1549301742-The_Theory_of_Difference_Schemes__Samarskii

18

and, consequently,

I

IF; 0 I

lyio I = O<i<N max IYi < - -=--D· < -

• 
  
  
  
  • io
  
  
  
  

F

D c

Preliminaries

Before going further, problem (25) with coefficients subject to condi-

tions (22) for 0 < i < N 1nay be of help in achieving the final aims:

A;> 0, B; > 0, D; = C; - A; - B; > 0.

When the condition D; > 0 fails to be true Yi arranges itself as a sum

0 0

Yi= Y; + 1li, where Yi solves

0 0 0 0

(28) B; (Yi+! -Yi)-A; (Yi - Yi-1) = -F;, O<i<N,

The statement of the problem for 1li is

(29)

0

L [u;] = B; 1l;+1 - Ci u; +A; -ui-1 = -D; Yi, O<i<N,

whose solution satisfies the inequality

0

(30) 11 u I I c = O<i<N max 1-ui I < - 11 y 11 c.

0

This result is a corollary to the next lem1na with tpi =Yi involved.

Lemma. For the solution of problen1 (25) with coefficients subject

to conditions (22) and the right part F, = D; <pi, i = 1, 2, ... , N - 1, the

estimate is valid:

(31) llYc < lllfc ·

Proof In the case D; = 0 the estimate is obvious, since Yi = 0 due to
Corollary 2 to Theorem 1. By relating D; > 0 at least at one point we have
occasion to use the function Y; > 0, being a solution to the problem

(32) i=l,2, ... ,N-1;