On the convergence of difference schemes approximating the second and third boundary value problems for elliptic equations

ON THE CONVERGENCE OF DIFFERENCE SCHEMES APPROXIMATINGTHE SECOND AND THIRD BOUNDARY VALUE PROBLEMS FOR ELLH?TICEQUATIONS* V. B.ANDREEV Moscow (Received 24 May 1967)

THE present paper considers difference schemes approximating the second and third boundary value problems for a self-adjoint elliptic equation without mixed derivatives in a rectangular region, and represents a direct continuation of [ll. We take the problem

Lu s

au

h (4 -

az, =

i L,u = --f(x), a=1

x-a(4

245 - g-cl

& =x+u(4 I.4-

kcCd

and G = {x = parallelepiped.

XEG,

(4,

g+,(x),

xa = 0, Xa =

(Zi, . . . . 3+):O
1

(1)

I

a=l,...,p,

,...,

p}

(2)

&a7

a=1

isarectangular

A difference approximation of conditions (2) was obtained in Ill for problem (l)_(2), giving an error O(lh(‘) on smooth solutions. It was shown that the difference scheme approximating the problem on an uniform mesh o (the operator * Zh. @Qhisl. Mat. mat.

Fiz. 8, 6, 1218-1231, 44

1966.

Second and third boundary value problems

45

L is approximated as usual with respect to (2~ + l&point pattern), is convergent in the mesh norm of L, to a reasonably smooth solution of the problem at a rate O(lh(‘), provided that the coefficients of the problem satisfy kc(x)

>Ci>O,

4&)

3

0,

X*&)

3

c2

>

0,

a=l,.

**, p. (3)

It was essential for the arguments of [II, not only that the conditions (3) are sufficient for the operator of the difference problem to be negative-definite uniformly with respect to jhl, but also that the coefficients x,,(x) of the boundary conditions are strictly positive everywhere, though this condition is unnecessarily strict from the point of view of the operator being negative-definite. We prove here that an a priori bound (stronger thanjn [ll) holds uniformly with respect to jh\ on the assumption that the operator A of the difference problem satisfies uniformly with respect to (hl - [Ku, u] > MI VII?, Some sufficient conditions more general (*) is satisfied. In particular, we show that merely need to be non-negative; only one of and that, only in part of the region in which

M > 0.

(4

than (3) are obtained, under which the coefficients K&~(X)as a whole them needs to be strictly positive, it is defined.

We also consider the case when qa(s) E 0,xs (2) E 0, a = 1,...,p, i.e. when the zero is a single eigenvalue of the operator. An a priori bound, uniform with respect to l/z\, is also obtained for this case, together with a convergence rate estimate O(\hj’). As distinct from [ll, all our discussion here refers to any non-uniform mesh. 1. The difference problem approximating problem (l)-(2) will be stated in the next section. Here we describe the difference mesh and introduce the necessary notation. We construct the mesh in G= G lJ S (S is the boundary of the parallelepiped) by means of one-dimensional meshes on intervals [O,ZJ. We subdivide each such interval by points 5:) = 0 < ~2) < . . . < xLNa’ = I, into N, parts. Denote by Oa = (3 (CZ): i, = 0 , . , . , Na} the set of all nodes in the closed interval [O, &I, & = {z&& : i, = 1,. . . , Na - 1) is the set of interior nodes in LO,I,], y - a is theleft-hand boundary node xa: = 0, and yka: the extreme righthand node X, = I,. The mesh step hl;za)= $J

_z>-1)

V. B. Andreev

46

is in general an arbitrary mesh function, which only needs to satisfy the $

normalization condition

hb’“’= I,.

put hbp’ = hhN,+l) = 0 and introduce

ipl

average step

the

(%z)= fia

@bia+l) + #a)

),& i,

(i,)

Notice that, if the mesh Zia is uniform, i.e. ha ( i,) ha

=

ha for i, = 1, 2, . . . , N, - 1 and hf’

mesh is constructed

w=

= ha, i, = 1, 2, . . . , N,, then (N,)

= ha

= h,/2.

Put

5 max _ Aaa. a=1 Oa

Ih12= The difference

0, 1, . . ., N,.

=

thus in G: O,
fi~i,={&s,,...,z,), a=1

is the set of all nodes of the difference interior nodes in i7, y =O\o

mesh in C, o =

fi w. a=1

is the set of

is the set of boundary nodes of the mesh i;i.

