1549301742-The_Theory_of_Difference_Schemes__Samarskii

658 Methods for Solving Grid Equations

Theoren1 1 If A is a self-adjoint. opeTatol' (A = A* > 0). then

( 11)

1

B >-TA

2

1

or (BJ:, x) > -T (Ax, x)

2

fol' all :v E H

is a sufficient condition fol' the conveTgence of the method of iterations (3')

in the space HA with the Tate of convergence of a geometric progression

( 12) k = 0, 1, ... ' p < 1)

where p = (1 - 2To.b/llB11^2 )^1 /^2 is its denominatol' and b =mink Ak(A),

o. =mink Ak (Bo - T A/2), B 0 = (B + B* )/2 is the symmetric part of the

opel'atol' B.

Proof Knowing frorn (3') z1.:+i = Sz1.: with the operator S = E - TB-^1 A,

we find that

= (A(E-TB-^1 A)z1.:i (E- TB-^1 A)zk))

=II z1.: II! - T [(AB-^1 Azk> zJ.;) + (B-^1 Az"' Az1.:)]

With the relation A= A* in view, we deduce upon substituting here Azk =

-Bvk and vk = -B-^1 Azk> where vk =; (zk+i - zk), that

(13)

Because of (11), by utilizing the fact that the operator P = B - TA/2 is
positive we establish its positive definiteness in a finite-di1nensional space
H (for more detail see Chapter 2, Section 1):

( 11 ')

1

B--TA>bE 2 - * ) o. > 0)

where o. is the s1nallest eigenvalue of the operator P 0 = Bo - ±TA, so that

( 13')