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On the use of conformal transformations for the numerical solution of harmonic boundary value problems
COMPUTER IN METHODS IN APPLIED TECHNICS 0 NORTH-HOLLAND PUBLISHING COMPANY
AND ENGINEERING 12 (1977) 201-218
~NTHEUSEOFCONF~RMALT~ANSFORMATIONS FORTHENUMERICALSOL'JTION OFHARMONICBOUNDARYVALUEPROBLEMS D. LEVIN,N.PAPAMrCHAEL andA.SIDERIDIS Department of Mathematics, Brunel University, Uxbridge, UK Manuscript received 17 January 1977; revised manuscript received 17 February 1977 Certain aspects concerned with the application of a conformal transformation method for the numerical solution of harmonic boundary value problems are considered. In particular, a procedure is decribed for removing the effects of singularities which are sometimes introduced by the transformations in the solution of the transformed problem. The transform problem is then solved by a collocation technique.
1. Introduction Let J-t be a simply connected domain with boundary 3sZ in the complex w-plane (w = x f iv) and consider the harmonic boundary value problem
AM, Y>= 0,
a,
kY)E
6,
Y)
E
(x, .Y) E In (1.1) A is the two-dimensional
as-t, ,
(1.1)
an,.
Laplacian operator, af2, and XZz are nonintersecting
subarcs of
an such that Xl = aSZ, u aQ2,, and a/au is the derivative in the direction of the outward normal to the boundary. The subarc aSZ, is always nonempty; when aa2, is empty, i.e. when afi = X2r, (1.1) is a Dirichlet problem. We consider the application of a conformal transformation method (CTM) which has been recently proposed in [ 1I for the numerical solution of problems of type (1.1). The CTM has been developed to deal primarily with problems involving boundary singularities for which standard numerical techniques, such as finite differences and finite elements, fail to produce accurate solutions. Although these standard techniques can be modified to improve the accuracy of the numerical solution (see e.g. Strang and Fix [ 2]), the CTM provides an alternative method which can, in many cases, produce solutions of high accuracy. Furthermore as it is shown in [ 1 I, even in the absence of singularities the CTM is often very well suited for the solution of harmonic boundary value problems involving curved bounda~es. The CTM consists essentially of three successive conformal transformations whose combined effect is to map the original region fi = Q, U &? in the w-plane onto the rectangle
202
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
in the w’-plane (w’ = g + in), thus transforming the original problem into a computationally simpler problem in fi’. Typically, by a suitable choice of the transformations the subarc aa2, is mapped onto the side ail; = ((0, 7): 0 <
Tj <
H}
of !%I,and the transformed problem is [ 1, p. 183 ]
AUK ‘17) =0
CL77)E 0’ ,
3
uct,7))= 4th
7) f
!y?p = $/(q)
)
(t, 77)E
aa;,
(0, rl) E
aa;.
(1.3)
In (1.3) ast’ = an; u aa;, #(t, 77)= f{x(.5 v), G(n) = -J$+(O,
77),.Y(O,77)1.
In the present paper we consider certain aspects concer_ning the numerical solution of the transformed problem. We show that when the original region s2 contains a comer A of internal angle arrr,0 < a! < 1, then application of the CTM introduces a singularity at the image of A which can seriously affect the numerical solution of the transformed problem. We show, however, that the effects of such a singularity can be easily removed, and we present numerical examples which illustrate this. We also use a new technique for the solution of the transformed problem. This technique involves the application of the modified collocation method of [ 31 to harmonic mixed boundary value problems and can be used to produce results of comparable accuracy to those obtained in [ 11 with much less computational effort. Its main advantage is that, unlike the finite difference technique used in [ 11, it computes the solution at any point of a’, and hence at specific points of Q without the need for bivariate interpolation.
2. The transformations Let aS2, = A,A, and a!$ = A,A,, where Ai -wi, i = 1,2. Then the three conformal transformations of the CTM are: (i) The transformation z = T(w)
)
(2.1)
where T(w) is a function of w, analytic in 0, so that (2.1) is a conformal mapping of fi onto the upper half z-plane. (ii) The bilinear transformation
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
t = M(z)
=k
203
(2.2)
where k is a constant. (iii) The Schwarz-Christoffel transformation w’
= S(t)
=
_-!_ K(m)
sn-* (t’l*, m) ,
(2.3)
where m < 1 is a positive real number, sn denotes the Jacobian elliptic sine, and K(m) is the complete elliptic integral of the first kind with modulus m. The combined effect of (2. l), (2.2) and (2.3) is to map 6 E w-plane onto the rectangle
in the w’-plane (w’ = t + iq) so that the points Ai 3 wi E aS2, i = 1, 2 are mapped respectively into the vertices (0,O) and (0, H) of fi’.The values of k in (2.2) and m in (2.3) may be chosen so that two further points, both on %I1 or both on aa,, are mapped into the other two vertices (1, 0) and (1, H) of tii: The form of the mapping function T(w) clearly depends on the geometry of the domain a. However, once the transformation (2.1) is constructed and the values of k and m are prescribed, the image P’ E (t, 7) E 32’ of the point P q (x, v) E fi is determined from (2.2) and (2.3) by a standard procedure. The main requirement of this procedure is the calculation of two incomplete elliptic integrals of the first kind for each transformed point [ 1, p. 1801.
3. Singularities introduced by the CTM Assume that part of the boundary &I of the original domain SI consists of two analytic arcs J?r and r2 which meet at the point w,, and form there a corner of interior angle ‘IT(Y, 0 < (YQ 2. Let wb E &?’ be the image under the CTM of the point we. We shall show that in certain cases, when 0 < (Y< 1, dw/dw’ becomes unbounded at wb and that, because of this, a singularity is sometimes introduced in the solution of the transformed problem. Let z0 E z-plane be the image of w,, under the transformation (2.1). Using the results of Lehmann [4] on the asymptotic expansion of a mapping function in the neighbourhood of a corner, we find that w
=
T-l (z) = w. + cl (Z - zo)a { 1 + O((Z - zo> lo&z - z,))) ,
as z -+ z. ,
(3.1)
and z = M-’ {S-l (w’)) = z. + c2 (w’ - why { 1 + U((w’ - wb) log(w’ - wb))) , as w’ + wb .
(3.2)
204
D. Levin, N. Paramichaeland
A. Sideridis, On the use of conformal transformations
In (3.2), p = 2 when wb is a corner of the rectangle fi’ and 0 = 1 when wb is any other point of a#. It follows from (3.1) and (3.2) that w =
T-’ {M-‘(S-‘(w’))]
= w. +
cl
cz(w’
-
wb)"O{1 + O((w’ - wb) log(w’ - wb))} , as w’ -+ wb ,
(3.3)
and, hence, -dw = c(w’ - ,b)@-1 {l +O((w’-wb)log(w’-wb))}, dw’
c#O,
asw’+wb.
(3.4)
Eq. (3.4) shows that dw/dw’ becomes unbounded at wb in the following two cases: (I) When /3= 2, i.e. when wb is a corner of 6’, if-0 < OL< l/2. (II) When 0 = 1, i.e. when wb is not a corner of a’, if 0 < (11< 1. Let t, and t, denote respectively the direction of the tangents to I’i and rz at the point wO. To be more specific, we assume that the arcs are labelled so that the direction t, goes into the direction t, by a positive rotation through the angle (YA.Consider the directional derivatives au/at,, i = 1, 2, of the solution u(x, v) of the original problem. These derivatives are connected to the corresponding directional derivatives av/a ti, i = 1, 2, of the solution of the transformed problem by the equation
AL at;
i=l,2.
