CONJUGATE FUNCTION METHOD AND CONFORMALMAPPINGS IN MULTIPLY CONNECTED DOMAINS
HARRI HAKULA
∗
, TRI QUACH
†
,
AND
ANTTI RASILA
‡
Abstract.
In this paper, we present a generalization of the algorithm from our earlier work, fornumerical computation of conformal mappings, on multiply connected domains. An implementationof the algorithm, along with several examples and illustrations are given.
Key words.
numerical conformal mappings, conformal modulus, multiply connected domains,canonical domains
1. Introduction.
Conformal mappings play an important role in both theoret
1
ical complex analysis and in certain engineering applications, such as electrostatics,
2
aerodynamics and ﬂuid mechanics. Existence of conformal mappings of simply con
3
nected domains onto the upperhalf plane or the unit disk follows from the Riemann
4
mapping theorem, a wellknown result in complex analysis [2], and there are gener
5
alizations of this result for doubly and multiply connected domains. However, con
6
structing such mappings analytically is usually very diﬃcult, and use of numerical
7
methods is required.
8
There exists an extensive literature on numerical construction of conformal map
9
pings for simply and doubly connected domains [23]. One popular method is based
10
on the SchwarzChristoﬀel formula [11], and its implementation SC Toolbox is due to
11
Driscoll [9, 10]. SC Toolbox itself is based on earlier FORTRAN package by Trefethen
12
[26]. A new algorithm involving a ﬁnite element method and the conjugate harmonic
13
function was presented by the authors in [13]. The aim of this paper is to generalize
14
this algorithm to multiply connected domains.
15
While the study of numerical conformal mappings in multiply connected domains
16
dates back to 1980’s [21, 24], recently there has been signiﬁcant interest in the subject.
17
DeLillo, Elcrat and Pfaltzgraﬀ [8] were the ﬁrst to give a SchwarzChristoﬀel formula
18
for unbounded multiply connected domains. Their method relies on Schwarzian re
19
ﬂection principles. Crowdy [4] was the ﬁrst to derive a SchwarzChristoﬀel formula for
20
bounded multiply connected domains, which was based on the use of SchottkyKlein
21
prime function. The natural extension of this result to unbounded multiply connected
22
domains is given in [5]. It should be noted that a MATLAB implementation of com
23
puting the SchottkyKlein prime function is freely available [6], and the algorithm is
24
described in [7]. A method involving the harmonic conjugate function is given in [20],
25
but the approach there diﬀers from ours.
26
The main result (Proposition 3.1) of this paper is the connection between integra
27
tion of the absolute value of gradient of harmonic solution
u
over equipotential curve
28
on domain Ω, the value of integration is denoted by
C
(Ω), and its conjugate problem
29
˜Ω. These values are connected by the identity
C
(Ω)
·
C
(˜Ω) = 1. Proposition forms
30
the theoretical foundation to our method. The presented algorithm itself is based on
31
solving numerically the Laplace equation subject to DirichletNeumann mixedtype
32
∗
Aalto University, Institute of Mathematics, P.O. Box 11100, FI00076 Aalto, FINLAND(
harri.hakula@aalto.fi
)
†
Aalto University, Institute of Mathematics, P.O. Box 11100, FI00076 Aalto, FINLAND(
tri.quach@aalto.fi
)
‡
Aalto University, Institute of Mathematics, P.O. Box 11100, FI00076 Aalto, FINLAND(
antti.rasila@iki.fi
)1
boundary conditions. This method is a generalization of the algorithm for simply
33
and doubly connected domains described in [13]. The foundation of the algorithm for
34
simply and doubly connected domains lies on properties of the (conformal) modulus,
35
which srcinates from the theory of quasiconformal mappings [1, 19, 23]. Our method
36
is suitable for a very general class of domains, which may have curved boundaries and
37
even cusps. The implementation of the algorithm is based on the
hp
FEM described
38
in [14], and in [15] it is generalized to cover unbounded domains. In [16], the method
39
has been used to compute a modulus of domains with strong singularities.
40
The rest of the paper is organized as follows: After a brief introduction to canon
41
ical domains, the Conjugate Function Method is discussed in detail in Section 3,
42
followed numerical version, and a series of numerical experiments highlighting dif
43
ferent features of the methodi in Section 5. Finally, some potential applications,
44
such as quadrilateral mesh generation and design of cubatures, are suggested before
45
concluding remarks.
46
2. Canonical Domains.
The so called canonical domains play a crucial role
47
in the theory of quasiconformal mappings (cf. [19]). These domains have a simple
48
geometric structure. Let us consider a conformal mapping
f
:
D
→
Ω, where
D
49
is a canonical domain, and Ω is the domain of interest. The choice of the canonical
50
domain depends on the connectivity of the domain Ω, and both domains
D
and Ω have
51
the same connectivity. It should be noted that in the simply and doubly connected
52
cases, domains can be mapped conformally onto each other if and only if their moduli
53
agree, which is given by the energy norm (see e.g. [19]). In this sense, moduli divide
54
domains into conformal equivalence classes. For simply connected domains, natural
55
choices for canonical domains are the unit disk, the upper halfplane and a rectangle.
