Classes of stable two-layer schemes 411are necessary and sufficient for the p-stability in the space H B of scheme
(la):
11 Yn 11 B < Pn 11 Yo 11 B ·
When, in addition, A is a positive operator, these are necessary and suffi-
cient for the p-stability in the space HA:II Yn llA < p" II Yo llA ·
In order to prove this theore1n, we beforehand reduce the implicit
scheme (la) to the explicit scheme (32) with the operator C = B-^1 /^2 AB-^112
(or C = A^112 B-^1 A^1 l^2 for A> 0) and then collect the results of Lemmas
3, 2 and 4.
Remark One fails to prove Theorem 3 on account of the energy identity
(20). However, the method developed in Section 6 may be of assistance in
achieving this aim. In the case D = B conditions (36) are equivalent to the
condition II E - rB-^1 Alls< p or to being nonnegative of the functionalSuppose now that H is a finite-dimensional space of the dimension N.
Substituting y = L~=l ck ~k> where ~k is an eigenelement of problem (27),
into the expression for the fuctional J B [y] yields
N
Js[y] = ~ c~ (p^2 - (1-r ,\k)^2 ) > 0l-p l+p
for < ,\k < --
k=l T Twhich is equivalent to condition (36).
- Stability with respect to the right-hand side. Recall that in Section 1 we
have established a Theorem 3. This is a way of saying that the stability in
the norm II · 11(1) "".ith respect to the initial data implies the stability with
respect to the right-hand side taken in the norn1II<p11( 2 ) =II B-^1 <p 11(1)' An
immediate implication of this is covered by the following assertion.
Theorem 4 If condition ( 14) is satisfied, then scheme ( 1) from the primary
family of schemes is stable with respect to the right-hand side and for a
solution of problem ( 1) the a priori estimate holds:
n
II Yn+1 llA <II Yo llA + ~ T II B-l<pk llA
k=O