boundary singularities in linear elliptic differential equations
TRANSCRIPT
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/ .
Inst. Maths Applies
(1969) 5, 340-350
Boundary Singularities in Linear Elliptic Differential Equations
L. Fox AND R. SANKA R
Oxford University Com puting Laboratory, Parks Road, Oxford
[Received 25 March 1968]
Finite-difference methods are relatively inefficient in the neighbourhood of boundary
singularities in elliptic problem s. A com bination of special treatment n ear the singularity,
based on local satisfaction of the differential equa tion and bound ary cond itions, is here
matched with finite-difference formulae in the rest of the field. The method is applied
to a general self-adjoint equation with either Dirichlet, Neumann or mixed conditions
on parts of the boundary consisting of two straight lines meeting at the singular point.
A practical problem , formerly solved by more extensive labou r, illustrates the pow er of
the method.
1.
Introduction
I T
IS WELL-KNOWN
that in many elliptic problems the presence of some forms of
boundary discontinuities or singularities may not be serious, in the sense that the
true solution is perfectly "well-behaved" at any interior point of the bounded region.
Even with the use of numerical methods, of the finite-difference type, the inaccuracies
of the computed solution due to the presence of the singularity are then usually
significant only in a certain "region of infection", and at sufficient distances from
the offending point our computed results are reasonably satisfactory.
On the other hand there is no doubt that in such circumstances we have three
significant problems. First, for success we need a small finite-difference interval, at
least in the region of infection. Second, we can never by this method get very accurate
results at points in the neighbourhood of the singularity. Third, our error analysis,
which is based on estimates of some derivatives of the true solution, breaks down or
becomes considerably more difficult if these derivatives become infinite at a point on
the boundary .
It appears that these problems are minimized if we use finite-difference methods
only in regions in w hich th e solution is sufficiently well behav ed in a num erical sense,
and combine this with special treatment in the neighbourhood of the diff icult point.
This special treatment effectively determines the nature of a function which satisfies
the differential equation and boundary conditions in the neighbourhood of the
singular point, and f inds any arbitrary constants involved by "matching" with the
finite-difference solution.
Such methods have previously been suggested for Laplace's equation in two
dimensions, for example by Motz (1946), Woods (1953), Wasow (1957) and Volkov
(1963). Here we extend the theory and its application to the treatment of a more
general self-adjoint elliptic problem in two dimensions, for which some part of the
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BOUNDA RY SINGULARITIES IN ELLIPTIC PROBLEMS 3 4 1
boundary consists of two straight lines, intersecting at a singular point. We consider
the effect of various forms of bou nda ry c onditions on these two lines.
To illustrate our suggestions we find the complete solution for the flow of in-
com pressible fluid past a screw prop ellor, a p roblem formerly solved by more extensive
labour by Goldstein (1929) and Wijngaarden (1956).
2. Solution in the Neighbourhood of a Singularity
W e treat the self-adjoint equ ation
V
2
u = -g(r,9)u, (1 )
where
g(r,ff)= g
n
{ey, (2)
n-0
-
d
2
u Idu
1
d
2
u
W u
=
6?
+
-r8-r
+
?W
( 3 )
and the origin of polar co-ordinates is taken at the intersection of the lines 0 = 0,
6\= co, which form part of the boundary. We seek the solution of (1) subject to three
different sets of boundary conditions on these two lines, given respectively by
(i) u = F(r) on 9 = 0, u = H(r) on 9 = co, (4)
u l P
e
=
F
W
oa e =
>
; P o =
H
W
oa e = }
>
1 3 M
(iii) u = F(r) on 0 = 0, - = H(r) on 9 =
co .
(6)
r
o
It is assumed that the functions of
r
in (4)-(6) have the convergent expansions
F(r)= t
fnr * ,
H(r)= h
n
^\ j8,y > 0, (7)
an d we seek a solution in one of the forms
= 1
KtfV*
1
,
u
= t
r
+
J{(logr)A
XiJ
(9)+B
ai
j(9)},
(8)
which appear to be sufficient to cover all possibilities.
