404 Stability Theory of Difference Sche1nesand relation (20) yields ll:iills < llYll 8 , meaning
II y(t) lls < II y(O) lls.
Remark If the operators A and B are commuting, then condition (14) is
necessary and sufficient for the stability of scheme (la) in the space Hn:II Yn lln < II Yo lln
where D = D* > 0 is any operator cmnmu ting with A and B. This is
acceptable if we involve, for instance, D = E, D = A^2 or D = B^2 for
B = B*, SO that II Yn II <II Yo II, All Yn II< All Yo II, Bii Yn II < Bii Yo II, etc.- Estimates of the norm of the transition operator. Stability considera-
tions are connected with the use of a new 1nethod based on the estimation
of the norm for the operator of transition frmn one layer to another. This
method actually falls within the category of energy 1nethods.
We may attempt the difference scheme (la) in the form
(21) :ii= Sy, s = E - T B-^1 A,
where S is the transition operator. Let D = D* be an arbitrary constant
operator in the space H. With these, it is plain to show that(22) ll:iilln = llSYlln < llSlln · llYlln,
where II S II~ is adopted as the s1nallest constant M subject to the inequality(DSy, Sy) < M(Dy, y).
As can readily be observed frmn (22), scheme (21) is stable in the space HD,
that is, 11 Yn lln < II Yo lln if the norm of the transition operator does not
exceed unity: II S lln < 1. This condition is equivalent to being nonnegative
of the functional ( M = 1)
, ] D [y] = ( D y, y) - ( D Sy, S y) > 0.
Of special interest is the case D = A, for which 11 S llA < 1 for B >
~r A. Indeed, upon substituting the expression for S into the functional J A
we obtain
J A [y l = (A y' y) - (A s y, s y)
= (Ay, y) - (A (E - T B-^1 A) y, (E - T B-^1 A) y)
= 2 T (A y' B-^1 A y) - r^2 (A B-^1 A y, B-^1 A y).