Perturbation method for the numerical solution of the Bernoulli problem

Abstract

We consider the numerical solution of the free boundary Bernoulli problem by employing level set formulations. Using a perturbation technique, we derive a second order method that leads to a fast iteration solver. The iteration procedure is adapted in order to work in the case of topology changes. Various numerical experiments confirm the efficiency of the derived numerical method.

Full text

Journal of Computational Mathematics, Vol.26, No.1, 2008, 23–36.
A PERTURBATION METHOD FOR THE NUMERICAL
SOLUTION OF THE BERNOULLI PROBLEM*
Fran¸cois Bouchon
Laboratoire de Math´ematiques, Universit´e Blaise Pascal (Clermont-Ferrand) and CNRS (UMR 6620)
Campus Universitaire des C´ezeaux, 63177 Aubi`ere cedex, France
Email: Francois.Bouchon@math.univ-bpclermont.fr
St´ephane Clain
Laboratoire MIP, UFR MIG, Universit´e Paul Sabatier Toulouse 3, 118 route de Narbonne, 31062
Toulouse cedex 4, France
Email: clain@mip.ups-tlse.fr
Rachid Touzani
Laboratoire de Math´ematiques, Universit´e Blaise Pascal (Clermont-Ferrand) and CNRS (UMR 6620)
Campus Universitaire des C´ezeaux, 63177 Aubi`ere cedex, France
Email: Rachid.Touzani@univ-bpclermont.fr
Abstract
We consider the numerical solution of the free boundary Bernoulli problem by em-
ploying level set formulations. Using a perturbation technique, we derive a second order
method that leads to a fast iteration solver. The iteration procedure is adapted in order to
work in the case of topology changes. Various numerical experiments confirm the efficiency
of the derived numerical method.
Mathematics subject classification: 35R35, 34E10, 65M06.
Key words: Bernoulli problem, Free boundary, Level sets.
1. Introduction
The Bernoulli problem stands for a prototype of a large class of stationary free boundary
problems involved in fluid dynamics and electromagnetic shaping (see [5, 6, 8] and the references
therein). This problem roughly consists in a Laplace equation with an additional boundary
condition that enables determining the solution of the equation as well as the unknown domain.
In order to obtain a reliable numerical approximation for this problem, a wide variety of
works have been produced. For instance, in Flucher and Rumpf [7], some numerical schemes
based on a local parametrization are developed. The authors prove in this work convergence
results and present some numerical examples. Nevertheless, due to the local parametrization,
the constructed methods cannot handle topological changes. In [3], we propose an extension of
the Flucher-Rumpf technique introducing a level set formulation to characterize the free bound-
ary. This approach enjoys the property of allowing topology changes as level sets generally do.
However, the scheme developed in [3] has the drawback to slowly converge and produces some
local oscillation of the computed boundary when the numerical solution approaches the steady
state. This drawback is removed in [11] where the authors consider an integral formulation of
the Bernoulli problem and where the level set equation is solved via the Fast Marching strategy.
The integral representation is however specific to partial differential equations for which this is
available.
*Received February 28, 2007 / Revised version received June 4, 2007 / Accepted July 19, 2007 /

