Boundary Value Problems in Some Ramified Domains with a Fractal Boundary: Analysis and Numerical Methods. Part I: Diffusion and Propagation problems.

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

problems.. 2004. �hal-00003628�

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

Part I: Diffusion and Propagation problems.

Yves Achdou

, Christophe Sabot

, Nicoletta Tchou

. December 16, 2004

Abstract

This paper is devoted to numerical methods for solving boundary value problems in self- similar ramified domains ofR²with a fractal boundary. Homogeneous Neumann conditions are imposed on the fractal part of the boundary, and Dirichlet conditions are imposed on the remaining part of the boundary. Several partial differential equations are considered.

For the Laplace equation, the Dirichlet to Neumann operator is studied. It is shown that it can be computed as the unique fixed point of a rational map. From this observation, a self-similar finite element method is proposed and tested. For the Helmholtz equation, it is shown that the Dirichlet to Neumann operator can also be computed as the limit of an inductive sequence of operators. Here too, a finite element method is designed and tested.

It permits to compute numerically the spectrum of the Laplace operator in the irregular domain with Neumann boundary conditions, as well as the eigenmodes. The repartition of the eigenvalues is investigated. The eigenmodes are normalized by means of a perturbation method and the spectral decomposition of a compactly supported function is carried out.

This permits to solve numerically the wave equation in the self-similar ramified domain.

1 Introduction

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

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 ambitions here are more modest, since the geometry of the problems (two dimen- sions 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

, called ω

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 [12, 9, 10]. A very nice theory on variational problems in fractal media is given in [13].

∗UFR Math´ematiques, Universit´e Paris 7, Case 7012, 75251 PARIS Cedex 05, France and Laboratoire Jacques- Louis Lions, Universit´e Paris 6. achdou@math.jussieu.fr

†CNRS, UMPA, UMR 5669, 46, Allee d’Italie, F-69364 Lyon Cedex 07, France. csabot@umpa.ens-lyon.fr

‡IRMAR, Universit´e de Rennes 1, Rennes, France, nicoletta.tchou@univ-rennes1.fr

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 the fractal construction after a finite number of generations.

We consider first Poisson problems (with the Laplace operator) with homogeneous Neumann conditions on the fractal part of the boundary. On the remaining part of the boundary, noted Γ

below, we impose a Dirichlet condition. In other words, we aim at computing the harmonic lifting of a function defined in Γ

. For that, it is possible to solve an equivalent boundary value problem in a subdomain obtained by interrupting the fractal construction after a finite num- ber of generations: this equivalent problem involves a non local Dirichlet to Neumann operator, which maps a function defined on Γ

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

can be computed very ac- curately by making use of the geometric self-similarity. The Dirichlet to Neumann operator is approximated as the limit of an inductive sequence, see (41) (42) below. When discretizing the problem with finite elements with self similar meshes, the same procedure can be implemented.

Next, turning to vibration problems in the domain described above leads to consider boundary value problems with Helmholtz equation. Here again, it is very natural to study the Dirichlet to Neumann operators (depending on the pulsation of the related harmonic wave), which, thanks to the self-similar structure of the set, can be approximated by iterations of a renormalization operator, see (77) (78) below. The discrete counterpart of this can be implemented with finite elements as soon as the mesh is self-similar. A related problem arises in the analysis of the spectrum of fractal domains such as Sierpinski gasket (the present paper does not consider a fractal domain, but a domain with a fractal boundary). The numerical method developed in this paper is very reminiscent of some of the techniques involved in the theoretical analysis of finitely ramified fractals (see [15],[20], [17], [16], and [2, 14, 5] 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). In this paper, the natural boundary is not so simple, but the numerical method is based on a similar strategy.