Substitution of the first of (8) into (1) gives the equation
U
+
2]r
+m
= - E 0 - A J K * . (9)
m = 0 I B - 0 V/ = O /
where the primes denote differentiation with respect to
9.
Then if the first of (8) is a
solution of (1) we must have
, = 0
9
m
-iK}>
m
= 0 ,1 ,2 , . . .
y-o
(10)
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3 4 2 L. FOX AND R. SANKAR
The same treatment with the second of (8) shows that A
at j
(9 ) should still satisfy
equations (10), while the equations for theB
at
fQ )coefficients are given by
(11)
We proceed to develop solutions of the relevant equations (10) and (11), for use
in the neighbourhood of the point of intersection of the boundary lines 0 = 0 and
9 =
(o ,
for each set of conditions (i) (Dirichlet conditions), (ii) (Neumann conditions)
and (iii) (mixed conditions) given in equations (4), (5) and (6) respectively.
Case
(0
The simplest solution of (10), given by the first of (8), which satisfies the simple
conditions
= / />+ / on
9 =
0, u = 0 on
0 = '
I+ p
satisfies (12), and the other terms in
the series vanish on9= 0 and 9=
a>.
From the first of(10)we then find
A (0)
4,oW - / .
s i n ( f I + / O a )
'
and we can easily solve for the other
Aaj(ff),j >
0, to satisfy the rest of (10) and the
second of
(13).
We denote the resulting solution by
where the symbolDrefers to th e Dirichlet case.
The possible vanishing of
the
denominator in (15) illustrates the need for a solution
of the type of the second of (8). Suppressing the details, we find that the solution
replacing (15) is then given by
""
(r
'
0)
a>cos(n+p)co
x D[r"
+
"{Gog r) sin
(
B
+ j8)(0-) cos (n +0 )(0 -a) )} ]. (16)
The corresponding solutions fortheboundary conditions
u= 0 on 0 = 0, u = /V
n + T
on 0 = co (17)
are given by
(18)
r) sin ( +y )0 +0 cos (n+y )0}]. (19)
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B O U N D A R Y S I N G U L A R I T I E S I N E L L I P T I C P R O B L E M S
343
We must also include solutions which vanish on both 0 = 0 and 6= a> . For these
we clearly take
a = , A
a0
= sin 0,
W CO
wheremisan integer, and the remaining
Aa,j,
for./> 0, follow as before.
The complete solution of case (i) is then given by
u(r,9)
= '
W
sin( - A ] ,
20)
(21)
where the c
ra
are arbitrary constants, and where the prime denotes that ui
l)
oru{
2)
is
replaced byu or i7
(b
= '
(h
30
= |
(see
-x
2
l + x
0,
0,
o ,
Fig.
x >
y =
X =
X -*
1) given
a, y
=
7T
0
0 0
by
0
y
= 0 (Une
(Une
(Une
(Une
(Une
O A )
AB)
DC)
OD)
B Q
(29)
Th e problem arises in the vortex theory of screw propellors, the num ber/? represent-
ing the number of blades and
a
their (non-dimensional) length. Goldstein (1929)
solved the casep = 2 by me thod s involving the separation of variables. W ijngaarden
FIG.
1.
(1956) for p = 3 first transformed the equation to a self-adjoint form, then applied
a conformal transformation to move the singular point A to infinity, and finally
computed the solution by finite-difference methods.
Finite-difference computation applied to (28) and (29), ignoring any singularity,
imm ediately reveals the presence of th e latter a t A , since the solutions s how , in the
behaviour of their differences, considerable "disturbance" in the neighbourhood of A
in Fig. 1.
