1549301742-The_Theory_of_Difference_Schemes__Samarskii

654 Methods for Solving Grid Equations

Before going further, we recall that any linear one-step iterative r11e-

thod can be written as

(2) k=0,1,2, .. .,

where Bk and Ck are linear operators from the space H into the space

H depending, generally speaking, on the iteration number k, Fk E H is a

known function of k and Yk is the kth iteration, under the agreen1ent that all

the inverses Bk^1 exist. A natural requirement in the further development

is that the exact solution 1l to e4uation (1), not depending on k, should

identically satisfy equation (2):

But it is possible only if (Bk - Ck) A -^1 f = Fk, implying that

• the inverse operator (B~, - Ck)-^1 exists;


• f =A (B~, - Ci.:)-^1 F'i,,.
  This is acceptable if we agree to consider

k=0,1,2, .. .,

where Tk+I > 0 is a numerical parameter. Under such an approach the

canonical forn1 of two-layer iteration schemes is

(3) k=0,1,2, ... ,

where the initial approximation Yo E H is free to be chosen in any con-

venient way. Since the inverse Bk^1 exists, it follows fr0111 the foregoing

that

(4)

or, what a.n1ounts to the sa.n1e,

where rk = Ayk - f is the residua.! and wk = Bk^1 1·k is the correction.
With knowledge of Yk the value of Yk+I can be recovered fr0111 equation
( 4). Knowing Yo, it is plain to determine successively y 1 , y 2 •... Of course,
it is 111ea.ningful only for c:onvergent i tera.l.i ve n1ethocls:

(5)