Computational method for a singular perturbation problem via domain decomposition and its parallel implementation

Computational Method for a Singular Perturbation Problem via Domain Decomposition and its Parallel Implementation I. P. Boglaev and V. V. Sirotkin Institute of Microelectronics

Russian Academy of Sciences Moscow District, 142432 Chernogolouka, Russia

Technology

ABSTRACT In this paper, a computation method for the nonlinear problem with a small perturbation parameter via domain decomposition is presented. We apply a combination of the two approaches to solving the problem: iterative algorithms for domain decomposition and a grid refinement technique on subdomains. Two iterative a&orithms are examined: the first one is the Schwarz alternating procedure and the second algorithm is highly suitable for parallel computing. Numerical examples are provided.

1.

INTRODUCTION We are interested

in iterative

algorithms

for domain

decomposition

that

reduce a given problem to sequences of boundary value problems on subdomains. The iterative algorithm highly suitable for parallel computing has been proposed in [2]. In this paper, we analyze and illustrate this algorithm solving the following singularly perturbed boundary value problem:

_&u(x)

= Ed =f(x,u),

fU>pi,

Elsevier Science

no,

&=const>O,

APPLIED MATHEMATICS

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x E

f& = (0, I>>

E fro x F

AND COMPUTATZON 56:71-95

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(f”

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72

I. P. BOGLAEV

AND V. V. SIROTKIN

where E = p2, p is a small positive parameter (the prime denotes differentiation) and the function f(~, U) is sufficiently smooth. It is well-known that under the given conditions there exists a unique solution to problem

(1)

and that the following estimates

hold:

n(x) = exp(-&X/p) + exp[-Pdl -x)/PI, where solution

C denotes

a positive constant

of problem

independent

(1) has the boundary

of p (see [l]).

layers at x = 0,l

Thus,

the

and the size of

these boundary layers is h, = pj ln( p>I/&,. The traditional numerical methods for solving singular perturbation problems require a fine mesh covering the whole domain in order to resolve local details. These methods are inefficient since the fine mesh is not needed in those parts of the domain where the solution has a moderate variation. The effective numerical method for singular perturbation problems is based on the special nonuniform grids [l, 41, which are constructed by using the estimates

of derivatives

of the

exact

solution

and the analysis

of the

consistency error. “Natural” decomposition

of the original domain KI, will be introduced: the boundary layers (i.e., the regions of rapid change of the solution) are localized in subdomains. We apply a combination of two approaches to solving problem (1): the iterative algorithms for domain decomposition and the grid refinement technique on subdomains. The structure of the paper is as follows. In Section 2, we present two iterative algorithms for domain decomposition in the case of decomposition of the original domain into two subdomains. The first algorithm is the Schwarz alternating procedure and the second one is highly suitable for parallel computing. In Section 3, we establish convergence properties of these algorithms. In Section 4, we analyze a version of the second algorithm with two inter-facial problems. In Section 5, we generalize the two algorithms from Section 2 to more than two subdomains. We end the paper by giving numerical results in Section 6. 2.

ITERATIVE

ALGORITHMS

For simplicity, we assume that the solution u(x) to (1) exhibits the boundary layer only at x = 0, i.e., the reduced solution satisfies the boundary condition at x = 1.

Computation Introduce

the overlap decomposition

into the two subdomains a, We means

73

by Domain Decomposition

fi,

= (0, XT),

emphasize

here,

of the original

fl,

Q, = (0,1)

O
= (Xl> I),

that in this case,

that the boundary

domain

and R,:

“natural’

layer is localized

decomposition

in the subdomain

of R,

CI,, and the

region, where the solution is smooth, is included in a,, i.e., xI > h,. We consider the two sequences of the function {u”(x>},{~“(x)}, rr > 1, satisfying the following problems: L@(x)

=f(x,n”),

x E R,,

u”(0)

L,w”(x)

=~(x,w”),

x E &,

UP

= Ua,

= w”(r,>,

= ur”;

w”(1) = ul,

= w;,

where L, and uO, ui are determined in (1). Now construct two iterative algorithms. The first one, Al, is the Schwarz alternating @+I

u”(xr)

procedure

&z”(x)

(2b)

[5]

wl” = nn(rl)>

(3)

where the initial guess u,! should be prescribed. As is clear from these formulas, algorithm Al is a serial iterative dure. The second algorithm,

(2a)

A2, is constructed

=f( x, Z”),

using the interfacial

x E w = (X,, XT>,

z”( X,) = u”( X,),

proce-

problem

(da)

z”( X,) = w”( X,),

where X, < xI < x, < X, (the scheme of the decomposition is indicated Figure 1). The boundary conditions in (2a) and (2b) are determined by urn+1 = Z”( XT), The initial guesses

WIn+l = z”( x1).