(3.5)
When wb is a corner of a’, the directions t’,, t; in (3.5) are simply the positive or negative t: and n directions. When wb is not a corner, then t; = -t; is the positive or negative [ or n direction. It follows from (3.5) that in either of the cases (I) or (II) the solution u(t, 71)of the transformed problem has unbounded derivatives at w’ = wb if au/at, # 0 at w = wO. This seriously affects the accuracy of the numerical solution of the transformed problem, particularly in the neighbourhood of wb ( see section 5, examples 1 and 2). However, we shall show that the effects of such a singularity can, in many cases, be removed by means of a simple procedure. We consider separately the details of such a procedure for three types of boundary conditions of r, u rz. In each case we let the parametric equations of the arcs I’i and I-‘2be respectively w=x(s)+iy(.s)=q(s),
j= 1,2,
where 1~1is the arc length, and choose the parameter s in such a way that I’i corresponds to the interval sr Q s Q 0 and rz to the interval 0 < s < sz (i.e. wa = x, (0) = xz (0)). We also assume that the angle ark is such that application of the CTM leads to one of the cases (I) or (II) and that the boundary conditions of the problem produce unbounded derivatives of u(t, r)) at w’ = wb. 3.1. Dirichlet boundary condition on lTI u rz Assume that the boundary condition on ri
U rz
is
D. Levin, N. Papamichaeland A. Sideridis, On the use of conformal transformations
f”‘(S)
(
'o,Y)=fw),Y(S))=
f'2'(s)
SE
’
205
b’, 01 ,
SE uxs,l ,
9
where the function fis continuous at w = we (i.e. when s = 0), and
dif(i)
i=l,2
I= 1,2,-N,
>
are bounded when s = 0. Let pn (x, y), where
(
P(')(S)
P&(ShY(SN
n P(2)(s)
=
n
s E b’, 01 ,
’
SE W,s,l
y
,
be a harmonic function such that
j= 1,2 ,...
$p(o)=~f(‘)(o),
n(n
i=l,2.
(3.6)
Define the harmonic function
m, Y> = Nx, Y)
-
P&,
Y)
3
(3.7)
and consider the application of the CTM to the.harmonic boundary value problem for the function 2(x, y). The Dirichlet boundary condition
of this problem is transformed into a Dirichlet boundary condition
where I” C G?‘. To ensure that the solution u(& q) of the transformed problem does not contain at wb singularities which seriously affect the accuracy of the numerical results, it is sufficient to demand that (for i = 1, 2) (a) &$/at; are bounded at wb when wb is a corner of fi’, (b) &#@t; is continuous at wb when wb is not a corner of 32’. In practice it is easier to satisfy condition (b) by demanding that a4 -=+$=O at,
atw’=wb. 2
By (3.6) we have that
206
g=
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
O(lsl”)
)
ass+O,
i.e. g=
O(lw-
WJy
)
as w + w0 along r1
U rz
.
Also, by (3.4) and (3.3), dw = O(lw - wol’-‘K@) ) ~dw’
l-l
as w + we ,
and hence, by (3.5),
a4 )Z at,
dw dy = O(l w _ wall-iI(@)+n 1, dw’ds
I I
as w + wa along ri
U rz
.
Therefore, in order to satisfy the conditions (a) and (b), we require that I2 + 1 - 1/(2Q) > 0 )
(3.8)
and n+l-l/a>O,
(3.9)
respectively. Thus, the main requirement of the procedure for removing the effects of a singularity introduced by the CTM is the construction of a harmonic function p,, (x, y) satis_fying (3.6). The y1in (3.6) must satisfy the inequality (3.8) if 0 < (Y< l/2 and wb is a corner of a’, and the inequality (3.9) if 0 < QI< 1 and wb is not a corner of 6’. In practice we take p,, = Zy=, a,qi, where the qi are simple harmonic functions (e.g. harmonic polynomials), and we satisfy (3.6) by solving the resulting system of equations for the coefficients a,, a*, . . . a,. It is always possible to choose the qi so that this system has a solution. We give below the details of the procedure for the cases where the boundary conditions on ri u Fz are respectively Neumann and mixed. In both these cases the procedure again consists of constructing a harmonic function p, (x, y) which satisfies conditions similar to (3.6) and of applying the CTM to the boundary value problem for the auxiliary harmonic function
w,
Y>= utx,Y) -
P&,
Y)
.
The derivation of the results stated in sections 3.2 and 3.3 below is similar to that used above and, for this reason, is omitted. 3.2. Neumann boundary conditions on rI u rz Let the boundary condition on rr u rz be
D. Levin, N. Papamichael and A. Sideridis, On the use
&w, Y) = gee),
8%) , s E bl, 0)
Y(S))
=
where
(
of conformal transformations
201
9
g(2)(s) , SE K4s,l ,
j = 0, 1, 2, .. . N ,
i=l,2.
In this case the harmonic function pn (x, v), where
5
Pn
MS), Y(S)) =
4%)
SE bJ-u n d2)w > SE
1
7
KAs,l
n
, ,
must satisfy the conditions lii[$q$Ys))=!\Y
($g”‘(s)),
j=O,l,...n,(n
i=l,2,
(3.10)
with (3.11)
Iz + 2 - 1/(2cV)2 0 if 0 < Q!< l/2 and wb is a corner of fi’, and
(3.12)
n+2-l/a>0 if 0 < CY < 1 and wb is not a corner of fi’. 3.3. Mixed boundary conditions on rl
U IT2
Let the boundary conditions on r1 and I-‘2be respectively U(X,Y)
=.Km,Yw>
=.I%),
s E [Sl, 01
and $ u(x,
Y) = gcesh
Y(S)) = g(s) ,
where the derivatives dif 7)
Cl.91
j= 1,2, ...N.
are bounded when s = 0, and
s E (0, s2 1
,
208
D. Levin, N. Papamichael and A. Sideridis, On the use of conform4
j=o,
transformations
1, . ..N.
It is important to observe that, for this type of boundary conditions, the procedure for removing the effects of the singularity can be used only when wb is a corner of a’. For this reason the parameters of the CTM must be chosen so that wb is a corner of fi’. In this case the harmonic function p,(x, v), where Pn (x(s), Y(S)> = Pn @I ,
s E bp SE
01 ,
W,s,l ,
must satisfy the conditions UP,
=
dim, dsJ
j= 1, 2, . . . It ) (3.13)
lim -K qn (s) = lim Kg(s) s-0 &i S-+0 dsi
j=
0,
1, .. . n-l
(n
with (3.14)
Jz+ 1 - 1/(2a) > 0.
4. Solution of the transformed problem In this section we consider the use of the modified collocation method of Levin [3] for the solution of the transformed problem. The method is explained fully in [3]. Here we only give a brief description of the technique by considering its application to the problem (1.3). Let hj(& n), j = 1, 2, . .. 2(Z + m), be simple harmonic functions (e.g. harmonic polynomials), and denote by ET,i = 1, 2, . . . 1, and nf, i = 1, 2, . . . m, respectively, the abscissas of the Gaussian quadrature formulas of order I on [ 0, 11 and order m on [ 0, HI. Also, let the Dirichlet boundary condition of (1.3) on as2; be
9(E,n)=
l,n=O,
$1(t) 9
o
&(r)),
r= 1, OGrlGH,
i @3(E)3
O<[<
l,q=H,
where d,(l)=&(O)
and
$,(H)=&(l).
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
209
Consider the expansion approximation 2wm)
(4.2) to the solution u(& q) of (1.3), obtained by collocating the boundary conditions of the problem at the 2(Z + m) boundary points (4‘,:,O),
i = 1, 2, . . . 1,
(0, nr),
(1, T$,
i= 1, 2, ... m .
(4.3)
It is shown in [3] that this approximation can be “corrected” by using the dominant singularities of the Green’s function G(& n; &,, q,) of the problem (1.3). To obtain such a corrected approximation, we consider the nine singularities of G(& q; .$, , qo) which lie in the rectangular region {(,$,7): -1 < g Q 2, -H < r) < 2H) and let 8
Wt,
7;
to,71~)= -
7&gci loglw’ -
41 ,
w’ = t + i77,
(4.4)
where w; =
wb = to + ino , w; =i2H+ii(,
,
w; =-w;,
2 - w; )
w;=2+i2H-w’
wb=i2H-wb,
I ws--Wb,
w; = q)
wrg = 2 - Wb .
