1549301742-The_Theory_of_Difference_Schemes__Samarskii

Mathematical apparatus in the theory of difference schemes 101

4) The second Green formula. In the integral calculus the fol-

lowing formula

1 1

Ju (kv^1 )1 dx - J v (ku^1 )1 dx = k (uv^1 - vu^1 )/6

0 0

is known as the second Green formula. Inserting u = y and v = azx in (3)

yields

(9)

Substracting (9) from (8) we derive the difference analog of the second

Green for1nula

In just the same way as was done on a non-equidistant grid, we find that

If y and z vanish at the points x = 0 and x = 1, the appropriate terms

in the expressions being substituted into (3) are equal to zero, thereby

reducing to

( 101 )

( 111 )

(Ay, z) = (y, Az),

(Ay,z). = (y,Az).,

Ay = (ayiJ.?;,

Ay = ( ayiJx.

These formulae confirm that the operator A is self-adjoint.

5) The Cauchy-Bunyakovski?: inequality and the €-inequality.

In the sequel we shall need the well-known Cauchy-Bunyakovsldi'. inequality

l(u, v)/ < II u II. II v 11,

where (, ) is the inner product in a vector space and II u II = ~ is
the associated norm. To be more specific, ( , ) designates one of the inner
products we have introduced above.
In what follows we will frequently employ the inequality