Once we know how to solve the boundary value problems with Helmholtz equation, it is natu-

ral to turn our attention to the spectral analysis of the Laplace operator in the domain under

consideration. The above mentioned Dirichlet to Neumann operators contain a lot of infor-

mation on the eigenvalues and their eigenfunctions. In particular, their construction permits

to compute numerically the spectrum, and the eigenmodes of the domain. Several important

problems concerning spectral analysis on domains with fractal boundaries motivate our numer-

ical computations. The first one concerns the eigenvalue repartition. Rigorous results about

the eigenvalue repartition have been obtained when the domain has a smooth boundary, but

when the boundary of the domain is fractal, we only have some bounds and conjectures (the

so-called Weyl-Berry formula, see [11]). In § 6.4.2, we present the numerical results relative to

this problem. Another important problem concerns the shape of the eigenfunctions: physicists

believe that they exhibit strong localization (cf [19], [18]), close to the fractal boundary. These

kind of properties are important to understand the geometry of the set, but from the numerical

point of view, they are difficult to analyze, due to the multiscale character of the fractal bound-

ary. The methods presented here, using self-similar meshes, takes into account fine scales in the

ramifications.

In order to solve time dependent problems in the irregular domain, the spectral information can be used, but for that, one needs to normalize the eigenmodes: in this paper, we propose a perturbation method for normalizing the eigenmodes. This permits to compute the spectral decomposition of any function compactly supported in the domain, and finally to solve numeri- cally time dependent equations like the wave equation.

It is also possible to develop numerical methods for boundary value problems with nonzero Neumann data on the fractal part of the boundary, by making use of the Dirichlet to Neumann operator. This is the topic of a forthcoming work, [1].

2 Geometrical setting of the model problem

Hereafter, we use the notation

s

=

i=0

2

. (1)

Consider the following T-shaped subset of

Q

= ( − 1, 1) × (0, 2]

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

The self-similar ramified domain Ω

is constructed as an infinite union of subsets of

obtained by translating/dilating Q

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

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

and are glued to Q

: more precisely, let F

and F

be the affine mappings

F

(x) = ξ

+ 1

2 x, where ξ

= ( − 3

2 , 3) and ξ

= ( 3

2 , 3), (2)

and let Q

be the set Q

= F

(Q

) ∪ F

(Q

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

for i = 1, . . . , 2

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

, ξ

= ξ

+

ξ

and ξ

= ξ

+

ξ

, and the following sets are introduced:

Q

= ∪

Q

, with Q

= ξ

+ 1

2

· Q

. (3)

For an integer n, n ≥ 1, calling A

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

} to { 1, 2 } , and for σ ∈ A

, M

(F

, F

) = F

◦ F

◦ · · · ◦ F

, (3) can also be written

Q

= ∪

M

(F

, F

)(Q

).

It will sometimes be convenient to agree that A

= { 0 } and that M

(F

, F

) is the identity.

Finally, the self-similar ramified Ω

is defined by

Ω

= ∪

Q

. (4)

The construction of Ω

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

⊂ ( − 3, 3) × (0, 6). Note that Ω

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

, thus Ω

is an open set.

It will be useful to define the truncated domain Ω

, which has also a fractal boundary:

Ω

= ∪

Q

. (5)

Figure 1: Left: the first step of the construction. Right: the self-similar ramified domain (only a few generations are displayed)

The following self-similarity property is true: Ω

is the union of 2

translated copies of

· Ω

, i.e.

Ω

= ∪

Ω

, (6)

where

Ω

= M

(F

, F

)(Ω

). (7)

Also, Ω

\ Ω

= Q

for any N ≥ 0.

We define the bottom boundary of Ω

by Γ

= (( − 1; 1) ×{ 0 } ) and Σ

= ∂Ω

∩{ (x, y); x ∈

, 0 <

y < 6 } . We have

∂Ω

∩ { (x, y); x ∈

, y < 6 } = Γ

∪ Σ

. (8) Similarly, the bottom boundary of Ω

is Γ

= ∪

Γ

, Γ

= ξ

+

· Γ

. In an equivalent manner,

Γ

= ∪

Γ

, (9)

where

Γ

= M

(F

, F

)(Γ

). (10)

For N > 0, Γ

is contained in the line y = y

= 3s

, see (1). We define also Σ

=

∂Ω

∩ { (x, y); x ∈

, 3s

< y < 6 } .

