1549301742-The_Theory_of_Difference_Schemes__Samarskii

Operator-difference schemes 395

for all tn nr and that the operators ofsche1ne (4) carry out B~^1 ) into

B(2). h.

(1) (2)

Ah , Bh : Bh f-> Bh.

Let B~

1

) and B~

2

) be normed spaces with norms II· ll(lo) and 11 · llc 2 o)' u(t) be

an abstract B~^1 l-valued function of the argument t E [O, T] and let f(t) be

an abstract B~^2 )-valued function of the argument t E [O, T]. Furthermore,

we refer to linear operators P~a), CY= 1, 2, projecting B~a) onto B~a):

uh= P~^1 ) u E B~^1 )

f h = P~^2 ) f E B~^2 )

if u E 3(1) 0 '

if f E B~^2 l.

We take for granted the condition of the norm concordance

lhl~o lim II 1/Jh (a)ll (ah) -- ll^01 'I" '(o:)ll (ao)'

Let Yhr(tj) be a solution ofproble1n (4) and u(t) be a continuous

function of the argument t, so that

In such a setting the error z~ 7 = ~ 7 - u~ needs investigation. We say that

scheme ( 4) converges on an abstract function u(t) E B~^1 ) if

lhl~o,r~o lim O~J~Jo max llY~^7 - u~llc^1 h ) = 0.

Scheme ( 4) converges with the rate 0 (I h Im + Tk) or is of accuracy

O(lhlm + rk) on an abstract function u(t) E B~l) if

where M = const > 0 is independent of h and T both.
An a priori characteristic. of a scheme is the error of approximation.
The approximation error on a function u(t) for scheme ( 4) is known as the
residual

1/Jlir = 1/J( tj 1 h, T).