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BOUN DARY SINGULARITIES IN ELLIPTIC PROBLEMS 3 4 5
To reduce the equation (28) to our required form (1) we apply the transformations
*pj-lj
, 30)
and obtain the self-adjoint form
d
2
u
,
d
2
u x
2
( 4 - x
2
)
d
2
dy
2
4 (1+x
2
)
3
The region OABCD becomes the upper half of the
(,y)
plane, with A as origin,
and in terms of polar co-ordinates
=
r
cos
9 , y
= r sin
9 ,
we have the boundary
conditions
u = 0 for 9= 0, ~ =
^ - 5 - ,
= H(r) = h
n
r for 9= n. (32)
This is a particular example of our case (iii), withco =
n
and / = 0 for all
n
and
y = 0 in (7). The solutions involving the log
r
terms are no t needed, and we find
that the solution in the neighbourhood of
A
can be taken in the form
u(r,9) =
Suppressing the details of the construction of the
M
solutions, we find, up to and
including terms in r
3
, the series
u(r,9) = -ho[r
sin
9+^g
o
(sin
3 0 - 3 sin ^ r ^ + ^ i r z sin
29-far*
sin
30
+
+
co(r*igor*) sin
i9+c\r*
sin | 0 +
c
2
r*
sin | 0 , (34)
with
h
o
=
The only unknown constants in (34) are CQ, C\ and
c
2
,
and we propose to compute
them by returning to the original plane, using in the neighbourhood of the singular
point the solution in the form
0 = (l+*2)-*H (r,0), (36)
(s
for "special") and m atching this to the
finite-difference
solution
W(g
for "general")
obtained at other points in the original rectangle OABCD.
4.
The Numerical Process
The nature of the numerical method is explained with reference to Fig. 2, which
shows a part of the "field" in the neighbourhood of the point A.
The solution
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346
L. FOX AND R. S NK R
to relate values at "m esh-points" with interval
h
in bo th d irections. The
(r,s)
notation
is that of Fig. 3.
At boundary points where the normal derivative is specified we use the simplest
central-difference formula involving one external point which is eliminated with the
use of
(37)
applied at the boundary poin t.
To connect the&with the we use the expressions (34) and (36) for the latter
at the three points Qi, Q2 and Q, of Fig. 2 (the choice being somewhat arbitrary),
which serve to express the constants CQ, a and C2 in (34) in terms of
(f>^
at Qi, Q2
P*
F I G . 2.
r . s+D- -
r.s-1) --
r.s) /+], s)
F I G .
3.
and
Q3.
These constan ts are then used to express the values at Pi to P6 (the value is
zero at Po) in terms of the
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BOUN DARY SINGULARITIES IN ELLIPTIC PROBLEMS 3 4 7
O(h
2
).
T o make these more compatible we apply the difference-correction method in
the
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3 4 8 L. FOX AND R. SANKAR
values of and 0
W
, computed by both methods, have a maximum difference of
00003.
The constants in (40) have the values
Co = l-2454{l-2485),
a =
1-2497(1-2243),
c
2
= 0-0236(0-1088), (44)
the bracketed
figures
representing the results of applying the difference corrections.
The important quantity in this problem is the Goldstein factor denned by
x,0).
(45)
7I X
This is easily calculable at mesh points on
y =
0, but values at other points must be
obtained from interpolated values of
the results obtained
from substitution in (36), which of course are the same in the special region. In Table 1
the singular point
is
just below the point
x \h
= 12.
TABLE 1
xlh
4
5
6
7
8
9
10
11
12
1288
679
34 5
166
74
30
10
0
0
>)
- 6 0 9
- 3 3 4
- 1 7 9
- 9 2
- 4 4
- 2 0
- 1 0
0
(52
+275
+155
+87
+48
+24
+
10
+ 10
- 1 2 0
- 6 8
- 3 9
- 2 4
- 1 4
0
,54
+52
+29
+15
+ 10
+
14
- 2 3
- 1 4
- 5
+ 4
+ 9
+ 9
+ 9
It was rather disconcerting to find that our computed values of
(x,y),
which we
confidently expect to be correct within a unit or so in the third decimal place, differ
from those of Goldstein by larger amounts, at some points even in the second figure.
