1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable three-layer schemes

is equivalent to the require1nents

(26*)

(27*)

B = B(t) > 0 for all t E W 7 ,

1

R>-A - 4 '

which assure us of the validity of the estimate

(28*)

where

2 1 2 2

llYn+1llA = 411 Yn+l + Yn llA + llYn+l - YnllR-tA ·

435

In concluding this section, let us stress that condition (26*) is not only

sufficient, but also necessary for the validity of estimate (28*).

4. Stability with respect to the right-hand side. We now consider problem
   (lb) under conditions (9), (10) and (27), which are put together for later
   use. Since A and Rare constant operators, identity (18) for (lb) is of the
   form
   ( 31) 2 T ( B Yo t 1 Yo t ) + 11 Y ( i + T) 112 = 11 Y ( i) 112 + 2 T ( <p 1 Yo t ) 1 t = nr.
   In the further development of a priori estimates of the form (20) or
   (21) a key role is played by the estimation of the functional 2r( <p, yo ). First
   t
   of all, we give below the obvious inequality

(32) 2 T (<p, yo)< t Teall yo t 112+-=-11<p11s^2 ,

0

where s 0 = const > 0 is independent of T and h both.

Theorem 2 Let A = A* > 0 and R = R* > 0 be positive operators. Then

under the conditions B > sE, R > A/4, c = const > 0, a solution of

problem ( 1) satisfies the a priori estimate

(33) II Y(t + r) II< II Y(r) II+~ [it T II <p(t') 11

2

]

112

It suffices to estimate only a solution of proble1n (lb), since Theorem
1 remains valid for B > sE. By merely setting in (32) s 0 = 2s we deduce
from (31) that

(34) II Y(t + r) 112 < II Y(t) 112 +

2

7

c II <p(t) 11

2

·

To finish the proof of the theorem, it remains to sum up this inequality with
respect to the variable t = r, 2r, ... , nr, exploit the fact that 11 Y ( T) 11 = 0
and apply Theorem 1.
\Ve cite here without proving the following assertion.