Other iterative methods^72910.5 OTHER ITERATIVE METHODS- Three-layer iteration sche111es. So far we have consicierecl two-layer
iteration schemes available for solving operator equations of the form Au =
f with a self-adjoint operator A under the assumption that the spectral
bounds / 1 and / 2 for the operator A are known in advance either in a space
Hor in a space JIB, where B = B* > 0 is some stabilizator. Other iterative
methods find a wide range of applications in some or other aspects.
An excellent start in this direction is to describe three-layer (two-step)
iteration schemes in the general setting due to which it is required to solve
the equation
(1) Au=f, A: Hf--+H,
with a self-adjoint positive definite operator A, whose spectral bounds are
already known:(2) A= A*, /1 > 0.
The links between three iterations Yk-l, 1h and Yk+l are provided by
three-layer iteration sche111es, by means of which Yk+I can be expressed
through the values Yk-l and Yk· The standard fonn of explicit schemes is(3) 1ik+l = (l+o:)S'1fk-CYY1.;-1 +(l+a)T 0 f, k = 1, 2, ... ,
y 1 = S Yo+ T 0 f, given Yo E H,
where S = E - T 0 A is the transition operator for the two-layer simple
iteration scheme with the optimal para111eter T 0
2
(4) To=
fl+ 121-~
l+~·<., c -- /1The two-layer sin1ple iteration scheme pern1its us to find the first iteration
1h.
In an attempt to create schen1e (3), equation (1) should be represented
in the so-called "preliminary" forn1
u=u-TAu+Tf=S(T)ll+Tf, S'(T)=E-TA,