24 F. BOUCHON, S. CLAIN AND R. TOUZANI
In order to improve the solver performances, we propose in this paper a second-order scheme
that can be viewed as a Newton-like method. The method is based on a perturbation of the
parametrization of the initial guess of the free boundary. It has, as will be shown, the advantage
of accelerating the convergence to the steady state solution, but as high-order methods require
additional regularity properties, the presented method fails to converge when a topology change
occurs during the iteration process. We then resort to switching to the first-order method while
a domain splits up or two subdomains collapse. Numerical experiments show that convergence
properties are dramatically improved when compared to the algorithm developed in [3].
The outline of the paper is as follows: In Section 2, we present a perturbation method
to derive a second-order formulation. Section 3 is devoted to the derivation of a numerical
scheme based on level sets and inspired by this perturbation technique. Section 4 presents
some numerical results for both a radial case for which the analytical solution is known and a
case with changing topology. Finally, a conclusion is drawn.
Let us mention that only the interior Bernoulli problem (see [3] for instance) is considered in
the present study. An analog analysis of the exterior problem can be deduced straightforwardly.
2. The Perturbed Problem
Let Ω be a bounded domain of R2with a C2-boundary ∂Ω. We seek a (not necessarily
connected) domain Awith ¯
A⊂Ω and a function udefined on Ω \Asuch that:
∆u= 0 in Ω \¯
A, (2.1)
u= 0 on ∂Ω,(2.2)
u= 1 on ∂A, (2.3)
∂u
∂n =λon ∂A, (2.4)
where λis a positive real number and nis the unit normal to the boundary ∂(Ω \¯
A) of Ω \¯
A
pointing inward A.
We propose, in this section, an alternative to the result obtained in by Flucher and Rumpf
(see [7], Theorem 2).
Proposition 2.1. Let ∂˜
A=∂A +ρn be a set close to A(in the sense that ρ≪1). Then the
function u— extended to Ω\˜
Aif necessary — is solution to the following problem:
∆u= 0 in Ω\¯
˜
A, (2.5)
u= 0 on ∂Ω,(2.6)
∂u
∂˜n−˜κu =λ−˜κ+O(ρ2)on ∂˜
A, (2.7)
where ˜κis the curvature of ∂˜
A.
To prove this result, we first need to consider some preliminary results.
2.1. Some technical results
Let γ: [0, L]→R2denote a parametric representation of the curve ∂A. We choose the
parametrization such that the unit normal vector nto ∂ A points inward A. The tangent vector

Numerical Solution of the Bernoulli Problem 25
A
Ω \ A
t
n
Fig. 2.1. Geometry of the domain.
tis chosen according to Fig. 2.1. We recall that if κ=κ(s) is the curvature of ∂A at γ(s), then
we have the Serret-Fr´enet formulae:
dγ
ds =t, dt
ds =κ n, dn
ds =−κ t.
To describe the perturbed boundary ∂˜
Aby the parametric function ˜γ(s) on [0, L], we assume
that there exists a smooth function ρdefined on [0, L] such that
˜γ(s) = γ(s) + ρ(s)n(s) 0 ≤s < L. (2.8)
We assume furthermore that
dρ
ds =O(ρ),(2.9)
˜κ=κ+O(ρ),(2.10)
which means that highly oscillating boundary perturbations are excluded. Let us prove a useful
technical result.
Lemma 2.1. We have the following identities:
˜
t=1
D(1 −ρκ)t+dρ
ds n,
˜n=1
D(1 −ρκ)n−dρ
ds t,
with
D=(1 −ρ κ)2+dρ
ds 2
1
2.
Proof. The first identity is obtained by differentiation of (2.8) and normalization. The
second identity is easily deduced from the first one.
Lemma 2.2. Let ube a smooth function defined on Aand satisfying Eqs. (2.1)-(2.4). Then
we have
nTH(u)n=κ∂u
∂n ,
where H(u)is the Hessian matrix of u.

