A domain decomposition method for conformal mapping onto a

Constr. Approx. (1991) 7:349-379

CONSTRUCTIVE APPROXIMATION .~ 1991Springer-V9

New York inc.

A D o m a i n Decomposition Method for Conformal Mapping onto a Rectangle N. Papamichael and N. S. Stylianopoulos Abstract. Let g be the function which maps conformally a rectangle R onto a simply connected domain G so that the four vertices of R are mapped respectively onto four specified points z I, z2, za, z4 on tgG.This paper is concerned with the study of a domain decomposition method for computing approximations to g and to an associated domain functional in cases where: (i) G is bounded by two parallel straight lines and two Jordan arcs. (ii) The four points zl, z2, z3, z4, are the corners where the two straight lines meet the two arcs.

1. Introduction Let G be a simply connected J o r d a n d o m a i n in the complex z-plane (z = x + iy), and consider a system consisting of G and four distinct points zl, z2, z3, z4, in counterclockwise order o n its b o u n d a r y aG, Such a system is said to be a quadrilateral Q and is denoted by

Q--={G; zl, z2, z3, z,}. The conformal module m ( Q ) of Q is defined as follows: Let R be a rectangle of the form (1.1)

R..= {(~, r/): a < ~ < b, c < r / < d},

in the w-plane (w = r + it/), and let h denote its aspect ratio, i.e., h := (d - c)/(b - a). T h e n m(Q) is the unique value of h for which Q is conformally equivalent to a rectangle of the form (1.1), in the sense that for h = m(Q) and for this value only there exists a unique conformal m a p R--* G which takes the four c o m e r s a + ic, b + ic, b + id, and a + id of R respectively onto the four points zl, z2, z3, and z4. In particular, with h = m(Q), Q is conformally equivalent to a rectangle of the form (1.2)

Rh{ct} ..= {(r t/): 0 < ~ < i, g < ~/< ~t + h}.

Date received: May 26, 1989. Date revised: September 11, 1989. Communicated by Dieter Gaier. AMS classification: 30C30. Key words and phrases: Numerical eonformal mapping onto a rectangle, Quadrilaterals, Conformal modules, Domain decomposition. 349

N . P a p a m i c h a e l a n d N . S. S t y l i a n o p o u l o s

350

z

1 +i(h-h 1 )

Za 4

Ib

.....

R

0

.......

z

1-ih I

........ G ........

~

7' 2 {a}

[b)

Fig. 1.1

Consider now the case where Q is of the form illustrated in Fig. 1.1(b) and let the arcs (z I, z2) and (z3, z4) have cartesian equations y = zl(x) and y = ~2(x), where ~ , j = 1, 2, are positive in [0, 1]. That is, let (1.3a)

Q,= {o; z~, z2, z~, z,},

where (1.3b)

G ..= {(x, y): 0 < x < 1, - , , ( x )

< y < r=(x)},

with %{x)>O,

j=l,

2

for x e [ O , l ] ,

and (1.3c)

zl = - i z l ( 0 ) ,

z 2 = 1 - i t 1 ( 1 ),

z3 = 1 + i'c2(1),

z4 = i,2(0).

Also let (1.4a)

G1 -'= {(x, y): 0 < x < 1, - z l ( x ) < y < 0}

and (l.5a)

G= := {(x, y): 0 < x < 1, 0 < y < r=(x)},

so that G = •1 to G a, and let Q1 and Qz denote the quadrilaterals

(1.4b)

Q~ ..= {6,; z,, z2, 1, 0}

and (1.5b)

{22 '= {G=; 0, 1, z 3, z,}.

