1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable two-layer schen1es 413

with C = A^1!^2 B-^1 A^1!^2 enables us to evaluate the smnmand

in inequality (41). Stability condition (14) has been used in Section 2 to

obtain through such an analysis estimate (24)

llE-rCll < 1,

implying that

where the norm ll<tJrllA-1 = j(A-^1 <pr,<f1f). Consequently, a solution of

problem ( 40) satisfies the estimate

n

II Vn+1 llA <II Vo llA + L rll ifJf,k IA-^1 1

k=l

which in combination with the inequality

gives
n
II Vn+1 llA <II Yo llA +II <fJo IA-^1 +LT II ifJf,k IA-^1 ·
k: = 1
Finally, by appeal to the triangular inequality from (38) we obtain (37).
Thus, Theorem 5 is completely proved.

Theorem 6 If the conditions

( 42) ' B >^1 + c
2

rA, B = B*

are satisfied, then for scheme ( 1) from the primary family of schemes the a
priori estimate holds:

( 43)

n

2 2 l+s~ 2

II Yn+1 llA <II Yo llA + 2 c L., T II ifJk lls-1'

k=l

where c > 0 is a constant independent of h and r both.