I

0

(4b)

u,! and w: are given.

%

x1

J, xl

%

I

w FIG. 1.

xr

t

‘r

in

1

74

I. P. BOGLAEV

AND V. V. SIROTKIN

On each iteration step of algorithm A2, problems (2a) and (2b) are solved concurrently. Thus, algorithm A2 can be carried out by parallel processing. 3.

CONVERGENCE Now we establish

RESULTS convergence

properties

Al and A2. In the

of algorithms

following lemmas, we obtain the required results necessary Introduce the linear two-point boundary value problem

where the coefficient

LEMMA 1.

b(x)

is smooth with b(x)

If y( x) is the solution to (5), then

below.

> /3,“.

fen-all

x E II, we haoe the

estimate

ly(x>l where

=sPfi(X)lYlI +

cp!Y(4lY,I~

cpk lz denote th e solutions of the following

linear problems

PROOF. Let Y(X) be the solution of the problem

L,Y( x) - P,;‘Y( x) = 0,

x E R,

Y( x*) = 12/J, Y( xp) = I yal.

From the maximum principle for the operator (L, Y(X) > 0, x E 0. The functions (Y + y 1 satisfy

L,(Yfy)

-b(Yky)

=(&-b)Y,

x~0,

-

P,‘),

we conclude

(Y f

that

y)q,x2 a 0

Using the inequalities Y(X) > 0 and b(x) 2 /3,“, again by the zaximum principle for the operator (L, - b), it follows that Y * y > 0, x E R. This is equivalent to 1y(x)1 6 Y(x), x E fin. Since Y(x) can be written in the exact

Computation by Domain Decomposition

75

form

Y(x) = Pofi(+j,l+ then we obtain the required

cpfi’(ahL

x E

a,

estimate.

LEMMA 2. The solutions ‘pk “(x)

n

to problems

(6)

satisfy the following

inequalities: 0 < cpk”( c$A(

x) < 1,

X) + crr~~‘( X) < max(c’,

X E n;

(7a) X E I=&

cl’),

(7b)

where constants cl, cl’ > 0;

d(x)
cpf(x)
PROOF. Proof of this lemma follows immediately

(7c)

from the exact expres-

sions for ‘p2 “(x):

d( ~1= sinh[L-$(Q- X)/rU]/Sinh[ k$(~, - x1)/~], pofi’( X) = sinh[ /3,( x - x,)/p]/sinh[

Now we formulate and A2.

and prove the convergence

&(Q

- “J//J].

results for algorithms

Al

THEOREM 1. Zfxl , (3) (i.e., the Schwarz alternating procedure) converges to the solution of problem (1) with

76

I. P. BOGLAEV

linear rate 0 < q < 1. The following

AND V. V. SIROTKIN

estimate on q holds

PROOF. We introduce W”(Z)

the functions 5 “(x) = u”(x) - V”-‘(X), 6 ?x) n > 2. From (21, (3) and the mean-value theorem,

- w*- l(x),

follows that 5 “(x)

and 5 “(x)

f;(x)

O”(x)),

where

=fu(x,

= it

satisfy the following problems:

O”(x)

E (v”-l,

IJ”). Denote n > 2.

6” = max[15”(x.)l,15n(~~)l],

From Lemma 1, we conclude that

Using the boundary

conditions for

l”(x),

t”(x)

and again Lemma

1, it

follows that

(8)

77

Computation by Domain Decomposition and using estimates (7a) f rom Lemma

2, we conclude

that 0 < q < 1. From

this, it follows that

where C = 6l. Thus, {b”(x)} + O,{t”(x)} + 0 as n + 00. We shall now show that {V”(X)} and {w”(x)} are convergent

sequences.