Finally, let (1.6)

h,=m(Q)

and

hj:=m(Qs),

j = 1,2,

Domain Decomposition

351

and let g and gi, J = 1, 2, be the associated conformal maps (1.7)

g:R ~G,

0.8)

gl: R1 ~ Gt,

and (1.9)

#2:R2 ~ G2,

where, with the notation (1.2), (1.10) (1.11)

R "-= Rh{-hl} = {(~, ~/): 0 < ~ < 1, - h ~ < r / < h - hi} , R 1 := R , , { - h l } = {(~, r/): 0 < ~ < 1, - h 1 < r/< 0},

and (1.12)

R 2 , = Rh2{0} = {(~, n): 0 < r < 1, 0 < n < h~}.

(The conformal map g is illustrated in Fig. 1.1.) This paper is concerned with the study of a domain decomposition method for computing approximations to h ,= re(Q) and to the associated conformal map #, defined by (1.7), in cases where the quadrilateral Q is of the form (1.3). More specifically, the method under consideration is based on decomposing Q into the two smaller quadrilaterals Qt and Q2, given by (1.4) and (1.5), and then approximating h, R, and g respectively by (l.13a)

~1..= h I + h2,

(l.13b)

,R:= Ri,{-hl} --- {(~, ~/): 0 < ~ < I, - h i < r / < h2} ,

and 0.14)

~g~(w): R2 ~ G2 ~(w) := (gl( w): RI ~ G1

for for

w ~ R2, w ~ R 1.

The motivation for considering this method emerges from the intuitive observation that if h* .-= min(hl, h2) is "large," then the segment 0 < x < 1 of the real axis is "nearly" an equipotential of the function u satisfying the following Laplaeian problem: Au-0 u=0

on an = 0

(Zl, Z2), on

inG, u=l

on

(z3, z,),

(z2, z3) u (z4, z 0.

This in turn indicates that if h* is large, then h -~ h I + h 2,

N. Papamichael and N. S. Stylianopoulos

352

and function (1.14) "approximates" the true conformal map g. (We note that (1.15)

h ~ hx + h 2,

and equality occurs only in the two trivial cases where: (a) G is a rectangle, and (b) zi(x) = z2(x), x e [0, 1]; seer e.g.p. 437 of [8].) The purpose of this paper is to provide a theoretical justification for the above decomposition method, and to show that (1.13), (1.14) are capable of producing close approximations to h and to 9, even when h* .-= min(hl, h2) is only moderately large. We do this by a method of analysis that makes extensive use of the theory given in Chapter 5, Section 3, of [2], in connection with the integral equation method of Garrick [6] for the conformal mapping of doubly connected domains. In particular, we derive estimates of (1.16a)

Eh,= I h - h l = h - ( h l

+ h2),

(1.16b)

E~'~ := max{Iv(w) - Vl(w)l : w e/~x},

(1.16,:)

E~2~"-=max{lv(w + iEh) - V2(w)[: w ~/~2},

and show that E h ~ O{e-2xh*}~

and Etl~ g = 0{e-~},

Et21 = ~g

0{e-~h,},

provided that the functions r~, j = 1, 2, in (1.3b), satisfy certain smoothness conditions. Although the main results of this paper are derived by considering quadrilaterals of the form (1.3), the domain decomposition method and the associated theory have a somewhat wider application. More specifically, it will become apparent from our work that both the method and the theory also apply to the mapping of quadrilaterals Q ..= {G; z~, z2, z3, z4}, in cases where the domain G and the crosscut c that decomposes Q into Q1 and Q2 are as described below: 9 G is of the form illustrated in Fig. 1.2. That is, G is bounded by a segment 11 -'= (z4, zl) of the real axis, a straight line 12 .'= (z2, za) inclined at an angle ~rr, 0 < ~ < 1, to 11, and two Jordan arcs Vl := (zt, z2) and ~2 '= (z3, z4) which are given in polar coordinates by (1.17)

7j '= {z: z = pj(O)e i~ 0 1 and 0 < p2(O) < 1, for 0 e [0, gn], and the crosscut c is the arc z = ei~ 0 < 0 < gn, of the unit circle. Although the results of this paper apply only to quadrilaterals that have one of

