1549301742-The_Theory_of_Difference_Schemes__Samarskii

Some properties of difference elliptic operators 285

for any v E H, where

0 0 2

A Y =-A Y = - L Yx" Xa

a:=!

and the constants f; > 0 and 6. > 0 arise from formulae (18) and (19).

As far as vi -Yh = 0, the first Green formula with regard to the operator

A°' v = (a°' vx ) implies that

" x"

where

N1 N2-I

(w,z]i =LL w(i 1 h 1 ,i 2 h 2 ) z(i 1 h 1 ,i 2 h 2 ) h 1 h 2 ,

i1 :=] i2=1

N 1 -l N2

(w, z] 2 = L L w(i 1 h 1 , i 2 h 2 ) z(i 1 h 1 , i 2 h 2 ) h 1 h 2 •

ii =I i2=1

With these representations in view, we have

2

(Av,v) = L (aavx,, ,vxJ°'.

a=!

On the other hand,

By condition (24) we are led to (26) and so it remains only to substitute

0

into (26) the bilateral estimate b II v 112 < (Av, v) < 6. II v 112, which has

been obtained in Section 2. These manipulations permit us to derive (27)

with the ingredients

sm. 2 --7r h°'

2! )

°'

2 4

(^6) =I: h2
a=! °'
In what follows inequalities (26) and (27) are adopted in operator form
and we agree to consider
(28)
(29)