that in many applied problems the data are nonregular.. The approximate methods for the problems of no linear differential equations with data belo ging the So olev spaces W i, G are p
Trang 1c hi Tin va Di'eu khi€ T.17, S.1 (200 1 ) , 1 - 1
Abstract It is known (see [1], [2], etc.) that in many applied problems the data are nonregular The approximate methods for the problems of no linear differential equations with data belo ging the So olev spaces W i, ( G ) are presented in [3 - 5] In this paper the finite -difference schemes of generalized solutio s
for a class of elliptic n nlinear differential equations are considered The theorem for the convergence of
appro imate solutio to generalized o e and error norm estimations is proved in the class of equatio s with the right-hand side defined by a continuo s linear functional in W J- (G)
Torn tlit N e u ba toan th u'c t~n du'oc dfin v'e gid cac bai toan d i vo'i phu' n trlnh vi p an r ieng voi
d ir kien kh6ng tro'n (xem [1 ]' [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 uyen vci ve phdi thucc cac 1 >ham kh d tich kh ac n hau (cac kho g gian Sobolev WI ; ( G l du'o'c ngh ien cu'u trong c.ic cong trlnh [3- 5] Bai nay xet luo'c do sai ph an, n hien crru su' h9i tu va din gii sai
so cd a ngh iem bai to an d i vo'i mot 161> ph ong trln vi phan phi t uyeri lcai ellip v 'i ve p rii kh ng twn d9
cao kie'u cac phiem ham tu en tinh lien tuc (cac khOng gian WJ - (G)
Let G be a retan le with the.boundary aGo Consider the following problem
6 u T X, U ,-,- =-f x ) xEa, u(x) = , xE aG,
where the given f (x) E W2-1(G) - the space of continuo s linear functio als on the space
being a nonegative integer, the function T(x, a), a = (ao, aI, a2 ) satsfies the conditions:
2
[ T( x,a ) - T( x, b) ao - bo) ~ e l 2 )ai - b i ) 2,
= 0
W~(G),1
[ T(x, a ) - T(x, b) [ < c , [2 :)ai - bi)2] ,
i = O
( 2)
where e1, J= 1, 2, are the p siive co stants (see [3,chap 3, sec 4 ) ).
We shall use the same notatio s a in [6] Consider the generalized solutio u(x) of the problem
o (1) in the space W ~ (G) satisfying the following equality:
P(u , v) = J J [6 u + T(x , u, : 1 ' :x:)] v(x)dx = - JJ f(x)v(x)dx, (3)
where v( x ) is a functio in the space D ( G ) of Schwartz basic functo s [7
One has v (x) E WHG). Then, by [3] (chap 3, sc 4), if the function ri-, u, - satisfie
aXl aX2
the condi ons (2), f x) E L (G) there exists uniquely a solution of integral equation (3) u(x) E
W~ ( G ) n W ~ (G)
• This w o rk i s pa rti a ll s u pported by the N a ti na l B asics R esea r c h Pr ogra m in N atu r a l S c ienc es
Trang 22 CONSTRUCTION OF DIFFERENCE SCHEMES
We first consider the case where J(x) E L ( G ) and let G be the unit square G = {x = (Xl, X2 )
° <X" < 1, n.= 1, 2}
Let us introduce in the region G a grid w with interior a d b undary grid points denoted by w
and, respectively [ 6
To co struct the differe c schemes one may ta e the test functio s v ( x ) in the form:
{ _lk-kexp {- I x k l2k } x·E e,
( 4)
wh re e= e ( x ) = { ~= ( ~1'~2 ) : k " - xn l < O h ", n = 1, 2} ,h" b ing the steple gths, k being a natural number
Let e ery gridpoint x E w be corresponding to a mesh e ( x ) The gen ralize solution (denoted
by the GS) u ( x ) ofth problem (1) in esatsfies the following integral equation:
:£ 1 + O , Sh l x 2 + 0 5h 2
J J [ ~U( ~ )+T( ~'U'U ( , :~,:~ ) ]a ( l ) dl
(6)
One may rewrite the equation (S) as follows
(7)
where
1
SiU ( X ) = h :
t
x,+O,5h,
J U ( Xl, ··· ,li , ,x,, ) d1 i u(± O Gi) ( x) -- U Xl,( · ··,Xt .±O 1Sh· tl • , Xn' )
Now, to obtain th difference schemes of the oper ator (7) pr e (u , a) o e ma approximate the mean integral operators S , by the quadrature formula of average retangles and th partial derivatives
by difference q otients as in [ 6 (see 2.1) Hence, one get th following difference approximations
corresponding to (7), (3):
K ( y ) = 1P l ( y, a ) = L ( a iY x , ) x, - SlS2 L aXi ( x ) Y x , + SlS2a ( dT ( I, y ( x ) , Yx " YX2) = -<p, x E w,
y ( x ) = o , xE"
(8)
and (ef [3, chap 3, sec 4]
2
L(y) ==2P~ (y , a ) = L Xi Xi + SlS2a ( T ( I, y ( x ) , fi x " fi x 2) = -<p, x Ew,
y ( x ) = 0, x E/,
Trang 3( ± 1,) - ( ± I ( ) - ( ±h, ) > 1
u == u x - u Xl) "" X l t, ,X n , 1 , _ ,
Note that b [3] (see chap.3, sec.4) there exists uniquely a solution of the operator equation
2P, : (y ) = - ' P and, then, of the equation IP: : (y , a )
3 ESTIMATION OF THE CONVERGENCE RATE
Estimate n w the method error and the approximate one of the scheme (8) and (9)
3 1 Consider the difference scheme ( 9 ) with ' P defined by (10), (7). Den te the metho error by
z = y - u, where y being the solutio of the problem ( 9 ) It follows from (9) that
L z = - t P( x ), x E w ; z ( x ) = 0, x E /, (11)
where t P( x ) is the approximatio error of the scheme ( 9 )
\{I( x ) = ' P+ Lu
From (10), ( 7) and by formulas (10), (11) in [6,sec 2], for the sufficiently small mesh sizes hi
and h2, o e has
"' [ (a U)- O 5 , )] (", a a U) ' P = - L S - aa;: x + S 1S 2 L 7 J 7 J.
