1549301742-The_Theory_of_Difference_Schemes__Samarskii

20 Preliminaries

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Proof Recall that the difference u; = Yi - Yi is the solution of proble1n

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(29) with the right part F; = D; Yi. The inequality

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llYc = II Y + uc < II Y c + ll^11 c

in combination with estimate (30) leads to esti1nate (34).

This provides reason enough to reduce proper evaluation of the solu-

tion of the general proble1n (25) to that of the simpler equation (28), whose

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solution Yi can be found in explicit form.

9. Maximum principle for the third kind boundary-value problem. The
   maximum principle and its corollaries remain valid for the general bounda-
   ry-value problem ( 6), ( 8'), whose statement is

[, [y;] = -F;' i = 0, 1, 2, ... , N;

(21 *)

Theorem 1 (The maximum principle). Let under the set of the restrictions

A; > 0, Bi > 0, i = 1, 2, ... , N - l;
(22*)
0 < - x? --< 1,

the function Yi, Yi :;:/:. canst, satisfy the conditions

[, [y;] > (^0) ( [, [Yi] < 0 ) , i = 0, 1, 2, ... , N.
Then Yi cannot take the maximal positive (the mini111al negative) value at
any node i = 0, 1, 2, ... , N, that is, Yi< 0 (Yi > 0).
The proof is analogous to that of Theorem 1 fr01n Section 8. It 1s
necessary only to consider, in addition, the following two cases:
a) ifi 0 = 0, that is, max; Yi= Yo= lvl 0 > 0 and y 1 < M 0 , then