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

, for N ≥ 0:

ω

= Int Ω

\ Ω

. (11)

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

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

3 A Poincar´ e inequality

Consider the function space H

(Ω

) = { v ∈ L

(Ω

) s.t. ∇ v ∈ (L

(Ω

))

} . Similarly, for all

positive integer p, it is possible to define H

(Ω

) as the space of functions whose partial deriva-

tives up to order p belong to L

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

, H

(Ω

) is defined

by interpolation between H

(Ω

) and H

(Ω

), where p is the integer such that p ≤ s < p + 1.

Likewise, it is possible to define the Sobolev spaces H

(ω

) for all nonnegative integers n.

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

(Ω

) to ω

belongs to H

(ω

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

. The trace operator on Γ

is bounded from H

(Ω

) to L

(Γ

), so one can define the closed subspace of H

(Ω

):

V (Ω

) = { v ∈ H

(Ω

) s.t. v |

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

, the notation h u i

will be used for the mean value of u on Γ

.

Theorem 1 (Poincar´ e’s inequality) For any u ∈ V (Ω

), k u k

≤ √

32 k∇ u k

. (13)

Proof. We proceed by proving first the Poincar´e inequality for functions in the space V (ω

) = { v ∈ H

(ω

) s.t. v |

= 0 } , with a constant independent of N . Since the function space { v ∈ C

(ω

) s.t. v |

= 0 } is dense in V (ω

), it is enough to prove the inequality for functions in that space.

The idea of the proof is to construct explicitly a change of variables which maps Ω

onto a fractured set contained in the rectangle ( − 1, 1) × (0, 8).

We define first a continuous and piecewise affine change of variables γ

mapping

ω

= (( − 1, 1) × (0, 4]) \ ( { 0 } × [3, 4]) onto Q

by

if x > 0, γ

(x, t) =





(x, t) for t ∈ (0, 3 − x]

(t − 3 + 2x, 3 − x) for t ∈ [3 − x, 4 − x]

(x + 1, t − 1) for t ∈ [4 − x, 4]

,

if x < 0, γ

(x, t) = ( − γ

( − x, t), γ

( − x, t)) for t ∈ (0, 4], γ

(0, t) = (0, t) for t ∈ (0, 3)

It is easy to check that γ

is one to one. The set ω

is fractured in the sense that it does not lye locally on one side of its boundary.

Note also that for each x ∈ ( − 1, 1), the trajectory { γ

(x, t), t ∈ (0, 4] } is made of at most three straight lines parallel to the axes, and that for x ∈ (0, 1), γ

(x, 4) = x + 1 so { γ(x, 4), x ∈ (0, 1) } = (1, 2) × { 3 } . Similarly, one can check that ∇ γ

is piecewise constant and can only take the values

∇ γ

= 1 0

0 1

or ∇ γ

=

2 1

− 1 0

.

Thus, at the points where it makes sense, det( ∇ γ

) = 1. Therefore the mapping γ

preserves the measure.

It is possible to define a one to one continuous and piecewise affine mapping γ

from ω

= (( − 1, 1) × (0, 6]) \ ( { 0 } × [3, 6])

( {

} × [

, 6])

( {−

} × [

, 6])

onto Q

∪ Q

, by

γ

(x, t) = γ

(x, t) for t ≤ 4,

γ

(x, t) = (

, 3) +

γ

(2x − 1, 2(t − 4)) for t > 4, x > 0,

γ

(x, t) = ( −

, 3) +

γ

(2x + 1, 2(t − 4)) for t > 4, x < 0.

It is very easy to check that det( ∇ γ

) = 1, at all the points where ∇ γ

is defined.

Call y

= 4s

− 2

and consider the doubly-indexed sequence (x

) for n ≥ 0 and 0 ≤ j < 2

defined by the recursion

x

= 0,

x

= −

+ j, j = 0, 1, x

= x

2

+ 2

x

, j = 0, . . . , 2

− 1,

where

and j%2 are respectively the quotient and remainder of the Euclidean division of j by 2.

