Tin học điều khiển tự động

TiJ-p chi Tin hoc va Di'eu khi€n boc,

T. 17, S.1 (2001), 10-16

DIFFERENCE
SCHEMES
OF GENERALIZED
SOLUTIONS
FOR A CLASS
OF ELLIPTIC
NON-LINEAR
DIFFERENTIAL
EQUATIONS
HOANG

Abstract.

It is known

(see [1], [2], etc.)

that

DINH DUNG

in many

applied

problems

the data

are nonregular.

The

approximate
methods for the problems of nonlinear differential equations with data belonging the Sobolev
spaces Wi, (G) are presented in [3 - 5]. In this paper the finite - difference schemes of generalized
solutions
for a class of elliptic nonlinear differential equations are considered.
The theorem for the convergence of
approximate
solution to generalized one and error norm estimations
is proved in the class of equations with
the right-hand

side defined

by a continuous

linear functional

in

WJ-I)(G).

Torn tlit. Nhie u ba.i toan t h u'c ti~n d u'oc dfin v'e gid.i cac bai t oan doi vo'i ph u'o'ng trlnh vi ph an r ien g voi
d ir kien kh6ng tro'n (xem [10]' [2)). Phuo-ng ph ap xfi p xl giai mot so b ai toan doi vo'i cac phtro'ng trlnh vi
ph an phi t uy en vci ve ph di thucc cac 161> ham k h d tich kh ac n h au (cac kh ong gian Sobolev WI; (Gll du'o'c

ngh ien cu'u trong c.ic cong trlnh [3- 5]. Bai nay xet luo'c do sai ph an, nghien crru su' h9i tu v a dinh gii sai
so cd a ngh iem bai t o an doi vo'i mot 161> phuong trlnh vi ph an phi t uyeri lcai ellip vo'i ve phrii kh ong twn d9

WJ-I)(G)).

c ao kie'u c ac ph ie m ham t uyen tinh lien tuc (cac khOng gian

1. INTRODUCTION
Let

G

be a rectangle

with

au
( X,U,-,aXl

6.u+T
where
being

the given
a nonegative

aGo

the.boundary

aU)
aX2

Consider

the following

=-f(x),xEa,

problem

u(x)=o,

(1)

xEaG,

E W 2-1 (G) - the space of continuous
linear functionals
on the space
integer,
the function
T(x, a), a = (ao, aI, a2), satisfies the conditions:

W~(G),1

f (x)

2

[T(x,a)

- T(x,b)](ao

- bo)

el2...)ai

~

- bi)2,

.=0

(2)

2

[T(x, a) - T(x, b)[ <

1/2

c, [2.:)ai

- bi)2]

,

i=O

where

e1,

J=

We shall

constants
(see [3, chap. 3, sec. 4)).
as in [6]. Consider
the generalized

1, 2, are the positive
use the same notations

solution

u(x)

of the problem

o

(1) in the space

W ~(G)

satisfying

the following

P(u, v)

=

JJ

+ T(x, u,

[6.u

c:
where

v( x)
One

the

is a function

has

conditions

v(x)
(2),

in the space

D (G)

equality:

::1'

:x:)]

v(x)dx

= -

WHG).
f(x)

E

f(x)v(x)dx,

of Schwartz

basic

[7].

functions
.

Then,

L2(G),

by [3] (chap.
there

exists

3, sec. 4), if the function
uniquely

a solution

au au
aXl aX2

ri-, u, --)

of integral

equation

W~(G) n W~(G) .

• This work is partially

supported

by the National

(3)

c

o

E

JJ

Basics Research

Program

in Natural

Sciences

(3)

.

satisfies

u(x)

E


DIFFERENCE

SCHEMES

OF GENERALIZED

2. CONSTRUCTION

°<

We first consider

OF DIFFERENCE

J(x)

the case where

SOLUTIONS

L2(G)

E

11

SCHEMES

and let G be the unit square

G

= {x =

X2)

(Xl,

n. = 1, 2}.
Let us introduce
in the region G a grid w with interior and boundary
grid points denoted
and,
respectively
[61.
To construct
the difference schemes one may take the test functions v (x) in the form:
X"

< 1,

_lk-kexp

where e = e(x)
natural number.