353

Domain Decomposition z 2

0 ~,::::'................. 'i z

t

z

4

Fig. 1.2

the two special forms illustrated in Figs. 1.1 and 1.2, we note that the mapping of such quadrilaterals has received considerable attention recently; see, e.g., El,i, [5,1 E1I,], [121, [15"], and [17]. In this connection the decomposition method is of practical interest for the following two reasons: (i) It can be used to overcome the "crowding" difficulties associated with the numerical conformal mapping of "thin" quadrilaterals of the forms illustrated in Figs. 1.1 and 1.2. (Full details of the crowding phenomenon and its damaging effects on numerical procedures for the mapping of "thin" quadrilaterals can be found in [121, [131, and [9]; see also p. 179 of [3,1 p. 428 of [81, and p. 4 of [16].) (ii) Numerical methods for approximating the conformal maps of quadrilaterals of the form (1.4) or (t.5) are often substantially simpler than those for quadrilaterals of the more general form (1.3); see, e.g., [5-] and Section 3.4 of [12,]. The paper is organized as follows: In Section 2 we state without proof some preliminary results which are needed for our work in Section 3. These concern wellknown properties of three integral operators that occur in the integral equations of the method of Garrick. In Section 3 we consider the Garrick formulations for the conformal maps of three closely related doubly connected domains. Hence, by making use of the theory given in Chapter V of [21, we derive a number of results that provide certain comparisons between the three conformal maps. Section 4 contains the main results of the paper. Here, we first identify certain well-known relationships between the conformal maps (1.7)-(1.9) and those considered in Section 3. Hence, by making use of the results of Section 3, we derive estimates of the errors (1.16) in the domain decomposition approximations (1.13)-(1.14). Finally, in Section 5 we present two numerical examples illustrating the theory of Section 4, and make a number of concluding remarks concerning this theory.

2. Preliminary Results In Section 3 we make extensive use of the properties of three linear integral operators in the real function space

(2.1)

L 2 ,= {u: u is 2It-periodic and square integrable in [0, 2n]}.

354

N. Papamichaeland N. S. Stylianopoulos

These operators are denoted by K, Rq, and Sq and are defined as follows (see pp. 194-195 of [2]): 9 K is the well-known operator for conjugation on the unit circle. That is, for u e L2, the function Ku is defined by the Cauchy principal value integral

K[u(cp,],=lpvf;ncot(~-~--~)u(t)dt. 9 The operators R a and S~ depend on a real parameter q, with 0 < q < 1, and are defined by RqEu(cp)] ,=

9q(q~- Ou(t) dr,

Sr

,= ~

hq(q~- t)u(t) dt,

where the kernels 9q and hq are given by the absolutely convergent series

~1 q2k

9q(~) ----4 = 1 Z q2k sin k~,

h~(~a)--

--4k~

= 1-

qk q2k sin k~0.

The properties of the above three operators are studied in detail on pp. 195-205 of [2], where in particular the following basic results can be found: 9 If u E L2, then

Ku, Rqu,Squ e L2,

(2.2) and

(2.3a)

IIKu II -< IIu II,

IIR~u II -< ~

2q ~

tl u II,

2q

IIS•U ]1 1 and

and as in Section 2, let f , | (3.2)

0 1, then the domain f~l reduces to a circular annulus of inner radius 1 and outer radius r 1. Thus, in this case, f ~ ( W ) = r~W and hence q~ = lift and Ol(q0 = O x ( ~ ) = q~. Therefore, since d t = ~ ] = 0 and ~ = e ~ , the results of Theorem 3.1 and 3.2 simplify to the following: (3.26)

Ilog q + log rt - log q21 < a(e2, e2)" e2dzq~,

and

l|

(3.27a) (3.27b)

- tPl -< x/l~fl(O, e2)" a2q,

I0(~) - O~(~)l -< 4~#(s~, ~,). e~q~,

where a(.,- ) and fl(-,. ) are given by (3.22) and (3.25). Similarly, if the inner boundary curve F2 is a circle of radius r z, i.e., if p~(O) = r z < t, then the results of the two theorems simplify as follows: (3.28)

