517.3
. . , . . , . .

. . . , , .
: , , .
Abstract. A Dirichlet boundary value problem for the Laplace equation in a threedimensional layer with a local perturbation of the boundary is solved by the method of boundary integral equation (IE). The unique solvability of the IE and its Fredholm property are proved. A Galerkin method of the IE numerical solution aimed at the use of parallel computations is developed and justified.
Key words: perturbed layer, boundary integral equation, Galerkin method.

. [1, 2], .
- , [3], [4], [5] . , , , , .
, , . [6] ( ) - - . , .
Q, Q. -
, .
, , . , , () . , (, ) ( ) , . , .
. , , , . , , . .
1.
:
u = 0, x = (1, X2, x) eU, (1)
u 1 = ^ u 1/ = 0 , (2)
|| = 0(-), |^| = ∞^ ^^, := V12 + 2 , (3)
= {: (1, 2) < 3 < 1} , ( = 0 3 = 1), , (1, 2). , 3 =(1, 2) , , 8 = , 0
2 - 00.
1 2
, 0 < (1, 2) < 1 ( ) ( ). = {: (1,2)0, 3 = (1,2)} .
( ) () .
:
( 1, 2, 3) 2() ( \ 0),

1) = 0 ;
2)
| =-(1,2,3),
(1, 2, 3) ( = 0 );
3) (3).
2.
1. (1)-(3) . . 0 = {: 0 < 3 < 1}
= {: 0 < 3 < 1, 2 + 2 < }
. 2, 1-3, , 0 = - 2 (1)-(3) . 0 ,
|0| < |0()|.

*
. (3) 0,
( ) < |() ^ 0,

*
, 0( ) = 0 0 () = 0 , , .
, .. .
, ( ), , ( ) .
2,
! = 0,, 2 = 0, ,
1 =^ 1 \\ = 0; 12 1 =^2, 21\ = 0,
, , 94
, , , . , :
|1 - \1 - ^
.
3.
(1)-(3) , .
= (, ), , 0, ={ :0 < 3 < 1} ,
[7]:
( \

(, ) =-! V 4
=-~
1
1
- -27|
- + 2 .
= (0, 0, 1) = (:, 2, -).

1

=0
= 1

=1
= 0
, , 0, 3 = 0 3 = 1.

= 7(1 - 1)2 + (2 - 2)2 + (3 - 3 - 2.)2

) = >/(1 -1)2 + (2 -2)2 + (3 + 3 - 2])2 . :
1
= V
4
.=-~
( \ 1 1
(4)
V, - :
( - uv)d = (V -

(5)

^
n - ; = cos !-------+ cos 2-----+ cos --------
dn 3xi dx2 x
n, al, a2, - Oxl, Ox2, Ox .
.
cos al = I
dxl ]
l+ ' 2 +
dxl j
dx2
.
cos 2 = I
dx2 }
l+
'2 + '2
dxl
dx2
cos = -lI
l+
'2 + '2
dxl
dx2
v = GU (5),
f (GU - uAGu )dx = )GU ^ - dGU '
U Q n
dn dn
d .
, , - (1)-(3). , 5- = . , ,
() = jjGU 4~d - jjd,
JJ dn dn
Q
Q
dG
dn
()
GU - ,
dG
U
dn
= cos al
dG
U
dG
U
dG
U
- + cos a2 ---------------+ cos -
dxl dx2 x

:=

(6), , , , ;
dG
n
U
-d .
[8]. , ^ (6)
-(),
() = ^-^() + 1 |(), .

,


= 2 ) + (), .
4.
(7) :
+ = /

=4 7- ;
:= 1+2 ; ,1^1
4
= -

- -
;
2 = 1^ ;
(7)
(8)
(9)
(10)
(11)
(12)

(13)


-1
(, ) = 4
] =-
1
+
4 ^
]=1
1
- -2]'|
- + 2 ]3 1
+
| - - 2 ]'|
- + 2 ]3
(14)
, - , 3. . , , ; 2. , - - .
Q = {} -
2
: ^ , {} - , . g () ga = g . 5 !() () - ,

=II
'(2) :
(,)' = {() (>(^), |12 = (,)'

, , , 2. , . , '() .
' ' () ' () , [9]:
' () = { | : ' ()}

' () = { ': 8 }.
(7) ~±- - 2 (); I 2 ();
__1__ 1 _ 1
: 2 () ^ 2(), : 2() ^ 2().
, ( ) .
(, ) -
( = ), .. ][ (), [7] [10] 2 .

~_1 - 1 1: 2()^2().
1
1 () ( 1 ()),
*
_
*
-

2
*
= . , 1 .
,
- 1
+ : 2 () ^ 2(),
( ) [11].
1 , , -
+ = 0 ( I = 0) . , -
^
, , = = 0 (6), -
(1)-(3), 1. : +
1 -
( + )-1: 2() ^ 2 ().
, .
~
2. 2 () + = I ( (7))
1
I 2().
(7), (1)-(3) . 0:
(, )()12 = (), = ( 2) 0; (15)
0
( 2) = 2 |( 2, ( 2) ) + ) ( 2, (1, 2) ); (16)
(1,2;1,2) = (1,2,(1,2);1,2,(1,2)); (17)
( 1, 2) =(1, 2, ( 1, 2))
1 +

2
(18)
(15)-(18) .
5.
(8) [12]:
(( + )N,) = (/,), = 1,...,N . (19)
n HN - , uN - , Hn = 12 - . (, ^ ' ,
' = 12 () = /2 (), (, /} = JJ/d .

[3] [12] , (19) ,
inf ||un | 0, N ^ (20)
uN N
.
(15) . = , := {: 0 < xi < ai,0 < 2 < a2}, .
(xi, 2), xi = ihi, 2 = ih2,
hi = h2 = , Ni > i, N2 > i, i = 0,...,Ni, j = 0,...,N2.
Ni N 2
:
u ij = (( xi xi)( xixi+i)) (( 22)( 22+i)),
%(t) = 0 t <0 %(t) = i t > 0. ,
nj
:= {: xi < xi < xi+i, x2j < x2 < x2+i}.
,
^-1N 2 -1
= /, = 0,...,N1 -1,1 = 0,...,N2 -1, (21)
=0 7=∞
1 / :
1 = // (1, 2; 1, 2 )<^^2<^1<^2 ; (22)
1
/1 = // ^(1, 2 ^1^2 . (23)

\|/ N1 N2
N -1N 2 -1
V^,N2 = ∞. (24)
=∞ 7=∞
, , , , .

, . . .

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- , , ,
E-mail: mmm@pnzgu.ru
,
E-mail: matematik3@mail.ru

,
E-mail: al.tsvetkov86@gmail.com
Smirnov Yury Gennadyevich Doctor of physical and mathematical sciences, professor, head of sub-department of mathematics and supercomputer modeling, Penza State University
Shcherbakov Anton Alekseevich Postgraduate student,
Penza State University
Tsvetkov Alexander Vitalievich
Postgraduate student,
Penza State University
517.3 , . .
/ . . , . . , . . // . . - . - 2012. - 1 (21). - . 92-102.