,
(4.5)
Then the correction term of the expansion approximation
+
sH$W,
rl;
to,vo) M2(rl) -
0’
at the point (to, qo) E AX’is
Oa T/(1,q)l dq + $ ,,s(t, H; to,r),) [$,W - 65
0
WI
dt
1
(4.6)
where the coefficients defining S(& 7; to, no) are
210
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
co =c2
=c5
=
cs
=
_c,
=
_c3
=
_c4
=
_C6
=
_c7
=
1 .
Thus
is the corrected expansion approximation to u(t,, r),). When &, = 0,O < r], < H, i.e. when (to, no) E &lk, the corrected approximation also given by (4.7), but the coefficients defining S(.!j, q; C;e,no) are now co =
c3 = c4 = c5 = C6 =
If (,$,, no) E aa;,
c, = 0 ,
Cl =
= -cg = -1
-c*
.
V,(&,, no) is
(4.8)
then it can be shown that (4.9)
where the coefficients co = c, =
c5
co =cl
=
cz =
co =
=
c2
Cl
= C6 =
c3
= c3
defining S(l, n, &-,, na) are c, = cs = 0 ,
=c,=c,=o, = CQ = c-5 =
Also, if (0, ne) E aa;,
0 ,
c2
= -c3
cq=cg
= -c4
=-c5=-1)
c6=c,=-cs=-1)
=
1 )
when n,, = 0 , when &, = 1 , when no = H .
then (4.10)
where the coefficients defining S(& 7); .$,, qO) are given by (4.8). The approximation V,(.$, , qo), (.$,, qo) E !X U a!Cl~, is said to be a “global near-best approximation of order 2(Z + m)” to the solution u(.$,, qO) of (1.3). If #i(g) E Cm[O,l I, &(q) E C” IO, H), &(t) E C”[O,l] and G(r)) E Cm[O, HI, then it is shown in [3] that
u(t,, 71~)- V,
(4.11)
where epbl denotes the error functional of the Gaussian quadrature formula of order k on [a, b] , and h, and h, are certain functions of continuity class C” [ 0,l ] and C” [0, H] , respectively. We note that the boundary conditions of the problems considered in this paper do not, in general, satisfy the continuity requirements for the “near-best” property (4.11) to hold. In fact, because of discontinuities in the higher derivatives of u at the corners of the rectangle, the choice of the Gaussian points as collocation points often does not produce the best possible accuracy. We found that the use of a non-Gaussian set, obtained by increasing the density of the collocation points near the corners of c’,_often imporves the accuracy of the numerical results. Such a set includes on the side r) = 0 of a’ the collocation points <&$,0), i = 1, 2, . . . 1, where
D. Levin, N. Papamichealand A. Sideridis, On the use of conformal transformations
t;: = (i ti:+
=
Llf12
211
i = 1, 2, . . . I, ,
l)h, )
l;yl+ ih, ,
i=l,2
) . . . I,,
= G,+,, + ih, 7
i=l,2
) . . . I, -
(4.12) 2)
with I= 2(Z, - 1) + 1, and h, < h,. On the three other sides of a’ the remaining collocation points are distributed in a similar manner. In practice, the integrals in the correction term (4.6) are evaluated numerically. In our applications the use of Simpson’s rule with step size h = 0.01 gives satisfactory overall results. We note, however, that the computation of these integrals presents some difficulties when (&,, no) is close to the boundary. To overcome these, it might be necessary, in some cases to decrease the step size of the quadrature rule used. The integers I and m in (4.3) or (4.12) are chosen so that Z/m - H. In our applications H = 1, and so the same number of collocation points is chosen on each side of the square, i.e. 1= m. If higher accuracy is required, then the approximation V, can be further improved as follows. The error
is approximated
by
2 (l+m) c j=l
P,h,(t,>
a,)
(4.13)
3
where the coefficients pi, j = 1, 2, . . . 2(Z + m), are determined by collocating the values
and Y$ v,(O, 7)) - rll(n) ,
6459
E aa;
,
given respectively by (4.9) and (4. lo), at the 2(Z + m) collocation points (4.3) or (4.12). We note that the matrix of coefficients of these collocation equations is the same as the matrix of the linear equations which determines the coefficients aj in (4.2). The new approximation to ~(8, 7)) is then given by 2(l+m)
5. Numerical results All the problems considered in this section are defined in the circular sector
212
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
s2 3 {(x, y): x2 f y2 < 1, arctan (y/x) Q cy7f)
(5. I )
for vario_us values of QI,0 < QI< 2. In all cases the conformal transformation to map LZonto the unit square
method (CTM) is used
For this we take (5.3) in (2. I), M(z) = +
(5.4)
or M(z) =
2 -
zT(w,>
’
w1 5%(0,
11,
(5.5)
in (2.2), and 1 S(t) = K( 1/vr~> sn-’ (t”‘, f id-3 in (2.3). The domain fi is mapped onto fi’ so that
1w-plane by (5.4)
to,O)-
w’-plane
i
(0, 0)
(l,O)-----+ (1,O)
and t
by (5.5)
w-plane
w’-plane
eiorn -
(0, 0)
(1, O)-
Cl,@
*
To illustrate the efficiency of the procedures for removing the effects of singularities introduced by the CTM, we list, where appropriate, the following results: (i) Values obtained by application of the CTM directly to the given boundary value problem for the function u(x, y). (ii) Values obtained by application of the CTM to the boundary value problem for the auxiliary function u”(x,Y> = w, Y) - P&G Y) 3
(5.6)
where p,(x, y) is a harmonic function satisfying the appropriate conditions given in section 3. (iii) Values computed from the analytic solution.
D. Levin, N, Papamichaeland A. Sideridis, On the use of conformal transformations
213
In all examples the transformed problem is solved by the modified collocation method described in section 4. In the expansion approximations (4.2) and (4.13) we use the set of harmonic polynomials h,=l, hj = Re(t + in)j-’ hi = Im(l
,
+ in)‘-*‘,
j=
2(1)(21+ 1))
j=
(21+ 2)(1)41,
and as collocation points we use, on each side of fi’, 1 points of the form (4.12) with I, = 3, I, = 4, h, = 0.05 and h, = 0.2. Example
I (Dirichlet boundary conditions)
Au@, Y) = 0,
(x,Y)EaT
U(X,y) = cos x sinh y ,
(x, Y) E x-i
(5.7)
where 6 is the circular section (5.1). Table 1. Example l(i) - points: r = 0.1(0.4)0.9,8 = O.O(n Notationa(h)b means {a, a + h, a f 2h, . .. b - h, b
Point 1 2 43 2 ii 9 10 11 12 13 14 15
ubyCTM
~byCTM
Analytic solution
o.ocQ1194
0.0320055 0.0257642 O.ol94167 0.0129891 0.0065081 0.1436561 0.1149282 0.0861858 0.0574470 0.0287202 0.1931110 0.1516858 0.1120789 0.0738988 0.0366997
0.0320055 0.0257641 0.0194165 0.0129889 0.0065078 0.1436561 0.1149282 0.0861858
0.0002281 0.0003199 0.0003884 0.0004293 0.1435895 0.1149460 o.o86m56 0.0574653 0.0287852 0.1931123 0.1516860 0.1120791 0.0738988 0.0366985
0.0574478
0.0287202 0.1931110 0.1516858 0.1120789 0.0738988 0.0366997
D. Levin, N, Papamichael and A. Sideridis, On the use of conformal transformations
214
We consider the two problems which correspond respectively to OL= 1IS and QI= 314 in (5.1). (i) (Y= l/8 The bilinear function (5.4) is-used in the CTM so that the corner wa = (0, 0) of G is mapped onto the corner wb = (0, 0) of 52’. The harmonic function p,(x, v) in (5.6) must satisfy the conditions (3.6) with y1satisfying the inequality (3.8), i.e. II > 3. Thus, an appropriate auxiliary function is u”(x,v) = u(x, v) - P&G v) 9 where p&,Y)=Y-~(x2Y
-iY”>.