[log q - log ql -- log r2l _< 0~(~l, el)" eidlq~,

and (3.29a)

IO(,e) - Olq~l < ,,/~/~(~,, ~,)" ~1ql,

(3.29b)

[~(q~) - q~l < ,v/nil(0, ~ l ) ' e t q 9

Furthermore, it is easy to see that the results (3.26)-(3.29) hold under the somewhat less restrictive assumptions obtained by replacing inequalities (3.13) of Assumption A3.1(iii) by (3.30a)

(3.30b)

2--

2) _--Z-~z 1 j" < 1,

when

pl(O) = rl,

S _---Z-~2j1 e~'.- d 1~1 < 1,

when

p2(0) = r2.

(This follows by modifying the analysis in an obvious manner, after first observing that in each of the two special cases under consideration the Garrick equations (2.14) take the simplified forms (3.10)-(3.11).) In addition, it is easy to see that the results (3.27) and (3.29) also hold in the limiting cases where r~ = 1 or r 2 = 1. Thus, in particular, (3.29b) and (3.27a) imply respectively that (3.31a)

I0~(~o)

-

~I -< ,r

~)~ql

and (3.31b)

IO~(q0 - q>l - v/-nfl( 0, e2)e2qz,

where ~)1 and 02 are the boundary correspondence functions defined by (3.8). Theorem 3.3. (3.32)

For any p, where q < p < 1, let f (pe i~') = P(p, r

~~ ~'J,

q~ ~ [0, 2r~].

Domain Decomposition

363

I f the curves Fj, j = 1,2, satisfy Assumptions A3.1 and, in addition, are both symmetric with respect to the real axis, then

(3.33)

I@(P, ~o) -- rgl --< x//-~fl(O, e){exp + E2(q/p)},

and

(3.34a)

Ilog P(p, q~) - logp + log xil --< 89

E){tIP + e2(q/p)} + dj~(cj, *){ejq2 + e 3_Jq},

j = 1, 2,

with

(3.34b)

xl '= ql

and

x2 '= q/q2,

where ~( . , . ) and fl( . , . ) are #iven by (3.22) and (3.25) and e ..= max (el, e2).

Proof. The symmetry of the curves Fj, j = 1, 2, implies that the Fourier series of the functions u(~o):= log px(| and fi(~o),= log p2(O(~0)) are of the form u(~P)= 89

~ akcosk~p

and

a(q~)= 89

k=l

~ akCOSk~o. k=l

The symmetry also implies that the function F(W) .'= log{f(W)/W} has the Laurent series expansion F(BO =

~. k ~

--

ckW k,

W ~ Aq,

oo

where the coefficients ck are all real and are related to the Fourier coefficients a~, dk by

1 f2,, Co = 89 = ~ Jo log pa(O(q~)) d~o, ck = {ak - dkq'}/(1 -- q2k) and

c_, = {dkqk -- akqZk}/(1 -- q2,),

k = 1, 2 .... ;

see p. 270 of 1-5]. It follows that, for any fixed p, q < p < 1, F(pe i~') = c o + Up(q~) + iVp(q~),

~o ~ l0, 2tO,

where the functions Up and Vv have the Fourier series representations Uv(r = ~ ~k cos k~o and k=l

W(~~ =

/~k sin kq~, k=l

with (3.35a)

% = {ak(p2k _ q2k) + dkqk(1 -- p2k)}/{pk(1 -- q2k)}

and (3.35b)

~k = {a~(P2k + q2k) _ dkqk(1 + p2k)}/{pk(1 -- q2k)}.

