1549301742-The_Theory_of_Difference_Schemes__Samarskii

312 Difference Schemes with Constant Coefficients

Putting these together with (26) we find that

N-l N-l N-1

fJ = L T',, x (k) = L qk Tk .X (k) + T L

k=l k=l k=l

With the aid of the well-known triangle inequality II v + w II < II v II+ II w II

we establish

or

(29)

Further, let

(30)

T N-1

( )

1/2

2= zp2

ll+O"r-\1 k=l k

h2

4r '

()" > 0

When this is the case, I qk I < 1, 1 + O"L\ > 1 and II fJ II < II Y II+ T 11 'P II or

11yj'+^1 11<11 y/ ll+r 11r'11· Summing over/= o, 1,2, ... ,j and exploiting

the fact that II y^0 II = 0 for the solution of problem (16b), we derive the

estimate

( 31)

. j ·I
11y^1 +^111 < 2= T 11 r 11.
j'=O

Let us stress here that estimate (31) was obtained under condition (30). In
subsequent reasonings we get rid of the bound O" > 0 and impose then the
constraint

(32)

'

()" € =

-^1 -^1 - E h2
2 4 T '

where E = const > 0 does not depend on h and T. True, it is to be shown
that

(l-E)h^2 AN-i (1-c)h^2 4

> 1- 4 >1 - 4 h 2 -c - J