{-

4rrh h

vx==

( )

12

{

12

0,
O,Sh",

Let every gridpoint
x E w be corresponding
by the GS) u(x) of the problem (1) in e satisfies

P

"(

u. a

)

+O,Shl

1
= hlh2

J

= -RJ,

x E w,

'

x·E e,

(4)
xEG\e,

== {~= (~1'~2) : k" - xnl <

:£1

Ix l2 }

4h k kh

by w

x2+0

J

n = 1, 2},h"

being

the steplengths,

to a mesh e(x). The generalized
the following integral equation:

k being

solution

a

(denoted

..5h2

[~U(~)+T(~'U'U(I),:~,:~)]a(l)dl
(S)

(6)
One may rewrite

the equation

(S) as follows

(7)
where
x,+O,5h,

SiU(X)

J

1

=

h:
t

U(Xl,···,li,

... ,x,,)d1i,

u (±O.Gi)(

x ) --

U ( Xl,···,Xt

. ±O

1

Sh·

tl

•••

,Xn'

)

Now, to obtain the difference schemes of the oper ator (7) pre (u, a) one may approximate
the mean
integral operators
S, by the quadrature
formula of average rectangles
and the partial derivatives
by difference quotients
as in [61 (see 2.1). Hence, one get the following difference approximations
corresponding
to (7), (3):
2

K (y)

== 1 Pl'(y, a) = L

2

(aiYx,)

x, -

SlS2

y(x)=o,

L

aXi (x)Yx,

+ SlS2a(dT(I,

y(x), Yx" YX2) =

-
x

E w,

i=l

i=l

xE"

(8)
and (ef. [3, chap. 3, sec. 4])
2

L(y) == 2 P~(y, a) = LYXiXi
i=l

y(x) =

0,

x

E /,

+ SlS2a(I)T(I,

y(x), fix" fix2) =

-
x

E w,

(9)


HOANG

12

DINH

DUNG

where

u

1

= _[u(+I,)
hi

z,

u (±1,)

==

u (±I,)

1

- u] u- = -[u}

( x) -

u ( Xl)""

= a(-O.5,)(X),

a;

.

Xl

=

'P

U(-I,)]

hi

x,

± h,t,

... ,

Xn

)

>

).,

1,

1., _

Rf.

(10)

Note that by [3] (see chap.3, sec.4) there exists uniquely
= -'P and, then, of the equation IP::(y,a).

a solution

of the operator

equation

2P,:(y,a)

3. ESTIMATION OF THE CONVERGENCE RATE
Estimate
Consider

3.1.

z

now the method

error and the approximate

one of the scheme (8) and (9).

scheme (9) with 'P defined by (10), (7). Denote the method

the difference

error by

= y - u, where y being the solution of the problem (9). It follows from (9) that.
= -tP(x),

Lz
where tP(x) is the approximation

x

E w;

z(x) =

x

0,

error of the scheme (9):

\{I(x) =
From (10), (7) and by formulas
and h2, one has

'P

+ Lu.

(10), (11) in [6,sec. 2], for the sufficiently

2

'P =

small mesh sizes hi

2

"'[
L

-

(11)

E /,

aa;:

(aU)-O.5,)]

S3-;

;= 1

+ S1S2 (",aaaU)
L 7J.7J.

x

'

i=1

,

- S1S2T (au
~,u(~), -,

-aU) , x

a~1 a~2

~,

~,

(12)

E w.

Thus,
2

?

\{I=L'"

[Ux,-S3-;

(

;=1

aa;:
au

)-0.5,)]

x

~ aa au
+SIS2La7J.

";=1

- SIS2

~I

[T(~, u(d, aau , ~)
~I

a~2

-

~,

T(~, u(x), UX1 (x), UX2 (x))].

By (9) one has
2

= -SIS2[T(~'Y(X),yxl'Yx,)]

LoY==LYx,x,

-'P=='Po,

(13)

xEw.

i=l

Then,

LoX = Loy - Lou

== -\{Io(x),

x E w;

z(x) =

0,

x

(14)

E T

From (12) - (14) it follows that
2

\{Io

= Lou -

'Po

= '"L

2
UXiX,

i=1

-

.'"L

aa:;

[S3-i (aU)(-O.5,)]

,=1

2

aa au
+ S1S2 '"L 7J.7J.

x

t

i=1

,

+ SIS2T(~, y(x), YXl' Yx2) - S1S2T (~, u(d,~,
Hence,

2

-Loz

= -

L
i=1

~,

~,

aU).

a~1 a~2

2

Zx,x,

=

L(1'/dx, + >"0 + (30,
i=1

x E W,

(15)