1549301742-The_Theory_of_Difference_Schemes__Samarskii

370 Difference Schemes with Constant Coefficients

and

(^2) Sill.^1 2 'Pk
T
the estirnate occurs:
J2 (1 - cos 'Pd
T T

( 19) SIIl!.pk T > v/\k ~ "I"+E

Observe that expansion (14) implies that

N-l ~ cos (. J - 2 1) 'Pk (k)

~ 1 Uok X

cos 2'Pk

N-1

2=

k=l

T sm J'Pk - X (k)

· Uok

sm 'Pk

Using estimates (18) and (19) behind we finally establish

We note in passing that the expression

is nothing than one admissible norm on the space H A-1 known as the

negative norm:

where A y = -A y = -Yxx in the space of all functions y defined on the
grid wh and vanishing at the points x = 0 and x = 1. Indeed,

A-I Uo - =

implying that

(A-1-Uo ''1lo - ) =

Thus, we have proved that under condition (16) scheme (15) admits the
estimate

(20)