We have n+k-1 Iv n+k(

x)

-

v”(

x)I

c

<

n+k-1 Id”(x)

-

d(x)I

<

i=n

c

I[i’l(

< q(l - q)-‘l<“(x)l<

x E ill,

C(1 - q)-lq”,

such that (V”(X)} is Cauhy’s sequence. If lim(u”(x)l = u*(x), ~0, then, proceeding to the limit as k + m, we obtain

1cE al,

I”*Cx) - u”(X)1 Q C(1 - q)-lq”, Consequently, {u”(x)} converges with the linear (8). A n al o g ousl y, 1‘t can be proved that {wn(x)] with the linear rate q. Now we prove that u*(x) and w*(x) satisfy boundary conditions u*(O) = ~a, u*(x,) = lim n-m w;, w*(l) = u,, respectively. We note that a function

f(& y>, x E gral equation

Y(X)

x)I

i=n

y(x)

0, = (x1, x2>, y(x,) (Green’s

formula)

= yl@&>

+

x E fin,, n +

n > 1.

rate q, where q is defined in converges to w*(x), x E fi, problems lim,,,u~”

(2a), (2b) with the and w*(x,) =

is the solution of the problem = yi, y(xz)

= yz, iff it satisfies

where (DA” are the solutions of the following linear problems: x E

c4,

a$(q)

= a$&)

@A(x2) = a$( q) =

0,

=

1,

=

the inte-

yz@;‘(4 + J*PGd~,~)f[~, y(s)] h, x1

L EcDz,zz = 0, II

L, y(x)

I. P. BOGLAEV

78

and G,(x, s> is the Green’s function homogeneous boundary conditions.

for

the

AND V. V. SIROTKIN

operator

Using this integral equation for V”(X) and proceeding integral term as n -+ 03 (here, we use the fact, that uniformly

to u* ( x) on fir),

we conclude

L,

on

IR with

to the limit in the {u”(x)} converges

that

From this, it follows that u*(x) satisfies the integral equation and thus, u*(x) is the solution of (2a) with u*(O) = u0 and u*(x,) = lim,,,u,!‘. Analogously, it is true for w*(x) on II,. Define the function

where x, < x* < x,. From (3) we have lim I.L$ = u* ( xl).

lim urn = w*( xr) ,

n-+m

n-+m Thus,

u*(x),

w*(x)

u*( x,), w*( x,.). u*(x) = w*(x),

on

w = (xl, x,)

This proves the convergence

COROLLARY

PROOF.

for 9.

satisfy the same boundary

conditions

From the uniqueness x E W. Consequently,

1.

property of the solution we obtain u*(x) is the solution of problem (1). n of algorithm Al.

For algorithm

From (8)

evaluating

(21, (3)

the following

bound on 4 holds

9 by (7b), (7~) we get the needed

estimate W

79

Computation by Domain Decomposition THEOREM 2. converges following

where

Zf x, < Xl < x, < x,, then the iterative algorithm (2), (4) to the solution of problem (1) with linear rate 0 < q < 1. The

estimate on q holds

cpA’$x>, cp&(x) from (6).

PROOF. Analogously to the proof of Theorem 1, we introduce the functions c”(x) = u”(x) - un-l(x), .$“(x) = w”(x) - w”-l(x), x”(x) = z”(x) - znel(x), n 2 2. From (2), (4) and the mean-value theorem, we conclude

that 5 “(xl,

e”(x),

l”(O)

&8”(x)

From Lemma

and x”(x)

= 0,

S”(q)

-f;l(x)5yx)

satisfy the problems

=

/p(Q

=

0,

x E R,,

1 we have

Denote n > 2.

I. P. BOGLAEV

80

By using conclude

1 and

Lemma

the

boundary

conditions

AND V. V. SIROTKIN for

5 “(x),

x”(x),

we

that

From (7b) it follows that

Analogously, 8” < q6”-‘,

the estimate

< q8 “-’

( (“(xl>1

be proved. Thus, we obtain

n > 2,

(9) Using Lemma

2 it follows that 0 < q < 1.