For later convenience, we distinguish boundary nodes on manifolds of different numbers of dimensions. This is done by introducing vectors r = (rl, . . . , rp), formed from zeros and numbers 4,

whose lengths are defined by 1r 1 =

‘2 or==L

(1 - 1ra I) 2 0. We shall only consider vectors in which at least one component is non-zero, i.e. )r\ < p - 1. Then, given any vector r with the components ra, = . . . =

0;

Tar=

rq+l

Yr-

=

fi

. .

@ajX

j-1

.=?-a,=-r

1; rar,+l=...=rap=+l,

b y-ajX ]i y+uj j-r-j-1

j=r'+l

is the set of nodes of the boundary y lying on an rdimensional face (a rib when r = 1 or vertex when r = 0) of the parallelepiped G, defined by Za,+l

=

. . . =

XC+, --0,

x+#+l=l~+l,...‘xa,=la

P

and not belonging to a face of fewer dimensions (set of “interior nodes” of the face),

Second

Yr E

and third boundary

value problems

r’

j) ciajx

P

III

Y-aj

111YTUj

x

j=r+1

j=l

47

j=r’+1

is the set of all boundary nodes lying on the face in question. We shall use the same notation as in Ll, 21for mesh functions specified on ZJ. To investigate the convergence of the difference scheme, we require the scalar products of functions specified on the meshes i3 and y, which we define thus :

(d,u, &u>

@,u,b,v),

z

2

ZE

&uba~H(~),

XER,

where

HEH(

XEG;

Hca) zHca)(5)

E

H&/&a, 5 E Ra;

(ra. are the components of the vector r), R, =

R (Da)

=o\yr

(Irl=p-‘-1,

is the domain of definition of the operator

-

D,,D,u

ra=

E u-

xgo

while it follow8 from

the definition of / , / that s ES(X)

EE

H/h,,,

i;

zczyr.

j-+1

Notice that, given this definition of H, H’ a) and s,

[I, I] = fi I, = mes G, a=1

P

/I, l/=2

Z;

(1, 1) = mes G,

1tP

n

Za=mSSS.

-1)

V. B. Andreev

We define the norms thus:

II* 110 - II* llL,(;T;) - r*, *I”,

n*Ilo’-

11 * [IL*(r) = l*,

*k

We indicate by writing ()v( ) L *(a, p, that the norm is only taken with respect to the variables xQ and xp, the other variables being fixed. Denote

,

~v*1102 - i Cl,=1

Il*ll?= U*l~~~~Il~*ll~~+I*ll~~~ IIf IL1= IIf 11,$-l)= ;g { I v9 ul I/ II24 Ill>. 2. We state the difference problem. The following difference scheme was constructed in [ll for the approximate solution of problem (l)-(2) (note that difference approximations of the normal derivative of the same type were used in [3, 41 when considering the second boundary value problem for two-dimensional Laplace and Poisson equations; my thanks to the author of [3, 41, E. A. Volkov, for drawing my attention to this): (4) a=1

where I hay Kay

=

E (@aY;a);a

-

ddi,

Xa E

ha-y

E

Ai1 [ara)yxa

-

(X-a

ha’y

F

fLil [--a&

-

(%+a +

m(x) while the coefficients

hada)

Yl

Yl,

Xa E y-a;

iFi)

Xa 63 Y+a;

zE:w;

@* =

_1- &da)

0,;

Icp+$v,

SEY,

(6)

and right-hand sides of the scheme are given e.g. by

a,(x)

= k,&b"@)),

cp(x)= f(s) + Ro,

da(x) = 4&(X),

(7)

v(x) = g(4 + Rv,

(3)

Second

and third boundary

value

problems

49

and the functions R, and R, do not exceed 0(lh12), being non-zero only when [a, 11 = 0 is required for the problem to be solvable. 3. The results to be obtained in subsequent fundamental theorems.

sections

are used to prove two

Theorem 1 If, given any function u(x), specified -[Xv,

4 2

on w, the relationship

J4,llnll?,

icll = const > 0,

(9)

is satisfied uniformly with respect to (hj , then the solution of problem (4)-(6) satisfies the a priori bound lly1112 <

~Z(ll~ll-12

+

ll~llo’2),

(10)

where M, = (1 + M,) / M,‘. Proof.

