1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference sche1nes as operator equations. General formulations 141

with the inner product (y, v) and associated norm II y II = V0J:0 and let

H 1 be a vector space with the inner product (y, v] and associated norm

II y II = yl(Y,YJ. Let, in addition, T be an operator from H into H 1 , S be

an operator from H 1 into H 1 and T* be an operator from H 1 into H. Then

the operator A really acts from H into H: A: H f--+ H. The operators T

and T* are mutually adjoint in the following sense:

(Ty, v] = (y, T*v) for all y EH, v E H 1.

Before going further, we give below several examples illustrating how

to construct factorized operators (60) for the simplest difference schemes.

Example 6 Of our initial concern is the first boundary-value problem

(61) (a Y:;; )r = -<p' O<x=ih<l,

Yo= 0,

In this case H E Dh is the space of all grid functions defined at the inner

nodes of the uniform grid wh ={xi= ih, i = 0,1,2, ... , N, hN = 1},

that is, for 0 < i < N and H 1 = Dt is the space of all grid functions defined

at the nodes of the uniform grid wt = {xi= ~ i h, i = 1, ~ 2, ... , N, h N =

1 }. The operator A: H f--+ H equals A y = -A y, where A y takes the form

a2 ( Y2 - Y1 ) - ai Y1

h2

i = 2, 3, ... , N - 2,

-aN YN-1 - aN-1 ( YN-1 - YN-2)

h2

The inner products in the spaces H and H 1 are defined by

N-1
( y, v) = L Yi vi h for Yi , vi E f{,
i=l
N
(y, v] = L yivih for Yi, vi E H 1.
i=l
Under these structures, we specify the operators Ty, T*v and Sv by the
relations

(Ty),·= Yi - Yi-1

h =YT ,, ' i'

·i = 2, 3, ... , N - 1,