26 F. BOUCHON, S. CLAIN AND R. TOUZANI
Proof. From equation (2.3), we have u(γ(s)) = 1. By differentiating, we obtain
∇u(γ(s)) ·t(s) = 0.
A second differentiation implies
tTH(u(γ(s))) t+κ(s)∇u(γ(s)) ·n(s) = 0.
From the identity
∆u=tTH(u)t+nTH(u)n= 0,
we get
−nTH(u)n+κ(s)∇u(γ(s)) ·n(s) = 0.
2.2. Proof of Proposition 2.1
For the sake of simplicity, we omit to mention the variable s.
Lemma 2.3. Let udenote a solution of problem (2.1)-(2.4), admitting a harmonic extension
in a neighborhood of ∂A. Then we have
∂u
∂˜n=λ+ρ κ λ +O(ρ2)on ∂˜
A. (2.11)
Proof. Using Lemma 2.1, we write
∂u
∂˜n(˜γ) = 1
D∇u(γ+ρ n)·(1 −ρ κ)n−dρ
ds t.(2.12)
Differentiating (2.3) in the tangential direction, we get ∇u(γ)·t= 0, which gives thanks to the
Taylor expansion,
∇u(γ+ρ n)·t=O(ρ).(2.13)
Assumption (2.9) implies
D=(1 −ρ κ)2+ (dρ
ds )2
1
2
=1−2ρ κ +O(ρ2)
1
2= 1 −ρ κ +O(ρ2),(2.14)
Furthermore, we have by using Lemma (2.2)
∂u
∂n (γ+ρ n) = ∂u
∂n (γ) + ρ nTH(u(γ)) n+O(ρ2)
= (1 + ρκ)∂u
∂n (γ) + O(ρ2).(2.15)
Combining (2.13)-(2.15), and Assumption (2.9) again, (2.12) yields
∂u
∂˜n(˜γ) = (1 −ρ κ)
D
∂u
∂n (γ+ρ n) + O(ρ2).

Numerical Solution of the Bernoulli Problem 27
We then obtain from Lemma 2.2, and Assumption (2.9)
∂u
∂˜n(˜γ) = 1−ρκ
1−ρ κ +O(ρ2)
∂u
∂n (γ)(1 + ρ κ) + O(ρ2)
= (1 + ρ κ)∂u
∂n (γ) + O(ρ2).
We conclude by using (2.4).
From Lemma 2.3 and Hypothesis (2.10), we deduce
∂u
∂˜n(˜γ) = (1 + ρ˜κ)λ+O(ρ2).
The Taylor expansion
u(˜γ) = u(γ) + ρ∂u
∂n (γ) + O(ρ2) = 1 + ρ λ +O(ρ2) (2.16)
yields then ∂u
∂˜n(˜γ) = λ+ (u(˜γ)−1) ˜κ+O(ρ2).
Whence ∂u
∂˜n(˜γ)−˜κ u(˜γ) = λ−˜κ+O(ρ2).
This completes the proof of Proposition 2.1.
Let us now use this material to derive an iterative process to solve problem (2.1)-(2.4): If
Akis a known approximation of the set A, we compute uksolution of
∆uk= 0 in Ω \¯
Ak,(2.17)
uk= 0 on ∂Ω,(2.18)
∂uk
∂nk
−κkuk=λ−κkon ∂Ak,(2.19)
where nk, κkare respectively the inward unit normal to ∂ Akand the curvature of ∂Ak. We
aim at computing an approximation of ρsuch that ∂A =∂Ak+ρnk.
Let us set u′=uk−u. Combining (2.5)-(2.7) with (2.17)-(2.19) shows that u′is solution
of: ∆u′= 0 in Ω \¯
Ak,(2.20)
u′= 0 on ∂Ω,(2.21)
∂u′
∂nk
−κku′=O(ρ2) on ∂Ak,(2.22)
which shows, at least in the case where κk≥0, that u′=O(ρ2) on Ω \¯
Ak, and then on ∂Ak.
Hence uk=u+O(ρ2). From (2.16), we have
ρ=u(˜γ)−1
λ+O(ρ2) = uk(˜γ)−1
λ+O(ρ2).
We deduce that the setting
Ak+1 := Ak−ρknk,(2.23)
with
ρk=uk(˜γ)−1
λ(2.24)
gives a “good” approximation to A.
In the following section, we present the numerical algorithm in the context of level set
methods.