Repeating the steps of the proof complete the proof of the convergence

COROLLARY 2.

4 <

47

For algorithm

formula

(21, (4), we have the following

(8),

we n

bound on q

4=max{exP[-Po(Xr-Xl)/r-Ll,exp[-p,(x,-X1)/~]}

PROOF. From (91, evaluating the required estimate. 4.

of Theorem 1 after of algorithm A2.

q with the estimates

A VERSION OF ALGORITHM INTERFACIAL PROBLEMS

A2 WITH

from Lemma

2, we get n

TWO

Here we give a convergence result for a version of algorithm (2), (4) with two interfacial problems. Introduce the following subdomains wi, wa as in

Computation

by Domain

Decomposition

81

Figure 2:

6J1

(XL xr’),

=

Instead of problem lems (i = 1,2):

(x12,xr”),

w2 =

(da), we consider

&zn( x) =f( x, q>

n

w1

w2

=

0,

the following two interfacial

x E

q=(xi,

prob-

xj),

( lOa) q( xi) = u”( Xl), The boundary

conditions

z”(Xf)=WyX;).

in (2a) and (2b) are determined

?Jrn+l= 22”( xr),

WI“+l

by

= z;(q).

( lob)

The initial guesses

uf, w: should be prescribed. It is clear, that for n-f=e problems (2a), (2b) can be solved concurrently,

the same is true for the problems from (lOa). Thus, algorithm (2), (10) can be carried out by parallel processing.

THEOREM3. If X1’ < xI < X,! < XF < x, < X,?, then dgurithm (2), (10) converges to the solution of problem (1) with linear rate 0 < q < 1. The following

where

estimate on q holds

cpz$ x>, cphz(x) from (6).

%

,

0

x;

x1

I I

w

1

I

Xi FIG.2

XI

,, x r 0 2

,

R2

x:

t

1

I. P. BOGLAEV AND V. V. SIROTKIN

82

PROOF. The proof is the same as that for Theorem functions

l”(x)

/y,“(x) = q?(x) value theorem,

= V’(X)

-

vnP1(x),

t”(x)

= W”(X)

2. Introducing - W”-‘(X),

- z:-yx>, i = 1,2, n 2 2, from (2), (10) and by the meanit follows that C”(r), [“(x), and xi”(~), i = 1,2, satisfy the

problems

x:(x;)

= y(x1),

xjyq = eyxr>.

Denoting 6”

and using Lemma

Using Lemma

I~yxr)I

=

the and

=

max[l~n(X.)l,15"(xI)Il~

1, we conclude

1 and the boundary Ixz”-l(x,)l

that

conditions

< Ix;-l(x~)lcp,l,(~.)

= I[“-‘(X~)lcp,I,(x,)

n > 2,

for C ‘Y x>, x;( x>, we have +

+ lsn-yxr2)bi:(Xr)

Ix2”-1RvaxJ

83

Computation by Domain Decomposition Applying estimate

(7b) from Lemma

2, it follows that

In the same way, we can obtain the estimate

Hence,

6” Q qS”-l,

where

= m=[

9 = m=(91,92)

cpfl(X1”>, cpA,(JC)]

because X: < Xl” and X,! < X,?. Repeating the steps of the proof of Theorem 1 after (B), we complete the proof of the convergence of algorithm (21, n (10).

COROLLARY 3.

PROOF.

5.

(21, (lo),

the following

bound on 9 holds

?=m={exp[-6(X,‘-~l)/~],exp[-Po(r,-X:)/IL]}.

9 <4>

estimate

For algorithm

From

Theorem

3, evaluating

9

by (74

we get the

for 9.

MULTIDOMAIN In this section,

needed W

DECOMPOSITION we generalize

algorithms

Al

and A2 from

Section

2 to

more than two subdomains. Here, we consider that the solution of problem (1) has the boundary layers at x = 0,l. Introduce the multidomain overlap decomposition of the domain R, = (0,l) into the subdomains Rj, j = 1,2,. , M:

Inj =

(x_1-‘,x’;), aj n .n,+, # 0, x0 1 = 0 >r> XM = 1

j=

1,2 ,...,

0



M - 1.

