736 Methods for Solving Grid Equationswith parameters( rk> wiJ
(29) Tk+l = (Awki wd 'Estimate (29) is still valid in that case under the following conditions:/1 > 0 J B = B' > 0.
- On solving equations with non-self-adjoint operators. In dealing with
an equation
Au= f, A:Hf-+H,
where A is a positive definite non-self-adjoint linear operator, the intention
is to use one of the stationary iterative methods associated with the two-
layer schen1e, the paran1eter of which is constant:
(30) Yk+l - Y1.: +A Y1.: _ -. , f"
Tk = 0, 1, 2, ... , given Yo EH,
The hmnogeneous equation for the error zk = Yk - u amounts to
zk+l = S zk, S = E - TA, k = 0, 1, 2, ... , z 0 EH,
permitting to establish the convergence rate of iterations. At first glance,
II z1.;+ 1 II ~ II S II · II z1.; II As before, a proper choice of t!JP para111el.er T is
stipulated by the condition n1in 7 11 S'(r) II·
In such a setting the assmnptions are niacle on the lower bounds of
the operators A and A -l:
A 2:: 11 E or (Ay,y) > l1 llYll^2 , /1 > 0,
(31) 1
A-^1 > - E or II Ay 113 < /2 (Ay, y), /2 > 0
/2
A simple observation that the second condition for the case where
A= A* is equivalent to the condition A < 12 E may be of help in further
elaboration on this subject. Provided the condition 2 - T / 2 > 0 holds, we
are able to arrive at the chain of the relations
II s y 112 = II y - T Ay 112 = II y 112 - 2 T (A y, y) + r^2 II Ay 112
~ llYll2 -2r(Ay,y) + T2/2 (Ay,y) = llYll2 - r(2-T/2)(Ay)
~II Yll^2 - r(2 - T/2ll1 llYll^2 = (1- 2T/1 + T^2 /1 l2) llYll^2 ,