1549301742-The_Theory_of_Difference_Schemes__Samarskii

414 Stability Theory of Difference Schemes

Proof The energy identity (13) is involved at the first stage. Estimation of

its right-hand side 2r( <p, y 1 ) is stipulated by successive use of the generalized

Cauchy-BunyakovskiY inequality and the s-inequality

2r(<p,Yi) < 2rll<flls-1 'llYills < 2Tcl II Yi II~+ 2T ll<fll~-1'

€1

Upon substituting this estimate into (13) we obtain

If condition ( 42) is satisfied, the number s 1 may be chosen so that 1/(1 -

si) = 1 + s, that is, s 1 = s/(1 + s). Under such an approach we arrive at

Summing up the latter inequality over k = 0, 1, 2, ... , leads to estimate

( 43).

What are the conditions under which the stability in the norm

II <p llc 2 l =II <p II reveals itself? The following assertion answers this question.

Theorem 7 Let the condition

( 44) B > c E + 0.5 r A

be satisfied, where c is a positive nwnber, and scheme (1) belongs to the

primary family of sche1nes. Then for a solution of problem ( 1) the a priori

estimate holds:

( 45) II Yn+l llA^2 < II Yo llA^2 +^1 ~^2

2 c k=O L.., T II <fk II ·

Proof With regard to identity (13) the Cauchy-BunyakovskiY inequality
and the s-inequality together give

I

2 T 2

2r(<p,Yi)<2rll<fll · llYill<2rsllYt I + 2 cll<fll ·