1,

I. P. BOGLAEV AND V. V. SIROTKIN

84

In the case of “natural” decomposition, we assume that the boundary layers at x = 0 and x = 1 are localized in the subdomains fi, and R,, respectively. On each subdomain Rj, j = 1,2,. n > 1 satisfying the following two-point

where

L,, uO, and ui are determined

A multidomain decomposition dure (algorithm Al from Section

n+l

“‘1 U

= gJ_l(r;-1),

n+l = y++l’( xi),

‘J

, define the sequence boundary value problem:

in (1).

version of the Schwarz 2) is given by

j = 2,3 1’.‘> M >

j=M-l,M-2,.

{V;(x)},

. . >1,

alternating

u;+l I u,!!+‘=u

proce-

=u”,

l>

n

2

1. (12)

The initial guesses

u(,, j = 2,3,.

, M, should be prescribed.

THEOREM 4. Zf Rj (7 flj,, # 0, (x/ < x!), j = 1,2,. . . , M - 1 and ‘j ” ‘j+S = 0, (x{ < x3,’ ‘1, j = 1,2, . . , M - 2, then the m&domain version (ll), (12) of the Sc h warz alternating procedure converges to the solution ofproblem q holds:

(1) with linear rate 0 < q < 1. The following

where cp&Cx 1, ~ofi:(x ) from

(6).

estimate on

by Domain Decomposition

Computation

2,.

85

PROOF. Introduce the functions 5j”(x) = vj”(x) - y”-‘(x), j = 1, . , M, n > 2. From (ll), (12), by the mean-value theorem, it is easy to

verify that .$n(x>, j = 1,2,

ujw q(,jP1)

.,M

satisfy

-f,:(ajn(x)

= q;+-‘),

5jn(X;)=5j;l(~!),

=

0,

x E Rj =

j = 2,3,...,

M,

(*:‘-1, d), [;(x;

j=M-l,M-2,...,I,

= 0) = 0,

l;(x,M=I)=o.

Denote

6” = max (max[ I$“( rc-‘)I; j Using Lemma

Analogously,

1 and the boundary

conditions

I qn( x!)l])

.

(121, we have

we conclude

From (13) and (7b), it follows that

Now we show by induction

I&(

d+l)l r

G

the following estimate

max[ an-l,

li;:k(x!‘k)l],

k z 2.

(15)

86

I. P. BOGLAEV

From (141, this is true for k = 2. Using the induction

AND

V. V. SIROTKIN

hypothesis

for k = k,

and using (14) for the index j + k,

< max

I ~n--l,l~t:k,+l(Z!+k”+l)l].

we obtain

= max The induction Since

[ a”-‘,

I~~~,+l(X~+ko+l)l]

is complete.

lG(x, M = 1) = 0, substituting

Substituting

This estimate

this estimate

k = M - j in (15), we conclude

that

in (141, it follows that

and (13) yield 6” < qS”-‘,

Repeating the steps of the proof of Theorem 1 after formula (81, we complete the proof of the convergence of algorithm (111, (12). I

Computation by Domain Decomposition COROLLARY4.

For algorithm

87

(111, (12) we have the following

PROOF. From Theorem 4, evaluating 2, we get the required estimate.

q with the estimates

x E

flj i-l .nj+1 c wj,

wj

as in Figure 3. The boundary

wj

n

=

(‘/>

wj+l

conditions

we introduce

1,2 , . I M -

=

x:‘
0,

1,


in (11) are given by

n+l = zpl( x_l-‘), “1, The initial guesses

decomposition,

j

‘i)>

=

from Lemma n

To generalize algorithm A2 to a multidomain the (M - 1) interfacial problems

LEZjy( X) = f( X, ZJ),

bound on q

II;+1 = z;( xi).

IJ~, I$, j = 1,2, . . . , M -

( 16b)

1, should be prescribed.

Algorithm (111, (16) dan be carried out by parallel processing because the M problems for (11) can be solved concurrently; the same is true for the (M - 1) inter-facial problems from (16a).