By proceeding recursively, we can define a one to one continuous and piecewise affine mapping γ

: ω

= (( − 1, 1) × (0, 4s

]) \

[N

n=0









2[ⁿ−1

j=0

{ x

}





× [y

, 4s

]





7→ ∪

Q

,

which preserve the measure. Finally, introducing the open set Ω

= (( − 1, 1) × (0, 8)) \

[∞

n=0









2ⁿ[−1

j=0

{ x

}





× [y

, 8)





,

we also have a one to one continuous and piecewise affine mapping from Ω

onto Ω

. The sets ω

and Ω

are displayed on Figure 2.

2 2

8 4

Figure 2: Left: the set ω

. Right: the open set Ω

(only the longest fractures are displayed) Let us define I

= 1 +

i=0

2

and call a

, 0 ≤ i ≤ I

the abscissa of the vertical boundaries of ω

, ordered increasingly. Consider a function u ∈ C

(ω

) such that u |

= 0.

Z

ω^N

u

=

ωd^N

u

(γ

(x, t)) =

IXN−1

i=0

Z ai+1

ai

dx

0

u

(γ

(x, t))dt

=

IXN−1

i=0

Z ai+1

ai

dx

0

Z t 0

d

ds (u(γ

(x, s))) ds

dt

≤

IXN−1

i=0

Z ai+1

ai

dx

0

dt t

0

∂u

∂x (γ

(x, s)) ∂γ

∂t (x, s)

+ ∂u

∂y (γ

(x, s)) ∂γ

∂t (x, s)

ds

by Cauchy-Schwarz inequality and because

= 0. Therefore

ω^N

u

≤ 32

IXN−1

i=0

Z _ai+1

ai

dx

0

∂u

∂x (γ

(x, s)) ∂γ

∂t (x, s)

+ ∂u

∂y (γ

(x, s)) ∂γ

∂t (x, s)

ds

≤ 32

IXN−1

i=0

Z _ai+1

ai

dx

0

∂u

∂x (γ

(x, s))

+ ∂u

∂y (γ

(x, s))

ds

because |

| ≤ 1 and |

| ≤ 1. Performing the inverse change of variables, we obtain that

ω^N

u

≤ 32

ω^N

|∇ u |

. (14)

By density, it is clear that (14) holds for u ∈ V (ω

). Since the constant in (14) does not depend of N , we obtain (13) by using Lebesgue’s theorem.

In what follows, we will use the notation

to indicate that there may arise constants in the estimates, which are independent of the index n in Ω

or ω

or on the mesh size when dealing with finite elements.

Corollary 1 There exists a positive constant C such that for all u ∈ H

(Ω

), k u k

≤ C

k∇ u k

+ k u |

k

. (15)

Proof. Define H

(Γ

) as the space of the traces on Γ

of the functions belonging to H

(ω

), endowed with the norm

k u k

= inf

v∈H¹(ω⁰),v|_Γ0=u

k v k

. It is a classical result that for all v ∈ H

(ω

),

k v |

k

k∇ v k

+ k v |

k

¹₂

. (16)

For u ∈ H

(Ω

), consider the function ˜ u ∈ H

(ω

) such that

∆˜ u = 0 in ω

, u ˜ |

= u |

, u ˜ |

= 0, ∂ u ˜

∂n = 0 on ∂ω

\ (Γ

∪ Γ

).

It can be checked that

k u ˜ k

k u |

k

. (17) Calling again ˜ u the extension by 0 of ˜ u in Ω

, we have that u − u ˜ ∈ V (Ω

), and from (13), (16) and (17),

k u − u ˜ k

≤ 32 k∇ (u − u) ˜ k

≤ 64

k∇ u k

+ k∇ u ˜ k

.

k∇ u k

+ k u |

k

H¹²(Γ⁰)

.

k∇ u k

+ k u |

k

.

We obtain (15) by using again (16) and (17).

