Iterational alternating direction schemes for the numerical solution of the third boundary value problem in a p-dimensional parallelepiped

ITERATIONAL ALTERNATING DIRECTION SCHEMES FOR TBE NUMERICAL SOLUTION OF THE TBIRD BDUNRARY VALUE PROBLEM IN A ~~D~~ENSrONAL PARALLELEPrP~D* t V. B.

~~~RR~EV Moscow

(Received

2 October

19841

IN this paper we construct for a selfconjugate elliptic equation given in a p-dimensional rectangular parallelepiped. aa ~~~oximation of boundary conditions of the third kind, which, for an ordinary (2~ + l)-point difference scheme in a rectangular network, ensures convergence with speed 0Clh121, where

For tne diffe~el~ceproXem Oblained an iteration&f variable direction process is constructed which, in the case of a problem with separable variables and the special choice of iteration parameters {?,I for decreasing the initial error by a factor of l/e. requires the same number of iterations as the similar process for the Dirichlet problem. 1. In the numerical integration of the boundary value problems for elliptic equations the problem of finding an approximate solution generally falls into two more or less independent parts: (1) the approximation of the differential equation and boundary coniiitionson a difference

*

Zh.

f

The basic results of this paper were presented by the author to the seminar of the faculty of Corn~~t~t~onal Mathematics of haoscow State University and in a joint paper with B.A. Samarskii at the “1964 Loaonosov Lectures” at Moscow State University.

uychisl. Mat. Rtat.Fit. 5, 4. 626 - 637, 1965,

56

alternating

Iterational

direct

ion

57

schemes

network and (2) the solution of the system of difference equations obtained with some particular degree of accuracy. If the question of the approximation of the differential equation is sufficiently well investigated, we cannot at present say this of the approximation of the bouudary conditions which contain the derivatives. References [II - [4 are concerned with the study of the ~proxima~ion of the normal derivative on the boundary of the rectangle (we snail not indicate here any literature relating to the approximation of the derivative on a curvilinear ooundary). In [II anti [23 outer points are used to approximate the tierivative anti in [?I, in the case of ~e~~1~’ s problem for Laplace’s equation. the uniform convergence of the difference scheme with speed O(h2 In h! is proved. In [?I a multi-point approximation of mixed boundary conditions is constructed whose error in the solution of Poisson’s equation is of the eth order of smallness in h, and the convergence of the difference scheme for Neumann’s problem with speed :)(h”-E) is proved. In 141the simplest approximation of order h is used. As for tne second part of this problem, a sufficiently large number of iteration methods exist for the solution of elliptic difference equain particular, a study is made of iteration protions. In [51 - [lo], cesses for the solution of the Dirichlet difference problem in a rectangular parallelepiped. possessing the greatest speed of convergence among the known iteration processes. In [ill the possibility of using tne methods of K.5: - [*I b to solve the third bo.adary value problem is noteu anti in fl21 the convergence of ES] - [S] is proved for ~eum~‘s problem. The question of the application of the method of E81 to the solution of the third boundary value problem with the simplest (0th.)) approximation of the boundary conditions in the case of sn equation of heat-conduction is investigated in f13’1. There the possibility of applying this method to solve elliptic problems is indicated. 2.

We shall

BP = (x = problem

consider

in a p-dimensional

(Xl,. * . ,xp)

: 0 < xa =g la, a =

1,. . . , p}

P

Lrs =

~LLalz= -f(x),

Lczu = -g a

a=1

%a- (x)

=

72 -

Farallelepiped

rectangular

with boundary r the

(kc&g1- qau, a

2a = 0,

vu- (2)

with

- %f(lr)

with

xa =

0,

2= 0,

xa+(x)

a =

a-

a%

= Xaf(t)U

ka(z)

2

ci >

!?a(4

(1)

1, . . . , p,

(2)

0.

(3)

b,, >

c2

>

V.B. Anclreyeu

58

With regard to the coefficients 01 problem fl) - f2S we assume that they possess the number of continuous derivatives which are needed to ensure the necessary smoothness of the solution of u(x) in $,. To ensure the necessary smoothness of u(x) near the bounaary we require for (1) (?I that the conditions of conjugacy of the necessary order be satisfied. 7. We shall

divide

tne p-dimensional

by (P - 1)“dimensional hyperplanes h a = &c I&CC, and i, are integers,

space of variables

~:a(‘~) = into

ha&, o =

x1, . . . , xp

1, I)- I +P,

where

~-ui:tl~ris~o~~~~lpsrallelepipeds,

The vertices of these parallelepipeds will be called the nodes of the difference network. The set of nodes, belonging the open region l$ = &I ‘\ r, we shall call inner nodes and denote by Opk = (22 = (2@), . . ..S#Pf). O
boundary nodes

the set of “inner” (P - r)-dimensional .‘.). T+r =I&, ad

if

0 6 “6

f’G

“angular the points

x0, = 1, (r:D = 0). Q figure).

