1549301742-The_Theory_of_Difference_Schemes__Samarskii

Some properties of difference elliptic operators 283

With this in n1ind, it remains only to evaluate the ren1aining sums

Ni-I N2-l

s = L Ak1

2

k2 = A~,i + L L Aki

2

k2.

k,=:1 k2=1

ki+k2>2

Because the function sin <pj<p is monotonically decreasing on the segment

0 < <p < 7r /2 and, therefore, sin <p > 2 <p/ 7r, formula ( 4) reduces in this case

to

A =7r

2

k;(sint.p 1 )2 7r

2

k;(sint.p 2 )2> 4 (k; k;)

ki k2 12 + 12 - 12 + 12 ,

I l.f!1 2 l.f!2 I 2

where 'Pa= 7rhaka/(2la), CY= 1, 2. Using this estimate behind, we obtain

Ni-I N2-l

LL

ki=I k2=l

ki+k2>2

IVIajorizing the sum J by the integral

4

and taking into account that A 1 , 1 > 00 = l~ + l~ > ~~,we deduce that

l 2 0

(22)

z4 7f z4 z4 1 z4

r:; < _o_ + o __ o (7f + -) < _o

L - 162 4 ' 16 - 64 4 - 16 '

Substitution of (22) into (21) yields inequality (20).

5. Equations with, variable coefficients. The Dirichlet problem for the
   elliptic equation in the don1ain G + r = G comes next:

(23) Lu=-f(x), x E G, u = μ(x),

with a rectangle G = {O < ~ra <le" CY= 1,2} and

2

Lu= L La H,

a=I

xEf '