N. Papamichael and N. S. Stylianopoulos

364

This implies that II U:,ll =

89

and

1k ' ~

II g;ll -- 89

k~l k2fl 2

9

Hence, by substituting the values (3.35) of ak and flk and applying the Minkowski inequality to each of the resulting right-hand sides, we find that [IU~,,< p{89k~= l k2a[}t/2 + (p){~ q I k~l co k2a2k} 112 1

< (1 -- e2) 112" {elp + e2(q/p)} and

IIv;It _< ,( ~--~.f.,t2 2

__l

: q2)tf "k2 { p(1 2q t1'2

dl

- 0, x E [13, i], i.e., if

G x'.= {(x,y):0 < x < 1, --c < y < 0} = Rc{-c}, then 9 z ( w ) = w, h 1 = c, dx = 8,1 = 0, and the results of Theorems 4.1-4.4 simplify respectively as follows: (4.21) (4.22a)

Eh,= h - (c + h2) ~ 7z-ld2Gt(~2, 8,2),~2e- 2~t~2 [X(~) - E l - '~-1/2fl( 0, e2)82 e-nh

and (4.22b) (4.23a)

12(~.) - 22(~)1 ~ 7r- x/2~(~2, e2)e2e-'~h2. IX(~, '7) - ~1 0, Eh = O{e-2~h'}, E~~ = O{e-~h'},

j = 1, 2,

and

E~~1 = O{e-~h'},

where h* := min(hl, h2). Remark 4.7. Let Q ~= {G; Zl, z2, z3, z4} be of the form illustrated in Fig. 1.2. That is, let Q consist of a domain G bounded by a segment 11 .-= (z4, zl) of the real axis, a straight line 12 -'= (z2, z3) inclined at an angle ccr~,0 < ~ _< 1, to 11, and two Jordan arcs ~1 := (Zl, z2) and 72 := (z3, z4) where

~ == {z: z = pj(O)e i~ 0 _ 0. In this ease d 1 = 0.3849, dz = 0.5830, and the largest values of ej, j = 1, 2, i.e., el = 0.3918 and e2 = 0.6064, occur when 1 = 0. Therefore, the functions Ti, j = I, 2, satisfy Assumptions A4.1, for all l > 0.

N. P a p a m i c h a e l a n d N. S. S t y t i a n o p o u l o s

376

T a b l e 5.1

(a) l 0.00 0.25 0.50 0.75 1.00

hI 1.565 1.815 2.065 2.315 2.565

514 515 515 515 515

h2 72 54 71 74 75

1.333 1.583 1.833 2.083 2.333

h

348 350 351 351 351

92 99 42 51 53

2.898 3.398 3.898 4.398 4.898

870 867 867 867 867

58 97 43 31 29

(b)

l

Es,

0.00 0.25 0.50 0.75 1.00

7.0E 1.5E 3.0E 6.3E 1.3E

-

T(Eh) 6 6 7 8 8

2.6E 5.1E 1.0E 2.1E 4.4E

-

E~lj 4 5 5 6 7

2.1E 9.6E 4.4E 2.0E 9.1E

-

T(F,~~) 3 4 4 4 5

4.2E 1.9E 8.4E 3.8E 1.TE

-

2 2 3 3 3

E~2~ 2.1E 9.6E 4.4E 2.0E 9.0E

-

T(E~2~) 3 4 4 4 5

5.5E 2.4E I.IE 5.0E 2.2E

-

2 2 2 3 3

The numerical results corresponding to the values l = 0.0(0.25)1.0 are listed in Table 5.1(a) and (b). This table contains respectively the computed values of the conformal modules, which are expected to be correct to seven significant figures, and the values of the error estimates Eh, T(Eh) and E~ ~, T(E~J3),j = 1, 2.

Example 5.2. Let Q and Q~,j = I, 2, be defined by (1.3) - (1.5) with zl(x) = c > 0

and

z2(x) = 0.25x 4 - 0.5x 2 + 2.0 + l,

l _> 0.

