1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable two-layer schemes 417

Theorem 8 Let A be a self-adjoint positive operator independent oft = nr;

A = A* > 0. Then for the weighted scheme ( 47) estimate (37) is valid for

<7 > <7 0 , estimate (43) for <7 >^1 fc - Tll~ll' c > 0, and estimate (45)

for <7 > <7 0 <J'c = ~ - (1 - s)/( r!! A!!), 0 < c < l, where a number c is

independent of h and T both.

The above statements will be proved if we succeed in verifying the

fulfilment of the conditions of Theorems 5, 6, 7 with the aid of the inequality

B - -rA^1 = E + (<7 - -)rA^1 > -( 1 + (<7 - -)r 1 ) A

2 2 - II A II 2.

Suppose now that A= A(t) > 0 and A* #A, that is, A= A(t) is a

positive non-self-adjoint variable operator.

Theore1n 9 Let A = A( t) > 0 be a positive non-self-adjoint variable

operator. If <7 > !, then for scheme ( 46) the estimate holds:

n

( 49) II Yn+l II <II Yo II+ II (A-^1 <tJ)o II+ II (A-^1 <tJ)n II+ L T II (A-^1 <tJh, k II·

k=l

Proof Consider scheme ( 48) with the right-hand side tjJ = A-^1 <p. When

<7 > !, the conditions of Theorem 5 are satisfied for this scheme and inequal-

ity (37) applies here together with the established relations: !!YI!;= II y II

and

ll\Ot,kllx-^1 =II 0t,k II= II (A-

1

<tJh,k II·

As a final result we obtain (49).

Remark If A*= A*(t) is a self-adjoint operator, then estimate (49) holds

true for <7 > <7 0 •

Theorem 10 Let' A(t)

estimate is valid:

A*(t) > 0 and <7 > <7 0. Then the following

(50)

N

2 2 1 ~ 2

II Yn+l II < II Yo II + + 2 c L__, T II <t1k !IA;;-' ·

k=l

Proof The energy identity (13) for scheme ( 48) takes for now the form