28 F. BOUCHON, S. CLAIN AND R. TOUZANI
3. Numerical Scheme
Let (A, u) be a smooth solution of the Bernoulli problem (Ais C3and uis C2, say). Our
aim is to build a sequence (Ak, uk)kof solutions of an approximate Bernoulli problem which
converges towards (A, u).
We first present the level set method, and then we derive from the previous analysis an
iterative scheme which converges provided that the initial guess is not too far from the solution.
Finally, we introduce what we will refer to as a mixed scheme.
3.1. The level set formulation
As we emphasized in the previous section, the scheme we have constructed is based on a
local description of Akgiven by the function ρk. If a topology change occurs, such a formulation
breaks down and this motivates the introduction of the level set formulation to characterize
the free boundary. To obtain a level set formulation, we use the principle that the level set
description and the local description with ρkmust coincide whenever this last one has a sense.
3.1.1. The level set definition
The level set formulation consists in characterizing the boundary of the domain Akas the zero
level set of a function φk. More precisely, we seek a function φksuch that
γk={x∈Ω; φk(x) = 0},
Ak={x∈Ω; φk(x)>0},
Ω\¯
Ak={x∈Ω; φk(x)<0}.
Since we state that φkis positive inside Akand negative outside, we get that the inward normal
vector on ∂Akis given by
nk=∇φk
|∇φk|.
3.1.2. The level set equation
Let φkand φk+1 be two level set functions associated to the domains Akand Ak+1 respectively
and assume that we have a local description of the boundary for both Akand Ak+1 . By
definition, the level set functions satisfy
φk+1(γk+1 ) = φk(γk) = 0,(3.1)
and the function γk+1 is given by relation (2.22),
γk+1 =γk−ρknk+1.
The Taylor expansion gives
φk+1(γk+1 ) = φk+1 (γk−ρknk+1 )
=φk+1(γk)−ρk∇φk+1 (γk)·nk+1 +O(ρ2
k).
We can write using identity (3.1),
φk+1(γk) = ρk∇φk+1 (γk)·nk+1 +O(ρ2
k).

Numerical Solution of the Bernoulli Problem 29
Using the expression of the inward normal
φk+1(γk) = ρk∇φk+1 (γk)·∇φk+1 (γk+1 )
|∇φk+1(γk+1 )|+O(ρ2
k).
Another Taylor expansion shows that
∇φk+1(γk+1 )
|∇φk+1(γk+1 )|=∇φk+1 (γk)
|∇φk+1(γk)|+O(ρk).
We finally obtain
φk+1(γk) = ρk|∇φk+1 (γk)|+O(ρ2
k).
Since the domain Akmoves to the domain Ak+1 thanks to the relation (2.24)
ρk=uk−1
λon ∂Ak,
and recalling that φk(γk) = 0, we obtain the level set equation for the function φk+1
φk+1(γk) = φk(γk) + uk−1
λ|∇φk+1(γk)|+O(ρ2
k).(3.2)
The function (1 −uk)/λ behaves like a speed of propagation to move the boundary but
equation (3.2) is only defined on the domain boundary ∂Ak. To complete the scheme we need
an extension of the normal velocity to obtain a level set equation on the whole domain Ω.
3.1.3. Extension of the normal velocity
Following [1], we construct a velocity vkby the Fast Marching method such that vk·nkcoincides
with the normal velocity (1 −uk)/λ on ∂Ak. To this end, for a given level set function φksuch
that |∇φk|= 1, we solve the equation
∇vk· ∇φk= 0,
with the condition
vk=1−uk
κk
on ∂Ak.
We define φk+1 by
φk+1 =φk−vk|∇φk+1|
=φk−vk|∇φk+O(ρk)|=φk−vk+O(ρ2
k),
where we have used the property vk=O(ρk). Now, we have by differentiation
∇φk+1 =∇φk− ∇vk+O(|∇ρ2
k|).
Since we have assumed |∇ρk|=O(ρk), then
|∇φk+1|2=∇φk· ∇φk+1 − ∇vk· ∇φk+1 +O(ρ2
k)
=∇φk·(∇φk− ∇vk)− ∇vk·(∇φk− ∇vk+O(ρ2
k)) + O(ρ2
k)
= 1 + O(ρ2
k).
The function φk+1 is then updated in order to satisfy |∇φk+1 |= 1 by using a Fast Marching
Method. Note that this correction does not modify the position of the free boundary. To
initialize the iterative process, we choose φ0as the signed distance function associated to the
initial guess A0.
We now design two numerical schemes based on the level set formulation.