Remark 2 Results similar to Corollary 1 can be proved, for instance: there exists a positive constant C such that for all u ∈ H

(Ω

),

k u k

≤ C

k∇ u k

+ h u i

. (18)

By a simple scaling argument, we obtain from (15) the

Corollary 2 There exists a positive constant C such that for all integer n ≥ 0, and for all i ∈ { 0, . . . , 2

} , for all u ∈ H

(Ω

),

k u k

≤ C

4

k∇ u k

+ 2

k u |

k

, (19)

and for all u ∈ H

(Ω

)

k u k

≤ C

4

k∇ u k

+ 2

k u |

k

. (20)

Lemma 1 There exists a positive constant C such that for all u ∈ H

(Ω

), for all N ≥ 0, k u k

≤ C2

k∇ u k

+ k u |

k

. (21)

Proof. We use a trace inequality on Q

: for a constant C independent on n, we have 2

k u |

2i−1

k

+ k u |

2i

k

≤ C k∇ u k

+ 2

k u |

k

. (22) Summing (22) over i, we obtain that

2

k u |

k

≤ C k∇ u k

+ 2

k u |

k

. (23) Multiplying (23) by 2

and summing up from n = 0 to N − 1, we obtain that

2

k u |

k

≤ C2

k∇ u k

+ k u |

k

Injecting this into (20), we obtain (21).

Theorem 2 (Compactness) The imbedding from H

(Ω

) in L

(Ω

) is compact.

Proof. From (21), we have

k u − 1

u k

≤ C2

k u k

.

On the other hand, the imbedding from H

(ω

) in L

(ω

) is compact. Combining the previous two remarks yields the desired result.

Remark 3 In [1], we give several other theoretical results on the space H

(Ω

), among which extension theorems, density results, and trace theorems on the top boundary of Ω

, namely Γ

= ( − 3, 3) × { 6 } .

4 Diffusion problems

The aim of this section is to study some Poisson problems in Ω

with Neumann boundary

conditions on Σ

.

4.1 Harmonic lifting of functions defined on Γ

For a function u ∈ H

(Γ

), we define the harmonic lifting H

(u) of u by H

(u) ∈ H

(Ω

), the trace of H

(u) on Γ

is u, and for all v ∈ V (Ω

),

Z

Ω⁰

∇H

(u) · ∇ v = 0. (24)

This is the weak form of the following problem

− ∆ H

(u) = 0, in Ω

, H

(u) = u, on Γ

,

∂H⁰(u)

∂n

= 0, on Σ

.

The existence and uniqueness of H

(u), and the fact that that H

is a bounded operator from H

(Γ

) to H

(Ω

) are consequences of Theorem 1.

Remark 4 All what follows holds when (24) is replaced by the more general problem

Ω⁰

χ ∇H

(u) · ∇ v = 0, (25)

where χ is a symmetric and positive definite tensor.

Remark 5 In [1], we study the boundary value problem with a nonzero Neumann data on Γ

. Similarly, for an integer n > 0, and for σ ∈ A

, one can define the lifting operator H

from H

(Γ

) to H

(Ω

): for all u ∈ H

(Γ

), the trace of H

(u) on Γ

is u and for all v ∈ V (Ω

),

Ω^σ

∇H

(u) · ∇ v = 0. It is easy to check that, for all v ∈ H

(Γ

),

H

(v) ◦ M

(F

, F

) = H

(v ◦ M

(F

, F

)). (26) Lemma 2 There exists a positive constant C such that, for all u ∈ H

(Γ

),

k∇H

(u) k

≥ C k∇H

(u) k

. (27) Proof. It is enough to prove (27) for all u ∈ H

(Γ

) such that

Γ⁰

u = 0, because H

(1

) = 1

. From the analogue of (18) for functions of H

(ω

) with mean value 0 on Γ

, we have that

k u k

k∇H

(u) k

. On the other hand, from the continuity of H

, we have that

k∇H

(u) k

k u k

. The desired result follows from the previous two estimates.

Lemma 3 There exists a constant ρ < 1 such that for all u ∈ H

(Γ

),

Ω¹

|∇H

(u) |

≤ ρ

Ω⁰

|∇H

(u) |

. (28)

Proof. The result is a direct consequence of Lemma 2.