We multiply (4) scalarly by y and use the form (6): -[KY,

By definition,

Yl =

C@, Yl = hJ7 Yl + h, Yl.

(11)

and using (18).

and from Lemma 2,

IIyllo’2< J45llyll?. substituting these inequalities in (11) and recalling (91, we get (101 when t = M,/ 2 (1 + M,). Hence the theorem is proved. Theorem 2 If

V. B. Andreev

50

d, =

~*a =

0,

(12)

l,...,p,

a=

and -

[XL’,u] > &II vulll12,

A43= const > 0,

(13)

for any function u(x), specified on ij, uniformlywith respect to Ih\,then problem (41_(6)has a solution provided that [CD,11 = [q+ I] -t Iv, Ii = 0

(14)

It is unique if I&

11= w,

(15)

where W is a fixed constant. The solution satisfies the u priori IIYII?

<

~4{llqAI-i2

+

II~IICP)

+

2&&A

bound

w,

0’3)

whereM4- (I+ MS)(I+ ~~~~~I ~~2. Proof. It is easily shown that the operator?%is self-adjoint and that, when conditions (12) are satisfied, zero is an eigenvalue of it, corresponding to a unique eigenfunction equal to a constant. Hence the existence of the solution of problem (41-(61follows from (141, and its uniqueness from (15) (see e.g. 151,pp. 226-222). To obtain the Q priori

bound we multiply (4) scalarly by y. We get -c&J*

VI d b-44 !/I + lv, Y/.

A bound is obtained for the right-handside precisely as in Theorem 1. Then, recalling (13), we get M3 II VY

lloa< e U + Mid IIy Ih’+

& ( 11 cpIL” + 11 v no”).

If we now use Poincar& inequality (Lemma5) with c = M,/2(1 + M, ) (l+ Ml,), we obtain (16). The theorem is therefore proved. 4. We now prove the inequalities used in the proofs of Theorems 1 and 2 and required for proving some sufficient conditions for the operator x to be negativedefinite. The ratio between the arithmeticand geometric means satisfies

Second and third boun&y

m

l/m

(n > GKI ii aa, a=1

51

value problems

aa

a=1

a,>O,

(17)

this is proved e.g. in 161.Since (17) will be used mostly for the case m = 2, we write it then in the more convenient form 1

[ ab I< 8a2 + 4e b2.

We shall use in conjunction with (181 the Cauchy inequality 1 Ir~,~lI~sn~/lo2+~ll~l102.

(19)

The special inequaiities that we require will be stated as lemmas. Lemma 1

Whatever the function v(x,), specified on a non-uniform mesh TV,, we have Na

maxu2(za) Be z)

XaGiqa

(DaOjzha

(20)

-t

i,=l

where E is an arbitrary positive constant and 1, is the length of the interval in which the mesh i3, is introduced. Proof. It is easily verified that ha/kzG2.

We write u%,) in the form (x, > r,)

u2 (2,)

=

v2 (Ea)

$

+

Dau2

(5a) ha

Ca+bfl)

xa =

ll’(Q

+

2 2 ,gi)

Ubaclha -

using this in conjunction with (21) and (191,

$ ++‘I a

ha (Bau)aha.

(21)

V. B. Andreev

52

Noting that this inequality holds whatever the x,, ta E I?!,, and summing it over 5, E ts,, (where as usual, we use the term summation in the sense of an analogue of integration, the value of the function being multiplied by a corresponding “elementary volume” prior to addition), we obtain (20). The lemma is proved. Lemma 2 Whatever the function V(X), specified on a non-uniform mesh ~5,

llullo’2< ~IIvvllo2+ J&i(E) llullo2,

llvllo’2< Mslldl*2,

(22)

where MS(E) =8p/c+mesS/mesG, MS = {mesS/mesG+

[( mes S / mes G)2 + 16~1 ‘h} / 2,

and t is an arbitrary positive number. Proof. Notice that (20) remains true whatever the x, E 8,, even when u(x,) is parametrically dependent on xp, /3 & a. Summing this inequality for arbitrary

rIo,;,

l_P xa

E

iZa

over the mesh

we get

:+a

maxIIv IlL(l,..., a-1, a+~ ,... ,P) 6

xa&a

El

(

(Bav)at

1) +

g-4-~)ll.ua".

