1549301742-The_Theory_of_Difference_Schemes__Samarskii

The maximum principle 265

Theore1n 4 Let the conditions

D(P) = 0

0

for P E w, D(P)>O for PEw*

hold, where~ is a connected grid. Then for a solution of problem ( 16) with

the right-hand side

F(P) = 0 for

0

P Ew, F(P) :f: 0 for PE w*,

the estimate

( 18) llvlle <II~ lie.

is valid in the norm II f lie• = maxPEw* I f(P) I·

Proof Let Y(P) be a majorant and.CY= I F(P) I on the grid w, YI,,= 0,

Y > 0. The function Y (P) should attain its maximum on a finite set w +I

at some node, not belonging to the boundary, because YI,, = 0. Also, it

does not enter the grid~ clue to the connectedness of~ and the 1naximum

principle. Hence,

max Y(P) = max Y(P) = Y(Po),

PEw PEw*

where Po is a node on the set w*.
By the initial hypothesis, D(Po) > 0. Arguing as in the proof of
Theorem 3 we arrive at (18). An analog of the remark to Theorem 3 is still
valid for that case.

4.3 STABILITY AND CONVERGENCE OF THE DIRICHLET

DIFFERENCE PROBLEM

1. Estimation of a solution of the Dirichlet difference problem. vVe make
   use of a priori estimates obtained in Section 2 for a grid equation of common
   structure for constructing a uniform estimate of a solution of the Dirichlet
   difference problem (24)-(26) arising in Section 1:

(1)

Ay = -<p

A*y= -<p

y = p(x)

at the regular nodes,

at the irregular nodes,

on the boundary,