1549301742-The_Theory_of_Difference_Schemes__Samarskii

130 Basic Concepts of the Theory of Difference Schemes

Let Yh be a solution of problem (21). We say that

1) scheme (21) converges if there exists an element u E 5(l) such that the

limit relation (27) occurs;

2) the scheme is of accuracy 0( I h 12 ) if there exists an element u E 5(l)

such that for I h I < h 0 relation (28) occurs.

Let us define the notion of the approximation error on an element

u E 8(^1 ). To this end, we must write down the equation for the difference

zh = Yh - uh. Substitution of Yh = zh +uh into (21) gives

(30) 1/J1i E 8h (2).

In this context, we call the right-hand side 1/J1i = 1/Jh( u) depending on the

choice of an element 1l from 5(l) the error of approximation on the element

u E 5(l) for scheme (21). Obviously, 1/Jh( u) is the residual emerging as a

result of replacing Yh by the element uh E P~^1 )u in (21).

We say that

1) scheme (21) generates an approximation on an element u E 5(l) if

(31)

2) scheme (21) is of the nth approximation order on an element u E 5(l) if

for sufficiently small I h I < h 0

(32) or

where M is a positive constant independent of h, h > 0.
Our next goal is to establish direct links between stability, approxi-
mation on an element 1l E 5(l) and convergence to u for scheme (21). If
scheme (21) is correct, then problem (30) for zh is well-posed. Because of
this fact, its solution obeys the estimate

(33)

In this respect, we obtain the profound result with the following the-
orem.

Theorem 1 If scheme (21) is correct and generates an approximation on
an element u E 5(l), then it converges. More precisely, a solution Yh
of problem (21) converges to this element u E 5(l) as I h I --+ 0 and, in
addition, the order of accuracy of scheme (21) coincides with the order