1549301742-The_Theory_of_Difference_Schemes__Samarskii

294 Difference Sche1nes for Elliptic Equations

2. Estimation of a solution of the difference boundary-value problem. Con-
   sider now the difference Dirichlet problem for the scheme of accuracy
   O(I h 14 ) in the rectangle G = {O < xo: <lo:, ex= 1, 2}:

(9)

{

A'y = -<p'

h2

<p=f+-1

12

YI -Yk =μ(x),

where A' y is given by formula (7). Each of the grid nodes is regular, because

the nine-point pattern belongs to the rectangle G (Fig. 16). The boundary

lh of the grid contains all the nodes on the boundary r including the

vertices of the rectangle. With this in mind, we set up the problem for the

error z = y - u:

(10) A' z = -1/;, x E wh, z = 0 on lh ,

where 1/J = A'u+<p = O(I h 14 ) for x E wh ifu E C(^6 l. To decide for yourself

whether the conditions of the n1aximum principle are satisfied, a first step

is to compare (8) with (1) from Section 2. As a final result we get

( 11) for

To evaluate the solution of problem (10), we should have at our disposal
the majorant of the type

Y(x) = 1{ (l; -x; + z; - x;).

Taking into account that AY = -41{, A 1 A 2 Y = 0, !!YI!< I<(l; + z;)
and accepting 41{ .= 111/J lie, we deduce by Theorem 3 for the solution of
problem (10) the estimate

z2 + z2

II z Ile <

1

4

2

111/J lie'

provided that the condition