Theorem 3 For all u ∈ H

(Γ

),

Ω^N

|∇H

(u) |

≤ ρ

Ω⁰

|∇H

(u) |

, (29)

where the constant ρ < 1 has been introduced in Lemma 3.

Proof. The desired result will be proved ounce we have established that

Ωⁿ⁺¹

|∇H

(u) |

≤ ρ

Ωⁿ

|∇H

(u) |

.

For that, we make use of (28); we consider the two bounded operators in H

(Γ

), L

, i = 1, 2:

L

(v) =

( H

v) |

◦ F

, (30)

where F

are defined in (2). Calling Ω

= F

(Ω

), i = 1, 2, it is easy to check that for all u ∈ H

(Γ

),

( H

◦ L

)(u) =

( H

(u)) |

◦ F

, (31)

and that

Ω⁰

|∇ ( H

◦ L

)(u) |

=

Ω¹_i

|∇H

(u) |

. (32)

Therefore, from (28),

X2

i=1

Z

Ω⁰

|∇ ( H

◦ L

)(u) |

≤ ρ

Ω⁰

|∇H

(u) |

. (33)

For σ ∈ A

, we use the notation M

( L

, L

) = L

◦ L

◦ · · · ◦ L

. For n > 1, we have, for all u ∈ H

(Γ

),

Z

Ωⁿ⁺¹

|∇H

(u) |

=

σ∈An

Z

Ω⁰

|∇ H

◦ M

( L

, L

) (u) |

=

i=1

X

σ∈An−1

Z

Ω⁰

|∇ H

◦ L

◦ M

( L

, L

) (u) |

, and from (33),

Z

Ωⁿ⁺¹

|∇H

(u) |

≤ ρ

σ∈An−1

Z

Ω⁰

|∇ H

◦ M

( L

, L

) (u) |

= ρ

Ωⁿ

|∇H

(u) |

, which yields (29).

From the general theory of boundary value problem, see [6] for example, we have the following regularity:

Lemma 4 (Local Regularity) For all u ∈ H

(Γ

), for all open bounded domain O strictly

contained in

× (0, 6), and for all ǫ, 0 < ǫ <

, the restriction of H

(u) to Ω

∩ O belongs to

H

(Ω

∩ O ).

Orientation We will try to solve (24) numerically. Of course, it is not possible to represent completely the domain Ω

in numerical simulations, because this would imply an infinite memory and computing time. Rather, for some n ∈

, we aim at computing as well as possible the restriction of H

(u) to ω

, n ∈

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

, σ ∈ A

. We will see later that these operators can be called Dirichlet to Neumann operators. They will be computed by using the geometric self-similarity.

4.2 The Dirichlet-Neumann operator Call

H

(Γ

)

the topological dual space of H

(Γ

) and consider the Dirichlet-Neumann op- erator T

: H

(Γ

) 7→

H

(Γ

)

h T

u, v i =

Ω⁰

∇H

(u) · ∇H

(v). (34)

We remark that

h T

u, v i =

Ω⁰

∇H

(u) · ∇ v, ˜ (35)

for any function ˜ v ∈ H

(Ω

) such that ˜ v |

= v.

When H

(u) is regular enough, T

u is the normal derivative of H

(u) on Γ

. This is why T

is called a Dirichlet-Neumann operator.

The operator T

is bounded, self-adjoint and positive semi-definite. It is clear that T

1 = 0.

