1549301742-The_Theory_of_Difference_Schemes__Samarskii

Iterative alternating direction methods

Ai = ~A~, A2 = ~A~, wC1l = ~r(ll and wC^2 l

rC^1 ) = rC^2 ) = T under the condition wC1l = wC^2 l.

717

~rC^2 l, we might have

For the model problem ( 37) posed completely in Section 2 we obtain

4

T)

16 1 1.6

7f2 h2 ~ h2 ,

giving in combination with (16)

• 1.27 4
  n(.:) ~ 0.'.2 In - In -.
  h .:
  Thus, for exan1ple, we find that n(.:) ~ 6 for h = 1/10, n(.:) ~ 9 for
  h = 1/50 and n(.:) ~ 11 for h = 1/100 with reasonable accuracy .: =
  2e-^10 ~10-^4.

4. ADM for the case of noncommutative operators. Of special interest is
   the equation of the form

with noncommutative operators A 1 and A 2 still subject to conditions (5)-

(6):

Aa = A~ > 0, DCYE < Aa < ~aE, DCY > 0, C\' = 1, 2.

These properties have had a significant impact on modifications of

iterative methods and provide the possibility of applications of the two-

parameter iterative ADM:

(17)

given y 0 E H , k = 0, 1, 2, ... ,
where a proper choice of paran1eters w 1 and w 2 will be substantiated a little
later for soine reason or another.
Consistent with zk+l = Y1.:+i - u and zk+i/ 2 = ~th+i; 2 - u is the idea
of setting up the homogeneous equations for the error such as

(18)

given z 0 E Ii, k = 0, 1, 2, ....