For reassurance we have performed the computations by Goldstein's method, and
find that these corrected results do agree with those of our method within at most
one unit in the third decimal place. It is interesting to observe that Goldstein's m ethod
involves the computation of an infinite series for each value of (x,y), and the co-
efficients of the terms of these series themselves involve the solution of an infinite
set of linear equations. These equations are ill-conditioned, and the solutions of
successive leading subsets converge very slowly indeed to the correct solution.
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BOUNDARY
SINGULARITIES IN ELLIPTIC PROBLEMS 3 4 9
Goldstein had observed this fact, but his suggestion for avoiding the difficulty is
apparently only partially successful and a considerable amount of computation is
necessary. In fact the computation of the coefficients of these equations, their solution,
and the computation of
4>{x,y)
at the mesh points, takes much more time (on the
KD F9 machine) than the method proposed in this paper.
In Table 2 we give some values of l(H(2/n)^(x,0) along the liney =0, in the range
OA in Fig. 1 with a = 2. The first six columns, computed by us using Goldstein's
method, show the behaviour of the results for different truncations of the infinite
series and the corresponding infinite set of algebraic equations. Convergence,
we
note,
is increasingly slow as we approach the singular point at x = 2. Column 7 gives
Goldstein's published results, and we believe that he has over-compensated for the
slow convergence. Column 8 contains the values computed by the methods of this
paper, and there is no reason to suppose that these results are less accurate near the
singular point than a t other parts of the region.
TABLE
2
From
Go ldstein's series withnterms Go ldstein's Our
x
n
= 5 10 15 30 45 60 80 values values
0-2 931 925 923 923 923 923
0-4 1782 1770 1766 1762 1760 1760
0-6 2493 2474 2467 2461 2459 2458
0-8 3031 3005 2996 2987 2984 2983
10
3384 3348 3336 3325 3321 3319
1-2 3543 3496 3481 3465 3460 3458
1-4 3497 3435 3414 3394 3387 3384
1-6 3215 3128 3100 3072 3062 3058 3054
1-8 2619 2474 2428 2384 2369 2362 2356
2 0 1485 1049 856 603 491 424 366
We also considered the three-bladed propellor,p = 3, and report thankfully that
our results agree very closely with those of Wijngaarden, and again are obtained with
considerably less effort.
7.
Conclusions
Any linear elliptic equation, given by
92
75
243
295
329
34
33
295
22
922
76
2456
2976
33 2
3449
3373
3 42
2339
p
2+
E^
+
F
e
JL
+
G u
= K, (46)
dy
2
dx dy
where the coefficients are functions of x and y and B
2
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3 5 0 L. FOX AND R. SANKAR
to the right-hand sides of equations like (10), causing no particular difficulty. Non-
zerop andq,however, mean that equations (10) have non-constant coefficients on the
left, so that the solutions of these equations will not usually be expressible in closed
mathematical form.
The form ofg(r,0) in (2) can be a little more general, since a term with n =
1
merely adds known terms on the right of (10). Terms withn = 2, however, have
effects similar to those of non-zero/> andq.
The choice of boundary conditions like (13) has no particular significance. They
are the simplest possible (for exampleAa,i in (10) is then always zero), and are valid
provided that the functions used in the series forw form a complete set.
This requirement, of course, is necessary for the success of our method in any case,
and though we believe that the chosen function set is complete we have noproof.
We are grateful for a generous grant from I.C.T. which enabled one of
us
(R. S.) to
take part in this investigation.
REFERENCES
COURANT, R. & HILBERT, D. 1962
Methods of Mathematical Physics,
H. New York:
Interscience.
GOLDSTEIN,
S. 1929
Proc. R. Soc. (A),
123,44