Numerical results obtained are given in table 1. (ii) a = 314 (a) The bilinear function (5.4) is used in the CTM so that the corner wa = (0, 0) of fi is mapped Table 2. Example l(ii) - points: r = 0.1(0.4)0.9,e
Point
(a) u by
CTM
1
0.100167
2
0.096710 0.086602 0.070593 0.049833 0.025765 0.521095 0.497761 0.432786 0.338639 0.229298 0.114928 1.026516 0.956495 0.775079 0.547071 0.331007 0.151686
t 2 i 9 10 11 12 13 14 15 16 17 18
(b) u by CTM 0.10012g
0.096676 0.086569 0.070559 0.049799 0.025761 0.521095 0.497760 0.432785 0.338639 0.229302 0.11&42 1.026514 0.956494 0.775095 0.547055 0.330989 0.151685
= n/12(~/12)n/2
ii by CTM
Analytic solution
0.100166 0.096710 0.086601 0.070593 0.049832 0.025764
0.100167 0.096710 o.o866oe 0.070593 0.049833 0.025764
(b)
0.521095 0.497761 0.432786 0.338639 0.229298 0.114928 1.026517 o-956495 0.775080 0.547069 0.331006 0.151686
0.521095 0.497761 0.432786 0.338639 0.229298 0.114928 1.026517 0.956495 0.775079 0.547071 0.331007 0.151686
D. Levin, N. Papamichaeland ASideridis, On the use of conformal transformation
215
onto the corner wb E (0,O) of 6’. No singularity is introduced by the CTM, and the results obtained by direct application of the method to problem (5.7) are accurate (see table 2). (b) The bilinear function (5.5) is used in the CTM so that the corner we = (0,O) of fi is mapped onto the point (t, 0), 0 < t < 1, of a’. In this case the function pn(x, Y) in (5.6) must satisfy the conditions (3.6) with n satisfying the inequality (3.9) i.e. n > l/3. Thus, an appropriate auxiliary function is Wx,y)=Mx,Y)-p,(x,y),
p2(x,y)=y
*
Numerical results obtained are given in table 2. Example 2 (Mixed boundary conditions)
Au(x, Y) = 0 ,
(x, Y) E a ,
U(X,y) = cosx sinhy , (x,y) E aCi2, , -%4(X, O)=cosx, ay
(x, Y) E aq
(5.8)
,
Table 3. Example 2(i) - points: r = 0.1(0.4)0.9,
u by CTM
iibyCTM
Analytic solution
1
0.059836
0.058630
0.058620
2
0.047770
Point
2
0.034279 0.019908 0.005285 0.274758
:
0.206760 0.137725
43
9 10 11
12 13 14 15
0.068110 0.003952 0.413565 0.292~96 0.184655 0.089017 0.000566
0.045247 0.030782 0.015585 0.000019 0.274080 0.206603 0.137912 0.068940 -0.000009 0.413548 0.292081 0.184664 0.089027 -0.000001
0.045234 0.030767 0.015568 0.000000 0.274080 0.206603 0.137913 0.068942 0.000000 0.413543 0.292081 0.184664 0.089027 0.000000
e = O.O(n/20)n/S
216
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
where 6 = !2 U XCI is the circular sector (5.1) with X2 = XL1 U XZ2,, i3S& 5 {(x, 0): 0 < x < 1). We consider the two problems which correspond respectively to (Y= l/4 and LY= 1 in (5.1). (i) cr = l/4 The bilinear function (5.4) is_used in the CTM so that the corner w,, q (0, 0) of fi is mapped onto the corner wb = (0, 0) of 52’. The harmonic function pn(x, v) must satisfy the condition (3.13) with n satisfying the inequality (3.14), i.e. n > 1. Thus, an appropriate auxiliary function is
iqx,y)
= u(x, Y> - P*(X, Y) > P&c Y) = Y
’
Numerical results obtained are given in table 3. We remark that when the CTM is applied to the boundary value problem for the auxiliary function f&x, v) = u(x, v) - P&, Y)
3
P&,Y)
then the computed approximations
=Y -
G2Y -
iv">,
at the points of table 3 are correct to seven significant figures.
(ii) (Y = I
The bilinear function (5.4) is used in the CTM so that the corner wa = (0,O) of L? is mapped onto the corner wb = (0, 0) of 52’. No singularity is introduced by the CTM, and the results obtained by direct application of the method to problem (5.8) are accurate (see table 4). Table 4. Example
2(ii) - points:
r = 0.1(0.4)0.9,0
Point
u by CTM
Analytic solution
1 z
o. 1001667 0.0866025 0.0498334 0.0000002
o. 1001667 0.0866025 0.0498334 0 .ooooooo
2 :
0.5210953 o. 4327858 -0.0000002 0.2292976
9 10
1.0265167 0.7750792
2
0.5210953
0.4327858 0.2292976 0.0000000
11
0.3310068
1.0265167 0.7756792 0.3310068
12
0.0000001
0.0000000
= O.O(n/6)n/2
D. Levin. N. Papamichaeland A. Sideridis, On the use of conformal transformations
217
Example 3 (Dirichlet boundary conditions) Au@, Y) = 0 , 0,
Y) = {(x + a2
+ (Y - da*m
,
(x9 Y) E m
3
where fi is the circular sector (5.1) with Q!= 3/2. The bilinear function (5.4) is used and thus no singularity is introduced by the CTM. In this case, for the solution of the transformed problem we use the collocation method with only four Gaussian collocation points on each side of fi’. Numerical results obtained are given in table 5. They are compared with the CTM results of [ 11, where a high-accuracy, finite difference approximation to Laplace’s equation is used to_compute the solution of the transformed problem at the grid points of a square mesh covering SY. The solution of the original problem at specific points Table 5. Example 3 - r = 0.1(0.4)(0.9),
Point
u by CTM-collocation
i 2 43 iI ii
9
10 11 12 13 14 15 16 17 18
0.486947 0.489772 0.497641 0.508851 0.520901 0.530909 0.342050 0.360270 0.411857 0.487943 0.574788
0.654879 0.169860
0.203971 0.301062 0.446148 o.616649 0.785226
0 = O.O(n/8)3n/4
u by
CTM-finite 0.486947 0.489772 0.497642 0.508852 0.520903 0.530992 0.342050 0.360271 0.411858 0.487943 0.574787 :*%::;I oh3973 0.301065 0.446148
difference
[l]
218
D. Levin, N. Papamichael and A. Sideridis, On the use
of conformal transformations
of a is then determined by cubic spline interpolation between the known values of the mesh points of fi’. Although the parameters of the interpolating spline are given explicitly in terms of these known values, we note that the technique of [ 11 requires the solution of a large linear system of equations. The results of table 5 show that the collocation method produces results of comparable accuracy with less computational effort.
6. Conclusions It has been shown that the effects of singularities, which are sometimes introduced by the CTM in the solution of the transformed problem, can be removed with little additional computational effort. It has also been shown that the modified collocation method of section 4 is computationally more efficient for the solution of the transformed problem than the finite difference technique used in [ 1 I. As it is remarked in [ 11, the main shortcoming of the CTM concerns the implimentation of the first transformation (2.1). If the transformation can be performed accurately, then in many cases the CTM overcomes all the difficulties associated with the numerical solution of harmonic problems involving curved boundaries and boundary singularities and produces results of high accuracy.
References [l] N. Papamichael and G.T. Symm, Numerical techniques for two-dimensional Laplacian problems, Comp. Meths. Appl. Mech. Eng. 6 (1975) 175-194. [2] G. Strang and G.J. Fix, An analysis of the finite element method (Prentice-Hall, New York, 1973). [ 31 D. Levin, Near-best approximations to some problems in applied mathematics, Report TR/66 (Dept. of Mathematics, Brunei University, 1976). [4] R.S. Lehman, Development of the mapping function at an analytic corner, Pacific J. Math. 7 (1957) 1437-1449.