30 F. BOUCHON, S. CLAIN AND R. TOUZANI
3.2. The “perturbation-method scheme”
Let us describe now the algorithm deduced from the analysis of the previous section. Assume
that we know the domain Akand a level set function φkassociated to Aksatisfying |∇φk|= 1.
(1) We compute ukon Ω \¯
Aksolving the elliptic problem with mixed condition on the
boundary
∆uk= 0 in Ω \¯
Ak,
uk= 0 on ∂Ω,
∂uk
∂nk
−κkuk=λ−κkon ∂Ak,
(3.3)
where
κk(x) = ∇ · ∇φk(x)
|∇φk(x)|.
(2) We compute the extended normal velocity vkon Ω by
∇vk· ∇φk= 0 in Ω,
vk=1−uk
λon ∂Ak,
using the Fast Marching method described in [1].
(3) We obtain the new level set function φk+1 by setting
φ∗
k+1(x) = φk(x)−vk(x),
which defines ∂Ak+1 ={x∈Ω; φ∗
k+1(x)>0}.
(4) We perform a correction step to compute φk+1 :
|∇φk+1|= 1 in Ω,
φk+1 = 0 on ∂ Ak,
φk+1φ∗
k+1 ≥0 in Ω.
(3.4)
Note that the last equation imposes that the sign of φk+1 remains the same as the sign
of φ∗
k+1. This step is performed using again the Fast Marching method ([1]).
3.3. The Neumann scheme
Since the previous scheme converges locally (if the initial guess is close enough to the solution
A), our aim is to improve it in order to extend its domain of convergence. For this end, we use
a scheme close to the Neumann scheme described in [3]: Let Akbe given and φkbe a level set
function associated to Ak. We consider the following algorithm.
(1) We compute ukon Ω \¯
Akby solving the elliptic problem
∆uk= 0 in Ω \¯
Ak,
uk= 0 on ∂Ω,
∂uk
∂nk
=λon ∂Ak.

Numerical Solution of the Bernoulli Problem 31
(2) We compute the extended normal velocity vkon Ω by
∇vk· ∇φk= 0 in Ω,
vk=1−uk
λon ∂Ak,
using the Fast Marching method described in [1].
(3) We obtain the new level set function φk+1 by setting
φ∗
k+1(x) = φk(x)−vk(x),
which defines ∂Ak+1 ={x∈Ω; φ∗
k+1(x)>0}.
The correction step (3.4) enables computing φk+1 .
The main advantage of this scheme is that we do not have to introduce the curvature. Its
drawback resides in its limitation to the context of elliptic solutions (see [7] and [3] for further
details).
3.4. The mixed scheme
As many “Newton-like” schemes, the perturbation-method scheme experiences a high rate
of convergence provided the initial guess is close enough to the solution. If the starting point is
too far (for example, if a topology change is necessary to reach the solution), then the Neumann
scheme shall be used.
Since the curvature of the set exhibits some singularity when facing a topology change, we
have chosen as criterion the maximum value of the curvature to determine which scheme to
use. Namely, at each iteration k, if the maximum value of the curvature κkis too large, then
we choose the second scheme (“Neumann Scheme”) and if it is small enough, then we choose
the “perturbation-method scheme”. In practice, the chosen criterion is given by
1
κ≤Ch
where his the grid size and Cis a given constant that ensures convergence of the numerical
scheme (see [4]).
4. Numerical Results
In order to solve the numerical problem, we resort to the classical five-point finite difference
scheme for the Laplace equation. We have implemented the obtained discrete problem in four
configurations. The first one aims to evaluate the convergence rate of the scheme. The second
one aims at showing that the scheme can converge in the both cases of elliptic and hyperbolic
solutions (depending on the initial guess). The notion of hyperbolic solution is the one defined
in [7].
In the two last configurations, we observe topology changes.