In this case Q is of the special form considered in Remark 4.5, i.e., gl(w) = w, h 1 .-= re(Q1) = c, and d 1 = e 1 = 0. Also, d 2 = 0.3849 and, for all l > 0, m 2 < 4.1 x 10-3. Hence, (4.25) gives

9 fl + ,2-'= a2 1

i, < 0.385,

vt >__o,

i.e., the simplified results (4.21)-(4.24) hold for all l > 0. The numerical results corresponding to the values c --- 1 and l = 0.0(0.5)2.0 are listed in Table 5.2(a) and (b). As in Example 5.1, the table contains respectively the computed values of the conformal modules h and h2, and the values of the error estimates Eh, T(Eh) and E~3, T(E~ j = 1, 2. We recall that h~ = 1, and observe that the values of h and h 2 listed in Table 5.2(a) are expected to be correct to the number of figures quoted. (The algorithms of [5] achieve this remarkable accuracy because, in this case, the curve F2: Z = p2(O)e~~corresponding to the arc y = zz(x) is analytic; see the comment made in Remark 3, p. 279, of 15].) We also observe that the estimates given in Table 5.2(b) remain unchanged for any value c > 0; see Remark 4.5.

377

Domain Decomposition

Table

5.2

(a) 1

h2

h

0.0

1.859 568 647 615

2.859 569 034 971

0.5

2.359 569 018 925

3.359 569 035 644

1.0

2.859 569 034 971

3.859 569 035 694

1.5

3.359 569 035 664

4.359 569 035 695

Z0

3.859 569 035 694

4.859 569 035 695

(b) t

E~

T(e~)

F~C,~

r(g")

E~,~j

r(E~2~)

0.0 0.5

3.9E - 7 1.7E - 8

t.4E - 6 6.1E - 8

5.0E - 4 1.0E - 4

2.9E - 3 6.0E - 4

5.0E - 4 1.0E - 4

4.8E - 3 1.0E - 3

1.0 1.5

7.2E - 10 3.1E - 11

2.6E - 9 1.1E - 10

2.2E - 5 4.5E - 6

1.3E - 4 2.6E - 5

2.1E - 5 4.5E - 6

2.1E - 4 4.4E - 5

2.0

1.4E - 12

4.9E - 12

9.3E - 7

5.4E - 6

9.3E - 7

9.0E - 6

We end this section by making the following concluding remarks: Remark5.1. The results of the two examples given above illustrate the remarkable accuracy that can be achieved by the domain decomposition method, even when the quadrilaterals involved are only moderately long. Furthermore, the results confirm the theory of Section 4 and show that the error estimates given in Theorems 4.1 and 4.4 reflect closely the actual errors in the domain decomposition approximations. R e m a r k 5.2. We recall the method used for computing the values Eg~) a n d Eg2j

listed in Tables 5.1(b) and 5.2(b), and note that in both examples the maxima of Ig(w) - #l(w)l and Ig(w + iEh) -- 02(w)l occur on the common boundary segment r/= 0 of R~ and R2. The errors on the sides !/= -h~ of Rt and r/= h 2 of R2 are much smaller, indicating that the estimates of E~ ) := max IX(C) - X~(r

and

E~ }.-= max I.,~(~) - .X'2(~)I,

given in Theorem 4.2, are pessimistic. In fact, there is strong experimental evidence which suggests that E~ ) and E~ ) are both O{e-2~h'}, rather than O{e -~h*} as predicted by (4.17). (Of course, exactly the same remark applies to the estimates referred to in Remark 4.2.) The very close agreement between the values _gE ~1) and E~2~ listed in the tables is related to the above observations, and can be explained by the results of Theorem 4.3 and those given in Remark 4.5.

N. Papamichael and N. S. Stylianopoulos

378

Remark 5.3. Since h >_ h 1 + h2, the results of Theorems 4.1-4.4 provide computable error estimates, i.e., estimates that can be computed easily once the approximations to the conformal modules h 1 and h 2 are determined. In addition the results of the theorems can be used to provide a priori error estimates, i.e., estimates that can be determined before the approximations to hi and h2 are computed. This can be done by observing that

hi___

min T~(x).'= b), j - 1, 2, O_