1549301742-The_Theory_of_Difference_Schemes__Samarskii

728 Methods for Solving Grid Equations

as required. By virtue of (37) it is plain to establish the relations

Here the parameter w is so chosen as to satisfy the 1nininmrn condition for

the function F ( w) by discovering that the functions Fi ( w) = (1-x 1 ) ( 1 + x 1 )

and F 2 (w) = (1 - x 2 ) (1 + x 2 ) attain the minimal values, respectively, for

w = 1/ ~ and w = 1/ ~· These points are are identical when

c5 1 f::..1 = c5 2 f::..2. In that case for w = l/~ = l/~ the inequality

(38)

o:=l,2,

1-/Th

1 + J77;'

holds and the error Z 11 = y 11 - 1l adn1its the estin1a.te

(39)

If c5 1 f::..1 f- c5 2 f::..2, then w = l/.;r;5., where c5 = min (c5 1 , c5 2 ) and f::.. =
max ( f::..1, f::..2).
In comparison with the case of self-adjoint opera.tors the number of the
iterations is being increased in two times. This is clearly seen, for example,
frmn (38) by setting c5 1 = c5 2 = c5 and f::.. 1 = f::..2 = f::..:

It should be compared with

= ( 1 - v0) 2

p l+J17 ,

e1nerging from estimate (39) with regard to self-adjoint operators A 1 and
A 2. One needs to exercise good judgement in deciding which to consider.