1549301742-The_Theory_of_Difference_Schemes__Samarskii

56 Basic Concepts of the Theory of Difference Schemes

For a difference function uh, we turn to the difference Yh - 1l h, which

gives a vector of the space H h. The nearness of Yh to u is well-characterized

by the number II Yh - Hh llh' where II · llh is the norm of the space Hh. It

seems natural to require that the norm II · llh should approximate the norm

II · llo in the following sense: for any vector u E Ho

This condition is known as the condition of concordance of the norms

on the spaces Hh and Ho.

Under the second approach mentioned above we proceed to the accu-

rate account of the errors of difference methods in the space of grid func-

tions. In the most cases the spaces involved appear to be finite-dimensional.

As we will see later, it is possible to present the principal aspects of the

theory of difference schemes with further treatment of H h as an abstract

vector space of arbitrary dimension.

After preliminary discussions of the simplest examples illustrating

some ways of producing grids and, thereby, of forming the spaces Hh of

grid functions we concentrate primarily on the problem of the difference

approximation of differential operators.

2. The difference approximation of elementary differential operators. Let
   a linear operator L assign the values to a function v = v( x). By replacing
   the derivatives built into Lv by their difference counterparts we derive the
   difference expression Lh vh, which is a linear combination of the values of
   the grid function vh on some set of grid nodes known as a pattern:

Lhvh(x)= ~ Ah(x,~)vh(~)

~EPatt(x)

or
(Lh vh)i = ~ Ah(xi, xj) vh(xj),
Xj EPatt(xi)
where Ah(x,~) are the coefficients, his a grid spacing and Patt(x) is a
pattern at a point x. Any replacement of Lv by Lh vh is called the ap-
proximation of a differential operator by a difference operator or
the difference approximation of an operator L.
The way this approach is used in practice is connected with prelimi-
nary studies of difference approximations of the operator L locally, that is,
at an arbitrary fixed point of the space. If v( x) is a continuous function,
then vh(x) = v(x). Before giving further constructions of a difference ap-
proximation of the operator L, it is necessary to choose a pattern of proper