THEOREM 5. Zf Rj and Qj n flj+2 = 0, verges to the solution

- y-1 xJ-’ 1

J-1 ,x1 ”

n LRj+l# 0, Rj n flj+, c oj,j = 1,2,. j = 1,2,. . , M - 2, then algorithm (10, (1)

of problem

linear

rate

(16) con0 < q < 1. The

RJ

xJ-l -I r, wJ-l

with

. , M - 1,

‘J+l-

x; xJ-1 r FIG. 3.

I W J

88

I. P. BOGLAEV

following

AND V. V. SIROTKIN

estimate on q holds:

4 Q 41

m=(ql~%)~

[+Pi> + ~fyP~)]

= ,$Z_,

q2=

7

max [ cp~i(V> 2
where cphl(x>, CPA:(X) from

2. Introducing G”(x)

using a procedure

similar the one used

the functions

= y”(x)

xJ”( x) = z,‘(x)

>

(6).

PROOF. This result is established for Theorem

+ ~qx!-‘>]

. .

-

y”-l(x),

- q-y

j=1,2

,...,

M

x),

j = 1,2,. ..)M - 1, n 2 2, and using the mean-value (16), we conclude that 6°C x), j = 1,2, ...,M satisfy

X) - ft(

L,q”(

5jn(,/-1)

q”(x!)

=

X)5jn( X) = 0,

x;:~+-~),

j

= x;-‘(xjr),

L,Xy(X) xj”(xlj)

Denote

=

-fi,(X)Xj”(

=

q(xj),

XE

2,3,...,

j = 1,2 ,...,

and ,yjn(x), j = 1,2, . . . , M -

and

aj

=

(

M,

M -

theorem,

x_l-‘>

&“(x;

1,

‘!)>

= 0) = 0,

&Y&x: = 1) = 0,

1, satisfy X) = OT

xj”(x!)

LXE =

from (ll),

Oj

=

(‘/>

L-j;1(x!).

‘!)

Computation

by Domain

From Lemma

Decomposition

1, it follows that

max x;~l~~(x)l .i [ I From Lemma conclude

1 and the boundary

G 6”.

I

conditions

for q”(x)

j = 1,2 >. . . >M - 2, Analogously,

the estimate

can be obtained:

and

(<;(x,M

xj”(x>,

= 1) =o).

we

I. P. BOGLAEV

90

Hence,

6” < 96”-‘,

AND V. V. SIROTKIN

where 9 = max(91,92),

Applying the estimate (7b), we obtain that 0 < 9 < 1. Repeating the steps of the proof of Theorem 1 after formula (B), we complete the proof of the convergence

of algorithm (111, (16).

COROLLARY 5.

For algorithm -

NUMERICAL

(111, (16) the following 4 = m49,,

91

=

92

= l;;z_l . .

(L’cosh[

Theorem

5, evaluating

PROOF. From estimate for q. 6.

n

l<;;;

1 (Wosh[

bound on 9 holds:

921,

P,( XF’ - +‘P])

>

P,( 4 - X:)/P])

.

q by (7d)

we get the

needed n

EXAMPLES

We present numerical results for two test problems using the iterative algorithms like Al and A2. We emphasize here, as is clear from Theorems 1-5, that the convergence results for the iterative algorithms are independent on the singularly perturbed character of problem (1). To construct effective numerical methods for algorithms Al and A2, it is necessary to take into account the fact that the solution to (1) has the boundary layers at x = 0, 1. As we noted before, the effective numerical methods for singular perturbation problems are based on the special nonuniform grids [l, 41. Th e main property of these methods is uniform in a small parameter convergence. The special grids are constructed in such a way that the number of grid points inside boundary layers is approximately the same as a number of grid points outside layers. In the case of problem (1) with the boundary layer only at x = 0, we assume that x2 > h, (h, is the size of the boundary layer). If this inequality is fulfilled and, on the subdomains R,, Sz,, the special nonuniform grids are

Computation

91

by Domain Decomposition

used, then the cost of the numerical method (2a) on fi2, is equivalent to the cost of the numerical method (2b) on fia. This property is very important for implementation of algorithm A2 on parallel computers since it permits the synchronization of computational times for problems (2a) and (2b) on A2iteration. Thus, we shall apply the combination of the two approaches to solving of (1): the iterative algorithms like Al and A2 for domain decomposition special nonequidistant grids on subdomains.

