1549301742-The_Theory_of_Difference_Schemes__Samarskii

450 Stability Theory of Difference Schemes

Proof To make our exposition more transparent, it is convenient to deal

with

J = ~ 11 y + y 11 ~ + ( ( D - :

2

A) Yt ' Yt).

To prove inequality (91), we concentrate mainly on

1

J = 4 (llYll~ +2(Ay,y)+l/YI/~)

-4 1 ( llY/IA-2(Ay,y)+ll1f/IA^2 2) +(Dyt,Yt)

= (Ay, Y) + llY1ll~

with respect to a new variable y = y + ry 1. With this relation established,

we arrive at

J =II y II~+ T (Ay, Yt) + llY1ll~ <II y II~+ T II y llA. II Yt llA + llY1ll~.

Condition (87) yields the estimate

2

II Yt llA < rJl + c: llYt lln '

so that

This implies the first inequality of the lemma.

A simple observation that J = ( Ay, y )+ 11Yt111 justifies the forthcoming

substitution y = y - ry 1 , leaving us with

Making use of the generalized Cauchy-Bunyakovskil inequality

and taking into account (87), we obtain

J>//y//~-rll1/llA · //Y1l/A +llY1ll~

2

>II Y II~ - Jl + c: II YI/A · llY1lln + llY1ll~ ·