We shall

where uj
. . . =xa,=O,

j G j *G r. We shall

all

(Xi E r}

also

denote

0 ( j ( j’
by Thp;ia ,,..., ++, _“r,tl ,_..,+..,

by q$: for = {xi E Tpk :2, = t,(0))-

points”,

of the boundary $,

situated

on the hyperplane

= a$ u T$, q&q = GE \ fi;;+@ u 7;; _%)

For network functions

given on &hl

we shall

z,@a)= x,, xi = x = (Xl, * . .,x*)* Yi = Y = (xp, * * . , x$y, xJi,*m), xy’ ) * * ., x$p) ),

&_.w%) =

[email protected])

denote by

points of the part of the boundary $, which is a paral~elepiped, situated on the hyperplane xbl = &,,...

x,~,+~=

p, the

the set of

yPh =

=

0,5(&'G + xp~)),

y;;= (y- y(-qd2,,

(see

use the notation Y

(x)1 m=0.5,

1,

$.p%) .- y (,m%l), jjxa =(y'+la'- Y)/ba*

4. We shall set the differential equation (11 in the difference network aph in correspondence with the (2~ f I)-point difference scheme

Iterat

whose coefficients in terms of the out dwalling in to calculate Q, that in tne case

ional

al ternat

ing

direct

ion

schemes

59

ao, do, ‘p are expressed by means of certain functions respective coefficients of the difference equation, withdetail on those properties which the functionals. used a’, and 9, must possess (see [I4], [El). We note only we are considering (with coefficients k,, qa aud f

-h Yr;*,

sufficiently smooth) to obtain a seconu order we can use the simplest expressions a, (z) =

k, (&=a))

)

k(x)

=

of

!?afzf,

approximation

in Ihl

(P(x) = f(s)-

In describing the houn.ky iiifference equations we bear in minu Liiac since the error of approximation of scheme (4) is a quantity of the second order of smallness in lhl, it is desiraole to obtain the anpro:;imation (2) also with an error not below Ot i hi 2f. For this we make use of the following ~ymptotic expressions:

aa(+la) =a u

_

k

where u is the solution

of

equation

(1).

Hence it

follows

that the

60

V.R.

boundary difference

with approximation

Andreyev

conditions

(2)

to tile solution

(1) have an error 1-P

Rut the differeuce problem iias not Yet been completely defined by the representation (4) and ((3: the boundary conditions (6) require for their description a knowledge of the solution at points, situated on the “edges” of the noundary, at which for the present no (P - 2)-dimensional conditions are given. To obtain tne missing equations we assume that conditions (2) hold not only at inner points of the (p - l)-dimensional hyperplanes of the bound~y 7, but also on all “edges” ~~~erpl~es of dimensionality less than (p - 1)). Hence on the “edge” of di~ensiona~ity (p - t-j. r 3 2, we obtain F conditions and on the difference network we shall approximate their sum directly with weights 2/haj Cj = 1, . . . , r). Then from (5) and (5 ‘I it follows that the difference boundary conditions, written for the Cp - r)-dimensional “edges” of the boundary yPh$ take the form

O,
r>2,

x E: r;;+a ,,...,

+a,,,

-a,‘+ls””

-=

P

’

where

and if h,=h, a=l, ,.., sum with the second order It iS easy to see that

p. theu (‘7) approximates in h.

conditions

(6)

are included

the corresponuin&

in (7)

for

r = 1

Iterat

ional

al ternat

ing

direct

ion

schelnes

61

and r’ = 0, 1. and equation (41 for r = 0. Therefore we rewrite (7) in the form of the operational equation

n’g+F=O, where for

A,

-h

x E oPtax)

for xE mph8

cp F

= i

( v$+j=$+lg;)

for x E $; .+cl. ,,....tar,, -i+.‘+l’...’ -y,’

q+2 i: j=l

and we shall sag that the difference problem is defined by relation (91, which combines the description of all 3P relations 141, (6) and (7). Note that the operator A* maps zph onto itself. 5. We now formulate conditions which are satisfied by the error of the solution of the given scheme. We shall consider the function z = Y - U, where y is a solution of problem (91, and u a solution of problem (11 - (3). Substituting y = z + u in (91 we find that A’2 = -?V,

(10)

where Y = h*u + F and by virtue of (5) and (5’)

y”,-’

~o(~aj)+

j=l

i”(hujz) for

j=r+l

xEYh,;+a,,...,+crrr’ -u,,+,,...,-q

(11).