AND ENGINEERING 12 (1977) 201-218
~NTHEUSEOFCONF~RMALT~ANSFORMATIONS FORTHENUMERICALSOL'JTION OFHARMONICBOUNDARYVALUEPROBLEMS D. LEVIN,N.PAPAMrCHAEL andA.SIDERIDIS Department of Mathematics, Brunel University, Uxbridge, UK Manuscript received 17 January 1977; revised manuscript received 17 February 1977 Certain aspects concerned with the application of a conformal transformation method for the numerical solution of harmonic boundary value problems are considered. In particular, a procedure is decribed for removing the effects of singularities which are sometimes introduced by the transformations in the solution of the transformed problem. The transform problem is then solved by a collocation technique.
1. Introduction Let J-t be a simply connected domain with boundary 3sZ in the complex w-plane (w = x f iv) and consider the harmonic boundary value problem
AM, Y>= 0,
a,
kY)E
6,
Y)
E
(x, .Y) E In (1.1) A is the two-dimensional
as-t, ,
(1.1)
an,.
Laplacian operator, af2, and XZz are nonintersecting
subarcs of
an such that Xl = aSZ, u aQ2,, and a/au is the derivative in the direction of the outward normal to the boundary. The subarc aSZ, is always nonempty; when aa2, is empty, i.e. when afi = X2r, (1.1) is a Dirichlet problem. We consider the application of a conformal transformation method (CTM) which has been recently proposed in [ 1I for the numerical solution of problems of type (1.1). The CTM has been developed to deal primarily with problems involving boundary singularities for which standard numerical techniques, such as finite differences and finite elements, fail to produce accurate solutions. Although these standard techniques can be modified to improve the accuracy of the numerical solution (see e.g. Strang and Fix [ 2]), the CTM provides an alternative method which can, in many cases, produce solutions of high accuracy. Furthermore as it is shown in [ 1 I, even in the absence of singularities the CTM is often very well suited for the solution of harmonic boundary value problems involving curved bounda~es. The CTM consists essentially of three successive conformal transformations whose combined effect is to map the original region fi = Q, U &? in the w-plane onto the rectangle
202
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
in the w’-plane (w’ = g + in), thus transforming the original problem into a computationally simpler problem in fi’. Typically, by a suitable choice of the transformations the subarc aa2, is mapped onto the side ail; = ((0, 7): 0 <
Tj <
H}
of !%I,and the transformed problem is [ 1, p. 183 ]
AUK ‘17) =0
CL77)E 0’ ,
3
uct,7))= 4th
7) f
!y?p = $/(q)
)
(t, 77)E
aa;,
(0, rl) E
aa;.
(1.3)
In (1.3) ast’ = an; u aa;, #(t, 77)= f{x(.5 v), G(n) = -J$+(O,
77),.Y(O,77)1.
In the present paper we consider certain aspects concer_ning the numerical solution of the transformed problem. We show that when the original region s2 contains a comer A of internal angle arrr,0 < a! < 1, then application of the CTM introduces a singularity at the image of A which can seriously affect the numerical solution of the transformed problem. We show, however, that the effects of such a singularity can be easily removed, and we present numerical examples which illustrate this. We also use a new technique for the solution of the transformed problem. This technique involves the application of the modified collocation method of [ 31 to harmonic mixed boundary value problems and can be used to produce results of comparable accuracy to those obtained in [ 11 with much less computational effort. Its main advantage is that, unlike the finite difference technique used in [ 11, it computes the solution at any point of a’, and hence at specific points of Q without the need for bivariate interpolation.
2. The transformations Let aS2, = A,A, and a!$ = A,A,, where Ai -wi, i = 1,2. Then the three conformal transformations of the CTM are: (i) The transformation z = T(w)
)
(2.1)
where T(w) is a function of w, analytic in 0, so that (2.1) is a conformal mapping of fi onto the upper half z-plane. (ii) The bilinear transformation
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
t = M(z)
=k
203
(2.2)
where k is a constant. (iii) The Schwarz-Christoffel transformation w’
= S(t)
=
_-!_ K(m)
sn-* (t’l*, m) ,
(2.3)
where m < 1 is a positive real number, sn denotes the Jacobian elliptic sine, and K(m) is the complete elliptic integral of the first kind with modulus m. The combined effect of (2. l), (2.2) and (2.3) is to map 6 E w-plane onto the rectangle
in the w’-plane (w’ = t + iq) so that the points Ai 3 wi E aS2, i = 1, 2 are mapped respectively into the vertices (0,O) and (0, H) of fi’.The values of k in (2.2) and m in (2.3) may be chosen so that two further points, both on %I1 or both on aa,, are mapped into the other two vertices (1, 0) and (1, H) of tii: The form of the mapping function T(w) clearly depends on the geometry of the domain a. However, once the transformation (2.1) is constructed and the values of k and m are prescribed, the image P’ E (t, 7) E 32’ of the point P q (x, v) E fi is determined from (2.2) and (2.3) by a standard procedure. The main requirement of this procedure is the calculation of two incomplete elliptic integrals of the first kind for each transformed point [ 1, p. 1801.
3. Singularities introduced by the CTM Assume that part of the boundary &I of the original domain SI consists of two analytic arcs J?r and r2 which meet at the point w,, and form there a corner of interior angle ‘IT(Y, 0 < (YQ 2. Let wb E &?’ be the image under the CTM of the point we. We shall show that in certain cases, when 0 < (Y< 1, dw/dw’ becomes unbounded at wb and that, because of this, a singularity is sometimes introduced in the solution of the transformed problem. Let z0 E z-plane be the image of w,, under the transformation (2.1). Using the results of Lehmann [4] on the asymptotic expansion of a mapping function in the neighbourhood of a corner, we find that w
=
T-l (z) = w. + cl (Z - zo)a { 1 + O((Z - zo> lo&z - z,))) ,
as z -+ z. ,
(3.1)
and z = M-’ {S-l (w’)) = z. + c2 (w’ - why { 1 + U((w’ - wb) log(w’ - wb))) , as w’ + wb .
(3.2)
204
D. Levin, N. Paramichaeland
A. Sideridis, On the use of conformal transformations
In (3.2), p = 2 when wb is a corner of the rectangle fi’ and 0 = 1 when wb is any other point of a#. It follows from (3.1) and (3.2) that w =
T-’ {M-‘(S-‘(w’))]
= w. +
cl
cz(w’
-
wb)"O{1 + O((w’ - wb) log(w’ - wb))} , as w’ -+ wb ,
(3.3)
and, hence, -dw = c(w’ - ,b)@-1 {l +O((w’-wb)log(w’-wb))}, dw’
c#O,
asw’+wb.
(3.4)
Eq. (3.4) shows that dw/dw’ becomes unbounded at wb in the following two cases: (I) When /3= 2, i.e. when wb is a corner of 6’, if-0 < OL< l/2. (II) When 0 = 1, i.e. when wb is not a corner of a’, if 0 < (11< 1. Let t, and t, denote respectively the direction of the tangents to I’i and rz at the point wO. To be more specific, we assume that the arcs are labelled so that the direction t, goes into the direction t, by a positive rotation through the angle (YA.Consider the directional derivatives au/at,, i = 1, 2, of the solution u(x, v) of the original problem. These derivatives are connected to the corresponding directional derivatives av/a ti, i = 1, 2, of the solution of the transformed problem by the equation
AL at;
i=l,2.
