the unique function in V (ω

⁰

) such that: for v ∈ V (ω

⁰

),

Z

ω⁰

∇ u

_F

· ∇ v + X

2

i=1

T

⁰

(u

_F

◦ F

_i

) , v |

Fi(Γ⁰)

◦ F

_i

= Z

Ω⁰

∇ e · ∇ v = − 1 2

X

2 i=1

y, v |

Fi(Γ⁰)

◦ F

_i

, (49) and from (41), y satisfies: for all v ∈ H

¹

(ω

⁰

),

h y, v i = Z

ω⁰

∇ u

_F

· ∇ v + X

2

i=1

T

⁰

(u

_F

◦ F

_i

) , v |

F_i(Γ⁰)

◦ F

_i

+ 1

2 X

2 i=1

y, v |

F_i(Γ⁰)

◦ F

_i

. (50) Note that (50) is trivially (for any y) satisfied if v is constant, because T

⁰

is symmetric and T

⁰

1 = 0. On the other hand, from (41), y satisfies also:

h y, 1 i = − 1

3 | Γ

^∞

| = − 2. (51)

Conversely, for z ∈ (H

¹²

(Γ

⁰

))

^′

, call U (z) ∈ V (ω

⁰

) the function uniquely defined by: for v ∈ V (ω

⁰

), 0 =

Z

ω⁰

∇ U (z) · ∇ v + X

2

i=1

T

⁰

(U (z) ◦ F

i

) , v |

Fi(Γ⁰)

◦ F

i

+ 1 2

X

2 i=1

z, v |

Fi(Γ⁰)

◦ F

i

. (52)

Regularity results for Laplace’s equation imply the following Lemma Lemma 3 For all z ∈ (H

¹²

(Γ

⁰

))

^′

, U (z) ∈ H

³²

(ω

⁰

∩ { x

2

< 1 } ).

Notice that, thanks to (52), for v ∈ H

¹

(ω

⁰

), Z

ω⁰

∇ U (z) · ∇ v + X

2

i=1

T

⁰

((U (z) ◦ F

i

) , v |

Fi(Γ⁰)

◦ F

i

+ 1

2 X

2 i=1

z, v |

Fi(Γ⁰)

◦ F

i

depends on v |

Γ⁰

only. From Lemma 3, we see that this quantity is in fact R

Γ⁰

∂U(z)

∂n

|

Γ⁰

v |

Γ⁰

. Therefore, one can define the operator B

⁰

bounded from (H

¹²

(Γ

⁰

))

^′

to L

²

(Γ

⁰

) by

B

⁰

z = ∂U(z)

∂n |

Γ⁰

, or in an equivalent manner, ∀ v ∈ H

¹

(ω

⁰

),

h B

⁰

z, v |

Γ⁰

i = Z

ω⁰

∇ U (z) · ∇ v + X

2 i=1

T

⁰

((U (z) ◦ F

i

) , v |

Fi(Γ⁰)

◦ F

i

+ 1

2 X

2 i=1

z, v |

Fi(Γ⁰)

◦ F

i

, (53) and we have

∀ z ∈ H

¹²

(Γ

⁰

), h B

⁰

z, 1 i = h z, 1 i . From (49), (50), (51), and Lemma 3, we have

Lemma 4 The distribution y defined in (41) satisfies the equations:

h y − B

⁰

y, v i =0, ∀ v ∈ H

¹²

(Γ

⁰

),

h y, 1 i = − 2, (54)

and y ∈ L

²

(Γ

⁰

).

Theorem 7 The normal derivative of u

F

on Γ

⁰

given by (40) (41) belongs to L

²

(Γ

⁰

) and is the unique solution to (54).

Proof. From Lemma 4, there remains only to prove uniqueness.

If z ∈ (H

¹²

(Γ

⁰

))

^′

is a solution to

h z − B

⁰

z, v i = 0, ∀ v ∈ H

¹²

(Γ

⁰

), and h z, 1 i = 0, (55) then z ∈ L

²

(Γ

⁰

). Let ˜ U (z) be the harmonic extension of U (z) in Ω

⁰

. For σ ∈ A

n

, call e

^σ

the function of V (Ω

⁰

) defined by:

e

^σ

|

Ω^σ

= ˜ U (z) ◦ M

⁻¹σ

(F

₁

, F

₂

), and e

^σ

|

Ω⁰\Ω^σ

= 0, and call u

⁽ⁿ⁾

the function of V (Ω

⁰

) defined by

u

⁽ⁿ⁾

= X

n p=0

X

σ∈Ap

e

^σ

.