But

where jr1 = p - 1, ra = + 1, SO that

We sum this inequality over all r for which /rj = p - 1:

Second

and third boundary

$-

53

value problems

($ + i +) IIv llo2. a a=1

If we now take t, = YXand note that 2 5

I;;’ = 2 i

. a=1

a=1

ls 1, [:.#a

I

P

n 1, =

mes S/mes

G,

a=1

we get (22). Selecting E = M, , the second part of the Lemma is proved. Lemma3

Let G, be a rectangular parallelepiped with faces parallel to the coordinate hyperplanes belonging to G. Let the non-uniform mesh 0 be such that 2 H ; nE0

> mes Go/2.

(23)

Then, whatever the function U(X), specified on .?X,

where 12 = 5 laZ,

-W6 = 4mes G/mes Go,

a=1

Proof.

As before, when xa &6, a = 1, . . . , p,

v2 (‘) = u2 (E) t- 2 i a=1

ii a=1

A%1, = max (M,, 1 + 2M6aZ2}.

z h

u (21,

. s ., Xa-1,

I&, &I+,, . . ., tp)

DaVha-

iti)

iit j2a (isaV)2 (+I) 4a

ha d

V2 (E) +

i [ 28, a-=1

iI V I/&a)

+ -$

,z la-l

(6,V)zha]

If we now observe that this inequality holds for all x, 4 E ?JJ,we can sum it over x E W:

.

V. B. Andreev

54

On now summing (26) over I$ E E n g,, and strengthening obtained, we get

the relationship

Taking to: = 2,/2Z’M,, we get (241. We obtain (25) by adding (1~ uI(,~, to both sides of (24). Hence the lemma is proved Lemma 4 Let S, be a (p - 1)dimensional rectangular parallelepiped with faces parallel to the coordinate hyperplanes, belonging to S fl lx, = 01. Let the non-uniform mesh ii be such that 2

s > mes S,/2.

(27)

w-G% Then, whatever V(X) specified on Z,

IIv Ilo2 B Mt3 2 v2s+ 2piJf82 11 vvllg2,

(28)

w-w0

II v Ill2

<

Jf9

(29)

(II vu llo2+ r; v2s) , m%

where

MS = 4mes G / mes So,

MS = max {MS, 1+

2pM3).

Proof.

Inequality (26) holds for any u(x) specified on i3. Let 5 = @,, . . . , . . ,p. Summing(26)over[Ey fl 5, and strengthening the resulting relationship, we get

c&,9

0, 5‘,+,,.. . ,$)andcp=cwhen/3=1,.

mes S,, x

v2s + mes G 2pe 11 v Ilo2+ $

vnS;,

1

IIPv /I,,2 .

[ Taking c = 1/2pM,, we get (28). Adding I)VuI(,Z toboth sides of (281, we get (29). The Lemma is proved. Lemma 5

Second and third boundary value problems

(Poincar@s inequality).

Whatever the u(x) specified

jlVl\$ <

55

on ;j,

A!f*ollvvllo2 + Jm-J7 119

where Ml1 =

Ml0 = plV2,

1 = max la.

[mes Cl-l,

a

A proof of Poincak’s inequality for functions of a continuous argument may be found e. g. in [7l. The proof of [71 may be repeated almost word for word for functions specified on the mesh 8. 5. We prove some properties of the operator A and obtain sufficient conditions for (*I. We shall assume that the coefficients of problem (1142) are bounded and satisfy one of the following conditions:

A- JCa@)>cl>O,

.

Q (5) =

5

Qa

(X) >O

for

xEG;

a=1

X&a(Z)>0

butq(x)>Ct>O

for xEGo,where

for xES; GocG

andmesGo>m P

>O

'

q(s) 20 for xEG; ~~(5) 20 for xES; B . k,(x) 2 CI > 0, but x-ar0(x+a0)~c3>0 for x~~o,whereSocSn{x:x,,=O} (So c ~S~{x:~~~=~~,})andrnesS~~m~_~>O. C. k(x) > Ct > 0, q(x) EO for 2 EC; u(x) =O for xES.

