1549301742-The_Theory_of_Difference_Schemes__Samarskii

260 Difference Schemes for Elliptic Equations

On the gridwhr = {(x; = ih,tj = jr), i = 0, 1, ... , N, hN = 1, j = 0, 1, ... }

this scheme takes the form

(8)

A Y = Yxx'

These difference equations are convenient to be presented in the canon-

ical form (1), giving Pas a node of the grid whr; P = P(xi,ij+ 1 ),

where Patt'(P) consists of the nodes Qi = (x;, tj), Q2 = (x;_ 1 , ij+ 1 ),

Q3 = (x;+ 1 , ij+ 1 ), Q4 = (x;_ 1 , tj), Qs = (xi+i, tj) and the boundary /

consists of the nodes (x;, 0) and (0, tj), (1, tj), i = 0, 1, ... , N, j = 0, 1, ....

Next, we fix some moment t = ij+l and rewrite (8) as

(

__!_ 2(o--1)) j

+ T + h2 Y;

1-0"(· ").

+ h2 Yf-1 + Y/+1 + l.f!/ ·

From here it is easily seen that B(P, Q) > 0 only if T < h^2 /2(1-o-) and

0 < o- < 1. By the same token, D(P) = 0.

2. The maximum principle.

Theorem 1 (the maximum principle) Let y( P) -:f- const be a grid function

defined on a connected grid w and let both conditions (2) and ( 4) hold.

Then the condition .Cy( P) < 0 ( .C y( P) > 0) on the grid w implies that

y(P) cannot attain the maximal positive (minimal negative) value at the

inner nodes P E w.

Proof Let .Cy( P) < 0 at all of the inner nodes P E w. On the contrary,

let y( P) attain its maximal positive value at an inner node P E w, so that

y(P) = m51x w y(P) =Mo > 0.

The theorem will be proved if we succeed in showing that there exists an
inner point P at which .C y(P) > 0, violating the condition .C y(P) < 0.