( a aU)
- S1S 2 ~ , u( ~ ), - , - , x Ew
Thus,
'" [ ( au ) -0.5,) ] ~ aa a u
\{I=L Ux ,-S3- ; aa;: x + SI S L a7J
- SIS2 [T( ~, u(d , a au , ~) - T( ~ , u(x), UX 1(x), UX2(x))].
~ I a 2
By ( 9 ) one has
2
L Y = = L x ,x, = - SI S 2[ T(~ 'Y (X),y l' Yx ,)] -'P= 'Po, x E w.
(13)
Then,
L o X = L o y - L ou ==- \ {I o ( x ), x E w; z( x ) = 0, x ET
From (12) - (14) it follows that
(14)
'" " [ ( a U) ( O.5,)] '" aa a u
\{Io = L u - 'Po = L U X i X, - L S 3-i aa:; x + S1S2 L 7 J.7J
+ S I S2T(~, y ( x ), YX l ' Y x 2 ) S1S 2 T ( ~, u(d , ~ , aU)
a~ 1 a 2
Trang 4DIFFERENCE SCHEMES OF GENERALIZED SOLUTIONS 13
(16)
-L ( zx , x" = L (l7i x , , z) + ( AO,Z ) +( Bo ,z) ,
i=l i = l
: E
(17)
IIZ t ] 1i2 -== (zx 1, ZXi 1i'
Nl N 2 -1 ~
( a, Z] l = L L a ( Jlhl,J 2h 2 )Z(Jlhl ,J 2h2)hlh 2 ,
Y1 =1 J ' 2= 1
N 1- l N 2
( a, zb = L L a ( Jlhl,J2h2)Z ( Jlhl,hh2 )hl h2 ,
VI = 1 ] 2 = 1
N , N 2
Il a l 2 = L L a2 ( Jl h l ,J2h2)hlh 2'
lt := 1i 2= 1
Then
(18)
where the constant C is independent of h ( l h l2 = h I +h~ ) and z ( x),
II I1 7.w = li z ll L + I V 'zI12, I l zllo w == I zll·
Now, we fist co sider the funct io al nj r ]defned by (16)
1
1 71 ( x ) = U X I - h
This expressio coincides with the one of 17 1( x ) (19) in [6] Hence, by (23) in [6]we have
117t x ) I ~ M l h l ( hlh2 ) - ~Il u I1 el,
where e is the followin mesh of the grid
Trang 5ei = ei (x) == {I = ( 11 , 12 ) : Xi - hi < Ii < Xi, 1 13 - i - X 3 l <0, 5h3- d,
II U II " , " l == II U ll w; n(c l ) = ( L J I Duu l2dx ) 1 /2.