Call V the closed subspace of H

(Γ

):

V = { v ∈ H

(Γ

), h v i

= 0 } . (36) From the definition of the norm in H

(Γ

) and from (18), we see that T

is coercive on V , i.e.

there exists a positive constant α such that

∀ v ∈ V, h T

v, v i ≥ α k v k

H¹²(Γ⁰)

. (37)

Similarly, for σ ∈ A

, one can define the operators T

, from H

(Γ

) (see (7) and (10)) to their respective duals by h T

u, v i =

Ω^σ

∇H

(u) · ∇H

(v) =

Ω^σ

∇H

(u) · ∇ ˜ v, for any function

˜

v ∈ H

(Ω

) such that ˜ v |

= v. From the self-similarity of Ω

, we have that

∀ u, v ∈ H

(Γ

), h T

u, v i = h T

(u ◦ M

(F

, F

)), (v ◦ M

(F

, F

)) i , (38) where the duality pairing in left (resp. right) hand side of (38) is the duality

H

(Γ

)

- H

(Γ

) (resp.

H

(Γ

)

- H

(Γ

)).

Lemma 5 For all u ∈ H

(Γ

), for n ≥ 1, the restriction of H

(u) to ω

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

(ω

) such that u ˆ |

= u and ∀ v ∈ V (ω

),

Z

ωⁿ⁻¹

∇ u ˆ · ∇ v +

σ∈An

T

(ˆ u |

◦ M

(F

, F

)) , v |

◦ M

(F

, F

)

= 0. (39)

Furthermore, ∀ v ∈ H

(ω

), h T

u, v |

i =

Z

ωⁿ⁻¹

∇ u ˆ · ∇ v +

σ∈An

h T

u ˆ |

, v |

i

=

ωⁿ⁻¹

∇ u ˆ · ∇ v +

σ∈An

T

(ˆ u |

◦ M

(F

, F

)), v |

◦ M

(F

, F

) .

(40)

Proof. Follows from (26) and (38).

Remark 6 Note that the boundary value problem (39) is well posed because the bilinear form in the left hand side is continuous, symmetric and coercive on V (ω

).

Orientation We see from (39) that once the nonlocal operator T

is known, the restriction of H

(u) to ω

can be computed exactly by solving a boundary value problem in ω

with a boundary condition involving T

. Thus, if T

or a good approximation of T

is available, then the restriction of H

(u) to ω

can be approximated by a standard discrete method for (39).

There remains to compute T

: for that, we will make use of (40), in the case n = 1.

Lemma 5, in the case n = 1, leads us to introduce the cone

of self adjoint, positive semi- definite, bounded linear operators from H

(Γ

) to its dual, vanishing on the constants, and the mapping

:

7→

defined as follows: for Z ∈

, define

(Z ) by

∀ u ∈ H

(Γ

), ∀ v ∈ H

(ω

), h

(Z)u, v |

i =

ω⁰

∇ u ˆ · ∇ v +

i=1

D

Z(ˆ u |

◦ F

), v |

◦ F

, (41) where ˆ u ∈ H

(ω

) is such that ˆ u |

= u and

∀ v ∈ V (ω

),

ω⁰

∇ u ˆ · ∇ v +

i=1

D

Z (ˆ u |

◦ F

), v |

◦ F

= 0. (42)

Lemma 5 tells that T

is a fixed point of

. In fact, we have the

Theorem 4 The operator T

is the unique fixed point of

. Moreover, for all Z ∈

, there exists a positive constant C independent of n such that, for all n ≥ 0,

k

(Z) − T

k ≤ Cρ

, (43) where ρ, 0 < ρ < 1 is the constant appearing in Theorem 3.

Proof. It is easy to check by induction that

∀ u ∈ H

(Γ

), ∀ v ∈ H

(ω

), h

(Z)u, v |

i =

Z

ωⁿ⁻¹

∇ u ˆ · ∇ v +

σ∈An

h Z (ˆ u |

◦ M

(F

, F

)), (v |

◦ M

(F

, F

)) i , (44)

where ˆ u ∈ H

(ω

) is such that ˆ u |

= u and

∀ v ∈ V (ω

),

ωⁿ⁻¹

∇ u ˆ ·∇ v +

σ∈An

h Z (ˆ u |

◦ M

(F

, F

)), (v |

◦ M

(F

, F

)) i = 0. (45)

Let H

(ˆ u) ∈ H

(Ω

) be defined by H

(ˆ u) =

u ˆ in ω

,

the harmonic lifting of ˆ u |

in Ω

.