1549301742-The_Theory_of_Difference_Schemes__Samarskii

110 Basic Concepts of the Theory of Difference Sche1nes

4. Difference analogs of the embedding theoren1s. In the estimation of var-
   ious properties of difference schemes such as stability, convergence, etc. we
   shall need yet inequalities corresponding to the simplest Sobolev embed-
   ding theorems. In this respect the appropriate results have been obtained
   with the following lemrna.s.

Lemma 1 For any grid function y( x) defined on the grid

wh = { X; = ih, 0 < i < N, Xo = 0, XN = 1}

and vanishing at the points x = 0 and x = 1 the inequality holds:

( 36)

Proof The function y(x) given on the grid wh can be expressed in the form

of the identity

( 37)

With the assigned values y(O) = y(l) = 0, one can write clown

Upon substituting these equalities into (37) we find that

y2(x) = (1-x) (x~h hyx(x')) 2 + x (",f+h hyx(x')) 2

Let us estimate the sums on the right-hand side of the preceding relation

with the a.id of the Cauchy-Bunya.kovskil inequality:

x x

y^2 (x) < (1-x) L h L y~(x')h+x 2=h2= y:(x')h x

x^1 =h x'=h

l

= x ( 1 - x) L y~ ( x') h = x ( 1 - x) 11 Vv] 12 ,

x'=h

where y; = (Yx)^2. The ma.xi1num of the expression x (l - x) on the segment

[ 0, 1] is attained at the point x = ~ and equals ~· Therefore,

Y2(x) -< l 4 11 Yx Jl2

and, consequently,