Boundary Singularities in Linear Elliptic Differential Equations

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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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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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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

Boundary Singularities in Linear Elliptic Differential Equations

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