I

To evaluate the solution of problem (10) we require scalar products with respect to sets of points which are situated on hyperplanes of dimensionality r, r = 0, ..., p. Since the r-dimensional faces of the parallelepiped $, belonging to r, are themselves parallelepipeds of dimensionality r, the set of nodes of the boundary uph Ez r, which are inner nodes with respect to the parallelepiped, situated on the left or right and of the paths xai, j = r + 1, . . . , p, we denote by qh, where r = (al,

. . . . aXr) and oj < oj’

if 1 < j < j ' < r. We shall denote by I’,“:+a

i

62

V.B.

Andreyev

the sets of points of the t-dimensional parallelepiped, which are analogous to the sets Tg. ia. Following 1141, we shall introduce the scalar products

Hr =

n

hUj,

r=h

“j<+,

. . ., a,),

if

l
Ho=l,

j=l

and the corrected norms

In addition we require Green’s first example 1141) ~

-

(hcrjZ,Z)r-Z=

fi

I(a"jz;aj,

difference

formula fsee for

Pj) + (&j& x),1 4

z;=

j

j=l r j=l +

2

I(a,jz;aj t Z)r-&,j=z,j

-

'"~jL~j,-

&-llr,j=ol

7

(9

j=l r -

1

=:

(al,..

., @j-l,

C4j+l,

6. We multiply each of the 3P relations

. - -7 C$).

!lO) scalarly

with respect

to

where (p - r) is the dimensionality of the by -2-p fi haj-*z, j=r+* hyperplaue on which the relation is given. Then using formula (12) and bearing in mind (8) we find orh

2~’ fi

h,,-l{2

j=r+l

i

h,i’

I(%jQaj7 ‘)$B-r+

(KZj+'S

')P-rl

+

j=l

+2

i j=r'+l

q-1

[-

(ayj%‘zxa., Z)p-r + (Gj- 2, z&l J

+

(13)

Iterat

+

ional

i: i=r+1

$j rcayj’

zxaj’

+

al ternat

)

[('"j',o,

ing

Z&l$t

direct

ion

63

schemes

@Ljz,Qp-rl +

+

j

3

qp-r-1- (aajz;ajl z,p-r-11} = 2-’ fi

j=r+l

r = (%+1,- *

P-

.7

ha,-l (Y, Z)p-r,

j=r+l

up>.

Here we have not yet indicated on which part of the boundary relation (13) holds if r # 0, although to each value of r there correspond Cpr2’ different parts of the boundary, formed by the (p - r-)-dimensional hyperplanes. This information can be obtained if we are given the coordinates of the vectors (1: - 1“) and F’ (0 < r’ < r). Let

(ii,zj’-$

+

- 2%) > +(iq-

, z=

2

p-r-1

22 ) p-r-1

and

5

Ip+ =

j&r+1

Now on adding all I$, we find that

3P relations

(13)

I?

P-r’

and multiplying

the sum obtained

by

P--l

I,+

Jf 2-’ fr hajr,&-r--(r r=1 +

(lb)!

2-’ fJ hajx[e8

5

where 1 denotes

(of dimensionality

evaluate

the arithmetic

5 $-‘\lz\~-r-(Y

&-r]

= 07

j=l

j=l

r=l

We shall

ZIP+

j=l

summation with respect to all the vectors r-‘, r - r'j with r' = 0, . . ., r:

(Y!, z)~_~

and geometric

for

r # 0.

mean [lfi,

On the basis

r’,

r - r’

of the theorem on

p. 291 which states

that

64

Y.3,

we find

Andreyev

that (Y,

z)p_.+-llY19-r+ ~IbllL

where co > 0 is an arbitrary

constant.

Let us assume that r

On the basis

of the sale

and from (17)

theorem

and (18)

and

Now substituting

f,+

P-l ~ 2-' fr r=1

To evaluate

(16) and (19) in (14) we obtain

aj~Jp-I-(y,

')P\<

j=l

I,,_= from below we require

Lenuna 1 For any function

z, given on the network w”~-~+,),

Iterational

F_r

alternating

<

M,I;_r

,

direction

Ma

schemes

=za(4ci4~~c2) . i

For a proof

65

of the lemma we use the following

(21)

2

identity

(see

[I?‘],

Lemma

(1)

(22)

+( I-;) Squaring

(22)

ZO+;~N,-

and using the inequalities

We now multiply (23) unity and obtain

scalarly

Also on the basis

(3)

of

with respect

and (6)

to the set

o~,--~(~~_,) by

66

Y,R.

On choosing

CO = Z,cJQcl

In the solution

holds,

where

In fact

M

is

and on the basis

we obtain

of problem (InI

a positive

on the basis

of

Andreyev

(21).

the

constant

a priori

evaluation

xvhicb does not depend on /hi.

