Classes of stable three-layer schemes 453
In order to understand the nature of this a little better, we take for
granted that
A and D are constant operators,
(102) A and D are self-adjoint positive operators.
Then under condition (87) for any schen1e of the form (101) estimate (99)
is valid with M 1 = 1. By 111erely setting x = D^112 y and C = n-^1!^2 An-^1!^2
we reduce (101) to the fonn
(103) xr 1 +C'x=(/J, x(O)=x 0 , x 1 (0)=x 0.
Applying the inverse operator c-^1 to scheme (103) yields
(104) c-^1 xtt + x = c-^1 0' x(O) = Xo' xt(O) = Xo.
Comparison with scheme (84) reveals the correspondences
c-I rv D ' E rv a ' c-^1 <p rv <p '
With this in mind, condition (87) acquires the fonn
l+c: l+c:
c-^1 - > 4 T^2 E or E -> E T^2 c '
Involving estimate (99) with M 1 = 1 and taking into account that C
is a constant operator, we arrive at
(105) llxn+1 II < {B!- ( llx(O)/I + llx1(0)llc-1 + ~ T llC'-
1
</Jllc)
with x = D^112 y and <p = n-^1 /^2 1.f! incorporated. What has been done is to
derive the desirable expressions
llx1(0)ll~-1 = (c-^1 x 1 (0), x 1 (0))
= (nlf2A-1Dlf2nl/2Yt(O),Dl/2Yt(O))
= llDYt (O)ll~ -1 ,
11c-^1 011~ = (c-^1 0,0)
= (nlf2A-1Dlf2n-1/2<p,D-1/2<p)
= (A-^1 t.p, l.f!) = lll.f!ll~-1 ,
so there is some reason to be concerned about estimate (105) in the original
variables:
(106) llYn+1lln < fB!- ( lly(O)/ln + llDY1(0)llA-1 + ~ T /ll.f!sl/A-1) ·
Thus, we have proved the useful assertion.