32 F. BOUCHON, S. CLAIN AND R. TOUZANI
4.1. Convergence tests
In this series of tests we consider first the adaptation of the described scheme to the exterior
Bernoulli problem. We next consider an interior Bernoulli problem.
We have chosen Ω = {x∈R2;|x−c|< ρ0}, ρ0= 0.2, c = (0.5,0.5). In this case, the only
solution is the circle centered in cof radius ρsuch that
λ=1
ρ(log ρ−log ρ0).
We have chosen λ= 7, for which the solution is the circle of radius ρ1≈0.3148 ···.
Table 4.1 shows the Hausdorff distance to this solution (denoted by A∞) when we take as
initial guess the circle centered in cof radius 0.30. Recall that the Hausdorff distance is defined
by (see [3])
D(A, B) := max sup
a∈A
(a, B),sup
b∈B
(b, A)A, B ⊂R2,
where d(a, B) := inf b∈B|a−b|. To calculate this distance we use the signed distance function
φk.
The stop criterion has been defined using the Hausdorff distance between two consecutive
sets Ak,Ak+1. Since we expect a second order scheme, the stop criterion must be finer than
the grid size h2. In our numerical tests, we have fixed the criterion
D(Akf−1, Akf)< h3.
Note that an erratic convergence behavior is observed for a coarse grid.
Table 4.1 also shows that the CPU time increases significantly within the grid size. This is
natural since each iteration requires the solution of the elliptic problem (3.3) with an increasing
number of unknowns.
Table 4.1: Convergence test for the elliptic solution.
Grid D(Ak, A∞) Rate Nb. Iter. Time (sec.)
40 ×40 4.17 ×10−412 30.6
60 ×60 1.73 ×10−42.2 13 52.4
80 ×80 1.63 ×10−40.2 4 23.7
120 ×120 6.10 ×10−52.4 5 49.2
160 ×160 3.50 ×10−51.9 5 80.0
240 ×240 1.82 ×10−51.6 5 190.8
320 ×320 9.47 ×10−62.3 6 510.5
480 ×480 4.39 ×10−61.9 6 2711.9
640 ×640 2.43 ×10−62.0 6 5743.4
We observe the convergence history for two particular grid sizes: 60 ×60 (solid line) and
80 ×80 (dotted line). The condition linking the curvature and the grid size is satisfied in this
the test for the grid 80×80, the algorithm converges then faster. For the 60×60 grid, the mesh
size h= 1/60 is too coarse and the first algorithm is used, this is the reason why we need more
iterations to reach convergence. Table 4.1 shows a second-order convergence rate behavior.
To consider an interior Bernoulli problem example, we choose Ω = {x∈R2;|x−c|<
ρ0}, ρ0= 0.42, c = (0.5,0.5). In this case, any circle centered in cof radius ρsuch that
λ=1
ρ(log ρ−log ρ0)