EXAMPLE 1. f(x,U)

We consider

problem

=exp[-(l-x)‘]

(l), where

-exp(-u),

It is easy to see that this test problem Introduce

a nonequidistant

layer [O, h,],

u,=O,

has the boundary

mesh {xi,

the mesh generating

layer only at x = 0.

1, . . . , N}. In the boundary

i = 0,

function

u,=O.

is the logarithmic

type function

from [l]: xi E [0, h,]:

xi =

pln[l

-

xi E (hE, 11: xi = xi-I

-

(1 -

+ (1 -

p)i/n,]/P,,

i = O,l,...,n,;

l,...,N=2n,.

i=n,+

h,)/n,,

If h, > 0.5, then we choose the following mesh:

X0

=

0,

xi=

xi_l

ho = min{0.5/n,;

+

i=1,2

hoYi-l,

-Pln[I

-

(I

,...,

n,,

- rcL)/n,]/P0j7

where y satisfies the condition n,-1

ho c

i=o

yi = 0.5;

i.e.,

x n.? = 0.5, and xi = Xi-i

+ 0.5/n E’

and the

i = n, + 1,. . , N = 2n,.

I. P. BOGLAEV

92

The differential

equation

from (I)

is approximated

AND

V. V. SIROTKIN

by a simple variable-

mesh formula. The nonlinear algebraic systems (after discretizations of (2), (4) and (10)) are solved by the Newton iterative method up to an accuracy of 10p5. The iterative algorithms (21, (3) - Al, (2), (4) - A2, and (2>, (10) A2M are finished to achieve an accuracy of 10p5. Let our domain decomposition conditions:

Xl -X/=X,I-x,-hh,,t, where

we denoted

as in Figures

1 and 2 satisfy the following

Xf-x1=x,-x+h X/ = X,,

X,? = X,.

We

x r

in,

choose

--,-h

x, = 0.5 - (h/Z).

We

suppose that the set {x2, x,, Xi, Xr, i = 1,2} belongs to our mesh. In Tables 1 and 2 we give the numerical results of algorithms Al and A2 for various values of Al. and h (the number of mesh points nE = 100). denote a number of iterations for Al and A2, respectively, to K.41, G2 achieve an accuracy of 10m5 (in the tables * denotes numbers KA1, K,, > 200). The tables contain the values of K,,, K,, for the two cases: bout = h/16 and bout = h. One can see that, in the first case, the inequality K,, 2 K,, is fulfilled if h/p < 1. In the case h,,,, = h, we have K,, - K,, for all values of p and h. If h/p s 1, then in both cases K,,, K,, = O(1). This fact is in agreement with Corollaries 1 and 2. Table 3 presents the numerical results hin

K,

=

bout

=

h/16.

for algorithm

We can see that this algorithm

A2M

is effective,

in the case i.e.,

K,,,

=

= K,,,

when h/p * 1. It is worthwhile to note here that when algorithm A2 (or A2M) is carried out on two parallel processors and the relationship K,, = K,, holds, then t, < t,,, where t Al, t,, are execution times for Al and A2, respectively.

EXAMPLE 2. We consider

problem

(1) with the discontinuity

function

fk u) f(x,u> = D(x) D(r)

=

exp

exp(-u),

[-(1 - 2x)“],

[ -(2 i exp

- 2x)““],

32E [o, 11, x E [0,0.5], x E (0.5,1],

where x = 0.5 is the point of discontinuity of the function D(x). The boundary conditions from (1) are chosen in the forms: u0 = 0, ui = 1. In this case, the solution to (1) has the two boundary layers are x = 0, 1 and the interior layer at x = 0.5 (see [3] for details).

93

Computation by Domain Decomposition TABLE 1 h 2r2 2-4 2-6 2-8

7; 11 21; 35 65; 111

2-lo

6 13 39 126

= h/16)

5; 6 6; 6 8; 13 23; 38 72; 123 2-6

*;*

2-2

introduce

function

5; 8; 23; 74;

1;:

El.