(3.5)
When wb is a corner of a’, the directions t’,, t; in (3.5) are simply the positive or negative t: and n directions. When wb is not a corner, then t; = -t; is the positive or negative [ or n direction. It follows from (3.5) that in either of the cases (I) or (II) the solution u(t, 71)of the transformed problem has unbounded derivatives at w’ = wb if au/at, # 0 at w = wO. This seriously affects the accuracy of the numerical solution of the transformed problem, particularly in the neighbourhood of wb ( see section 5, examples 1 and 2). However, we shall show that the effects of such a singularity can, in many cases, be removed by means of a simple procedure. We consider separately the details of such a procedure for three types of boundary conditions of r, u rz. In each case we let the parametric equations of the arcs I’i and I-‘2be respectively w=x(s)+iy(.s)=q(s),
j= 1,2,
where 1~1is the arc length, and choose the parameter s in such a way that I’i corresponds to the interval sr Q s Q 0 and rz to the interval 0 < s < sz (i.e. wa = x, (0) = xz (0)). We also assume that the angle ark is such that application of the CTM leads to one of the cases (I) or (II) and that the boundary conditions of the problem produce unbounded derivatives of u(t, r)) at w’ = wb. 3.1. Dirichlet boundary condition on lTI u rz Assume that the boundary condition on ri
U rz
is
D. Levin, N. Papamichaeland A. Sideridis, On the use of conformal transformations
f”‘(S)
(
'o,Y)=fw),Y(S))=
f'2'(s)
SE
’
205
b’, 01 ,
SE uxs,l ,
9
where the function fis continuous at w = we (i.e. when s = 0), and
dif(i)
i=l,2
I= 1,2,-N,
>
are bounded when s = 0. Let pn (x, y), where
(
P(')(S)
P&(ShY(SN
n P(2)(s)
=
n
s E b’, 01 ,
’
SE W,s,l
y
,
be a harmonic function such that
j= 1,2 ,...
$p(o)=~f(‘)(o),
n(n
i=l,2.
(3.6)
Define the harmonic function
m, Y> = Nx, Y)
-
P&,
Y)
3
(3.7)
and consider the application of the CTM to the.harmonic boundary value problem for the function 2(x, y). The Dirichlet boundary condition
of this problem is transformed into a Dirichlet boundary condition
where I” C G?‘. To ensure that the solution u(& q) of the transformed problem does not contain at wb singularities which seriously affect the accuracy of the numerical results, it is sufficient to demand that (for i = 1, 2) (a) &$/at; are bounded at wb when wb is a corner of fi’, (b) &#@t; is continuous at wb when wb is not a corner of 32’. In practice it is easier to satisfy condition (b) by demanding that a4 -=+$=O at,
atw’=wb. 2
By (3.6) we have that
206
g=
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
O(lsl”)
)
ass+O,
i.e. g=
O(lw-
WJy
)
as w + w0 along r1
U rz
.
Also, by (3.4) and (3.3), dw = O(lw - wol’-‘K@) ) ~dw’
l-l
as w + we ,
and hence, by (3.5),
a4 )Z at,
dw dy = O(l w _ wall-iI(@)+n 1, dw’ds
I I
as w + wa along ri
U rz
.
Therefore, in order to satisfy the conditions (a) and (b), we require that I2 + 1 - 1/(2Q) > 0 )
(3.8)
and n+l-l/a>O,
(3.9)
respectively. Thus, the main requirement of the procedure for removing the effects of a singularity introduced by the CTM is the construction of a harmonic function p,, (x, y) satis_fying (3.6). The y1in (3.6) must satisfy the inequality (3.8) if 0 < (Y< l/2 and wb is a corner of a’, and the inequality (3.9) if 0 < QI< 1 and wb is not a corner of 6’. In practice we take p,, = Zy=, a,qi, where the qi are simple harmonic functions (e.g. harmonic polynomials), and we satisfy (3.6) by solving the resulting system of equations for the coefficients a,, a*, . . . a,. It is always possible to choose the qi so that this system has a solution. We give below the details of the procedure for the cases where the boundary conditions on ri u Fz are respectively Neumann and mixed. In both these cases the procedure again consists of constructing a harmonic function p, (x, y) which satisfies conditions similar to (3.6) and of applying the CTM to the boundary value problem for the auxiliary harmonic function
w,
Y>= utx,Y) -
P&,
Y)
.
The derivation of the results stated in sections 3.2 and 3.3 below is similar to that used above and, for this reason, is omitted. 3.2. Neumann boundary conditions on rI u rz Let the boundary condition on rr u rz be
D. Levin, N. Papamichael and A. Sideridis, On the use
&w, Y) = gee),
8%) , s E bl, 0)
Y(S))
=
where
(
of conformal transformations
201
9
g(2)(s) , SE K4s,l ,
j = 0, 1, 2, .. . N ,
i=l,2.
In this case the harmonic function pn (x, v), where
5
Pn
MS), Y(S)) =
4%)
SE bJ-u n d2)w > SE
1
7
KAs,l
n
, ,
must satisfy the conditions lii[$q$Ys))=!\Y
($g”‘(s)),
j=O,l,...n,(n
i=l,2,
(3.10)
with (3.11)
Iz + 2 - 1/(2cV)2 0 if 0 < Q!< l/2 and wb is a corner of fi’, and
(3.12)
n+2-l/a>0 if 0 < CY < 1 and wb is not a corner of fi’. 3.3. Mixed boundary conditions on rl
U IT2
Let the boundary conditions on r1 and I-‘2be respectively U(X,Y)
=.Km,Yw>
=.I%),
s E [Sl, 01
and $ u(x,
Y) = gcesh
Y(S)) = g(s) ,
where the derivatives dif 7)
Cl.91
j= 1,2, ...N.
are bounded when s = 0, and
s E (0, s2 1
,
208
D. Levin, N. Papamichael and A. Sideridis, On the use of conform4
j=o,
transformations
1, . ..N.
It is important to observe that, for this type of boundary conditions, the procedure for removing the effects of the singularity can be used only when wb is a corner of a’. For this reason the parameters of the CTM must be chosen so that wb is a corner of fi’. In this case the harmonic function p,(x, v), where Pn (x(s), Y(S)> = Pn @I ,
s E bp SE
01 ,
W,s,l ,
must satisfy the conditions UP,
=
dim, dsJ
j= 1, 2, . . . It ) (3.13)
lim -K qn (s) = lim Kg(s) s-0 &i S-+0 dsi
j=
0,
1, .. . n-l
(n
with (3.14)
Jz+ 1 - 1/(2a) > 0.
4. Solution of the transformed problem In this section we consider the use of the modified collocation method of Levin [3] for the solution of the transformed problem. The method is explained fully in [3]. Here we only give a brief description of the technique by considering its application to the problem (1.3). Let hj(& n), j = 1, 2, . .. 2(Z + m), be simple harmonic functions (e.g. harmonic polynomials), and denote by ET,i = 1, 2, . . . 1, and nf, i = 1, 2, . . . m, respectively, the abscissas of the Gaussian quadrature formulas of order I on [ 0, 11 and order m on [ 0, HI. Also, let the Dirichlet boundary condition of (1.3) on as2; be
9(E,n)=
l,n=O,
$1(t) 9
o
&(r)),
r= 1, OGrlGH,
i @3(E)3
O<[<
l,q=H,
where d,(l)=&(O)
and
$,(H)=&(l).
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
209
Consider the expansion approximation 2wm)
(4.2) to the solution u(& q) of (1.3), obtained by collocating the boundary conditions of the problem at the 2(Z + m) boundary points (4‘,:,O),
i = 1, 2, . . . 1,
(0, nr),
(1, T$,
i= 1, 2, ... m .
(4.3)
It is shown in [3] that this approximation can be “corrected” by using the dominant singularities of the Green’s function G(& n; &,, q,) of the problem (1.3). To obtain such a corrected approximation, we consider the nine singularities of G(& q; .$, , qo) which lie in the rectangular region {(,$,7): -1 < g Q 2, -H < r) < 2H) and let 8
Wt,
7;
to,71~)= -
7&gci loglw’ -
41 ,
w’ = t + i77,
(4.4)
where w; =
wb = to + ino , w; =i2H+ii(,
,
w; =-w;,
2 - w; )
w;=2+i2H-w’
wb=i2H-wb,
I ws--Wb,
w; = q)
wrg = 2 - Wb .
