1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference sche1nes as operator equations. General formulations 143

and work in the space H 1 of all grid functions defined on the grid wk under

the inner product structure ( y, v] = L~l hyi-l/ 2 vi_ 1 ; 2 +yN vN+Yo V 0 and

the norm 11 y JI = J( y, y ]. As before, H is the space of all grid functions

defined on the uniform grid wh ={xi= ih, i = 0, l, ... , N,h = 1/ N}

with the inner product ( y, v) = L~~^1 Yi vi h+ ~ h (Yo v 0 + yN vN) and the

associated norm II y II = J( y, y ). We specify the operators T: H f--+ H 1

and T*: H 1 f--+ H as follows:

(TY)o =Yo, ( )

Yi - Yi-1

Tyi-1/2= h , i:= 1,2, ... ,N,

(T* ) = _ v 1/2 - v a

y^0 0.5 h 0.5 h

(T* v )· ' = _ Vi+l/2 -h Vi-1/2 ' i = 1, 2, ... , N - 1, y EH,

This provides support for the view that A = T* ST, where the operator
S: H 1 f--+ H 1 is defined by the relations

(Sy)o = CJ'1 Yo, (SY )i-1/2 = ai Yi-1/2,

(Sy)N = CJ'2YN ·

Obviously, S is a self-adjoint operator in H 1 and (Sy, y] > c 1 11yJ1^2 with
constant c 1 = min ( ai, CJ' 1 , CJ' 2 ). Let us show that the operators T and T
are mutually adjoint in the following sense: ( T v, y) = ( v, Ty] for y E H,
v E H 1. Indeed,

N-1

( T*v, Y) = - L ( Vi+l/2 - vi-l/2) Yi - Yo ( V1;2 - Va)

i=l

N

= L Vi-1/2 (Yi - Yi-1) +Yo Vo - YN VN

i=l

=(v,Ty].

vVe are now in a position to derive some a priory estimates for the equation
Ay =T'S Ty= <p. Under the natural premise S > c 1 E, c 1 > 0, we obtain