1549301742-The_Theory_of_Difference_Schemes__Samarskii

284 Difference Schemes for· Elliptic Equations

where ko:(x) are sufficiently smooth functions, 0 < c 1 < ko:(x) < c 2 , CY =

1, 2, with constants c 1 > 0 and c 1 > 0. We con1pose in just the same way

the grid with steps h 1 and h 2 and denote it by wh = wh + lh as we did in

Section 1.

An excellent start in this direction is to approxi1nate the operator Lo:

by the difference operator

}_3!_)

ox 0: ,

where a 0 J x) is so chosen as to satisfy the relations

(24) 0 < c 1 < a o: ( x) < c 2.

This means that Ao: provides an approximation of order 2, for instance, for

ao: = k~-^0 ·^5 a) or

The next step is to put the operator Au = L!=l Ao: u in correspondence

with the operator Lu due to which the difference Dirichlet problem

Ay = -<p,

(25)

A O' y -- (a a~Xa ·I/- ) ~t'n'

having the approxi111ation error (residual) 1/J = Au+ <p = 0(1h1^2 ) will be
associated with problem (23).
Such an approxirnation is the result of a natural generalization of ho-
mogeneous conservative schemes frorn Chapter 3 for one-dimensional equa-
tions to the multidimensional case. These schemes can be obtained by
means of the integro-interpolational inethod without any difficulties.
0
We now investigate the properties of the operator A in the space 0.1i
of all grid functions with the inner product in the sense of (15). In what
0
follows we accept Ay = -Ay for any y E 0,.

Lemma The operator A y = -A y = - La=!^2 Aa y, y E H, is self-adjoint,

positive definite and admits the estimates

(26)

(27)

0 0

c 1 (Av,v) < (Av,v) < c 2 (Av,v),