1549301742-The_Theory_of_Difference_Schemes__Samarskii

120 Basic Concepts of the Theory of' Difference Sche1nes

In this case we write, as before, A = Ah with further reference that h is a

vector of the dimension N, that is, an element of the space Qt consisting

of all functions defined on the grid

wt= { X; E ( 0, 1], i = 1, 2, ... , N, xN = 1}.

Example 2. The third boundary-value problem. Given the same

grid wh as in Example 1, we now consider the difference boundary-value

problem of the third kind

Ay = Y:tx = -J(x)' O<x=ih<l,

(9)

Let rth be a set of all functions defined on the grid wh = { J:; = i h, i =

0, 1, ... , N }. We begin by specifying the operator A by the relations

1

0.5h (Y~·.o - cr1 Yo)= A_- y, i --^0 '

(A Y); = A y = Y:fJo:' i=l,2, ... ,N-1,

• 0.5 1 ( h Y:r:,N + CT2YN) -- A + y' i = N.

By merely setting A= -A problem (9) is recast as

(10)

where

'Pi =

Ay=<p,

1

0.5 h μ^1 '

f;'

1

-0 .u r: I 7, μ^2 '

i = 0'

i = 1, 2, ... , N - 1,

i = N.

The linear operator A 1naps (th onto (th. To 1nake our exposition 1nore
transparent, it will be convenient to introduce the inner product

N-1

[y, v] = L Y;V;h+ ~h(y 0 v 0 +yNvN)

i=l

and the associated norm I [y] I = J[Y,Y]. The operator A is self-adjoint,
that is, [y, Av] = [v, Ay], where

1

[y, Av] = -(y, Av) - 2 h (Yo A - v + YN A+ v)