of Lemma 1

(15)

(26) Substituting

Let

and

and (26)

in (20) we

Iterat

at ternating

ionat

Then assuming that Iv = ~/ml,

direction

schemes

from CR) we obtain (24).

Theorem 2

If the solution u(x) of problem (1) * (3) satisfies the conditions in which the error of the approximation of scheme (9) to this solution can be put in the form (11). then scheme (9) converges in the mean with speed O( / h( 2l so that on an arbitrary tion IIY($1 holds, work,

sequence of networks $h the rela-

a(z) Ilp* 4 JqhJ2,

constant which does not depend on the net-

where M’ is a positive

For a proof of the theorem we use the a priori the basis of (11) ~~u’~~P2=~(~?z(4), P

P

/I

x

hajY lk-r = II ,_T+, kj

evaluation (24). On

and since P

ir

( 2 0 (f&f + 2 0 (ha))I?= 0 (1h I*), j=r+1

j=l

j=r+-1

the evaluation

(33)

(28) is proved.

8. For an approximation to the solution of problem (91 we use the method of alternating directions. Let u = vCn+lf be the (n + l)-th iteration, later, form*

G = uCn), T = f,+l and i,

the iteration

parameter, which will be chosen

= (V - i/j/~. We shall use the iteration A?+=

R*j + F,

scheme in the

P(O)(XT:) = u&X),

(2%

where A” =

fi A@.“, OS=1

AaC = E -

ITzJla*,

Eu = Y,

and VO(X? is the zero approximation. Note that the operator A* which we are using maps Gph onto $h, as distinct from the operator A of iI81 and [ISI which operates from GPh into aph. Such a definition of the operator A* enables us without difficulty to construct one-dimensional variable direction algorithms at a time when, for the operator A of [131,

l

A similar iteration [91, I101*

scheme was put forward in [81 (see also [5] - [?I.

68

V.S.

Andreyev

the a~gorit~ put forward there works only in the cient b, (K~* in’ our notation) does not depend on . . . , xp and its extension to the case of arbitrary work. For the determination of v from (29) we can one-dimensional alternating direction algorithms. algorithm similar to that of (3.2) of [IS]: ”

Ai”w(i) = A*u +

ease where the coeffithe variables xatl, b requires additional construct a series of Here for example is an

F, A,‘W@) = W(U-1), a = 2,. * .p,

Y

u = u + zqp). (30)

Algorithm (30) differs

from (3.2) of

[181 only in the fact that here for

WC,) on yPh; +a fuld on uph; - o boundary conditions of the third kind are given and the equation for w(a) is written in the region &,hta). including the boundary points in the direction ‘“p, fi # a, at the time when in [la] acal vanishes on yph,. *a and in the same way the eqnation for w(,) is written only in mph. We now describe in more detail

9. Let us investigate

formulae (30) in the case where p = 2:

the speed of the iteration

process (29) which

alternating

Iterational

is applicable k, (5)

riirection

to the case of separable variables. G

f@(z) f

ka(ra),

%a* (z)

Q&a),

We csn show that on tnese ass~ptions the theorems on convergence from [81 sufficient to establish only negative and the presence of a general system A* and A*. Let ita =

schemes

59

Let

= %i* = cm&.

for the iteration process (Zo) all and [lOI are valid. For this it is definiteness for the operator A* of eigenfunctions for the operators

and Aa = kko, k, = 0,. . . , NCG, a = 1,

pk,ba)

the eigenf~ctions and eigenvalues of the one-Dimensions Liouville difference problem.

are eigenfunctions A*pk

fi. pk, Cr=t

(d

1

be

(31’)

Then by the ordinary procedure of separation of the variables to see that the functions =

p,

Sturm -

A a*Fla+- &&a = 0.

pk (2)

. . . ,

it is easy

k=(ki,...,W,

of the problems +

b+k

=

0,

A*pk

-

qkpk

=

0

and

hk =

$)

hk,,

a=i

The negative definiteness of the operator A* follows from the positiveness of the eigenvalues of problem (31) established in 1191. By the same token the convergence of the iteration process (29) is established. In particular, on the basis of the results of [81 for finding a solution of system (9) to within E it is sufficient to perform 0 !1, ; IL), 0 e

ho =

min !a, 4

interations

by method (29) with special

choice of the parameters T,. Note. It is easy to see that the negative definiteness of the operator A* is not violated if, instead of the requirement of strict posiwe require the positiveness of only tiveness for the coefficients “nf, one K and equate the rest to zero.

70

V.R.

.4ndreyev

In conclusion I wish to thank A.A. Samarskii for discussing the results, and for his valuable advice and constant interest in my work, and E.G. D’ yakonov for a number of suggestions on the editing of the paper.

Translated

H. F. Cleaves

by

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