1549301742-The_Theory_of_Difference_Schemes__Samarskii

138 Basic Concepts of the Theory of Difference Schemes

All this enables us to derive for this norm the estimate

( 53)

For its proof the right-hand side of equation ( 48) is expressed in the

form 'Pi= Sx , i> i = 1, 2, ... , N-l, where Si is specified by conditions (52).

The equation (a Yx )x + <p = (a Yx + S )x = 0 implies that ai Yx, i +Si = c =

const for all i = 1, 2, ... , N. Whence Yi - Yi-l = h ( c - Si)/ ai. Summing

up over i = 1, 2, ... , N and taking into account that y 0 = yN, we deduce

that

(

N h )2(N h)-1

c= 2= -a s, 2= -a

i= 1 I 1= 1 I

On the other hand,

ll'Pll~-1 = (A-^1 <p, 'P) = (y, 'P) = (y, Sx) = -(S, Yx]

= - "'L.,i=l^1 " h Yx i S' i ·

'

Substituting Y-x) i. = ( c - Si)/ ai yields

as required. Estimate (53) is simple to follow:

N

2= hs;

i=l

N-1^2

= ~ c "'"" ~ h S1+1^2 = 2_ c

1 i= 1 1

Example 5 Consider now the third boundary-value problem (9). As in
Example 2 of Section 1 it will be convenient to introduce the space H h = Dh
of the dirnension N + 1 consisting of all grid functions defined on the uniforrn
grid wh = {xi= i h, i = 0, 1, 2, ... , N, h = l/N~} under the inner product