1549301742-The_Theory_of_Difference_Schemes__Samarskii

Other iterative methods^733

Adopting those ideas to the case of a. non-self-adjoint opera.tor A, we

derive from the foregoing the a. priori estimate

(17) II rk+1 II ~Po 111'1.; II, 1neanmg II A Y11 - .f II ~ P~' II A Yo - .f II,

where p 0 = (1 - ~)/(1 + ~), ~ = fl/f 2 and fl, ~( 2 a.re the accurate bounds

of the opera.tor A= A*> 0.

Indeed, for the value Tk+l assigned by (14) the right-hand side of (16)

is minima.I for fixed 1·k E H. Due to this property it is being increased for

a.II other values and, in particular, for T = T 0. The meaning of this is that

we should have

11rk+l11

2

< 11rk11^2 - 2 To(A rk> rd+ T; II Ark 112

< II 7'1.: - To A T'k 112 < II E - To A 112 · II r1.: 112,

On the other hand, we learn frmn Section 2 that 11 E - T 0 A 11 = Po for

T 0 = 2/ (fl + f 2 ), thereby justifying estimate (17) and the convergence of

the 1ninin1a.l residua.I method with the sa.n1e rate as occuned before for the

simple iteration method with the exact values fl and f· 2.

During the course of MRM the same procedures (13') and (14) a.re

workable with increased volun1e of c:a.lcula.tions in connection with forn1ula.

( 14) for T1.;+i as cmnpa.red with the s1n1ple iteration method.

The implicit mini1nal residual method can be designed in line

with established practice:

( 18)

which is referred to as the minimal correction 1nethod. In that case
instead of the governing equation Au= .f we are dealing with

( 19) C v = cp, v = B^1 l^2 u, C = B-^1!^2 AB-^1!^2 , cp = B-^112 .f.

Applying here the explicit n1inin1a.l residua.I 111ethod yields

where

T'k - = ,-, v x k - cp.