l(l~"Lf:
The functio al T / dx ) is estimated similarly Then,
x
The expressio of Ao coincides with the o e of T/ o (15) in 1 ].Then, b (26) in 1 ]we have
where H( h ) - + °as hl ' h - + 0
Consider now f3 (~ in (18) The difference of the form f 3(~ is estimated in 1 ] see chap.3, see 4) ,
one has
11f 3( ~ < C l h III U I12( ; From the last inequality and (16) it follows that
Finaly, combining (18) (2 ) we get
3.2. Consider the following difference scheme
1
2
where y = ~ ( y + 1 ) y and 11are defined (8) and (9) respectively Then,
2
M y = Z [ (1 +a i ) Y x , L , - Z3132 L ax , y , +
+ Z3132 a ( \ "lT( I ,Y ( X ) , Y x "Y x 2 ) + T( I , Y( X )'Y x "Y x , ) J
(23)
Thus,
2
MoY == L [(1 + ai )Y x , L ,
2
= 3132 L a x , Yx, - 3132 [ aT ( ' y ( x ) , Y l' Y x , ) + T ( s" , y ( x ) , u z , Y X2 )] - 2 p
- <Po, X E w,
y ( x ) = 0, X ET
From the last equality, (7) and (12) one has
Trang 6DIFFERENCE SCHEMES OF GENERALIZED SOLUTIONS 1
\II(X) = - L [(1 +a ; z x , L , = L [ 7) i + f l )x, + ) 0 + (30 + q o,
i = 1 i = 1
(25)
By (24), (25), in the same way as in 3.1 o e has
2
[[Z[ w < C ( L (1 1 7)il l + Ilflil l i) + 11 ) 011 + 1 1(30 11 + Il q O II )
=
(26)
In (26) 7)i has the form (16), then one has the estimatio (19) for 7) i
The expression of fl i coincides with the one of I i (31) in [6)' then by (39) in [6) o e has
(27)
where H i ( h ) t = 1, 2 tend to z ro as h - O
) has the form (33) of ~ in [6)' then by (44) in [6)'
( 28 ) ( 30 has the form (16), then by (21) o e has
(29)
Consider the last summand q ill (26) The form of q is analogous to (3() and o e may easiy
verify that
Il q ll < C ! h H(h) lll u llv;
Now, combining (19), (2) ( 3 0) yields
I li I + f j - 2 ll lw :::: C : lhl ", - l iul l rn ( ;, m= 2 ,3
( 3 0)
(31)
Finally, by (22) and (31) we get the estmation of metho error for the difference scheme (8):
( 32 )
R ema rk In a manner analogous to the proof of the inequal ies (22) and (32), o e may verify that
these ineq alites are also valid if in the formula of the GS u(x) (5), (7), v(x) ( = a (hx) ) is a Schwarz
basic functio
3.3. The estimates (22) an (3 ) are obtained with the assumptio f E L2( G ) n w wesh w that the
results may be generalized to the eq atio s with rg t-hand side f E W J - I ) (G) , W J- I ) G ) being the
is the Dirac delta functio 6
Indeed, by our assumpto , f ( x ) E D'(G) , D'(G) bein the space of Schwartz distributions
Therefore, b the theorem on local structure of the distrib to s (see [7,chap 3, sec 6) there exists
Trang 7f(x) = D~ D~ g ( x ) , (33)
where x Ee,the set eis compact in G ER", D i = a / aXi
Let v ( x ) E D( e ) , By (S) and (33) one has
// [6u( x ) +T(x, u, : [ , :XU2 ) ] v ( x ) dx = - IIg ( x ) v ( x ) dx, (34)
where
We see that v ( x ) is also a test functio : v ( x ) E D ( e ) c W~ ( e ) an g (x) E L 2 ( e ). Thus, the
equatio (34) has the form (S) Hence, one may repeat the procedure used above for the difference
schemes (8), (9) and obtaines the following
Theorem. L e t in the probl e m ( 1 ) the [uric o T ( ) s ati s fy the condition s (2 ) and the right - hand
s id e f E W ~ - I) ( G) Then the s olu t on y of the diffe r ence s cheme (8) or ( 9 ) ( y = y or 1j) conve r ges to
th e GS (S) u( x ) of the problem ( 1 ) In the grid norm Wi(w) with t h~ rate O ( I ~ I ) ' that I S, one has the
follo wing err o r e timation
Il - U ll l W :s: C lhlll u ll 2 ( ;,
w here the con s tan t C is independe n t of h and u ( x )
REFEREN C E S
[ G.1 Marchuk, Mathematical M odelling in the Env i ronme n t Problems , Nau ka, Moscow, 1982 (Russian)
[2] V S Vlad irniro , Generalized F unction s in M athematica l Ph qsic s , Mir, Moscow, 197
[3] A A Sam arsk ii,R D Laz arov ,V 1.Makarov, D ifference Scheme s for Gene r alized Solution s of Diffe r ential Equation s, Vus Univ., Moscow, 1987
[4] C Padr a, A posterior error estimators for nonconforming approximation of some quasi-Ne wto-nian flows, S I AM J Num e r , Anal 34 (4) (1997) 1600-161S
[S] C N Davson ,M.F Wheeler, C S Woodward, A two-grd finite difference scheme for non-linear parabolc equations, S I AM 1.Nurner Anal 35 (2) (1998) 43S-4S2
[6] Hoang Dinh Dung, Difference schemes for generalized soluto s of some elliptic differental
equations, I, Journal of Compute r Science and Cuberneiics 15 (1) (1999) 49-61
[7] L Schwartz, Th eorie de s Di s t ri but i ons, Hermann, Paris, 1 78
R eceived M a r h 2 , 20 00
Revi s ed January 5, 2001
I n s t itu t e o f M a th em atic s, N CS T o f Vi e t n am