,
(4.5)
Then the correction term of the expansion approximation
+
sH$W,
rl;
to,vo) M2(rl) -
0’
at the point (to, qo) E AX’is
Oa T/(1,q)l dq + $ ,,s(t, H; to,r),) [$,W - 65
0
WI
dt
1
(4.6)
where the coefficients defining S(& 7; to, no) are
210
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
co =c2
=c5
=
cs
=
_c,
=
_c3
=
_c4
=
_C6
=
_c7
=
1 .
Thus
is the corrected expansion approximation to u(t,, r),). When &, = 0,O < r], < H, i.e. when (to, no) E &lk, the corrected approximation also given by (4.7), but the coefficients defining S(.!j, q; C;e,no) are now co =
c3 = c4 = c5 = C6 =
If (,$,, no) E aa;,
c, = 0 ,
Cl =
= -cg = -1
-c*
.
V,(&,, no) is
(4.8)
then it can be shown that (4.9)
where the coefficients co = c, =
c5
co =cl
=
cz =
co =
=
c2
Cl
= C6 =
c3
= c3
defining S(l, n, &-,, na) are c, = cs = 0 ,
=c,=c,=o, = CQ = c-5 =
Also, if (0, ne) E aa;,
0 ,
c2
= -c3
cq=cg
= -c4
=-c5=-1)
c6=c,=-cs=-1)
=
1 )
when n,, = 0 , when &, = 1 , when no = H .
then (4.10)
where the coefficients defining S(& 7); .$,, qO) are given by (4.8). The approximation V,(.$, , qo), (.$,, qo) E !X U a!Cl~, is said to be a “global near-best approximation of order 2(Z + m)” to the solution u(.$,, qO) of (1.3). If #i(g) E Cm[O,l I, &(q) E C” IO, H), &(t) E C”[O,l] and G(r)) E Cm[O, HI, then it is shown in [3] that
u(t,, 71~)- V,
(4.11)
where epbl denotes the error functional of the Gaussian quadrature formula of order k on [a, b] , and h, and h, are certain functions of continuity class C” [ 0,l ] and C” [0, H] , respectively. We note that the boundary conditions of the problems considered in this paper do not, in general, satisfy the continuity requirements for the “near-best” property (4.11) to hold. In fact, because of discontinuities in the higher derivatives of u at the corners of the rectangle, the choice of the Gaussian points as collocation points often does not produce the best possible accuracy. We found that the use of a non-Gaussian set, obtained by increasing the density of the collocation points near the corners of c’,_often imporves the accuracy of the numerical results. Such a set includes on the side r) = 0 of a’ the collocation points <&$,0), i = 1, 2, . . . 1, where
D. Levin, N. Papamichealand A. Sideridis, On the use of conformal transformations
t;: = (i ti:+
=
Llf12
211
i = 1, 2, . . . I, ,
l)h, )
l;yl+ ih, ,
i=l,2
) . . . I,,
= G,+,, + ih, 7
i=l,2
) . . . I, -
(4.12) 2)
with I= 2(Z, - 1) + 1, and h, < h,. On the three other sides of a’ the remaining collocation points are distributed in a similar manner. In practice, the integrals in the correction term (4.6) are evaluated numerically. In our applications the use of Simpson’s rule with step size h = 0.01 gives satisfactory overall results. We note, however, that the computation of these integrals presents some difficulties when (&,, no) is close to the boundary. To overcome these, it might be necessary, in some cases to decrease the step size of the quadrature rule used. The integers I and m in (4.3) or (4.12) are chosen so that Z/m - H. In our applications H = 1, and so the same number of collocation points is chosen on each side of the square, i.e. 1= m. If higher accuracy is required, then the approximation V, can be further improved as follows. The error
is approximated
by
2 (l+m) c j=l
P,h,(t,>
a,)
(4.13)
3
where the coefficients pi, j = 1, 2, . . . 2(Z + m), are determined by collocating the values
and Y$ v,(O, 7)) - rll(n) ,
6459
E aa;
,
given respectively by (4.9) and (4. lo), at the 2(Z + m) collocation points (4.3) or (4.12). We note that the matrix of coefficients of these collocation equations is the same as the matrix of the linear equations which determines the coefficients aj in (4.2). The new approximation to ~(8, 7)) is then given by 2(l+m)
5. Numerical results All the problems considered in this section are defined in the circular sector
212
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
s2 3 {(x, y): x2 f y2 < 1, arctan (y/x) Q cy7f)
(5. I )
for vario_us values of QI,0 < QI< 2. In all cases the conformal transformation to map LZonto the unit square
method (CTM) is used
For this we take (5.3) in (2. I), M(z) = +
(5.4)
or M(z) =
2 -
zT(w,>
’
w1 5%(0,
11,
(5.5)
in (2.2), and 1 S(t) = K( 1/vr~> sn-’ (t”‘, f id-3 in (2.3). The domain fi is mapped onto fi’ so that
1w-plane by (5.4)
to,O)-
w’-plane
i
(0, 0)
(l,O)-----+ (1,O)
and t
by (5.5)
w-plane
w’-plane
eiorn -
(0, 0)
(1, O)-
Cl,@
*
To illustrate the efficiency of the procedures for removing the effects of singularities introduced by the CTM, we list, where appropriate, the following results: (i) Values obtained by application of the CTM directly to the given boundary value problem for the function u(x, y). (ii) Values obtained by application of the CTM to the boundary value problem for the auxiliary function u”(x,Y> = w, Y) - P&G Y) 3
(5.6)
where p,(x, y) is a harmonic function satisfying the appropriate conditions given in section 3. (iii) Values computed from the analytic solution.
D. Levin, N, Papamichaeland A. Sideridis, On the use of conformal transformations
213
In all examples the transformed problem is solved by the modified collocation method described in section 4. In the expansion approximations (4.2) and (4.13) we use the set of harmonic polynomials h,=l, hj = Re(t + in)j-’ hi = Im(l
,
+ in)‘-*‘,
j=
2(1)(21+ 1))
j=
(21+ 2)(1)41,
and as collocation points we use, on each side of fi’, 1 points of the form (4.12) with I, = 3, I, = 4, h, = 0.05 and h, = 0.2. Example
I (Dirichlet boundary conditions)
Au@, Y) = 0,
(x,Y)EaT
U(X,y) = cos x sinh y ,
(x, Y) E x-i
(5.7)
where 6 is the circular section (5.1). Table 1. Example l(i) - points: r = 0.1(0.4)0.9,8 = O.O(n Notationa(h)b means {a, a + h, a f 2h, . .. b - h, b
Point 1 2 43 2 ii 9 10 11 12 13 14 15
ubyCTM
~byCTM
Analytic solution
o.ocQ1194
0.0320055 0.0257642 O.ol94167 0.0129891 0.0065081 0.1436561 0.1149282 0.0861858 0.0574470 0.0287202 0.1931110 0.1516858 0.1120789 0.0738988 0.0366997
0.0320055 0.0257641 0.0194165 0.0129889 0.0065078 0.1436561 0.1149282 0.0861858
0.0002281 0.0003199 0.0003884 0.0004293 0.1435895 0.1149460 o.o86m56 0.0574653 0.0287852 0.1931123 0.1516860 0.1120791 0.0738988 0.0366985
0.0574478
0.0287202 0.1931110 0.1516858 0.1120789 0.0738988 0.0366997
D. Levin, N, Papamichael and A. Sideridis, On the use of conformal transformations
214
We consider the two problems which correspond respectively to OL= 1IS and QI= 314 in (5.1). (i) (Y= l/8 The bilinear function (5.4) is-used in the CTM so that the corner wa = (0, 0) of G is mapped onto the corner wb = (0, 0) of 52’. The harmonic function p,(x, v) in (5.6) must satisfy the conditions (3.6) with y1satisfying the inequality (3.8), i.e. II > 3. Thus, an appropriate auxiliary function is u”(x,v) = u(x, v) - P&G v) 9 where p&,Y)=Y-~(x2Y
-iY”>.
