1549301742-The_Theory_of_Difference_Schemes__Samarskii

126 Basic Concepts of the Theory of Difference Schemes

operator) in rectangular domains will be studied by means of similar meth-

ods. If the initial differential operator is self-adjoint and positive definite,

one should construct the difference operator also to possess these proper-

ties in the grid space. This can be achieved, for example, by employing the

balance method (the integro-interpolation method from Chapter 3) or the

variational method in designing difference schemes.

We learn from the examples under consideration tha.t the difference

equations can be treated as operator equations with operators in a finite-

dimensional normed vector space. A feature of such operators is that they

map the entire space into itself as further developments occur.

vVe proceed to the description of the theory of difference schemes

treated as operator equations, the 111eaning of which we have discussed

above.

2. Stability of a difference scheme. Let two normed vector spaces BP) and
   B~^2 ) be given with parameter h being a vector of some normed space with
   the norn1 I h I > 0. In dealing with a linear operator Ah with the domain
   V( Ah)= BP) and range R( Ah) ~ B~^2 ) we consider the equation

(21) 'Ph E Bh (2) '

where 'Ph is a given vector. Varying the parameter h, we obtain the set

of solutions { Yh} to equation (21). We call the operator equation (21)

depending on the parameter h a difference scheme.

Let II · ll(lh) and II · ll( 2 h) be the norms on the spaces B 1 ~

1

) and B~

2

),

respectively. Scheme (21) is said to be correct or problem (21) is said to be

well-posed if for all sufficiently small I h I < h 0

(1) a solution Yh of (21) exists and is unique for all 'Ph E B~^2 ) (scheme

(21) is uniquely solvable),

(2) this solution continuously depends on 'Ph and this dependence is

uniform in h (scheme (21) is stable). In other words, there exists a

positive constant M independent of h and 'Ph such that a solution

to equation (21) admits for any 'Ph E B 1 ~^2 ) the estimate

The meaning of the solvability of scheme (21) is that there exists an inverse
operator AJ;"^1 such that

(23) Yh = A-l h 'Ph ·