We shall show that conditions A and conditions B are both sufficient to hold, while conditions C are sufficient for (13) to hold.

for (9)

Lemma6. (Green’s first difference formula.) Whatever the functions u(x) and w(x) specified identically

on the mesh ‘i3, we have

P -[KU,

W]

=

2

(Qau,

Baw>

+

[dv,

w]

$

IXV,

WI,

a=1

where

d= i a=1

as g, i.e.

da, and the function x is defined by means of the same equation

V. B. Andreev

56

Proof. First consider the operators - &. We multiply - xau scalarly by w and transform the result by summation by parts. We get

-

$ Xavwfia= - 5-l

A,vwh,

ia=o

+ [-

acla) v,,w

+

ia=l

X-aVw + kadawvl

Ixa=o + [aav;aw

+ %aVW +

Na fiadauw] /xa=la =

2 aabauDawha ia=l

+

Na 2 davwfia f ia=

x-avow0 $ x+auN,wN,.

1fP

If we sum this equation over the mesh

n Go, l+a

then over all a from 1 to p, we

get the statement of the Lemma. Theorem 3.

(First sufficient

condition.)

Let conditions A be satisfied and let the non-uniform mesh i3 be such that (23) is satisfied. Then, whatever the u(x) specified on 75, -[Xv,

v] b

MI2llUII12,

where M,, = min lc,, c,! /M,. Proof.

Conditions A, [d, v2] >

2 dv2H >c2 2 v2H. GnG +-)I%

From Lemmas 6 and 3 and conditions A.

Second and third boundary value problems

Hence the theorem is proved. Theorem 4. (Second sufficient

condition.)

Let conditions B be satisfied and let the non-uniform mesh E be such that (271 is satisfied. Then, whatever the v(x) specified on 5, - [Al;, v] > where

Jf13 -

Proof.

M13llVll13,

min {co, q} / 1~~. From conditions

B,

lx, v8J >

2 xl+% > CS 2 vs. Yk5, Ml0

From Lemmas 6 and 4 and conditions

B,

cB)1vP. - [xv, VI> CO IIVuIt2+ c3 2 ~3 > K1 min{co, vfl~, Hence the theorem is proved. Lemma 7. If conditions C are satisfied,

we have

-I& whatever the u(x) specified

VI >

ctllvvllo~.

on W.

The proof folloes almost at once from Lemma 6. 6. We turn to an error estimate for the solution of problem (4M61. Theorem 5 Let the coefficients ka(s)

E

cqq,

of problem (l)_(2) satisfy the conditions

!744 f(s) 3

E

C(O)(Q 7

x-(5),

v*a(r)

EC(O)(S).

V. B. Andreev

58

Let u (x1 E error

C(“) Ccl be the solution of problem (l)-(2).

involved by taking problem (4)~(6) with coefficients solution of (11 as

Then the approximation

(7)~(8) can be written on the

where

the constant M,, > 0 is independent of i3, and 4a @)I ra=!,P) = *a (X1lXaEla= O*

(32)

Proof. The fact that Y (3~)can be written as (30) for x E G when conditions (31) are satisfied follows form [Sl. If +,(x) do not satisfy conditions (321, their values at xcr = h:‘) and x, = I, can be removed e.g. by means of linear functions, in such a way that conditions (31) remain valid. It follows from r13 that Y(x) = 0 (1hl) when x E y, so that it only needs to be shown that H / s = O(Jhl ) in order for the last of (31) to be satisfied. It is easily shown that H/s = [i&h;;j”]-’ and applying (17) twice to this, we get

The proof is now complete. Lemma 8 If qS(x), specified on 8, can be written as

for 5 E Yr,

Second

and third boundary

value

problems

59

(33)

and (34) then

Proof. By definition, (35) Using (33)-(34) and the formula for summation by parts,

Applying the usual Cauchy inequality to this and racalling (21), (36) The required inequality follows from (36) and (35). Hence the lemma is proved. We now have everything needed for estimating the rate of convergence problem (4)-(6).

of

Theorem 6 If the conditions of Theorems 3 and 5, or 4 and 5, are satisfied, the solution of problem (4)~(6) is convergent in the sence of the norm of space Wztl) to the solution of problem (l)-(2) at a rate O(lh(‘), i.e.

where M,, is a positive constant independent of 8. The proof easily follows from Theorems 1 and 5.

60

V.. B. Andreev

Consider problem (l)_(2) when conditions C are satisfied. exist, we have to assume that 5 f(z)as+Sg(s)ds G

For the solution to

= 0.

