1549301742-The_Theory_of_Difference_Schemes__Samarskii

Mathematical apparatus in the theory of difference schemes 111

Remark 1 Lemma 1 remains valid on an arbitrary non-equidistant grid

wh·

Remark 2 In the sequel we will also use an inequality of the type (36) for

segments of arbitrary length l. Such an inequality can be derived from (36)

by merely substituting x' = l x. Then x' varies on the interval (0, /) and

y_, x = Y-x I l' h^1 = h l.

By inserting yx = Yx' l and h = h' / l in (24) we arrive at the chain of the

relations

2 N 2 N 2 2

llY-JI X = .. L (y_,) Xz l^2 l-^1 h^1 =l L. (y_,) Xi h'=lllY-,]1. X

1.=l i=l

Consequently, on any interval of length l

(38)

Re1nark 3 Inequalities (36) and (38) are valid for all the functions vanish-

ing at both ends of the interval in view. Being concerned with the function

y( x) vanishing only on the boundary, one can derive another inequality

(39)

Inequalities (36), (38) and (39) fail to be true, in general, for arbitrary
functions. However, it is plain to show that in this case the inequalities of
alternative forms occur:

Le1nma 2 For a.ny function y( x) defined on a.n a.rbitra.ry grid

a.nd va.nishing a.t the points x = 0 a.nd x = l the inequa.lity holds:

( 40)

2 12 2

llYll < 4 llYx]I, Yo = YN = 0 ·

Indeed, it is easy to check that II y 112 < l II y II~. Substituting this inequality

into (36) yields (40). In the case of an equidistant grid estimate (40) can
be improved and the reader is invited to do it on his/her own.