1549301742-The_Theory_of_Difference_Schemes__Samarskii

270 Difference Schemes for Elliptic Equations

we establish the relations

1 1 p 2

A( x) = h a J la+ + -h2 a + 2= h2 (3 '

(3= l

(3 t-C<

1 p 2

2= B(x,~) = J;2 + 2=

EEPatt'(.-c) er (3=1

/3 t-C<

1 1

D(x)=h h > h2.

c> er+

h2 (3 '

If x is a regular near-boundary node only with respect to x a, then

1 1

D(x)= h2>12·

C< 1

When, in addition, it turns out to be near-boundary in other directions,

the function D(x) can only increase. Just for this reason estimate (10)

continues to hold on the same footing.

To evaluate a solution of problem (8), we apply Theorem 4 of Section

2 due to which

(13)

Collecting estimates (6), (9), (13) and then involving the well-known in-
equality II y lie < II fJ lie+ II v lie+ II w lie, we establish the following theo-
rem.

Theorem 1 For a solution of the Dirichlet difference problem the estimate

(14)

R2

II Y lie< IIμ lie.,+ 2 P II <p lie. + h

2

II <p lie·

holds with

llflle= max IJ(x)I,

xEwh +l'h

llJlle^0 =max 0 IJ(x)I,

.7JEWh

II f lie· = ~~~ I f(x) 1,

h

11 f 11 c -, = x max E/'h I f ( x) I.

This theorem expresses the stability of the Dirichlet difference problem
(1) with respect to the boundary data and the right-hand side.