1549301742-The_Theory_of_Difference_Schemes__Samarskii

Iterative alternating direction methods 727

In trying to solve the equation Au = f the iterative ADM with itera-

tion parameter w is good enough for our purposes:

(36)

By exactly the same reasoning as in Section 4 with regard to the error

zk+l = Yk+l - u we deduce for V1;+ 1 = (E + wA2) zk+l that

o:=l,2,

We are going to show that under conditions (34) the following estimates

hold true:

( 37) o:=l,2.

Indeed, from the obvious identities

II (E - wAa) x 112 =I/ (E + wAa) X 1/^2 - 4w (Aax, x),

which are put together with inequalities (34)-(35), it follows that

By inserting here x = ( E + w Aa )-^1 we find that