Numerical Solution of the Bernoulli Problem 33
0 5 10 15
k
10-8
10-6
10-4
10-2
100
D(Ak-1,Ak)
Fig. 4.1. Convergence history for the 60 ×60 grid (solid line) and 80 ×80 grid (dotted line).
is a solution of the problem. We have chosen λ= 7, the circle of radius ρ1≈0.218285 ··· is the
elliptic solution, and the circle of radius ρ2≈0.098528 ··· is the hyperbolic solution.
Table 4.2: Convergence test for the elliptic solution.
Grid D(Ak, A∞) Rate Nb. Iter. Time (sec.)
40 ×40 1.63 ×10−317 45
60 ×60 1.01 ×10−31.2 19 73
80 ×80 7.63 ×10−41.0 23 127
120 ×120 1.61 ×10−43.8 7 82
160 ×160 1.04 ×10−41.5 7 160
240 ×240 5.77 ×10−51.4 7 581
320 ×320 3.36 ×10−51.9 7 2253
480 ×480 1.33 ×10−52.3 8 17167
640 ×640 7.73 ×10−61.9 8 51783
Table 4.3: Convergence test for the hyperbolic solution.
Grid D(Ak, A∞) Rate Nb. Iter. Time (sec.)
40 ×40 (no convergence)
60 ×60 (no convergence)
80 ×80 (no convergence)
120 ×120 (no convergence)
160 ×160 (no convergence)
240 ×240 1.02 ×10−46 626
320 ×320 4.99 ×10−52.5 6 2262
480 ×480 4.01 ×10−50.5 5 19397
640 ×640 1.38 ×10−53.7 5 59394
Table 4.2 shows the Hausdorff distance to the solution when we take as initial guess the
circle centered in cof radius 0.32. Note that, in this case, the algorithm converges to the
elliptic solution (which has been taken as reference solution A∞). Here also, a second-order
convergence rate is observed.
Table 4.3 shows the Hausdorff distance to the solution when we take as initial guess the circle

34 F. BOUCHON, S. CLAIN AND R. TOUZANI
centered in cof radius 0.1. Note that, in this case, the algorithm converges to the hyperbolic
solution (which has been taken as reference solution A∞). No convergence rate can however be
deduced from numerical experiments.
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0.9
1
X
Y
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 4.2. Domains A0to A30.
For the coarse grids, the criterion on the curvature is not satisfied. Then, the first (Neumann)
scheme is used, and since it works only in the elliptic context, then we have no convergence of
our algorithm (namely, the set Akis empty for some k).
4.2. Topology change tests
In this test, Ω is the union of four disks of radius 0.11 centered at P1= (0.3125,0.3125),
P2= (0.6875,0.3125), P3= (0.3125,0.6875) and P4= (0.6875,0.6875) respectively. The value
of λhas been taken equal to 25, and the initial guess is a disk centered at (0.5,0.5) of radius
0.42 (containing Ω). The exact solution is given by the union of four disjointed disks centered
at P1,P2,P3and P4.
We have run this test on a 240×240 grid, with an initial guess consisting in a disk containing
¯
Ω. Convergence has been reached after 32 iterations (1272 sec.).
Fig. 4.2 shows the evolution of the boundary A0,A5,A10 ,A15 ,A20,A23 ,A24 ,A25 and A30.

Numerical Solution of the Bernoulli Problem 35
X
Y
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Y
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Y
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Y
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Y
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Y
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Y
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Y
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Y
0 0.5 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 4.3. Domains A0to A55.
To consider an interior topology change test, Ω is a set similar to the set used by Flucher
and Rumpf [7]. λhas been taken equal to −18, and the initial guess is the union of two disks
centered at (0.25,0.5) and (0.75,0.5) of radius 0.12 (included in Ω).
We have run this test on a 240 ×240 grid; convergence has been reached after 55 iterations
(1374.1 sec).
Fig. 4.3 shows the evolution of the boundary A0,A10 ,A20,A30 ,A40 ,A46,A47 ,A50 and
A55.
5. Conclusion
In this paper we have presented various extensions of the Flucher and Rumpf numerical
method to allow topological change. The method is based on a level set formulation coupled
with an elliptic equation derived by asymptotic analysis. Two schemes have been proposed:
The first one is devoted to the computation of an accurate solution but requires regularity and
does not allow topological changes. The second one is designed to overcome this difficulty but
is less accurate. A hybrid technique based on both schemes yields a good convergence speed
and a robust solver for the Bernoulli problem.

36 F. BOUCHON, S. CLAIN AND R. TOUZANI
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