Numerical results obtained are given in table 1. (ii) a = 314 (a) The bilinear function (5.4) is used in the CTM so that the corner wa = (0, 0) of fi is mapped Table 2. Example l(ii) - points: r = 0.1(0.4)0.9,e
Point
(a) u by
CTM
1
0.100167
2
0.096710 0.086602 0.070593 0.049833 0.025765 0.521095 0.497761 0.432786 0.338639 0.229298 0.114928 1.026516 0.956495 0.775079 0.547071 0.331007 0.151686
t 2 i 9 10 11 12 13 14 15 16 17 18
(b) u by CTM 0.10012g
0.096676 0.086569 0.070559 0.049799 0.025761 0.521095 0.497760 0.432785 0.338639 0.229302 0.11&42 1.026514 0.956494 0.775095 0.547055 0.330989 0.151685
= n/12(~/12)n/2
ii by CTM
Analytic solution
0.100166 0.096710 0.086601 0.070593 0.049832 0.025764
0.100167 0.096710 o.o866oe 0.070593 0.049833 0.025764
(b)
0.521095 0.497761 0.432786 0.338639 0.229298 0.114928 1.026517 o-956495 0.775080 0.547069 0.331006 0.151686
0.521095 0.497761 0.432786 0.338639 0.229298 0.114928 1.026517 0.956495 0.775079 0.547071 0.331007 0.151686
D. Levin, N. Papamichaeland ASideridis, On the use of conformal transformation
215
onto the corner wb E (0,O) of 6’. No singularity is introduced by the CTM, and the results obtained by direct application of the method to problem (5.7) are accurate (see table 2). (b) The bilinear function (5.5) is used in the CTM so that the corner we = (0,O) of fi is mapped onto the point (t, 0), 0 < t < 1, of a’. In this case the function pn(x, Y) in (5.6) must satisfy the conditions (3.6) with n satisfying the inequality (3.9) i.e. n > l/3. Thus, an appropriate auxiliary function is Wx,y)=Mx,Y)-p,(x,y),
p2(x,y)=y
*
Numerical results obtained are given in table 2. Example 2 (Mixed boundary conditions)
Au(x, Y) = 0 ,
(x, Y) E a ,
U(X,y) = cosx sinhy , (x,y) E aCi2, , -%4(X, O)=cosx, ay
(x, Y) E aq
(5.8)
,
Table 3. Example 2(i) - points: r = 0.1(0.4)0.9,
u by CTM
iibyCTM
Analytic solution
1
0.059836
0.058630
0.058620
2
0.047770
Point
2
0.034279 0.019908 0.005285 0.274758
:
0.206760 0.137725
43
9 10 11
12 13 14 15
0.068110 0.003952 0.413565 0.292~96 0.184655 0.089017 0.000566
0.045247 0.030782 0.015585 0.000019 0.274080 0.206603 0.137912 0.068940 -0.000009 0.413548 0.292081 0.184664 0.089027 -0.000001
0.045234 0.030767 0.015568 0.000000 0.274080 0.206603 0.137913 0.068942 0.000000 0.413543 0.292081 0.184664 0.089027 0.000000
e = O.O(n/20)n/S
216
D. Levin, N. Papamichael and A. Sideridis, On the use of conformal transformations
where 6 = !2 U XCI is the circular sector (5.1) with X2 = XL1 U XZ2,, i3S& 5 {(x, 0): 0 < x < 1). We consider the two problems which correspond respectively to (Y= l/4 and LY= 1 in (5.1). (i) cr = l/4 The bilinear function (5.4) is_used in the CTM so that the corner w,, q (0, 0) of fi is mapped onto the corner wb = (0, 0) of 52’. The harmonic function pn(x, v) must satisfy the condition (3.13) with n satisfying the inequality (3.14), i.e. n > 1. Thus, an appropriate auxiliary function is
iqx,y)
= u(x, Y> - P*(X, Y) > P&c Y) = Y
’
Numerical results obtained are given in table 3. We remark that when the CTM is applied to the boundary value problem for the auxiliary function f&x, v) = u(x, v) - P&, Y)
3
P&,Y)
then the computed approximations
=Y -
G2Y -
iv">,
at the points of table 3 are correct to seven significant figures.
(ii) (Y = I
The bilinear function (5.4) is used in the CTM so that the corner wa = (0,O) of L? is mapped onto the corner wb = (0, 0) of 52’. No singularity is introduced by the CTM, and the results obtained by direct application of the method to problem (5.8) are accurate (see table 4). Table 4. Example
2(ii) - points:
r = 0.1(0.4)0.9,0
Point
u by CTM
Analytic solution
1 z
o. 1001667 0.0866025 0.0498334 0.0000002
o. 1001667 0.0866025 0.0498334 0 .ooooooo
2 :
0.5210953 o. 4327858 -0.0000002 0.2292976
9 10
1.0265167 0.7750792
2
0.5210953
0.4327858 0.2292976 0.0000000
11
0.3310068
1.0265167 0.7756792 0.3310068
12
0.0000001
0.0000000
= O.O(n/6)n/2
D. Levin. N. Papamichaeland A. Sideridis, On the use of conformal transformations
217
Example 3 (Dirichlet boundary conditions) Au@, Y) = 0 , 0,
Y) = {(x + a2
+ (Y - da*m
,
(x9 Y) E m
3
where fi is the circular sector (5.1) with Q!= 3/2. The bilinear function (5.4) is used and thus no singularity is introduced by the CTM. In this case, for the solution of the transformed problem we use the collocation method with only four Gaussian collocation points on each side of fi’. Numerical results obtained are given in table 5. They are compared with the CTM results of [ 11, where a high-accuracy, finite difference approximation to Laplace’s equation is used to_compute the solution of the transformed problem at the grid points of a square mesh covering SY. The solution of the original problem at specific points Table 5. Example 3 - r = 0.1(0.4)(0.9),
Point
u by CTM-collocation
i 2 43 iI ii
9
10 11 12 13 14 15 16 17 18
0.486947 0.489772 0.497641 0.508851 0.520901 0.530909 0.342050 0.360270 0.411857 0.487943 0.574788
0.654879 0.169860
0.203971 0.301062 0.446148 o.616649 0.785226
0 = O.O(n/8)3n/4
u by
CTM-finite 0.486947 0.489772 0.497642 0.508852 0.520903 0.530992 0.342050 0.360271 0.411858 0.487943 0.574787 :*%::;I oh3973 0.301065 0.446148
difference
[l]
218
D. Levin, N. Papamichael and A. Sideridis, On the use
of conformal transformations
of a is then determined by cubic spline interpolation between the known values of the mesh points of fi’. Although the parameters of the interpolating spline are given explicitly in terms of these known values, we note that the technique of [ 11 requires the solution of a large linear system of equations. The results of table 5 show that the collocation method produces results of comparable accuracy with less computational effort.
6. Conclusions It has been shown that the effects of singularities, which are sometimes introduced by the CTM in the solution of the transformed problem, can be removed with little additional computational effort. It has also been shown that the modified collocation method of section 4 is computationally more efficient for the solution of the transformed problem than the finite difference technique used in [ 1 I. As it is remarked in [ 11, the main shortcoming of the CTM concerns the implimentation of the first transformation (2.1). If the transformation can be performed accurately, then in many cases the CTM overcomes all the difficulties associated with the numerical solution of harmonic problems involving curved boundaries and boundary singularities and produces results of high accuracy.
References [l] N. Papamichael and G.T. Symm, Numerical techniques for two-dimensional Laplacian problems, Comp. Meths. Appl. Mech. Eng. 6 (1975) 175-194. [2] G. Strang and G.J. Fix, An analysis of the finite element method (Prentice-Hall, New York, 1973). [ 31 D. Levin, Near-best approximations to some problems in applied mathematics, Report TR/66 (Dept. of Mathematics, Brunei University, 1976). [4] R.S. Lehman, Development of the mapping function at an analytic corner, Pacific J. Math. 7 (1957) 1437-1449.