From (52), (55) and (53), the function u

⁽ⁿ⁾

satisfies: for v ∈ V (Ω

⁰

) Z

Ω⁰

∇ u

⁽ⁿ⁾

· ∇ v + 1 2

ⁿ⁺¹

X

σ∈An+1

h z, v |

Γ^σ

◦ M

σ

(F

₁

, F

₂

) i = 0. (56) But

1 2

ⁿ⁺¹

X

σ∈An+1

h z, v |

Γ^σ

◦ M

^σ

(F

₁

, F

₂

) i ≤



 1 2

ⁿ⁺¹

X

σ∈An+1

k v |

Γ^σ

◦ M

^σ

(F

₁

, F

₂

) k

L²(Γ⁰)



 k z k

L²(Γ⁰)

≤



 1 2

ⁿ⁺¹

X

σ∈An+1

k v |

Γ^σ

◦ M

^σ

(F

₁

, F

₂

) k

²L²(Γ⁰)





1 2

k z k

L²(Γ⁰)

= k v |

Γⁿ⁺¹

k

L²(Γⁿ⁺¹)

k z k

L²(Γ⁰)

From (15) in the case q = 2, we know that k v |

Γⁿ⁺¹

k

L²(Γⁿ⁺¹)

≤ C k∇ v k L

²

(Ω

⁰

). Therefore, the sequence u

⁽ⁿ⁾

is bounded in V (Ω

⁰

), and up to the extraction of a subsequence, we can assume that u

⁽ⁿ⁾

converges weakly to some function w in V (Ω

⁰

). It is clear that w coincides with U (z) in ω

⁰

.

Let I be an extension operator bounded from W

^1,q

(Ω

⁰

) to W

^1,q

( Ω b

⁰

), for each number q, 1 <

q < 2. For any v ∈ V (Ω

⁰

), 1

2

ⁿ⁺¹

X

σ∈An+1

h z, v |

Γ^σ

◦ M

σ

(F

₁

, F

₂

) i = X

σ∈An+1

Z

Γⁿ⁺¹

1

_Γ^σ

z ◦ M

⁻¹σ

(F

₁

, F

₂

)

( I v) |

Γⁿ⁺¹

Let ¯ z be the function defined on R , periodic of period 6, and such that

¯ z(x

₁

) =

 



0 if x

₁

∈ ( − 3, − 1) z(x

₁

, 0) if x

₁

∈ ( − 1, 1) 0 if x

₁

∈ (1, 3)

,

and let z

n

be the function defined on ( − 3, 3) by z

n

(x

₁

) = ¯ z( − 3 + 2

ⁿ

(x

₁

+ 3)). The integral above

can be written Z

3

−3

z

_n+1

(x

₁

) I f v(x

₁

, y

_n+1

)dx

₁

,

where y

_n+1

is the second coordinate of the points contained in Γ

ⁿ⁺¹

, and where I f v denotes the extension of I v by 0 out of Ω c

⁰

. But it is easy to prove that, for p ∈ R , p ≥ 1,

n→∞

lim k I f v(., 6) − I f v(., y

n+1

) k

L^p(−3,3)

= 0.

On the other hand, we know that z

n

converges weakly to 0 in L

²

( − 3, 3), since R

3

−3

z ¯ = 0.

We have proved that, for all v ∈ V (Ω

⁰

),

n→∞

lim 1 2

ⁿ⁺¹

X

σ∈An+1

h z, v |

Γ^σ

◦ M

σ

(F

₁

, F

₂

) i = 0. (57) This implies that w satisfies

Z

Ω⁰

∇ w · ∇ v = 0, ∀ v ∈ V (Ω

⁰

),

yielding that w = 0. Therefore U (z) = 0, which implies that B

⁰

z = 0 and finally that z = 0.

Uniqueness is proved.

Therefore, the normal derivative y of u

F

is characterized by (54). Once y is known, the restriction of u

F

to ω

⁰

is found by solving (49). Similarly, for any integer n, n ≥ 1, the restriction of u

F

to ω

ⁿ

can be found by solving the variational problem: u

F

|

ωⁿ

∈ V (ω

ⁿ

) and for all v ∈ V (ω

ⁿ

),

Z

ωⁿ

∇ u

F

· ∇ v + X

σ∈An+1

T

⁰

(u

F

◦ M

σ

(F

₁

, F

₂

)) , v |

Γ^σ

◦ M

σ

(F

₁

, F

₂

)

= − 1 2

ⁿ⁺¹

X

σ∈An+1

h y, v |

Γ^σ

◦ M

σ

(F

₁

, F

₂

) i .

(58)

We see that the knowledge of the operator T

⁰

enables to compute exactly the restriction of u

F

to any domain ω

ⁿ

, n ≥ 0.

Note also that u

F

can be computed by the following induction on n: for all n ≥ 1 and σ ∈ A

n

, u

F

|

Mσ(F1,F₂)(ω⁰)

∈ H

¹

( M

σ

(F

₁

, F

₂

)(ω

⁰

)),

• u

F

|

Mσ(F1,F2)(Γ⁰)

in known from step n − 1,

• ∀ v ∈ H

¹

( M

σ

(F

1

, F

2

)(ω

⁰

)) such that v |

Mσ(F1,F2)(Γ⁰)

= 0, Z

Mσ(F1,F₂)(ω⁰)

∇ u

F

· ∇ v + X

2

i=1

T

⁰

(u

F

◦ M

σ

(F

₁

, F

₂

) ◦ F

i

) , v |

Mσ(F1,F2)◦Fi(Γ⁰)

◦ M

σ

(F

₁

, F

₂

) ◦ F

i

= − 1 2

ⁿ⁺¹

X

2 i=1

y, v |

Mσ(F1,F₂)◦Fi(Γ⁰)

◦ M

σ

(F

₁

, F

₂

) ◦ F

i

.

Note that the domains involved by the induction above are all deduced from ω

⁰

by affine map-

pings. Therefore, when implementing the discrete analogue of this induction by using the finite

element method, no other mesh than that of ω

⁰

will be needed.