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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 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 c.ic cong trlnh [3- 5] Bai xet luo'c 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 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) [T(x,a) - T(x,b)](ao - bo) el2 )ai ~ - bi)2, =0 (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 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 = 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 = h: t U(Xl,···,li, ,x,,)d1i, u (±O.Gi)( x ) U ( Xl,···,Xt ±O 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): K (y) == Pl'(y, a) = L (aiYx,) x, - SlS2 y(x)=o, L aXi (x)Yx, + SlS2a(dT(I, y(x), Yx" YX2) = -