ABSTRACT
So we have the domain G with sub domains GI and G2 : G = G1 U G2 , GI n G2 = 0 , the domains G, GI , G2 have Lipschitz boundaries . We will study the boundary value problem
where
in G, (3) on aG ,
in GI in G2
and "-2/ "- I � 1. Sometimes the results are true also in the more general case, when GI and G2 are subset , which consist of a finite number of simply connected subdomains. We suppose that the boundary value problem
can be solved with a small computational effort . The method with the preconditioner suggested by D 'jakonov is often used for solving (3) :
(4)
The convergence rate of this method is determined by the upper and lower estimates of the value
on the subspace r E HHG) ; here and below we denote Vr = ( ::1 , . . . , :: ) , (Vr)2 = t:. t:,. If 0 < m ::; A(r) ::; M < 00 then for r = m�M the iteration error tends to zero as
There are obvious upper and lower estimates 1 ::; A(r) ::; � and this estimates cannot be improved. So for the problem (2) we have the convergence rate as
which is not good for big values of w . Further we discuss some methods without such a deficiency which were proposed in the latest years at the Moscow University and the Institute of Numerical Mathematics of the Russian Academy of Sciences [2]-[7] . The main idea of these methods is to iterate in subspaces. This idea was suggested by Marchuk, Lebedev, Kuznetsov . There were studied four methods. 1 ) (See [5] . ) For the problem ( 1 ) the iteration error rn = un - u satisfies the relation
and in the sub domain GI we have
Let uO be the solution of boundary value problem:
(5)
where fo = f in GI ; from (5) it follows that Lorn = 0 in GI for every n . So the convergence rate is determined by estimates of A(r) on the subspace of functions R: r E HHG), Lo r = 0 in GI ; that means
(6)
then
7 = ( "'2 1 + 1') -1 , "'1 2
(7 )
Practically, the value l' and another such constant values which will appear in what follows are not known. If we want to obtain the rate of convergence which do not depend upon the ratio � , we must take 7 = M1 = .!U. in the case (4) . Or we must " 1 #£2 find 7 from variational principles , using the steepest descent method. We can also use the conjugate gradient method. We do not use anywhere, that ",(x) = const in G2 • If "'2 :::;",( x) :::;"'2f3 then we can obtain the estimates
and for the optimal value of 7, instead of (7 ) , the estimate
Let us take the equation (2) , 7 = � , divide (4) by w and find the limit of (4) for w � 00 . We obtain the iteration process
where
un+1 _ un a ( aun ) � + - {} - = 0, 70 ax; ax;
2) After multiplication of (3) by the factor :1 ' we can rewrite (3 ) in the form �u + div p = F,
where F = 2"'11 f and grad u = w-1p,
{ � - 1 w = 1< 1 1 in G1 •
The iteration method which was suggested in [2] has the form
un+1 _ un A +�un+div pn+l-F=0 ,
(8 )
(9 )
3 ) The method introduced in [3) is some modification of ( 10 ) . We can transform (9) to the form
Bp = w-1 + grad �-l div p = grad �-I F
and study the iteration process
pn+! _ pn =--------'=--- = Bpn _ grad � -1 F.