56
In the case of doubly connected domains an annulus is used as the canonical domain.
57
For
m
connected domains,
m >
2, we have 3
m
−
6 diﬀerent moduli, which leads to
58
various choices of canonical domains. These domains have been studied in [12, 22].
59
The generalization of Riemann mapping theorem onto multiply connected domains is
60
based on these moduli, see [12, Theorem 3.9.12, 3.9.14].
61
f
Fig. 2.1: Conformal mappings of a triply connected domain onto an annulus withradial slits on the positive real axis and radial slits.
3. Conjugate Function Method.
In this section we introduce the algorithm
62
to construct a conformal mapping from a multiply connected domain onto an annulus
63
with
n
radial slits on the positive real axis.
64
2
Let
E
0
,E
1
,...,E
m
be disjoint and nondegenerate continua in the extended com
65
plex plane
C
∞
=
C
∪ {∞}
. Suppose that
E
j
,
j
= 1
,
2
,...,m
are bounded, and that
66
the a set Ω
m
+1
=
C
∞
\
mj
=0
E
j
is connected. Then Ω
m
+1
is an (
m
+ 1)
connected
67
domain
and its
(conformal) capacity
is deﬁned by
68
cap Ω
m
+1
= inf
u
0
Ω
m
+1
∇
u
0

2
dxdy,
where the inﬁmum is taken over all nonnegative, piecewise diﬀerentiable functions
69
u
0
with compact support in
mj
=1
E
j
∪
Ω
m
+1
such that
u
0
= 1 on
E
j
,
j
= 1
,
2
,...,m
.
70
Suppose that a function
u
0
is deﬁned on Ω
m
+1
with 1 on
E
j
,
j
= 1
,
2
,...,m
and
71
0 on
E
0
. Then if
u
0
is harmonic, it is unique, and it minimizes the above integral.
72
Then a modulus of Ω
m
+1
is deﬁned by M(Ω
m
+1
) = 2
π/
cap Ω
m
+1
. If the the degree
73
of connectivity does not play an important role, the subscript will be omitted and we
74
simply write Ω.
75
3.1. Cutting Process.
In order to explain the cutting process in multiply con
76
nected domains similar to our work on doubly connected domains [13], we ﬁrst describe
77
the process for triply connected domains. Let Ω =
C
∞
\
(
E
0
∪
E
1
∪
E
2
) and suppose
78
that
u
is a harmonic solution in Ω such that it attains 0 on
E
0
and 1 on
E
1
∪
E
2
.
79
First we cut the domain Ω from
∂E
1
to
∂E
2
along the steepest descent curve, which
80
is a ﬂowline of the gradient of
u
, such that it goes through the saddle point that lies
81
between the sets
E
1
and
E
2
. Then we cut from
∂E
0
to
∂E
1
, again along a ﬂowline
82
of the gradient of
u
. In triply connected domains, the cut between
E
1
and
E
2
always
83
exists. In general, if such a cut exists between
E
i
and
E
j
,
i
=
j
, then we say that
84
E
i
and
E
j
are
conformally visible
to each other. It is easy to see that in a triply
85
connected domain, the sets
E
1
and
E
2
are always conformally visible.
86
In general, we cannot, a priori, say which of the sets
E
j
,
j
= 1
,
2
,...,m
are
87
conformally visible to each other. Fortunately, without loss of generality, we can
88
rename
E
j
’s as
E
′
j
such that
E
′
j
can be considered as a binary tree of conformally
89
visible sets (cf. Subsection 3.2). The exact numbering of
E
′
j
is done by using preorder
90
method of tree search, see Figure 3.1 for an example. For more information about tree
91
search methods, we refer to [3, Sec. 12]. Note that, the rearrangement is not unique,
92
and any arrangement can be used in our algorithm. Obviously, a careful choice leads
93
to easier geometrical conﬁgurations and thus reduce the computational workload. For
94
the rest of the article, we assume that
E
j
,
j
= 1
,
2
,...,m
are arranged such that
E
j
95
is a binary tree of conformally visible sets.
96
For
j
= 1
,
2
,...,m
, let
γ
j
denote the cut between
E
j
and its parent node, see
97
Figure 3.1. For
γ
1
, we use a following deﬁnition,
γ
+1
is the cut when travelling from
98
E
0
to
E
1
and likewise
γ
−
1
when we are travelling from
E
1
to
E
0
. This distinction is
99
used when the travelling direction between
E
0
and
E
1
does matter and becomes clear
100
when the DirichletNeuman problem is deﬁned in Section 3.4.