(37)

S

The unique solution can be distinguished

by the condition

j U(Z) dz = W = con&.

(38)

Theorem 7

If conditions C are satisfied, together with (37) and (14)-(15), the solution of the difference problem (4)~(6) is convergent in the sense of the norm of space Wz(‘) to the solution of problem (l)-(2), satisfying (381, at a rate O(lhl’), i.e.

where M,, is a positive constant independent of w’. Proof. U’riting z(x) = y(x) - u(x), where y is the solution of problem (4)-(6), and u the solution of problem (l)-(2), we have AZ = - Y. From Lemma 7 and Theorems 2 and 5,

ll~lli2% ~i8{ll~ll-i2 + ll~&? + [z, I]“}. But

O=Gj f(~)ds+Sg(S)ds=[f,~~+~g,~i+~(~h~2), s * since [f, 11 and /g, I/ are the trapezoid quadrature formulae for the integrals rf(x)dx and Jgklds [91. From (14) and (7)~(8), 8 % 0 =

[@, 11 =

[Cp,11 + 1% I/ =

[!, 11 + /g, I/ +

[R,,11+

/Rv, I/

so that

[R~,1l+lR,,1/=-{[f,1]+/g,1/}=0(Ih12). If we now take e.g. RY = 0, & = -{[f, satisfied, and we find from Theorem 5 that

11 +/g,

I/} /mes G, (14) will be

ll~ll-12+ ll*vllo’2= O( pal”). Further,

61

Second and third boundary value problems

[2,1]=[y-u,1]=w-[[u,1]

=

1 uaz-~u,1]=O(pip), G

since the error of the trapezoid quadrature formula [u, 11 is O((hl’1. Combining all the inequalities, we obtain the statement of the theorem. Note. Friedrichs and Keller DO1 investigate the convergence of difference schemes obtained by the variational method, for second-order elliptic equations in a domain with a piecewise smooth boundary. In particular, it is shown by taking the example of Neumann’s problem for the two-dimensional Poisson equation that the resulting difference scheme is convergent in the integral norm of W?(l) (multilinear completion of the mesh functions is employed) at a rate O((hl) if h,/h, = O(1). Our Theorem 7 imposes no restrictions on the ration between the steps and gives a stronger result for rectangular regions. It should also be mentioned that our approximation of the boundary conditions with a normal derivative can be obtained by a variational method [ill. In conclusion I wish to than A. A. Samarskii for his interest and discussion of the results, and E. A. Volkov for valuable comments.

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ANDREEV, solution

Zh. 2.

V. B. Iterative schemes with alternating directions for the numerical of the third boundary value problem in a p-dimensional parallelepiped. Vy’chisl. Mat. mat. Fiz. 5, 4, 626-637, 1965.

SAMARSKII,

vychisl.Mat.

A. A. Locally

mat. Fiz.

uniform

difference schemes 1963.

on non-uniform

meshes.

Zh.

3, 3, 431-466.

3.

VOLKOV, E. A. A mechanical instrument for solving other elliptic equations. Vestn. MGU. 7, 10, 3-17,

4.

VOLKOV, E. A. On improvement methods using higher order differences extrapolation. Dokl. Akad. Nauk SSSR. 150.3, 455-456, 1963.

and h2

5.

SHILOV, G. E.,Mathematical analysis Moscow, Fizmatgiz, 1960.

course,

6.

HARDY,

7.

MIKHLIN, S. G. Variational methods in mathematical v matematicheskoi fizike), Moscow. Gostekhizdat,

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TIKHONOV, A. N. and SAMARSKII. A. A. Univorm difference meshes. Zh. vychisl. Mat. mat. Fiz. 2, 5, 812432. 1962.

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(Matematicheskii

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methods (Metody vychisleniil,

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FRIEDRICHS, K. 0. and KELLER, H. B. A finite difference scheme for generalized Neumann problems, in Collection, Numerical solution of partial differential equations, New York - London, Acad, Press, 1966, 1-19.

11.

OGANESYAN, Collection,

L. A. Numerical computation of slabs (Chislennyi raschet plit), in solution of engineering problems by electronic computer (Reshenie

inzhenernykh zadach na elektronnykh Lensovnarkhoz, TsBTI, 1963, 85-97.

vychislitel’nykh

mashinakh),

Translated

by

Leningrad,

D. E. Brown