To

(h,,,

Ku; KA2

2-4

a nonequidistant

mesh,

we

3; 3 4; 5 6; 6 8; 12 19; 32 2-a

firstly

define

3; 3 3; 3 4; 4 6; 6 7; 10 2-'0

the

following

on [x1, x2]:

k(i7

Xl>

3) = i

kJE(i,

Xl>

h,,(i,

x1,

x2),

*,,(i

x

x

’

lp

i = O,l,. ..,n,;

)

2

)

i =n, + 1,...,2n E'

x2) = (x1 + h,) + (i - n,)(

x2

-

x1

-

hE)/rtE.

The logarithmic type function A,, generates the mesh points on [x1, x1 + h,] and A2E generates the mesh points with uniform density on (x, + h,, x2]. Now we divide the original domain a,, = (0,1) into the four equal subintervals

I, = (x”-‘,x’),

xp =0.25p,

p = 1,2,3,4,

x0=0,

x4 =

TABLE 2 h 2-2 2-4 2-6 2-s 2-10 P

K,,; KM2 7; 6 21; 19 65; 64 *; * *; * 2-2

5; 5 8; 7 23; 22 74; 72 *; * 2-4

(h,,, 5;s 6; 6 8; 7 23; 21 72; 70 2-6

= h) 3;3 4; 4 6; 6 8; 7 19; 18 2-s

3; 3 3; 3 4; 4 6; 6 7; 7 2-10

1

I. P. BOGLAEV AND V. V. SIROTKIN

94

TABLE 3 h 2-z 2-4 2-6 2-8 2-10 CL

23 63 193 * * 2-z

Using the mesh generating points: xi = h,(i, XZn,-i = xp -

xp-1,

XP),

h,(i,

xp-l,

K,,,(h,,

= hi, =

14 26 71 * * 2-4

9 14 26 69

4 8 14 24 59 2-x

2-*6

function

A,, we introduce

i = 0,l >. . . , 2n,, XP),

h/16)

the following

and

p=l

2nE,

i=O,l,...,

3 5 8 13

mesh

p=3;

p=2and

p=4.

Thus, we chose the special mesh points inside the boundary and interior layers at x = 0, 1 and at x = 0.5 and the uniform mesh points outside these layers. Discretization of problem (1) and the numerical method algebraic equations are chosen similarly as in Example 1.

for nonlinear

We use the multidomain decomposition: the domain &, = (0,1) is decomposed into the five subdomains Rj = (x-l- ‘, xi), j = 1,. . . ,5, where x; = O.ZS(j

- 0.5)

hj,,,=xj-X:=X;-&

- h/2,

hi = xi - x;,

hj = h, j=l

hiut=hout,

,...,

j = 1,...,4, 4.

We suppose that the set {xi, rj,, XI, Xi, j = 1, . . . ,4} belongs to our mesh. Tables 4 and 5 present the numerical results for algorithms (111, (12) - Al and (ll),

(16) - A2 (the number

of mesh points

n, = 25).

TABLE 4

h

K,,; K,

2-4 2-6 2rs 2-10

lr.

37;67 122; * *; * *;

2-z

*

10;16 30;52 98; 171 *;

*

2-4

(h,, = h/16) 7; 7 9; 15 27;47 89; 153 2-6

7; 7 7; 8 10;15 27;45 2-s

7; 7 7; 7 7; 8 8; 11 2-10

by Domain

Computation

95

Decomposition TABLE

h

As

h/p K,,,

5 (h,,t

K.4,; KAZ

= h)

2-4

37; 35

10; 8

7; 7

7; 7

7; 7

2-”

122; 121

30; 28

9; 8

7; 7

7; 7

2-s

*; *

98; 97

27; 26

10; 8

7; 7

2-10 CL

*; * 2-2

*; * 2-4

89; 87 2-6

27; 25 2-s

8; 8 2-10

in Example 1, for the case bout = h/16, we have K,, > K,, if < 1 and for the case bout = h, K,, 2: K,,. If h/p S- 1, it follows that K,, - O(1).

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Approximate

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Physics

I. P. Boglaev perturbed 1MACS

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I. P. Boglaev

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ings of the workshop

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boundary

value problem

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A uniform

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R. Vichnevetsky,

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solution

for the highest-order

Dresden,

‘91, (H.-G.

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