1549301742-The_Theory_of_Difference_Schemes__Samarskii

128 Basic Concepts of the Theory of Difference Schemes

may be true.

Here the meaning of weak stability is that condition (25) should be

valid. In other words, this asserts that there exists a d0111ain for I h I, for

example, 0 < h. < I h I < h 0 , where (22). is satisfied with constant M,

depending on h •.

The definition of well-posedness and ill-posedness of a scheme is closely

connected with the selection rules for norms II · ll(h) and 11 · ll( 2 h)' It may

happen that for some choices of these nonns estimate (25) is fulfilled, while

for the others estimate (22) is true. It is worth noting here that for the

0

scheme A y = -yxx = <p for y E Q h, as stated in Section 3, estimate (22) is

still valid in the norms

M-1 - '

II 'Ph 11(2h) = [ ~til h ( :~il h 'Pk)

2

]

112

3. Convergence and approximation. Let B(l) and B(^2 ) be normed vector
   spaces with norms II · 11(1) and II · 11( 2 ), respectively. One assumes, in
   addition, that
   (1) there exist linear operators P~^1 ) from 5(^1 ) into B~^1 ) and P~^2 ) from B(^2 )
   into B~^2 ) known as projectors such that
   P~^1 )1l=1thEB~^1 ) ifuEB(l) and P~^2 )f=fhEB~^2 ) if/EB(^2 l;
   (2) the conditions of concordance of norms are satisfied:

(26) lim II p~l) 1l 11(1 ) = II u 11(1)' lim II P~

2
lhl--+0 h lhl--+0 ) f 11(2 h ) = II f 11(2).
Given a vector Yh of the space B ~^1 ), we study the convergence of { Yh }
as I h I -+ 0 to a fixed element 1l from B (l).

1. A sequence { Yh }, where Yh E BP), is said to be convergent to an element
   1l E B(l) if

(27)

2. A sequence { Yh} is said to be convergent to 1th E B(l) with the rate
   0( I h In), n > 0, or { Yh} approximates 1l with accuracy 0( I h In) if for
   all sufficiently small I h I < h 0 the esti111ate is valid:

(28)

where M > 0 is a constant independent of h.