101
3.2. Common Saddle Point.
In some domains, three or more sets have one
102
common saddle point. Thus, for these sets, the cutting process will produce a cycle
103
and not the binary tree, which we would be expecting. This kind of anomaly can
104
be seen, e.g., in fully symmetric domains, see Figure 3.2, where the common saddle
105
point lies in the middle. Note that, nodes around the common saddle point can form
106
a binary tree or even a cycle of their own depending on their child nodes.
107
3.3. Jump Between Cutting Curves.
Integrating the absolute value of gra
108
dient of
u
through equipotential curves starting from
γ
+1
to
γ
−
1
, we encounter two
109
3
E
0
E
1
E
2
E
3
E
4
E
5
E
6
E
7
E
8
renaming
γ
1
γ
4
γ
8
γ
5
γ
7
γ
6
γ
2
γ
3
E
′
0
E
′
1
E
′
2
E
′
3
E
′
6
E
′
5
E
′
4
E
′
7
E
′
8
Fig. 3.1: In above illustration, on the right hand side we have a binary tree of conformally visible sets
E
j
. By renaming the sets, we have
E
′
j
shown on the left hand sidewith the cutting curves
γ
j
.
E
0
E
1
E
2
E
3
E
4
E
5
Fig. 3.2: Example of a fully symmetric domain. A simple symmetric case and insteadof a binary tree, we obtain a cycle, which is due to the common saddle point in themiddle.situations. If we take an equipotential curve near
∂E
0
then it will enclose all the
E
j
,
110
j
= 1
,
2
,...,m
. On the other hand, if we choose an equipotential curve near
∂E
1
then
111
it will enclose only
E
1
. In the latter case, to overcome this enclosing problem we will
112
integrate only to cutting curves deﬁned in Section 3.1. If the integration process hits
113
a cutting curve, we will move along the cutting curve past the saddle point and start
114
the integration again. For (
m
+ 1)connected domain, the process has up to 2
m
−
1
115
jumps, see Figure 3.3 for an example. In the process we form a curve which encloses
116
E
j
,
j
= 1
,
2
,...,m
. Note that, e.g., in the case of symmetric domains, we have less
117
than 2
m
−
1 jumps.
118
Let Γ
0
be an equipotential curve near the
∂E
0
, then we deﬁne a constant
d
as
119
follows
120
C
(Ω) =
d
=
Γ
0
∇
u

ds.
(3.1)Let us assume that we have
n
jumps, then Γ
j
,
j
= 1
,...,n
are each a part of an
121
4
γ
3
γ
2
γ
1
Γ
1
Γ
2
Γ
3
Γ
4
Γ
5
γ
1
Γ
1
Γ
2
Γ
3
Γ
4
Fig. 3.3: Jump process with 5 total of jumps. The number of jumps depends onsymmetricity of the domain, and jump between the cutting curves depends on thebranching of the cutting process.equipotential curves and we deﬁne
122
d
j
=
Γ
j
∇
u

ds.
(3.2)By the properties of
u
, we obtain
123
d
=
n
j
=1
d
j
.
3.4. DirichletNeumann Problem.
Let Ω be a domain in the complex plane
124
whose boundary
∂
Ω consists of a ﬁnite number of regular Jordan curves, so that at
125
every point, except possibly at ﬁnitely many points of the boundary, a normal is
126
deﬁned. Let
∂
Ω =
A
∪
B
where
A,B
both are unions of regular Jordan arcs such
127
that
A
∩
B
is ﬁnite. Let
ψ
A
,
ψ
B
be realvalued continuous functions deﬁned on
A,B
,
128
respectively. Find a function
u
satisfying the following conditions:
129
1.
u
is continuous and diﬀerentiable in Ω.
130
2.
u
(
t
) =
ψ
A
(
t
)
,
for all
t
∈
A
.
131
3. If
∂/∂n
denotes diﬀerentiation in the direction of the exterior normal, then
132
∂ ∂nu
(
t
) =
ψ
B
(
t
)
,
for all
t
∈
B.
Suppose that
u
is the (unique) harmonic solution of the DirichletNeumann problem
133
with mixed boundary values of
u
equal to 0 on
∂E
0
, equal to 1 on
∂E
j
,
u
=
u
0
on
γ
j
,
134
j
= 2
,
3
,...,m
, and
∂u/∂n
= 0 on
γ
1
. Let
v
be a conjugate harmonic function of
u
135
such that
v
(Re ˜
z,
Im ˜
z
) = 0, where ˜
z
is the intersection point of
E
0
and
γ
+1
.
136
Then
ϕ
1
=
u
+
iv
is an analytic function, and it maps Ω onto a rectangle
R
d
=
137
{
z
∈
C
: 0
<
Re
z <
1
,
0
<
Im
z < d
}
minus
n
−
2 linesegments, parallel to real axis,
138
between points (
u
(˜
z
j
)
,d
j
) and (1
,d
j
), where ˜
z
j
is the saddle point of the corresponding
139
j
th jump. In the process we have total of
n
jumps. See Figure 3.4 for an illustration
140
of a triply connected example.
141
5